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PAT H I N T E G R A L S A N D H A M I LTO N I A N S

Providing a pedagogical introduction to the essential principles of path integralsand Hamiltonians, this book describes cutting-edge quantum mathematical tech-niques applicable to a vast range of fields, from quantum mechanics, solid statephysics, statistical mechanics, quantum field theory, and superstring theory to fi-nancial modeling, polymers, biology, chemistry, and quantum finance.

Eschewing use of the Schrödinger equation, the powerful and flexible combina-tion of Hamiltonian operators and path integrals is used to study a range of differ-ent quantum and classical random systems, succinctly demonstrating the interplaybetween a system’s path integral, state space, and Hamiltonian. With a practicalemphasis on the methodological and mathematical aspects of each derivation, thisis a perfect introduction to these versatile mathematical methods, suitable for re-searchers and graduate students in physics and engineering.

B e l a l E . Ba aq u i e is a Professor of Physics at the National University ofSingapore, specializing in quantum field theory, quantum mathematics, and quan-tum finance. He is the author of Quantum Finance (2004), Interest Rates andCoupon Bonds in Quantum Finance (2009), and The Theoretical Foundations ofQuantum Mechanics (2013), and co-author of Exploring Integrated Science (2010).

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PATH INTEGRALS ANDHAMILTONIANS

Principles and Methods

B E L A L E . BA AQU I ENational University of Singapore

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University Printing House, Cambridge CB2 8BS, United Kingdom

Published in the United States of America by Cambridge University Press, New York

Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit ofeducation, learning, and research at the highest international levels of excellence.

www.cambridge.orgInformation on this title: www.cambridge.org/9781107009790

© Belal E. Baaquie 2014

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2014

Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication Data

ISBN 978-1-107-00979-0 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.

www.cambridge.org/9781107009790

www.cambridge.org

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This book is dedicated to the memory ofKenneth Geddes Wilson (1936-2013).

Intellectual giant, visionary scientist, exceptional educator, altruistic spirit.

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Contents

Preface page xvAcknowledgements xviii

1 Synopsis 1

Part one Fundamental principles 5

2 The mathematical structure of quantum mechanics 72.1 The Copenhagen quantum postulate 72.2 The superstructure of quantum mechanics 102.3 Degree of freedom space F 102.4 State space V(F) 11

2.4.1 Hilbert space 142.5 Operators O(F) 142.6 The process of measurement 182.7 The Schrödinger differential equation 192.8 Heisenberg operator approach 222.9 Dirac–Feynman path integral formulation 232.10 Three formulations of quantum mechanics 252.11 Quantum entity 262.12 Summary: quantum mathematics 27

3 Operators 303.1 Continuous degree of freedom 303.2 Basis states for state space 353.3 Hermitian operators 36

3.3.1 Eigenfunctions; completeness 373.3.2 Hamiltonian for a periodic degree of freedom 39

3.4 Position and momentum operators x and p 403.4.1 Momentum operator p 41

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viii Contents

3.5 Weyl operators 433.6 Quantum numbers; commuting operator 463.7 Heisenberg commutation equation 473.8 Unitary representation of Heisenberg algebra 483.9 Density matrix: pure and mixed states 503.10 Self-adjoint operators 51

3.10.1 Momentum operator on finite interval 523.11 Self-adjoint domain 54

3.11.1 Real eigenvalues 543.12 Hamiltonian’s self-adjoint extension 55

3.12.1 Delta function potential 573.13 Fermi pseudo-potential 593.14 Summary 60

4 The Feynman path integral 614.1 Probability amplitude and time evolution 614.2 Evolution kernel 634.3 Superposition: indeterminate paths 654.4 The Dirac–Feynman formula 674.5 The Lagrangian 69

4.5.1 Infinite divisibility of quantum paths 704.6 The Feynman path integral 704.7 Path integral for evolution kernel 734.8 Composition rule for probability amplitudes 764.9 Summary 79

5 Hamiltonian mechanics 805.1 Canonical equations 805.2 Symmetries and conservation laws 825.3 Euclidean Lagrangian and Hamiltonian 845.4 Phase space path integrals 855.5 Poisson bracket 875.6 Commutation equations 885.7 Dirac bracket and constrained quantization 90

5.7.1 Dirac bracket for two constraints 915.8 Free particle evolution kernel 935.9 Hamiltonian and path integral 945.10 Coherent states 955.11 Coherent state vector 965.12 Completeness equation: over-complete 985.13 Operators; normal ordering 98

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Contents ix

5.14 Path integral for coherent states 995.14.1 Simple harmonic oscillator 101

5.15 Forced harmonic oscillator 1025.16 Summary 103

6 Path integral quantization 1056.1 Hamiltonian from Lagrangian 1066.2 Path integral’s classical limit �→ 0 109

6.2.1 Nonclassical paths and free particle 1116.3 Fermat’s principle of least time 1126.4 Functional differentiation 115

6.4.1 Chain rule 1156.5 Equations of motion 1166.6 Correlation functions 1176.7 Heisenberg commutation equation 118

6.7.1 Euclidean commutation equation 1216.8 Summary 122

Part two Stochastic processes 123

7 Stochastic systems 1257.1 Classical probability: objective reality 127

7.1.1 Joint, marginal and conditional probabilities 1287.2 Review of Gaussian integration 1297.3 Gaussian white noise 132

7.3.1 Integrals of white noise 1347.4 Ito calculus 136

7.4.1 Stock price 1377.5 Wilson expansion 1387.6 Linear Langevin equation 140

7.6.1 Random paths 1427.7 Langevin equation with potential 143

7.7.1 Correlation functions 1447.8 Nonlinear Langevin equation 1457.9 Stochastic quantization 148

7.9.1 Linear Langevin path integral 1497.10 Fokker–Planck Hamiltonian 1517.11 Pseudo-Hermitian Fokker–Planck Hamiltonian 1537.12 Fokker–Planck path integral 1567.13 Summary 158

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x Contents

Part three Discrete degrees of freedom 159

8 Ising model 1618.1 Ising degree of freedom and state space 161

8.1.1 Ising spin’s state space V 1638.1.2 Bloch sphere 164

8.2 Transfer matrix 1658.3 Correlators 167

8.3.1 Periodic lattice 1688.4 Correlator for periodic boundary conditions 169

8.4.1 Correlator as vacuum expectation values 1718.5 Ising model’s path integral 171

8.5.1 Ising partition function 1728.5.2 Path integral calculation of Cr 173

8.6 Spin decimation 1758.7 Ising model on 2×N lattice 1768.8 Summary 179

9 Ising model: magnetic field 1809.1 Periodic Ising model in a magnetic field 1809.2 Ising model’s evolution kernel 1829.3 Magnetization 183

9.3.1 Correlator 1849.4 Linear regression 1859.5 Open chain Ising model in a magnetic field 189

9.5.1 Open chain magnetization 1909.6 Block spin renormalization 191

9.6.1 Block spin renormalization: magnetic field 1959.7 Summary 196

10 Fermions 19810.1 Fermionic variables 19910.2 Fermion integration 20010.3 Fermion Hilbert space 201

10.3.1 Fermionic completeness equation 20310.3.2 Fermionic momentum operator 204

10.4 Antifermion state space 20410.5 Fermion and antifermion Hilbert space 20610.6 Real and complex fermions: Gaussian integration 207

10.6.1 Complex Gaussian fermion 20910.7 Fermionic operators 211

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Contents xi

10.8 Fermionic path integral 21110.9 Fermion–antifermion Hamiltonian 214

10.9.1 Orthogonality and completeness 21610.10 Fermion–antifermion Lagrangian 21710.11 Fermionic transition probability amplitude 21910.12 Quark confinement 22010.13 Summary 222

Part four Quadratic path integrals 223

11 Simple harmonic oscillator 22511.1 Oscillator Hamiltonian 22611.2 The propagator 226

11.2.1 Finite time propagator 22711.3 Infinite time oscillator 23011.4 Harmonic oscillator’s evolution kernel 23011.5 Normalization 23311.6 Generating functional for the oscillator 234

11.6.1 Classical solution with source 23411.6.2 Source free classical solution 236

11.7 Harmonic oscillator’s conditional probability 23911.8 Free particle path integral 24011.9 Finite lattice path integral 241

11.9.1 Coordinate and momentum basis 24311.10 Lattice free energy 24311.11 Lattice propagator 24511.12 Lattice transfer matrix and propagator 24611.13 Eigenfunctions from evolution kernel 24911.14 Summary 250

12 Gaussian path integrals 25112.1 Exponential operators 25212.2 Periodic path integral 25312.3 Oscillator normalization 25412.4 Evolution kernel for indeterminate final position 25612.5 Free degree of freedom: constant external source 26012.6 Evolution kernel for indeterminate positions 26112.7 Simple harmonic oscillator: Fourier expansion 26412.8 Evolution kernel for a magnetic field 26712.9 Summary 270

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xii Contents

Part five Action with acceleration 271

13 Acceleration Lagrangian 27313.1 Lagrangian 27313.2 Quadratic potential: the classical solution 27513.3 Propagator: path integral 27713.4 Dirac constraints and acceleration Hamiltonian 28013.5 Phase space path integral and Hamiltonian operator 28313.6 Acceleration path integral 28613.7 Change of path integral boundary conditions 28913.8 Evolution kernel 29113.9 Summary 293

14 Pseudo-Hermitian Euclidean Hamiltonian 29414.1 Pseudo-Hermitian Hamiltonian; similarity transformation 29514.2 Equivalent Hermitian Hamiltonian HO 29714.3 The matrix elements of e−τQ 29814.4 e−τQ and similarity transformations 30114.5 Eigenfunctions of oscillator Hamiltonian HO 30414.6 Eigenfunctions of H and H† 305

14.6.1 Dual energy eigenstates 30714.7 Vacuum state; eQ/2 30914.8 Vacuum state and classical action 31214.9 Excited states of H 313

14.9.1 Energy ω1 eigenstate �10(x, v) 31414.9.2 Energy ω2 eigenstate �01(x, v) 315

14.10 Complex ω1, ω2 31714.11 State space V of Euclidean Hamiltonian 318

14.11.1 Operators acting on V 31914.11.2 Heisenberg operator equations 321

14.12 Propagator: operators 32214.13 Propagator: state space 32414.14 Many degrees of freedom 32614.15 Summary 328

15 Non-Hermitian Hamiltonian: Jordan blocks 33015.1 Hamiltonian: equal frequency limit 33115.2 Propagator and states for equal frequency 33115.3 State vectors for equal frequency 334

15.3.1 State vector |ψ1(τ )〉 33415.3.2 State vector |ψ2(τ )〉 335

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Contents xiii

15.4 Completeness equation for 2× 2 block 33615.5 Equal frequency propagator 33715.6 Hamiltonian: Jordan block structure 33915.7 2×2 Jordan block 340

15.7.1 Hamiltonian 34215.7.2 Schrödinger equation for Jordan block 34315.7.3 Time evolution 344

15.8 Jordan block propagator 34415.9 Summary 347

Part six Nonlinear path integrals 349

16 The quartic potential: instantons 35116.1 Semi-classical approximation 35216.2 A one-dimensional integral 35316.3 Instantons in quantum mechanics 35516.4 Instanton zero mode 36216.5 Instanton zero mode: Faddeev–Popov analysis 364

16.5.1 Instanton coefficient N 36816.6 Multi-instantons 37016.7 Instanton transition amplitude 371

16.7.1 Lowest energy states 37216.8 Instanton correlation function 37316.9 The dilute gas approximation 37416.10 Ising model and the double well potential 37616.11 Nonlocal Ising model 37716.12 Spontaneous symmetry breaking 380

16.12.1 Infinite well 38116.12.2 Double well 381

16.13 Restoration of symmetry 38116.14 Multiple wells 38316.15 Summary 383

17 Compact degrees of freedom 38517.1 Degree of freedom: a circle 386

17.1.1 Poisson summation formula 38717.1.2 The S1 Lagrangian 388

17.2 Multiple classical solutions 38817.2.1 Large radius limit 391

17.3 Degree of freedom: a sphere 39117.4 Lagrangian for the rigid rotor 393

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xiv Contents

17.5 Cancellation of divergence 39517.6 Conformation of DNA 39717.7 DNA extension 39917.8 DNA persistence length 40117.9 Summary 403

References 405Index 409

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Preface

Quantum mechanics is undoubtedly one of the most accurate and importantscientific theories in the history of science. The theoretical foundations of quantummechanics have been discussed in depth in Baaquie (2013e), where the main focusis on the interpretation of the mathematical symbols of quantum mechanics andon its enigmatic superstructure. In contrast, the main focus of this book is on themathematics of path integral quantum mechanics.

The traditional approach to quantum mechanics has been to study theSchrödinger equation, one of the cornerstones of quantum mechanics, and whichis a special case of partial differential equations. Needless to say, the study ofthe Schrödinger equation continues to be a central task of quantum mechanics,yielding a steady stream of new and valuable results.

Interestingly enough, there are two other formulations of quantum mechanics,namely the operator approach of Heisenberg and the path integral approach ofDirac–Feynman, that provide a mathematical framework which is independent ofthe Schrödinger equation. In this book, the Schrödinger equation is never directlysolved; instead the Hamiltonian operator is analyzed and path integrals for differ-ent quantum and classical random systems are studied to gain an understanding ofquantum mathematics.

I became aware of path integrals when I was a graduate student, and what in-trigued me most was the novelty, flexibility and versatility of their theoretical andmathematical framework. I have spent most of my research years in exploring andemploying this framework.

Path integration is a natural generalization of integral calculus and is essentiallythe integral calculus of infinitely many variables, also called functional integration.There is, however, a fundamental feature of path integration that sets it apart fromfunctional integration, namely the role played by the Hamiltonian in the formalism.All the path integrals discussed in this book have an underlying linear structure thatis encoded in the Hamiltonian operator and its linear vector state space. It is this

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xvi Preface

combination of the path integral and its underlying Hamiltonian that provides apowerful and flexible mathematical machinery that can address a vast variety andrange of diverse problems. Path integration can also address systems that do nothave a Hamiltonian and these systems are not studied. Instead, topics have beenchosen that can demonstrate the interplay of the system’s path integral, state space,and Hamiltonian.

The Hamiltonian operator and the mathematical formalism of path integrationmake them eminently suitable for describing quantum indeterminacy as well asclassical randomness. In two chapters of the book, namely Chapter 7 on stochasticprocesses and Chapter 17 on compact degrees of freedom, path integrals are ap-plied to classical stochastic and random systems. The rest of the chapters analyzesystems that have quantum indeterminacy.

The range and depth of subjects that come under the sway of path integrals areunified by a common thread, which is the mathematics of path integrals. Prob-lems seemingly unrelated to indeterminacy such as the classification of knots andlinks or the mathematical properties of manifolds have been solved using path in-tegration. The applications of path integrals are almost as vast as calculus, rangingfrom finance, polymers, biology, and chemistry to quantum mechanics, solid statephysics, statistical mechanics, quantum field theory, superstring theory, and all theway to pure mathematics. The concepts and theoretical underpinnings of quantummechanics lead to a whole set of new mathematical ideas and have given rise to thesubject of quantum mathematics.

The ground-breaking and pioneering book by Feynman and Hibbs (1965) laidthe foundation for the study of path integrals in quantum mechanics and is alwaysworth reading. More recent books such as those by Kleinert (1990) and Zinn-Justin(2005) discuss many important aspects of path integration and cover a wide rangeof applications. Given the complex theoretical and mathematical nature of the sub-ject, no single book can conceivably cover the gamut of worthwhile topics thatappear in the study of path integration and there is always a need for books thatbreak new ground. The topics chosen in this book have a minimal overlap withother books on path integrals.

A major field of theoretical physics that is based on path integrals is quantumfield theory, which includes the Standard Model of particles and forces. The studyof quantum field theory leads to the concept of nonlinear gauge fields and to theconcept of renormalization, both of which are beyond the scope this book.

The purpose of the book is to provide a pedagogical introduction to the essentialprinciples of path integrals and of Hamiltonians, as well as to work out in full de-tail some of the varied methods and techniques that have proven useful in actuallyperforming path integrations. The emphasis in all the derivations is on the method-ological and mathematical aspect of the problem – with matters of interpretation

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Preface xvii

being discussed only in passing. Starting from the simplest examples, the variouschapters lay the ground work for analyzing more advanced topics. The book pro-vides an introduction to the foundations of path integral quantum mechanics and isa primer to the techniques and methods employed in the study of quantum finance,as formulated by Baaquie (2004) and Baaquie (2010).

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Acknowledgements

I would like to acknowledge and express my thanks to many outstanding teachers,scholars, and researchers whose work motivated me to study path integral quantummechanics and to grapple with its mathematical formalism.

I had the singular privilege of doing my Ph.D. thesis under the guidance of NobelLanreate Kenneth G. Wilson; his visionary conception of quantum mechanics andof quantum field theory – rooted in the path integral – greatly enlightened andinspired me, and continues to do so today. As an undergraduate I had the honor ofmeeting and conversing a number of times with Richard P. Feynman, the legendarydiscoverer of the path integral, and this left a permanent impression on me.

I thank Frederick H. Willeboordse for his consistent support and Wang Qinghai,Kang Hway Chuan, Zahur Ahmed, Duxin and Cao Yang for many helpful discus-sions.

I thank my wife Najma for being a wonderful companion and for her uplift-ing approach to family and professional life. I thank my precious family membersArzish, Farah, and Tazkiah for their delightful company and warm encouragement.They have made this book possible.

I am deeply indebted to my late father Muhammad Abdul Baaquie for being alife long source of encouragement and whose virtuous qualities continue to be abeacon of inspiration.

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1

Synopsis

This book studies the mathematical aspect of path integrals and Hamiltonians –which emerge from the formulation of quantum mechanics. The theoretical frame-work of quantum mechanics provides the mathematical tools for studying bothquantum indeterminacy and classical randomness. Many problems arising in quan-tum mechanics as well as in vastly different fields such as finance and economicscan be addressed by the mathematics of quantum mechanics, or quantum mathe-matics in short. All the topics and subjects in the various chapters have been specif-ically chosen to illustrate the structure of quantum mathematics, and are not tied toany specific discipline, be it quantum mechanics or quantum finance.

The book is divided into the following six parts, in accordance with the Chapterdependency flowchart given below.

• Part one addresses the Fundamental principles of path integrals and (Hamilto-nian) operators and consists of five chapters. Chapter 2 is on the Mathematicalstructure of quantum mechanics and introduces the mathematical framework thatemerges from the quantum principle. Chapters 3 to 6 discuss the mathematicalpillars of quantum mathematics, starting from the Feynman path integral, sum-marizing Hamiltonian mechanics and introducing path integral quantization.

• Part two is on Stochastic processes. Stochastic systems are dissipative and areshown to be effectively modeled by the path integral. Chapter 7 is focused on theapplication of quantum mathematics to classical random systems and to stochas-tic processes.

• Part three discusses Discrete degrees of freedom. Chapters 8 and 9 discuss thesimplest quantum mechanical degree of freedom, namely the double valuedIsing spin. The Ising model is discussed in some detail as this model contains allthe essential ideas that unfold later for more complex degrees of freedom. Thegeneral properties of path integrals and Hamiltonians are discussed in the con-text of the Ising spin. Chapter 10 on Fermions introduces a degree of freedom

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2 Synopsis

4. Feynman pathintegral

6. Path integralquantization

7. Stochasticsystems

5. Hamiltonianmechanics

3. Operators

2. Mathematical structure ofquantum mechanics

11. Simple harmonicoscillator

16. Quartic potential:instantons

8. Ising model 10. Fermions

9. Ising model:magnetic field

12. Guassian pathintegral

13. AccelerationLagrangian

14. Pseudo-HermitianEuclidean Hamiltonian

15. Non-HermitianHamiltonian:

Jordan blocks

17. Compact degreesof freedom

Chapter dependency flowchart

that is essentially discrete – but is represented by fermionic variables that aredistinct from real variables. The calculus of fermions and the key structures ofquantum mathematics such as the Hamiltonian, state space, and path integralsare discussed in some detail.

• Part four covers of Quadratic path integrals. Chapter 11 is on the simple har-monic oscillator – one of the prime exemplars of quantum mechanics – and itis studied using both the Hamiltonian and path integral approach. In Chapter 12different types of Gaussian path integrals are evaluated using techniques that areuseful for analyzing and solving path integrals.

• Part five is on the Acceleration action. An action with an acceleration term isdefined for Euclidean time and is shown to have a novel structure not present inusual quantum mechanics. In Chapter 13, the Lagrangian and path integral are

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Synopsis 3

analyzed and shown to be equivalent to a constrained system. The Hamiltonianis obtained using the Dirac constraint method. In Chapter 14, the accelerationHamiltonian is shown to be pseudo-Hermitian and its state space and propagatorare derived. Chapter 15 examines a critical point of the acceleration action andthe Hamiltonian is shown to be essentially non-Hermitian, being block diagonaland with each block being a Jordan block.

• Part six is on Nonlinear path integrals. Chapter 16 studies the nonlinear quarticLagrangian to illustrate the qualitatively new features that nonlinear path inte-grals exhibit. The double well potential is studied in some detail as an exemplarof nonlinear path integrals that can be analyzed using the semi-classical expan-sion. And lastly, in Chapter 17 degrees of freedom are analyzed that take valuesin a compact manifold; these systems have a nonlinearity that arises from thenature of the degree of freedom itself – rather than from a nonlinear piece inthe Lagrangian. Semi-classical expansions of the path integral about multipleclassical solutions, classified by a winding number and path integrals on curvedmanifolds, are briefly touched upon.

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Part one

Fundamental principles

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2

The mathematical structure of quantum mechanics

An examination of the postulates of quantum mechanics reveals a number offundamental mathematical constructs that form its theoretical underpinnings.Many of the results that are summarized in this Chapter will only become clearafter reading the rest of the book and a re-reading may be in order.

The dynamical variables of classical mechanics are superseded by the quantumdegree of freedom. An exhaustive and complete description of the indeterminatedegree of freedom is given by its state function, which is an element of a state spacethat, in general, is an infinite dimensional linear vector space. The properties ofthe indeterminate degree of freedom are extracted from its state vector by the linearaction of operators representing experimentally observable quantities. Repeatedapplications of the operators on the state function yield the average value of theoperator for the state [Baaquie (2013e)].

The conceptual framework of quantum mechanics is discussed in Section 2.1.The concepts of degree of freedom, state space and operators are briefly reviewedin Sections 2.3–2.5. Three distinct formulations of quantum mechanics emergefrom the superstructure of quantum mechanics and these are briefly summarized inSections 2.7–2.9.

2.1 The Copenhagen quantum postulate

The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohrand Werner Heisenberg, provides a conceptual framework for the interpretation ofthe mathematical constructs of quantum mechanics and is the standard interpreta-tion that is followed by the majority of practicing physicists [Stapp (1963), Dirac(1999)].

The Copenhagen interpretation is not universally accepted by the physics com-munity, with many alternative explanations being proposed for understandingquantum mechanics [Baaquie (2013e)]. Instead of entering this debate, this book

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8 The mathematical structure of quantum mechanics

is based on the Copenhagen interpretation, which can be summarized by thefollowing postulates:

• The quantum entity consists of its degree of freedom F and its state vectorψ(t,F). The foundation of the quantum entity is its degree of freedom, whichtakes a range of values and constitutes a space F . The quantum degree of free-dom is completely described by the quantum state ψ(t,F), a complex valuedfunction of the degree of freedom that is an element of state space V(F).

• The quantum entity is an inseparable pair, namely, the degree of freedom andits state vector.

• All physically observable quantities are obtained by applying Hermitian opera-tors O(F) on the state ψ(t,F).

• Experimental observations collapse the quantum state and repeated observationsyield Eψ [O(F)], which is the expectation value of the operator O(F) for thestate ψ(t,F).

• The Schrödinger equation determines the time dependence of the state vector,namely of ψ(t,F), but does not determine the process of measurement.

It needs to be emphasized that the state vector ψ(t,F) provides only statisticalinformation about the quantum entity; the result of any particular experiment isimpossible to predict.1

The organization of the theoretical superstructure of quantum mechanics isshown in Figure 2.1.

The quantum state ψ(t,F) is a complex number that describes the degree offreedom and is more fundamental than the observed probabilities, which are alwaysreal positive numbers. The scheme of assigning expectation values to operators,such as Eψ [O(F)], leads to a generalization of classical probability to quantumprobability and is discussed in detail in Baaquie (2013e).

To give a concrete realization of the Copenhagen quantum postulate, consider aquantum particle moving in one dimension; the degree of freedom is the real line,namely F = � = {x|x ∈ (−∞,+∞)} with state ψ(t,�). Consider the positionoperator O(x);2 a measurement projects the state to a point x ∈ � and collapsesthe quantum state to yield, after repeated measurements

P(t, x) ≡ Eψ [O(x)] = |ψ(t, x)|2, P (t, x) > 0,∫ +∞

−∞dxP (t, x) = 1. (2.1)

Note from Eq. 2.1 that P(t, x) obeys all the requirements to be interpreted asa probability distribution. A complete description of a quantum system requires

1 There are special quantum states called eigenstates for which one can exactly predict the outcome of someexperiments. But for even this special case the degree of freedom is indeterminate and can never be directlyobserved.

2 The position projection operator O(x) = |x〉〈x|; see Chapter 3.

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2.1 The Copenhagen quantum postulate 9

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V(�)

O (�)

EV[O (�)]

Quantum Entity

Figure 2.1 The theoretical superstructure of quantum mechanics; the quantumentity is constituted by the degree of freedom F and its state vector, which is anelement of state space V(F); operators O(F) act on the state vector to extractinformation about the degree of freedom and lead to the final result EV [O(F)];only the final result, which is furthest from the quantum entity, is empiricallyobserved.

specifying the probability P(t, x) for all the possible states of the quantum sys-tem. For a quantum particle in space, its possible quantum states are the differentpositions x ∈ [−∞,+∞].

The position of the quantum particle is indeterminate and P(t, x) = |ψ(t, x)|2is the probability of the state vector collapsing at time t and at O(x) – the pro-jection operator at position x. The moment that the state ψ(t,�) is observed atspecific projection operator O(x), the state ψ(t,�) instantaneously becomes zeroeverywhere else. The transition from ψ(t,�) to |ψ(t, x)|2 is an expression of thecollapse of the quantum state. It needs to be emphasized that no classical wave un-dergoes a collapse on being observed; the collapse of the state ψ(t,�) is a purelyquantum phenomenon.

The pioneers of quantum mechanics termed it as “wave mechanics” since theNewtonian description of the particle by its trajectory x(t) was replaced by thestate ψ(t,�) that looked like a classical wave that is spread over (all of) space �.Hence the term “wave function” is used by many physicists for denoting ψ(t,�).

The state ψ(t,F) of a quantum particle is not a classical wave; rather, the onlything it has in common with a classical wave is that it is sometimes spread overspace. However, there are quantum states that are not spread over space. For ex-ample, the up and down spin states of a quantum particle exist at a single point;such quantum states are described by a state that has no dependence on space andhence is not spread over space.

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In the text, the terms state, quantum state, state function, or state vector arehenceforth used for ψ(t,F), as these are more precise terms than the term wavefunction.

The result given in Eq. 2.1 is an expression of the great discovery of quantumtheory, namely, that behind what is directly observed – the outcome of experimentsfrom which one can compute the probabilities P(t, x) = |ψ(t, x)|2 – there liesan unobservable world of the probability amplitude that is fully described by thequantum state ψ(t,F).

2.2 The superstructure of quantum mechanics

The description and dynamics of a quantum entity require an elaborate theoreticalframework. The quantum entity is the foundation of the mathematical superstruc-ture that consists of five main constructs:

• The quantum degree of freedom space F .• The quantum state vector ψ(F), which is an element of the linear vector state

space V(F).• The time evolution of ψ(F), given by the Schrödinger equation.• Operators O(F) that act on the state space V(F).• The process of measurement, with repeated observations yielding the expecta-

tion value of the operators, namely Eψ [O(F)].

The five mathematical pillars of quantum mechanics are shown in Figure 2.2.

2.3 Degree of freedom space FRecall that in classical mechanics a system is described by dynamical variables,and its time dependence is given by Newton’s equations of motion. In quantummechanics, the description of a quantum entity starts with the generalization of theclassical dynamical variables and, following Dirac (1999), is called the quantumdegree of freedom.

Degree of freedom

� V(�)

State space Operators Observation

O (�) [O (�)]Eψ

Dynamics

ψ∂ (t,� )∂t

Figure 2.2 The five mathematical pillars of quantum mechanics.

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2.4 State space V(F) 11

The degree of freedom is the root and ground on which the quantum entity isanchored. The degree of freedom embodies the qualities and properties of a quan-tum entity. A single quantum entity, for example the electron, can simultaneouslyhave many degrees of freedom, such as spin, position, angular momentum and soon that all, taken together, describe the quantum entity. The symbol F is taken torepresent all the degrees of freedom of a quantum entity.

A remarkable conclusion of quantum mechanics – validated by experiments –is that a quantum degree of freedom does not have any precise value before itis observed; the degree of freedom is inherently indeterminate and does not havea determinate objective existence before it is observed. One interpretation of thedegree of freedom being intrinsically indeterminate is that it simultaneously hasa range of possible values; the collection of all possible values of the degree offreedom constitutes a space that is denoted by F ; the space F is time independent.

The entire edifice of quantum mechanics is built on the degree of freedom and,in particular, on the space F .

2.4 State space V(F)

In the quantum mechanical framework, a quantum degree of freedom is inherentlyindeterminate and, metaphorically speaking, simultaneously has a range of possiblevalues that constitutes the space F .

Consider an experimental device designed to examine and study the propertiesof a degree of freedom. For a quantum entity that has spin �, the degree of freedomconsists of 2� + 1 discrete points. A device built for observing a spin � systemneeds to have 2�+1 possible distinct outcomes, one for each of the possible valuesof the degree of freedom.

The experiment needs to be repeated many times due to the indeterminacy ofthe quantum degree of freedom. The outcome of each particular experiment iscompletely uncertain and indeterminate, with the degree of freedom inducing thedevice to take any one of its (the device’s) many possible values.3 However, thecumulative result of repeated experiments shows a pattern – for example, withthe device pointer having some positions being more likely to be observed thanothers.

How does one describe the statistical regularities of the indeterminate and uncer-tain outcomes of an experiment carried out on a degree of freedom? As mentionedin Section 2.3, the subject of quantum probability arose from the need to describequantum indeterminacy. A complex valued state vector, also called the state func-tion and denoted by ψ , is introduced to describe the observable properties of the

3 It is always assumed, unless stated otherwise, that a quantum state is not an eigenstate.

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degree of freedom. The quantum state ψ maps the degree of freedom space F tothe complex numbers C, namely

ψ : F → C.

In particular, for the special case of coordinate degree of freedom x ∈ � = F thestate vector ψ is a complex function of x and hence

x ∈ � ⇒ ψ(x) ∈ C.

Noteworthy 2.1 Dirac’s formulation of the quantum state.

• The foundation of the quantum entity is the degree of freedom F ; the quantumstate (state, state vector, and state function) provides an exhaustive description ofthe quantum entity.

• The term state or state vector refers to the quantum state considered as a vector instate space V(F), usually denoted by ψ(t,F).

• In Dirac’s bracket notation, a state vector is denoted by |ψ(t,F)〉 or |ψ〉 in short,and is called a ket vector.

• The dual to the ket vector is denoted by 〈ψ(t,F)| or 〈ψ | in brief and is called abra vector.

• The scalar product of two state vectors |χ〉, |ψ〉 is a complex number ∈ C and isdenoted by the full bracket, namely 〈χ |ψ〉.

• The term state function refers to the components of the state vector and is denotedby 〈x|ψ(t,F)〉 ≡ 〈x|ψt 〉 ≡ ψ(t, x), where x ∈ F , namely x is a representation ofthe degree of freedom F .

For degrees of freedom taking discrete values, Dirac’s bra and ket vectors are nothingexcept the row and column vectors of a finite dimensional linear vector space, withthe bracket of two state vectors being the usual scalar product of two vectors.

When the degree of freedom becomes continuous, Dirac’s notation carries overinto functional analysis and allows for studying questions of the convergence ofinfinite sequences of state vectors that go beyond linear algebra.

One of the most remarkable properties of the quantum state vector |ψ〉 is that itis an element of a state space V that is a linear vector space. The precise structureof the linear vector space V depends on the nature of the quantum degree of free-dom F . From the simplest quantum system consisting of two possible states, to asystem having N degrees of freedom in four dimensional spacetime, to quantumfields having an infinite number of degrees of freedom, there is a linear vector spaceV and a state vector defined for these degrees of freedom.

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2.4 State space V(F) 13

Euclidean space �N is a finite dimensional linear vector space; the linear vectorspaces V that occur in quantum mechanics and quantum field theory are usuallystate spaces that are an infinite dimensional generalization of �N . Infinite dimen-sional linear vector spaces arise in many applications in science and engineering,including the study of partial differential equations and dynamical systems andmany of their properties, such as the addition of vectors and so on, are the general-izations of the properties of finite dimensional vector spaces.

The state vector is an element of a time independent normed linear vector space,namely

|ψ〉 ∈ V(F).

The following are some of the main properties of a vector space V:

1. Since they are elements of a linear vector space, a state vector can be addedto other state vectors. In particular, ket vectors |ψ〉 and |χ〉 are complex valuedvectors of V and can be added as follows

|η〉 = a|ψ〉 + b|χ〉, (2.2)

where a, b are complex numbers ∈ C, and yield another element |η〉 of V .Vector addition is commutative and associative.

2. For every ket vector |ψ〉 ∈ V , there is a dual (bra) vector 〈ψ | that is an elementof the dual linear vector space VD. The dual vector space is also linear andyields the following

〈η| = a∗〈ψ | + b∗〈χ |.The collection of all (dual) bra vectors forms the dual space VD.

3. More formally, VD is the collection of all linear mappings that take elements ofV to C by the scalar product. In mathematical notation

VD : V → C.

The vector space and its dual are not necessarily isomorphic.4

4. For any two ket |ψ〉 and bra 〈η| vectors belonging to V and VD, respectively, thescalar product, namely 〈η|ψ〉, yields a complex number and has the followingproperty:

〈η|ψ〉 = 〈ψ |η〉∗,

4 Two spaces are isomorphic if there is an invertible mapping that maps each element of one space to a(unique) element of the other space.

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where ∗ stands for complex conjugation. The scalar product is linear and yields

〈η|ζ 〉 = a∗〈ψ |ζ 〉 + b∗〈χ |ζ 〉.In particular, 〈ψ |ψ〉 ≡ |ψ |2 is a real number – a fact of far reaching conse-quence in quantum mechanics.

5. One of the fundamental properties of quantum states is that two states are dis-tinct if they are linearly independent. In particular, two states |ψ〉 and |χ〉 arecompletely distinct if and only if they are orthogonal, namely

〈χ |ψ〉 = 0 : orthogonal. (2.3)

2.4.1 Hilbert space

Starting in the 1900s, Hilbert space was studied by David Hilbert, Erhard Schmidt,and Frigyes Riesz as belonging to the class of infinite dimensional function space.The main feature that arises in a Hilbert space is the issue of convergence of aninfinite sequence of elements of Hilbert space, something that is absent in a finitedimensional vector space.

To allow for the probabilistic interpretation of the state vector |ψ〉, all state vec-tors that represent physical systems must have unit norm, that is

〈ψ |ψ〉 ≡ |ψ |2 = 1 : unit norm.

The restriction of the linear vector space V to be a normed vector space defines aHilbert space. For a Hilbert space, the dual state space is isomorphic to the Hilbertspace, namely V � VD, shown in Figure 2.3.

The state space of quantum entities is a Hilbert space. However, there are classi-cal random systems, for example that occur in finance and for quantum dissipativeprocesses, where the state space is not a Hilbert space and in particular leads to adual state space: VD is not isomorphic to the state space V [Baaquie (2004)].

For the continuous degree of freedom F = �, an element of |ψ〉 of Hilbertspace has unit norm and hence yields

〈ψ |ψ〉 ≡ |ψ |2 =∫ +∞

−∞dx|ψ(x)|2 = 1 : unit norm.

2.5 Operators O(F)

The connection of the quantum degree of freedom with its observable and measur-able properties is indirect and is always, of necessity, mediated by the process of

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2.5 Operators O(F) 15

ψ ψ

V =State Space VD � V : isomorphic

Figure 2.3 Hilbert space is a unit norm state space with V � VD .

measurement. A consistent interpretation of quantum mechanics requires that themeasurement process plays a central role in the theoretical framework of quantummechanics.

In classical mechanics, observation and measurement of the physical propertiesplays no role in the definition of the classical system. For instance, a classical parti-cle is fully specified by its position and velocity at time t and denoted by x(t), v(t);it is immaterial whether a measurement is performed to ascertain the position andvelocity of the classical particle; in other words, the position and velocity of theclassical particle x(t), v(t) exist objectively, regardless of whether its position orvelocity is measured or not.

In contrast to classical mechanics, in quantum mechanics the degree of freedomF , in principle, can never be directly observed. All the observable physical prop-erties of a degree of freedom are the result of a process of measurement carried outon the state vector ψ . Operators, discussed in Chapter 3, are mathematical objectsthat represent physical properties of the degree of freedom F and act on the statevector; the action of operators on the state vector is a mathematical representationof the process measuring the physical properties of the quantum entity.

The degree of freedom F and its measurable properties – represented by the op-erators Oi – are separated by the quantum state vector ψ(t,F) [Baaquie (2013e)].An experiment can only measure the effects of the degree of freedom – via the statevector ψ(t,F) – on the operators Oi . Furthermore, each experimental device isdesigned and tailor made to measure a specific physical property of the degree offreedom, represented by an operator Oi .

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ψ

Hilbert Space

O ψ

O

V V

Figure 2.4 An operator O acting on element |ψ〉 of the state space V and mappingit to O|ψ〉.

Every degree of freedom F defines a state space V and operators O that act onthat state space. All operators O are mathematically defined to be linear mappingsof the state space V into itself, shown in Figure 2.4, and yield, for constant a, b

O : |ψ〉 → O|ψ〉 ⇒ O : V → V

O(a|ψ1〉 + b|ψ2〉

)= aO|ψ1〉 + bO|ψ2〉 : linear.

Operators are the generalization of matrices; an arbitrary element of an operatorO is given by

〈χ |O|ψ〉 with |ψ〉 ∈ V, 〈χ | ∈ VD.

The diagonal matrix element of an operator is given by

〈ψ |O|ψ〉 with |ψ〉 ∈ V, 〈ψ | ∈ VD.

Important physical quantities associated with a particle such as its position, mo-mentum, energy, angular momentum, and so on are physical observables that arerepresented by Hermitian operators, discussed in Section 3.3. Physical quantitiesare indeterminate; the best that we can do in quantum mechanics is to measure theaverage value of a physical quantity, termed as its expectation value.

For example, a quantum particle, in general, has no fixed value for its observ-able properties, but only has an average value. For example, the expectation value(average value) of the particle’s position x is given by

E[x] =∫

dxxP (x) =∫

dx x |ψt(x)|2 . (2.4)

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2.5 Operators O(F) 17

The quantum particle’s average value of the position is interpreted as thediagonal values of the position operator x since Eqs. 2.4 and 3.29 yield thefollowing:

E[x] =∫

dx〈ψt |x〉x〈x|ψt〉= 〈ψt |x|ψt〉.

All (Hermitian) operators are linear mappings of V onto itself. Let O be anobservable, which could be the position operator x, or momentum operator p, orthe Hamiltonian operator H and so on. Generically, for an operator O we have

O : V → V .

Hence, an operator is an element of the space formed by the outer product of Vwith its dual VD, that is

O ∈ V ⊗ VD. (2.5)

A fundamental postulate of quantum mechanics that follows from Eq. 2.1 is thefollowing: on repeatedly measuring the value of the observable O in some state|χ〉, the expectation value (average value) of the observable is given by

E[O] ≡ 〈χ |O|χ〉. (2.6)

In other words, the expectation value of the observable is the diagonal value ofthe operator O for the given state |χ〉. The expected value of a physical quantityis always a real quantity, and this is the reason for representing all observables byHermitian operators.

Consider some physical quantity, such as a particle’s position, and let it be repre-sented by a Hermitian operator O with eigenvalues λi and eigenstatesψi defined by

O|χi〉 = λi |χi〉, 〈χi |χj 〉 = δi−j , (2.7)

where, for Hermitian operators the eigenvalues λi are all real. A typical physicalstate can always be expressed as a superposition of the eigenstates of a Hermitianoperator with amplitude ci and can hence be written as

|ψ〉 =∑i

ci |χi〉.

The result of measuring the physical quantity O for the state ψ(x) always resultsin the state function ψ(x) “collapsing” (being projected), with probability |ci |2, toone of the eigenstates of the operator O, say |χi〉 – whose eigenvalue λi is thenphysically observed.

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After repeated measurements on the system – each made in an identical mannerand hence represented by |ψ〉 – the average value of O is given by

Eψ [O] = 〈ψ |O|ψ〉 =∑i

|ci |2〈χi |O|χi〉 =∑i

|ci |2λi. (2.8)

The measured values of the position, energy, momentum, and so on of a quantumparticle are always real numbers. Hence, all physical quantities such as the averageposition, momentum, energy, and so on must correspond to operators that haveonly real eigenvalues, namely, for which all λi are real; this is the reason why allphysical quantities are represented by Hermitian operators.

2.6 The process of measurement

Ignore for the moment details of what constitutes an experimental device. Whatis clear from numerous experiments is that the experimental readings obtainedby observing a quantum entity by the experimental device cannot be explainedby deterministic classical physics and, in fact, require quantum mechanics for anappropriate explanation.

Consider a degree of freedom F ; the existence of a range of possible values ofthe degree of freedom is encoded in its state vector ψ(F). Let physical operatorsO(F) represent the observables of the quantum degree of freedom. Recall the de-gree of freedom cannot be directly observed; instead, what can be measured is theeffect of the degree of freedom on the operators mediated by the state vector ψ(F).

The preparation of a quantum state yields the quantum state ψ(F), which isthen subjected to repeated measurements.

Operators O(F) are the mathematical basis of measurement theory. The ex-perimental device is designed to measure the properties of the operator O(F).Measurement theory requires knowledge of special quantum states, namely theeigenstates χn of the operator O(F), which are defined in Eq. 2.7.

The process of measurement ascertains the properties of the degree of freedomby subjecting it to the experimental device. The measurement is mathematicallyrepresented by applying the operator O(F) on the state of the system ψ(F) andprojecting it to one of the eigenstates of O(F), namely

|ψ(F)〉 → measurement = O(F)|ψ(F)〉 → χn : collapse of state ψ(F).

Applying O(F) on the state vector causes it to collapse to one of O(F)’s eigen-states. The projection of the state vector ψ to one of the eigenstates χn of theoperator O(F) is discontinuous and instantaneous; it is termed as the collapse ofthe state vector ψ . The result of a measurement has to be postulated to lead to

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2.7 The Schrödinger differential equation 19

the collapse of the state vector and is a feature of quantum mechanics that is notgoverned by the Schrödinger equation.

Unlike classical mechanics, where the same initial condition yields the samefinal outcome, in quantum mechanics the same initial condition leads to a widerange of possible final states. The result of identical quantum experiments is in-herently uncertain.5 For example, radioactive atoms, even though identically pre-pared, decay randomly in time precisely according to the probabilistic predictionsof quantum mechanics.

After many repeated observations performed on state ψ(F), all of which in prin-ciple are identical to each other, the experiment yields the average value of thephysical operator O(F), namely

O → measurements on ψ(F) → Eψ [O(F)].The process of measurement cannot be modeled by the Schrödinger equation,

and this has long been a point of contention among physicists. Many theorists holdthat the fundamental equations of quantum mechanics should determine both theevolution of the quantum state as well as the collapse of the state caused by the pro-cess of measurement. As of now, there has been no resolution of this conundrum.

2.7 The Schrödinger differential equation

The discussion so far has been kinematical, in other words, focused on the frame-work for describing a quantum system. One of the fundamental goals of physicsis to obtain the dynamical equations that predict the future state of a system. Thisrequirement in quantum mechanics is met by the Schrödinger partial differentialequation that determines the future time evolution of the state function ψ(t,F),where t parameterizes time. The Schrödinger equation is time reversible.

To exist, all physical entities must have energy; hence, it is reasonable that theHamiltonian operator H should enter the Schrödinger equation. The Hamiltonianoperator H represents the energy of a quantum entity; H determines the form andnumerical range of the possible allowed energies of a given quantum entity. Fur-thermore, energy is the quantity that is conjugate to time, similar to position beingconjugate to momentum and one would consequently expect that H should play acentral role in the state vector’s time evolution. However, in the final analysis, thereis no derivation of the Schrödinger equation from any underlying principle and onehas to simply postulate it to be true.

The Schrödinger equation is expressed in the language of state space and oper-ators and determines the time evolution of the state function |ψ(t)〉, with t being

5 Except, as mentioned earlier, for eigenstates.

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the time parameter. One needs to specify the degrees of freedom of the system inquestion, that in turn specifies the nature of the state space V; one also needs tospecify the Hamiltonian H .

The celebrated Schrödinger equation is given by

−�

i

∂|ψ(t)〉∂t

= H |ψ(t)〉. (2.9)

For the case of the degree of freedom being all the possible positions of a quantumparticle, F = �, in the position basis |x〉, the state vector is

〈x|ψ(t)〉 = ψ(t, x)

and the Schrödinger equation given in Eq. 2.9, yields the following

− �

i〈x| ∂

∂t|ψ(t)〉 = 〈x|H |ψ(t)〉

⇒ − �

i

∂ψ(t, x)

∂t= H(x,

∂

∂x)ψ(t, x), (2.10)

where we note that the Hamiltonian operator acts on the dual basis.For a quantum particle with mass m moving in one dimension in a potential

V (x), the Hamiltonian is given by

H = − �

2m

∂2

∂x2+ V (x) (2.11)

and yields Schrödinger’s partial differential equation

−�

i

∂ψ(t, x)

∂t= − �

2m

∂2ψ(t, x)

∂x2+ V (x)ψ(t, x).

A variety of techniques has been developed for solving the Schrödinger equa-tion for a wide class of potentials as well as for multi-particle quantum systems[Gottfried and Yan (2003)].

Let |ψ〉 be the initial value of the state vector at t = 0 with 〈ψ |ψ〉 = 1. Equation2.9 can be integrated to yield the following formal solution

|ψ(t)〉 = e−itH/�|ψ〉 = U(t)|ψ〉. (2.12)

Similar to the momentum operator translating the state vector in space, as in Eq.3.39, the Hamiltonian H is an operator that translates the initial state vector in time,as in Eq. 2.12. The evolution operator U(t) is defined by

U(t) = e−itH/�, U †(t) = eitH/�

and is unitary since H is Hermitian; more precisely

U(t)U †(t) = I.

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2.7 The Schrödinger differential equation 21

The unitarity of U(t), and by implication the Hermiticity of H , is crucial for theconservation of probability. The total probability of the quantum system is con-served over time since unitarity of U(t) ensures that the normalization of the statefunction is time-independent; more precisely

〈ψ(t)|ψ(t)〉 = 〈ψ |U †(t)U(t)|ψ〉 = 〈ψ |ψ〉 = 1.

The operator U(t) is the exponential of the Hamiltonian H that in many cases, asis the case given in Eq. 2.11, is a differential operator. The Feynman path integralis a mathematical tool for analyzing U(t) and is discussed in Chapter 4.

The Schrödinger equation given in Eq. 2.9 is a linear equation for the state func-tion |ψ(t)〉. Consider two solutions |ψ1(t)〉 and |ψ2(t)〉 of the Schrödinger equa-tion; then their linear combination yields yet another solution of the Schrödingerequation given by

|ψ(t)〉 = α|ψ1(t)〉 + β|ψ2(t)〉, (2.13)

where α, β are complex numbers. The quantum superposition of state vectors givenin Eq. 2.13 is of far reaching significance and in particular leads to the Dirac–Feynman formulation of quantum mechanics discussed in Section 2.9.

The mathematical reason that state vector |ψ(t)〉 is an element of a normed lin-ear vector space is due to the linearity of the Schrödinger equation and yields theresult that all state vectors |ψ(t)〉 are elements of a linear vector space V .

The fact that |ψ(t)〉 is an element of a linear vector space leads to many nonclas-sical and unexpected phenomena such as the existence of entangled states and thequantum superposition principle [Baaquie (2013e)].

The Schrödinger equation has the following remarkable features:

• It is a first order differential equation in time, in contrast to Newton’s equationof motion that is a second order differential equation in time. At t = 0, theSchrödinger equation requires that the initial state function be specified for allvalues of the degree of freedom, namely |ψ(�)〉, whereas in Newton’s law, onlythe position and velocity at the starting point of the particle are required.

• At each instant, Schrödinger’s equation specifies the state function for all valuesof the indeterminate degree of freedom. In contrast, Newton’s law of motionspecifies only the determinate position and velocity of a particle.

• The state vector |ψ(t)〉 is complex valued. In fact, the Schrödinger equation isthe first equation in natural science in which complex numbers are essential andnot just a convenient mathematical tool for representing real quantities.

Quantum mechanics introduces a great complication in the description of Natureby replacing the dynamical variables x, p of classical mechanics, which consistof only six real numbers for every instant of time, by an entire space F of the

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indeterminate degree of freedom; a description of the quantum entity requires, inaddition, a state vector ψ that is a function of the space F . According to Dirac(1999), the great complication introduced by quantum indeterminacy is “offset”by the great simplification due to the linearity of the Schrödinger equation.

2.8 Heisenberg operator approach

Every physical property of a degree of freedom is mathematically realized by aHermitian operator O. Generalizing Eq. 2.8 to time dependent state vectors andfrom Eq. 2.12, the expectation value of an operator at time t , namely O(t), isgiven by

Eψ [O(t)] = 〈ψ(t)|O|ψ(t)〉 = 〈ψ |eitH/�Oe−itH/�|ψ〉= tr

(O(t)ρ) : ρ = |ψ〉〈ψ |. (2.14)

The density matrix ρ is a time-independent operator that encodes the initial quan-tum state of the degree of freedom.

From Eq. 2.14, the time-dependent expectation value has two possible interpre-tations: the state vector is evolving in time, namely, the state vector is |ψ(t)〉 andthe operator O is constant, or equivalently, the state vector is fixed, namely |ψ〉 andinstead, the operator is evolving in time and is given by O(t). The time-dependentHeisenberg operators O(t) are given by

O(t) = eitH/�Oe−itH/�

i�∂O(t)

∂t= [O(t),H ]. (2.15)

In the Heisenberg formulation of quantum mechanics, quantum indeterminacyis completely described by the algebra of Hermitian operators.

All physical observables of a quantum degree of freedom are elements of theHeisenberg operator algebra, and so are the density matrices that encode the initialquantum state of the degree of freedom. Quantum indeterminacy is reflected in thespectral decomposition of the operators in terms of its eigenvalues and projectionoperators (eigenvectors), as given in Eq. 3.21. For example, the single value ofenergy for a classical entity is replaced by a whole range of eigenenergies of theHamiltonian operator for a quantum degree of freedom, with the eigenfunctionsencoding the inherent indeterminacy of the degree of freedom.

The time dependence of the state vector given by the Schrödinger equation isreplaced by the time dependence of the operators given in Eq. 2.15. All expectationvalues are obtained by performing a trace over this operator algebra, namely bytr(ρO(t)) as given in Eq. 2.14.

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2.9 Dirac–Feynman path integral formulation 23

From the aspect of quantum probability, Heisenberg’s operator formulation goesfar beyond just providing a mathematical framework for the mechanics of the quan-tum, but instead, also lays the foundation of the quantum theory of probability[Baaquie (2013e)].

2.9 Dirac–Feynman path integral formulation

The time evolution of physical entities is fundamental to our understanding of Na-ture. For a classical entity evolving in time, its trajectory exists objectively, regard-less of whether it is observed or not, with both its position x(t) and velocity v(t)having exact values for each instant of time t .

We need to determine the mode of existence of quantum indeterminacy for thecase of the time evolution of a quantum degree of freedom.

Consider a quantum particle with degree of freedom x ∈ � = F . Suppose thatthe particle is observed at time ti , with the position operator finding the particle atpoint xi and a second observation is at time tf , with the position operator findingthe particle at point xf . To simplify the discussion, suppose there are N -slits be-tween the initial and final positions, located at positions x1, x2, . . . , xN , as shownin Figure 2.5.

There are two cases for the quantum particle making a transition from xi, ti toxf , tf , namely when the path taken at an intermediate time t is observed and whenit is not observed. For the case when the path taken at an intermediate time t isobserved, one simply obtains the classical result.

Time

tf

tixi

xf

Space

x1t x2 x3 � xN

Figure 2.5 A quantum particle is observed at first at initial position xi at timeti and a second time at final position xf at time tf . The quantum particle’s pathbeing indeterminate means that the single particle simultaneously exists in all theallowed paths.

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What is the description of the quantum particle making a transition from xi, ti

to xf , tf when it is not observed at an intermediate time t? The following is asummary of the conclusions:

• The quantum indeterminacy of the degree of freedom, together with the linearityof the Schrödinger equation, leads to the conclusion that the path of the quantumparticle is indeterminate.

• The indeterminacy of the path is realized by the quantum particle by existing inall possible paths simultaneously; or metaphorically speaking, the single quan-tum particle simultaneously “takes” all possible paths.

• For the case of N -slits between the initial and final positions shown in Figure2.5, the quantum particle simultaneously exists in all the N -paths.

The concept of the probability amplitude, which is a complex number, is usedfor describing the indeterminate paths of a quantum system.

To start with, a probability amplitude is assigned to each determinate path. Inthe case of no observation being made to determine which path was taken, allthe paths are indistinguishable and hence the particle’s path is indeterminate, withthe particle simultaneously existing in all the N -paths, as shown in Figure 2.5.The probability amplitude for the quantum particle having an indeterminate pathis obtained by combining the probability amplitudes for the different determinatepaths using the quantum superposition principle.

Let probability amplitude φn be assigned to the determinate path going througha slit at xn with n = 1, 2, . . . N , as shown in Figure 2.5, and let φ(xf , tf |xi, ti)be the net probability amplitude for a particle that is observed at position xi attime ti and then observed at position xf at later time tf . The probability ampli-tude φ(xf , tf |xi, ti) for the transition is obtained by superposing the probabilityamplitudes for all indistinguishable determinate paths and yields

φ(xf , tf |xi, ti) =N∑n=1

φn : indistinguishable paths. (2.16)

Once the probability amplitude is determined, its modulus squared, namely |φ|2yields the probability for the process in question. For the N -slit case

|φ(xf , tf |xi, ti)|2 = P(xf , tf |xi, ti),∫

dxf P (xf , tf |xi, ti) = 1,

where P(xf , tf |xi, ti) is the conditional probability that a particle, observed at po-sition xi at time ti , will be observed at position xf at later time tf .

Quantum mechanics can be formulated entirely in terms of indeterminate paths,a formulation that is independent of the framework of the state vector and the

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2.10 Three formulations of quantum mechanics 25

Schrödinger equation; this approach, known as the Dirac–Feynman formulation,is discussed in Chapter 4.

2.10 Three formulations of quantum mechanics

In summary, quantum mechanics has the following three independent, but equiva-lent, mathematical formulations for describing quantum indeterminacy:

• The Schrödinger equation for the state vector postulates that the quantum statevector encodes all the information that can be extracted from a quantum degreeof freedom. The degree of freedom forever remains indeterminate since all mea-surements only encounter the quantum state vector, causing it to collapse to anobserved manifestation.

• The Heisenberg operator formalism. The state vector is completely dispensedwith and instead a density matrix, which is an operator, represents the quantumentity. All observations consist of detecting the collapse of the density matrix,which makes a transition from the pure to a mixed density matrix; the detectionof the mixed density matrix by projection operators results in the experimentaldetermination of the probability of the various projection operators detecting thequantum entity.

Quantum probability assigns probabilities to projection operators. The in-determinate nature of the degree of freedom is reflected in that it is neverdetected by any of the operators. The violation of the Bell-inequality showsthat the quantum indeterminacy cannot be explained by classical probabilitytheory; in particular, the degree of freedom has no determinate value beforean observation – and hence no objective existence – showing its indeterminatenature.6

• The Dirac–Feynman path integral formulation. The path integral is the sumover all the indeterminate (indistinguishable) paths, from the initial to the finalstate, and reflects quantum indeterminacy which is at the foundation of quan-tum mechanics. The state vector appears as initial and final conditions for theindeterminate paths that are being summed over.

In the path integral approach, the quantum degrees of freedom appear as in-tegration variables and provide the clearest representation of the indeterminatenature of the degree of freedom. An integration variable has no fixed value but,rather, takes values over its entire range; for the degree of freedom this meansthat the entire degree of freedom space F is integrated over. The freedom tochange variables for path integration is equivalent to changing the representation

6 Quantum probability is fundamentally different from classical probability. The difference was crystallized bythe ground-breaking work of Bell (2004) and is discussed in detail by Baaquie (2013e).

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chosen for the degree of freedom, and is similar to the freedom in choosing basisstates for Hilbert space.

Each framework has its own advantages, throwing light on different aspects ofquantum mechanics that would otherwise be difficult to express. For example, theSchrödinger equation is most suitable for studying the bound sates of quantum en-tities such as atoms and molecules; the Heisenberg formulation is most suitable forstudying the measurement process; and the Feynman path integral is most appro-priate for studying the indeterminate quantum paths.

2.11 Quantum entity

In light of the mathematical superstructure of quantum mechanics, what is a quan-tum entity? A careful study of what is an entity, a thing, an object leads to theremarkable conclusion that the quantum entity is intrinsically indeterminate andits description requires a framework that departs from our classical conception ofNature.

The quantum entity’s foundation is its degree of freedom F and quantum inde-terminacy is due to to the intrinsic indeterminacy of the degree of freedom. A land-mark step, taken by Max Born, was to postulate that quantum indeterminacy canbe described by a state vector ψ(F) that obeys the laws of quantum probability.The state vector is inseparable from the degree of freedom and encodes all the in-formation that can be obtained from the indeterminate degree of freedom, and isillustrated in Figure 2.6.

The state vector ψ(F) encompasses the degree of freedom, but does not do soin physical space; rather, Figure 2.6 illustrates the fact that all observations carried

� ψ( � )

Figure 2.6 A quantum entity is constituted by its degree of freedom F and thestate vector ψ(F) that permanently encompasses and envelopes its degrees offreedom.

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2.12 Summary: quantum mathematics 27

out on the degree of freedom always encounter the state vector and no observationcan ever come into direct “contact” with the degree of freedom itself. All “contact”of the measuring device with the degree of freedom is mediated by the state vector.

In brief, quantum mechanics provides the following as a definition of the quan-tum entity: A quantum entity is constituted by a pair, namely the degree(s) of free-dom F and the state vector ψ(F) that encodes all of its properties. This inseparablepair, namely the degree of freedom and the state vector, embodies the condition inwhich the quantum entity exists.

2.12 Summary: quantum mathematics

Classical physics is based on explaining the behavior of Nature based on attribut-ing mathematical properties directly to the observed phenomenon; for example,a tangible force acts on a particle and changes its position. The logic of quan-tum mechanics is quite unlike classical physics. An elaborate mathematical super-structure connects the experimentally observed behavior of the particle’s degree offreedom – enigmatically enough the degree of freedom can never in principle everbe empirically observed – with its mathematical description [Baaquie (2013e)].

All our understanding of a quantum entity is based on theoretical and mathe-matical concepts that, in turn, have to explain a plethora of experimental data. Inthe case of quantum mechanics, the mathematical construction has led us to inferthe existence of the quantum degree of freedom. The theoretical constructions ofquantum mechanics are far from being arbitrary and ambiguous; on the contrary,given the maze of links from the quantum entity to its empirical properties, it ishighly unlikely that there are any major gaps or redundancies in the theoreticalsuperstructure of quantum mechanics.

Quantum mechanics and quantum field theory – bedrocks of theoretical physicsand of modern technology – synthesize a vast range of mathematical disciplinesthat constitutes its mathematical foundations and has given rise to the disciplineof quantum mathematics. Quantum mathematics includes such diverse mathemati-cal fields as calculus, linear algebra, functional analysis and functional integration,probability and information theory, dynamical systems, logic, combinatorics andgraph theory, Lie groups and representation theory, differential and algebraic ge-ometry, topology, knot theory, and number theory, to name a few.

The relation of quantum mathematics to quantum mechanics is analogous to theconnection of calculus to Newtonian mechanics: although calculus was discoveredby Newton for explaining classical mechanics, calculus as a discipline goes farbeyond Newtonian mechanics – having applications in almost every branch of sci-ence. Similarly, it is worth noting that quantum mathematics is a discipline that

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is far greater than quantum mechanics – with possible applications in all fields ofscience as well as the social sciences that are based on uncertainty and randomness.

Quantum mathematics describes random, uncertain and indeterminate systemsusing the concept of the degree of freedom, which in turn defines a linear vectorstate space; the dynamics of the degrees of freedom is determined by the analogof the Hamiltonian or the Lagrangian, which are defined on the state space. Theexpectation values of random quantities – which are functions of the degrees offreedom – can be obtained by using either the techniques of operators and statespace or by employing the Feynman path integral (functional integration) that en-tails summing over all possible configurations of the degrees of freedom.

A leading example of quantum mathematics is the explanation of critical phe-nomena. Classical random systems undergoing phase transitions – such as a pieceof iron becoming a magnet when it is cooled – are examples of critical phenom-ena and are described by classical statistical mechanics. Wilson (1983) solved theproblem of classical phase transitions by describing it as a system that has infinitelymany degrees of freedom and which is mathematically identical to a (Euclidean)quantum field theory. Experiments later validated the explanation of critical clas-sical systems by quantum mathematics, and in particular by the mathematics ofquantum field theory.

In fact, based on the common ground of quantum mathematics, there is a twoway relation between classical random systems and quantum mechanics. For ex-ample, the work of Wilson (1983) showed that all renormalizable quantum fieldtheories, in turn, are mathematically equivalent to classical systems that undergosecond order phase transitions.

Phase transitions are mathematically described by quantum field theories inEuclidean time. If one restricts quantum mathematics to quantum mechanics, thenone may ask questions such as “is probability conserved in phase transitions?” –questions that are clearly meaningless since systems undergoing phase transitionsare in equilibrium and hence there is no concept of time evolution in phase tran-sitions. Instead, using quantum mathematics, Wilson (1983) computed classicalquantities such as critical exponents that characterize phase transitions, exponentsthat can be experimentally measured [Papon et al. (2002)].

From the example of phase transitions it can be seen that the symbols of quan-tum mathematics, when applied to other fields such as finance [Baaquie (2004),Baaquie (2010)], the human psyche [Baaquie and Martin (2005)], the social sci-ences [Haven and Khrennikov (2013)] and so on, have interpretations that are quitedifferent from quantum mechanics The interpretations of quantum mathematics inthese diverse fields have no fixed prescription but, instead, have to be arrived atfrom first principles [Baaquie (2013a)].

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2.12 Summary: quantum mathematics 29

The main thrust of the remaining chapters is on the mathematics of quantummechanics, leaving aside questions of how these mathematical results are appliedto physics, finance, and other disciplines. Various models are analyzed to developthe myriad and multi-faceted principles and methods of quantum mathematics.

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3

Operators

Operators represent physically observable quantities, as discussed in Section 2.5.The structure and property of operators depend on the nature of the degree of free-dom; operators act on the state space and in particular on the state vector of a givendegree of freedom. The significance of operators in the interpretation of quantummechanics has been discussed in Baaquie (2013e).

The operators discussed in this chapter are mostly based on the continuous de-gree of freedom, which is analyzed in Section 3.1. Hermitian operators representphysically observable properties of a degree of freedom and their mathematicalproperties are defined in Section 3.3. The coordinate and momentum operators arethe leading exemplar of a pair of noncommuting Hermitian operators and these arestudied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, afinite dimensional example of the shift and scaling operators; Section 3.8 providesa unitary representation of the coordinate and momentum operators.

The term self-adjoint operator is used for Hermitian operators when there is aneed to emphasize the importance of the domain of the Hilbert space on which theoperators act – a topic not usually discussed in most books on quantum mechanics.Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particularthe crucial role played by the domain for realizing the property of self-adjointness.It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.

3.1 Continuous degree of freedom

Continuous and discrete degrees of freedom occur widely in quantum mechanics.An in-depth analysis of a discrete degree of freedom is presented in Chapter 8. Inthis chapter, the focus is on analysis of a continuous degree of freedom and its statespace and operators. The structure of the continuous degree of freedom is seen to

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3.1 Continuous degree of freedom 31

88 +– a

−2a 2a−a 0 a

Figure 3.1 Discretization of a continuous degree of freedom space F = �.

emerge naturally by taking the continuum limit of an underlying system consistingof a discrete degree of freedom.

Consider a quantum particle that can be detected by the position projection oper-ators at any point of space; to simplify the discussion suppose the particle can movein only one dimension and hence can be found at any point x ∈ [−∞,+∞] = �.Hence, the degree of freedom is F =� and the specific values of the degree of free-dom x constitute a real continuous variable. Since there are infinitely many pointson the real line, the quantum particle’s degree of freedom has infinitely many pos-sible outcomes.

As shown in Figure 3.1, let the continuous degree of freedom x, −∞ ≤ x ≤+∞, take only discrete values at points x = na with lattice spacing a and withn = 0,±1,±2, . . .; in other words, the lattice is embedded in the continuous line� and the lattice point n identified with the point na in �. To obtain the continuousposition degree of freedom F , let a → 0 and the allowed values of the particle’sposition x can take any real value, that is, x ∈ �, and hence F → �.

The discrete basis vectors of the quantum particle’s state space V are representedby infinite column vectors with the only nonzero entry being unity in the nth posi-tion. Hence

|n〉 : n = 0,±1,±2, . . .±∞,

where, more explicitly

|n〉 =

⎡⎢⎢⎢⎢⎣. . .

010. . .

⎤⎥⎥⎥⎥⎦ : nth position.

The basis vectors for the dual state space VD are given by

〈n| = [· · · 0 1 0 · · · ]⇒ 〈n|m〉 = δn−m. (3.1)

The completeness of the basis states yields the following:+∞∑

n=−∞|n〉〈n| = diagonal(. . . , 1, 1, . . .) = I : completeness equation,

where I above is the infinite dimensional unit matrix.

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32 Operators

The limit of a → 0 needs to be taken to obtain a continuous x; in terms of theunderlying lattice, the continuous point x is related to the discrete lattice point n by

−∞ ≤ x ≤ +∞ : x = lima→0[na], n = 0,±1,± . . . .±∞.

The state vector for the particle is given by the ket vector |x〉, with its dual vectorgiven by the bra vector 〈x|. The basis state |n〉 is dimensionless; the ket vector|x〉 has a dimension of 1/

√a since, from Eq. 3.14, the Dirac delta function has

dimension of 1/a. Hence, due to dimensional consistency

|x〉 = lima→0

1√a|n〉, 〈x| = lim

a→0

1√a〈n|. (3.2)

The position projection operator is given by the outer product of the position ketvector with the bra vector and is given by

|x〉〈x| = lima→0

1

a|n〉〈n|. (3.3)

The scalar product, for x = na and x ′ = ma, in the limit of a → 0, is given,from Eqs. 3.1, 3.2, and 3.14, by the Dirac delta function

〈x|x ′〉 = lima→0

1

aδm−n ⇒ 〈x|x ′〉 = δ(x − x ′). (3.4)

The completeness equation above has the following continuum limit:

I =+∞∑

n=−∞|n〉〈n| = lim

a→0a

+∞∑n=−∞

|x〉〈x| (3.5)

⇒∫ ∞

−∞dx|x〉〈x| = I : completeness equation. (3.6)

Equation 3.5 shows that the projection operators given in Eq. 3.3 are complete andspan the entire state space V .I is the identity operator on state space V; namely for any state vector |ψ〉 ∈ V ,

it follows from the completeness equation that

I|ψ〉 = |ψ〉.The completeness equation given by Eq. 3.6 is a key equation that is central to

the analysis of state space, and yields

〈x|I|x ′〉 =∫ ∞

−∞dz〈x|z〉〈z|x ′〉 =

∫ ∞

−∞dzδ(x − z)δ(z− x ′) = δ(x − x ′),

that follows from the definition of the Dirac delta function δ(x − x ′). The aboveequation shows that δ(x − x ′) is the matrix element of the identity operator I forthe continuous degree of freedom F = � in the x basis.

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3.1 Continuous degree of freedom 33

The state space V(F) of a continuous degree of freedom F is a function spaceand it is for this reason that the subject of functional analysis studies the mathe-matical properties of quantum mechanics.

For the case of F = �, the state vector |f 〉 is an element of V(�) and yields astate function f (x) given by f (x) = 〈x|f 〉; hence all functions of x, namely f (x),can be thought of as elements of a state space V(�). Being an element of a statespace endows the function f (x) with the additional property of linearity that needsto be consistent with all the other properties of f (x). It should be noted that not allfunctions are elements of a (quantum mechanical) state space.

Noteworthy 3.1 Dirac delta function

The Dirac delta function is useful in the study of continuum spaces, and some of itsessential properties are reviewed. Dirac delta functions are not ordinary Lebesguemeasureable functions since they have support set with measure zero; rather they aregeneralized functions also called distributions. In essence, the Dirac delta function isthe continuum generalization of the discrete Kronecker delta function.

Consider a continuous line labelled by coordinate x such that −∞ ≤ x ≤ +∞,and let f (x) be an infinitely differentiable function. The Dirac delta function, denotedby δ(x − a), is defined by the following:

δ(x − a) = δ(a − x) : even function,

δ(c(x − a)) = 1

|c|δ(x − a),∫ +∞

−∞dxf (x)δ(x − a) = f (a), (3.7)∫ +∞

−∞dxf (x)

dn

dxnδ(x − a) = (−1)n

dn

dxnf (x)|x=a. (3.8)

The Heaviside step function �(t) is defined by

�(t) =⎧⎨⎩

1 t > 012 t = 00 t < 0

. (3.9)

From its definition �(t)+�(−t) = 1. The following is a representation of theδ-function: ∫ b

−∞δ(x − a) = �(b − a), (3.10)

⇒∫ a

−∞δ(x − a) = �(0) = 1

2, (3.11)

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34 Operators

where the last equation is due to the Dirac delta function being an even function.From Eq.(3.10)

d

db�(b − a) = δ(b − a).

A representation of the delta function based on the Gaussian distribution is

δ(x − a) = limσ→0

1√2πσ 2

exp{− 1

2σ 2 (x − a)2}. (3.12)

Moreover

δ(x − a) = limμ→∞

1

2μ exp

{−μ|x − a|}.The definition of Fourier transform yields a representation of the Dirac delta functionthat is widely used in various chapters for representing the payoff of financialinstruments. It can be shown that

δ(x − a) =∫ +∞

−∞dp

2πeip(x−a). (3.13)

A proof of Eq. 3.13 is found in the book on quantum mechanics by Landau andLifshitz (2003).One can perform the following consistency check of Eq. 3.13.Integrate both sides of Eq. 3.13 over x as follows:

L.H.S =∫ +∞

−∞dxe−ikxδ(x − a) = e−ika,

R.H.S =∫ +∞

−∞dxe−ikx

∫ +∞

−∞dp

2πeip(x−a)

=∫ +∞

−∞dp

2πe−ipa2πδ(p − k) = e−ika,

where Eq. 3.13 was used in performing the x integration for the right hand side.Hence, one can see that Eq. 3.13 is self-consistent.

To make the connection between the Dirac delta function and the discreteKronecker delta function consider the discretization of the continuous line into adiscrete lattice with spacing a. As shown in Figure 3.1, the continuous degree offreedom x, −∞ ≤ x ≤ +∞, takes only discrete values at points x = na withn = 0,±1,±2, . . . The discretization of Eq. 3.7, for x = na and y = ma, yields

δn−m ={

0 n �= m

1 n = m.

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3.2 Basis states for state space 35

Discretize continuous variable x into a lattice of discrete points x = nε, and leta = mε; then f (x)→ fn. Discretizing Eq.(3.7) gives∫ +∞

−∞dxf (x)δ(x − a)→ ε

+∞∑n=−∞

fnδ(xn − am) = fm =+∞∑

n=−∞δn−mfm

⇒ δ(x − a)→ 1

εδn−m.

Taking the limit of ε → 0 in the equation above yields

δ(x − a) = limε→0

1

εδn−m =

{0 x �= a

∞ x = a. (3.14)

3.2 Basis states for state space

The bra and ket vectors 〈x| and |x〉 are the basis vectors of VD and V respectively.For the infinite dimensional state space, a complete basis set of vectors must satisfythe completeness equation, which for the co-ordinate basis |x〉 is given by Eqs. 3.6and 3.4, namely ∫ ∞

−∞dx|x〉〈x| = I, 〈x|x ′〉 = δ(x − x ′).

In general, state vectors |ψn〉 – with components given by ψn(x) = 〈x|ψn〉 – forma complete basis if

+∞∑n=−∞

|ψn〉〈ψn| = I ⇒+∞∑

n=−∞ψn(x) ψ

∗n (x

′) = δ(x − x ′).

The completeness equation is also referred to as the resolution of the identity sinceonly a complete set of basis states can yield the identity operator on state space.

An element of the state space V is a ket vector |ψ〉, and can be thought of as aninfinite dimensional vector with components given by ψ(x) = 〈x|ψ〉. The vector|ψ〉 has the following representation in the |x〉 basis:

|ψ〉 =∫ ∞

−∞dx|x〉〈x|ψ〉 =

∫ ∞

−∞dxψ(x)|x〉, ψ(x) = 〈x|ψ〉. (3.15)

The vector |ψ〉 can be mapped to a unique dual vector denoted by 〈ψ | ∈ VD; incomponents ψ∗(x) = 〈ψ |x〉 and

〈ψ | =∫ ∞

−∞dx〈ψ |x〉〈x| =

∫ ∞

−∞dxψ∗(x)〈x|, ψ∗(x) = 〈ψ |x〉.

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36 Operators

Note that the state vector and its dual are related by complex conjugation, namely

〈χ |ψ〉 = 〈ψ |χ〉∗ ⇒ 〈x|ψ〉 = 〈ψ |x〉∗. (3.16)

The scalar product of two state vectors is given by1

〈χ |ψ〉 ≡∫

dxχ∗(x)ψ(x).

The vector |ψ〉 and its dual 〈ψ | have the important property that they define the“length” 〈ψ |ψ〉 of the vector. The completeness equation, Eq. 3.6, yields

〈ψ |ψ〉 =∫ ∞

−∞dxψ(x)∗ψ(x) ≥ 0.

3.3 Hermitian operators

An operator acting on state space defines a linear mapping of the state space V ontoitself, and for a Hilbert space is an element of the tensor product space V ⊗ VD.

For a two-state system, the state space is a two dimensional Euclidean spaceand operators are 2 × 2 complex valued Hermitian matrices. Operators on linearvector state space are infinite dimensional generalizations of N ×N matrices, withN →∞ and have new properties that are absent in finite matrices.

The Hermitian conjugate of a matrix M is defined by M†ij ≡ M∗

j i . Similar toa matrix, the Hermitian conjugate of an arbitrary operator O, denoted by O†, isdefined by

〈ψ |O†|χ〉 ≡ 〈χ |O|ψ〉∗ : Hermitian conjugation.

An operator is Hermitian if the Hermitian conjugate operator is equal to the oper-ator itself, that is, if

O† = O ⇒ 〈ψ |O†|χ〉 ≡ 〈χ |O|ψ〉∗. (3.17)

One of the reasons for studying the Hermitian conjugate operator is because onecan ascertain the state space on which its conjugate acts. It is not enough thatthe form of a Hermitian operator be invariant under conjugation, as in Eq. 3.17.For self-adjoint (Hermitian) operators, it is also necessary that the domains of theoperator and its conjugate be isomorphic, and this is discussed in Section 3.10.

1 A more direct derivation of the completeness equation is the following:

〈χ |ψ〉 = 〈χ |{∫ ∞−∞

dx|x〉〈x|}|ψ〉 ⇒ I =∫ ∞−∞

dx|x〉〈x|.

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3.3 Hermitian operators 37

However, for non-Hermitian Hamiltonians, and these are the ones that occur inthe acceleration (higher derivative) Lagrangian discussed in Chapter 14, as well asfor non-Hermitian Hamiltonians describing classical random systems, the differ-ence between the domains of the operator and its conjugate is non-trivial.

Note that all the diagonal elements of a Hermitian operator O are real, since forany arbitrary state vector |ψ〉, the diagonal element is

〈ψ |O|ψ〉 = 〈ψ |O†|ψ〉 = 〈ψ |O|ψ〉∗ : real.

Furthermore, similar to matrices, the Hermitian adjoint of a sum and products ofoperators is given by

(A+ B + . . .)† = A† + B† . . . , (AB . . .)† = . . . B†A†.

The trace operation for an operator O, similar to matrices, is defined as a sumof all its “diagonal elements.” To make this statement more precise, one needs aresolution of the identity operator on state space V . Consider for concreteness, thecontinuous degree of freedom with the completeness equation given by Eq. 3.6 asfollows:

I =∫ ∞

−∞dx|x〉〈x|.

Trace is a linear operation on O and is defined by

tr(O) = tr(OI) =∫ ∞

−∞dx tr

(O|x〉〈x|

)=∫ ∞

−∞dx〈x|O|x〉. (3.18)

Trace operation is defined in Eq. 3.18, and is summarized below.

tr(A+ B + . . .) = tr(A)+ tr(B) . . .

tr(A†) = tr∗(A)tr(ABC) = tr(CBA) : cyclic.

A unitary operator, the generalization of the exponential function exp iφ, is givenin terms of a Hermitian operator O by

U = eiφO ⇒ UU † = I

V = 1− iaO1+ iaO ⇒ VV † = I.

3.3.1 Eigenfunctions; completeness

Consider a Hermitian operator O. All Hermitian operators have the following im-portant properties:

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38 Operators

• The eigenfunctions are a very special set of state functions that are only rescaledwhen the operator acts on them and are given, as in Eq. 2.7, by

O|ψn〉 = λn|ψn〉,where λn are eigenvalues of the operator O.

• The eigenfunctions are orthonormal and complete, namely

〈ψn|ψn〉 = δn−m,∑n

|ψn〉〈ψn| = I. (3.19)

Every state vector |ψ〉 has an expansion in terms of the basis states given by

|ψ〉 = I|ψ〉 =∑n

〈ψn|ψ〉|ψn〉. (3.20)

• The Hilbert space and its dual are isomorphic and hence O ∈ V ⊗VD ≡ V ⊗V;the spectral resolution of the Hermitian operator yields

O =∑n

λn|ψn〉〈ψn|, (3.21)

O† =∑n

λ∗n (|ψn〉〈ψn|)† = O.

In other words, an operator is completely equivalent to the set of all of its eigen-functions and eigenvalues.

• Hermitian operator O is represented by an experimental device. Experimentalobservations carried out on the state |ψ〉 function are represented by the operatoracting on the quantum state, namely O|ψ〉. Repeated observations cause thestate |ψ〉 to collapse to one of the eigenfunctions |ψn〉 of O with probability|〈ψn|ψ〉|2. This procedure is the mathematical basis for making measurementson a quantum entity, as discussed in Section 2.6.

Functions of Hermitian operators are fundamental to a quantum system; for anarbitrary operator valued function f (O), the spectral resolution given in Eq. 3.21yields

f (O) =∑n

f (λn)|ψn〉〈ψn|, (3.22)

where f (λn) is an ordinary numerical valued function of the eigenenergies λn. Ifthe function f (λ) = f ∗(λ) : real, then from its definition, for Hermitian operator O

f †(O) = f (O) : Hermitian.

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3.3 Hermitian operators 39

3.3.2 Hamiltonian for a periodic degree of freedom

The eigenfunctions of every Hermitian operator yield a resolution of the identity asgiven in Eq. 3.19. This result is derived for the Hamiltonian operator of a particlefreely moving on a circle S1 with radius L.

The free particle Hamiltonian operator H , with the potential V (x) = 0, is given,from Eq. 2.11, by

H = − �

2m

∂2

∂x2. (3.23)

Since x ∈ S1 is defined on a periodic domain [0, 2πL], the degree of freedom hasthe periodicity

x = x + 2πL.

Consider the normalized eigenfunctions of H given by

Hψn(x) = EnψE(x),

∫ 2πL

0dx|ψn(x)|2 = 1.

The ground state (having lowest energy) is nondegenerate and is given by

ψ0(x) = 1√2πL

, E0 = 0.

All the other energy eigenfunctions are two fold-degenerate and given by

Hψ±n(x) = Enψ±n(x), En = n2

2mL2,

ψ±n(x) = 1√2πL

e±inxL , n = 1, 2, . . .+∞.

From the spectral decomposition of Hermitian operators given in Eq. 3.21, theHamiltonian has the representation

H =∑n

En|ψn〉〈ψn| = 1

4πmL3

+∞∑n=1

n2(|ψ+n〉〈ψ+n| + |ψ−n〉〈ψ−n|

). (3.24)

The general result given in Eq. 3.19 yields that the eigenfunctions of H arecomplete and hence provide the completeness equation

I = |ψ0〉〈ψ0| ++∞∑n=1

(|ψ+n〉〈ψ+n| + |ψ−n〉〈ψ−n|). (3.25)

Note that the completeness equation requires that all the eigenfunctions, includingall the degenerate eigenfunctions, of the Hermitian operator, be included in the

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40 Operators

resolution of the identity operator. To prove the completeness equation, considerthe following expression:

〈x ′|{|ψ0〉〈ψ0| ++∞∑n=1

(|ψ+n〉〈ψ+n| + |ψ−n〉〈ψ−n|)}|x〉

= 1

2πL(1+

+∞∑n=1

(einL(x′−x) + e−i

nL(x′−x)))

= 1

2πL

+∞∑n=−∞

einL(x′−x) =

+∞∑n=−∞

δ(x − x ′ + 2πnL

)= δ

(x − x ′

)since x, x ′ ∈ [0, 2πL].

To obtain the final expression requires Poisson’s summation formula

1

2πL

+∞∑n=−∞

einL(x′−x) =

+∞∑n=−∞

δ(x − x ′ + 2πnL

). (3.26)

Taking the limit ofL→∞ yields the result for x ∈ [−∞,+∞], namely that thecompleteness equation given in Eq. 3.25 for a circle converges to the unit operatoron the real line � given in Eq. 3.6,

limL→∞

1

2πL

+∞∑n=−∞

einL(x′−x) = 1

2π

∫ +∞

−∞dpeip(x

′−x) = δ(x ′ − x). (3.27)

The limit of L → ∞ shows that the eigenfunctions of the Hamiltonian H forthe degree of freedom being a real line � have a divergent normalization. To makethe normalization finite for �, one can use various methods, including making thedegree of freedom periodic to obtain a finite normalization for the eigenfunctions;one then needs to take the limit of the circle having an infinite radius to obtain thereal line �.

3.4 Position and momentum operators x and p

A quantum particle has a continuous (real) degree of freedom x ∈ �. The statespace consists of all functions of the single variable x, namely V = {ψ(x)|x ∈ �},where 〈x|ψ〉 = ψ(x).

One of the most important observables is the Hermitian coordinate operator xthat represents the coordinate degree of freedom on the state space of the quantumparticle. The operator notation x is often simplified to x if there is no ambiguity inits meaning.

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3.4 Position and momentum operators x and p 41

The action of operator x is defined by the multiplication of the state functionψ(x) ∈ V by x, that is

xψ(x) ≡ xψ(x).

The operator x has a continuous spectrum of eigenvalues and eigenstates.Similarly to an N × N matrix M that is fully specified by its matrix elements

Mij , i, j = 1, . . . , N , an operator is also specified by its matrix elements. In thenotation of Dirac

xψ(x) ≡ 〈x|x|ψ〉 = x〈x|ψ〉 = xψ(x).

In other words, the matrix element 〈x|x|ψ〉 of the operator x is given by xψ(x).Let us choose the function |ψ〉 = |x ′〉 that yields

〈x|x|x ′〉 = x〈x ′|x ′〉 = xδ(x − x ′)⇒ x|x ′〉 = x ′|x ′〉. (3.28)

From the above it follows that the observable x has eigenfunctions |x〉 with eigen-values x ∈ �; hence the spectral resolution of observable x and its completenessequation are given by

x =∫ ∞

−∞dx|x〉x〈x|,

∫ ∞

−∞dx|x〉〈x| = I. (3.29)

The second equation above is the completeness equation given earlier in Eq. 3.6.For N particles in three-dimensions, one has the following straightforward gen-

eralization of the coordinate operator x and completeness equation:

x = [x1 ⊗ y1 ⊗ z1

]⊗ [x2 ⊗ y2 ⊗ z2]. . .⊗ [xN ⊗ yN ⊗ zN

](3.30)

IN =∫ ∞

−∞dx1dy1dz1 . . . dxNdyNdzN

× |x1, y1, z1〉 ⊗ 〈x1, y1, z1| . . . |xN, yN, zN 〉 ⊗ 〈xN, yN, zN |,where |x1, y1, z1〉 = |x1〉|y1〉|z1〉, 〈x1, y1, z1| = 〈x1|〈y1|〈z1| and so on.

3.4.1 Momentum operator p

Momentum is a central concept in classical physics and important classicalquantities such as energy and angular momentum depend on momentum. Since thequantum particle’s state function ψ(x) depends only on x, how do we define thequantum generalization of classical momentum p = mdx/dt , where a particle’smass is m?

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42 Operators

In Section 6.7, it is shown that in the path integral framework, the momentumof a particle continues to be p = mdx/dt , but with the caveat that p is an inde-terminate (uncertain) quantity. The path integral is one way of defining a quantumsystem; another equivalent method is to define momentum to be an operator p onstate space V .

There are many ways of motivating the definition of the momentum operatorbut in the final analysis one has to postulate the definition, as there is no way of“deriving” this result from classical physics. Of course, the final test of whether thepostulate is correct is experiment, and the definition adopted has been rigorouslytested experimentally.

The momentum operator p is postulated to be2

p = −i� ∂

∂x. (3.31)

Note Planck’s constant � enters due to dimensional consistency, but its actual em-pirical value is fixed by Nature and has to be obtained by doing an appropriateexperiment.

Consider a particle moving in one dimension. The differential operator ∂/∂xmaps ψ(x) ∈ V to its derivative ∂ψ(x)/∂x ∈ V . The momentum operator p =−i�∂/∂x, in Dirac’s notation, is given by

〈x|p|ψ〉 = −i�〈x| ∂∂x|ψ〉 = −i�∂ψ(x)

∂x. (3.32)

From Eq. 3.17, a Hermitian operator satisfies the following

p† = p ⇒ 〈ψ |p†|χ〉 ≡ 〈χ |p|ψ〉∗ = 〈ψ |p|χ〉. (3.33)

Doing an integration by parts yields the following proof that p is Hermitian:

〈χ |p|ψ〉∗ =[−∫ +∞

−∞dxχ∗(x)i�

∂ψ(x)

∂x

]∗= −

∫ +∞

−∞dxψ∗(x)i�

∂χ(x)

∂x= 〈ψ |p|χ〉.

The eigenfunctions of p, in the notation of Eq. 3.31, are given by

p|p〉 = p|p〉with completeness equation ∫ +∞

−∞dp

2π�|p〉〈p| = I. (3.34)

2 From Eq. 3.30, since the coordinate operator for the 3N degrees of freedom is a tensor product of the singledegree of freedom, it is sufficient to define the momentum operator for one dimension and build up themomentum for the 3N degrees of freedom by an appropriate tensor product.

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3.5 Weyl operators 43

The operator p, from the completeness equation of Hermitian operators given inEq. 3.21, has the representation

p =∫ ∞

−∞dp

2π�|p〉p〈p|. (3.35)

The eigenfunctions of p, in the notation of Eq. 3.31, are given by

〈x|p〉 = eipx/� ⇒ 〈x|p|p〉 = peipx/�. (3.36)

The completeness equation for degree of freedom x, using Eq. 3.36, yields thefollowing transformation between the momentum and coordinate basis states:

|p〉 =∫ ∞

−∞dx|x〉〈x|p〉 =

∫ ∞

−∞dxeipx/�|x〉. (3.37)

The momentum operator, acting on the state function, shifts it in space. Moreprecisely, using Eq. 3.31, for a constant a consider the following shift operator:

T (a) = eia�p = ea

∂∂x , T (a)T (b) = T (a + b), (3.38)

〈x|T (a)|ψ〉 = ea∂∂x ψ(x) = ψ(x + a) = 〈x + a|ψ〉, (3.39)

⇒ T (a)|x〉 = |x − a〉, 〈x|T (a) = 〈x + a|. (3.40)

One can define the momentum operator p, from first principles, as a translationoperator using Eqs. 3.39 and 3.40.

Three Hermitian operators that play a central role in quantum mechanics are thecoordinate operator x, the momentum operator p and the Hamiltonian operator H ;the Hamiltonian, as given in Eq. 2.11, has the form

H = 1

2mp2 + V (x) = − 1

2m

∂2

∂x2+ V (x) = H †.

3.5 Weyl operators

Both the coordinate and momentum operators, x and p respectively, are infinitedimensional Hermitian operators. Since trx = ∞ = trp, these operators cannothave any finite dimensional matrix representation. In contrast, when acting on aperiodic lattice, both

• the exponential of the momentum operator, namely, the shift operator T (a) givenin Eq. 3.38

• and the exponential of the coordinate operator

have a finite dimensional matrix representation.The finite dimensional realizations of the exponential of the coordinate and mo-

mentum operators are called the Weyl operators.

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44 Operators

N1 2

N-1 3

Figure 3.2 A finite periodic lattice, with N lattice sites.

Consider the periodic chain being embedded in real space and let its length beL. Let the lattice points be located at x = na with Na = L. The periodic lattice –with sites n = 1, 2, . . . .N – is shown in Figure 3.2 and has position eigenstates |n〉that are periodic, namely

|n+N〉 = |n〉.The shift operator V , the finite dimensional analog of the momentum shift operatorT (a) given in Eq. 3.38, is defined by

V |n〉 = |n+ 1〉.The coordinate operator U , the finite dimensional analog of the exponential of thecoordinate operator x, is a multiplication operator given by

U |n〉 = λn|n〉.For the periodic chain the following N -dimensional orthonormal vectors form acomplete coordinate basis:

|1〉 =

⎛⎜⎜⎜⎝

10...

0

⎞⎟⎟⎟⎠ , |2〉 =

⎛⎜⎜⎜⎝

010...

⎞⎟⎟⎟⎠ , . . . |N − 1〉 =

⎛⎜⎜⎜⎝

0...

10

⎞⎟⎟⎟⎠ , |N〉 =

⎛⎜⎜⎜⎝

0...

01

⎞⎟⎟⎟⎠ .

The shift operator is given as V |n〉 = |n+1〉; for the state |N〉, due to periodicity,one has V |N〉 = |N + 1〉 = |1〉; hence

V |N〉 = |N + 1〉 = |1〉 ⇒ V

⎛⎜⎝0...

1

⎞⎟⎠ =

⎛⎜⎝1...

0

⎞⎟⎠ .

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3.5 Weyl operators 45

A matrix representation of V is, consequently, given by

V =

⎛⎜⎜⎜⎜⎜⎜⎝

0 . . . . . . 0 11 0 . . .

0 1 0 . . .

. . .. . .

0. . . 1 0

⎞⎟⎟⎟⎟⎟⎟⎠ . (3.41)

It can be verified that the matrix representation of V given in Eq. 3.41 yields, asexpected,

V N |n〉 = |N + n〉 = |n〉 ⇒ V N = I.

For the coordinate representation, note U is a diagonal matrix with the constraintthat

U |N + 1〉 = λN+1|N + 1〉 = U |1〉 = λ|1〉⇒ λN = 1 ⇒ λ = e

2πiN .

Hence, one obtains the following matrix representation of the U and V operators:

U =

⎛⎜⎜⎜⎜⎜⎝

λ 0 . . .

0 λ2 0 . . .

0 0 λ3 0

0 . . .. . . 0

0 . . . 1

⎞⎟⎟⎟⎟⎟⎠ , V =

⎛⎜⎜⎜⎜⎜⎜⎝

0 . . . . . . 0 11 0 . . .

0 1 0 . . .

. . .. . .

0. . . 1 0

⎞⎟⎟⎟⎟⎟⎟⎠ .

Note, as required by a consistent representation, one has the following

UN =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

λN 0 . . .

0 λ2N 0 . . .

0 0 λ3N 0

0 . . .. . . 0

0 . . . λ(N−1)N 00 . . . 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎠= I.

The finite Fourier transform of the basis states |n〉 yields the following:

|k〉 =N∑n=1

e−2πiknN |n〉

⇒ V |k〉 =N∑n=1

e−2πiknN |n+ 1〉 = e

2πiNk

N∑n=1

e−2πiknN |n〉 = λk|k〉

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46 Operators

U |k〉 =N∑n=1

e−2πiknN U |n〉 =

N∑n=1

e−2πiknN e

2πinN |n〉 = |k − 1〉.

Hence the ket vectors |k〉, k = 1, 2, . . . N are eigenstates that are dual to |n〉, being(momentum) eigenstates of the operator V and on which U is a shift operator.

The finite version of the commutator of U and V is given as

UV |n〉 = U |n+ 1〉 = λn+1|n+ 1〉VU |n〉 = λnV |n〉 = λn|n+ 1〉⇒ UV = λVU. (3.42)

In Section 3.8, the matrix operators U and V on the periodic lattice are representedby the coordinate and momentum operator x and p and yield an exponential (uni-tary) version of the Heisenberg commutation.

3.6 Quantum numbers; commuting operator

A classical system has conserved quantities such as energy, momentum, angularmomentum and so on; in fact, one usually characterizes a classical system by itsconserved quantities, called constants of motion. There is a quantum mechanicalgeneralization of classically conserved quantities.

The Hamiltonian H is the most important operator for a quantum system sinceit evolves the state function in time. The commutator of any two operators A andB is defined by

[A,B] = AB − BA.

Consider a collection of commuting operators Oi; i = 1, 2, . . . N that also com-mute with H , namely

[Oi , H ] = 0, i = 1, 2, . . . N[Oi ,Oj

] = 0, i, j = 1, 2, . . . N.

Hermitian operators have an important property that all commuting operatorscan be simultaneously diagonalized with eigenfunctions |ψn,n1,n2,...nN 〉 that obey

H |ψn,n1,n2,...nN 〉 = En|ψn,n1,n2,...nN 〉, n = 1, 2, . . .

Oi |ψn,n1,n2,...nN 〉 = λini |ψn,n1,n2,...nN 〉, i = 1, 2, . . . N, ni = 0,±1,±2, . . .

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3.7 Heisenberg commutation equation 47

Suppose |ψn1,n2,...nN 〉 is the initial state function and is an eigenfunction of all theoperators Oi; i = 1, 2, . . . N – but not necessarily an eigenfunction of H ; then,from Eq. 2.12 and due to the commutativity of Oi with H

|ψt;n1,n2,...nN 〉 = e−itH/�|ψn1,n2,...nN 〉⇒ Oi |ψt;n1,n2,...nN 〉 = e−itH/�Oi |ψn,n1,...ni ,...nN 〉

= λini e−itH/�|ψn1,...ni ,...nN 〉

= λini |ψt;n1,...ni ,...nN 〉.The result above shows that the set of observables which commute with theHamiltonian and with each other provides eigenvalues and eigenfunctions that areconserved in time.

The integers ni, i = 1, 2, . . . N are called quantum numbers – and replace theclassical constants of motion. The time independent eigenvalues λini are quantumconstants of motion and are the generalization of classically conserved quantities.3

Fixing the values of the various nis fully specifies a particular eigenstate of theobservables Oi; i = 1, 2, . . . N . An arbitrary state function for such a system canbe expressed by the following eigenfunction expansion:

|ψt〉 =∑

n1,n2,...nN

cn1,n2,...nN (t)|ψn1,n2,...nN 〉.

3.7 Heisenberg commutation equation

Operators that do not commute with the Hamiltonian and with each other occurwidely in quantum mechanics; that is the reason that operator algebras that occurin quantum mechanics are nontrivial. One of the most important cases of noncom-muting observables is that of position and momentum, for which[

x, p] = i�I �= 0. (3.43)

Equation 3.43 defines the famous Heisenberg commutation equation, also calledthe Heisenberg algebra.

To explore the concept of noncommuting operators, Eq. 3.43 is derived from firstprinciples. Consider the following representation of the coordinate and momentumobservables in one dimension given by Eqs. 3.29 and 3.35:

x =∫ ∞

−∞dx|x〉x〈x|, p =

∫ ∞

−∞dp

2π�|p〉p〈p|.

3 There are quantum systems for which observables like momentum and position have continuous eigenvalues;the discussion for integer quantum numbers can be generalized for these systems.

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48 Operators

The commutator of the coordinate and momentum of a quantum particle, from theabove equations, is given by4

[x, p

] ≡ xp − px =∫ ∞

−∞dx

dp

2π�xp

[|x〉〈x|p〉〈p| − |p〉〈p|x〉〈x|

]. (3.44)

Expressing the commutator entirely in the coordinate basis by transforming themomentum basis using Eq. 3.37 yields[

x, p] = ∫ ∞

−∞dxdx ′

dp

2π�xp[eipx/�e−ipx

′/�|x〉〈x ′| − eipx′/�e−ipx/�|x ′〉〈x|

]= �

i

∫ ∞

−∞dxdx ′x

[|x〉〈x ′| ∂

∂xδ(x − x ′)− |x ′〉〈x| ∂

∂x ′δ(x ′ − x)

], (3.45)

= �

i

∫ ∞

−∞dxdx ′

[(x − x ′)

∂

∂xδ(x − x ′)

]|x〉〈x ′|

= i�

∫ ∞

−∞dx|x〉〈x| = i�I, (3.46)

where Eq. 3.45 and the last equation follow from the identity5

(x − x ′)∂

∂xδ(x − x ′) = −δ(x − x ′).

Hence [x, p

] = i�I : Heisenberg commutation equation. (3.47)

For N particles moving in three space dimensions, the degrees of freedom arexai, pai with a = 1, 2, . . . N and i = 1, 2.3. The Heisenberg commutation equa-tion is given by [

xai, pbj] = i�δa−bδi−j I[

xai, xbj] = 0,

[pai, pbj

] = 0.

3.8 Unitary representation of Heisenberg algebra

The Weyl operators discussed in Section 3.5 are re-examined in light of theHeisenberg algebra given in Eq. 3.47.

The periodic lattice given in Figure 3.2 is defined as being embedded in contin-uous space, and with the radius of the periodic lattice being R. In other words the

4 There is an elementary derivation of [x, p] using the chain rule of differentiation; the derivation givenexamines the operator structure of the momentum and coordinate operators.

5 The identity follows from the equation

∂

∂x

[(x − x′)δ(x − x′)

] = 0.

.

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3.8 Unitary representation of Heisenberg algebra 49

lattice sites of the periodic lattice correspond to a discrete set of points in continu-ous space. The length of the lattice is 2πR = L and lattice spacing a is given byNa = L.

The position eigenstates are

|n〉 = |na〉 = |x〉, n = 1, 2, . . . N

〈n|m〉 = δ(an− am) = 1

aδn−m : continuum normalization.

Let us define the continuous realization of the Weyl operators by (infinite dimen-sional) unitary operators U,V given for the continuous coordinate basis as

V = e−i�ap = e−a

∂∂x , U = e

2πixL .

The continuous degree of freedom yields, from Eq. 3.40

V |x〉 = |x + a〉 ⇒ V |na〉 = |na + a〉and

U |x〉 = e2πixL |na〉 = e

2πinaL |na〉 = λn|na〉

⇒ λ = e2πi( aL ) = e2πiN , L = aN.

Let U = eA and V = eB . Since

[x, ∂x] = x∂x − ∂xx = −1,

the Campbell–Hausdorff formula yields

eAeB = eA+B+12 [A,B].

Hence

UV = e2πixL e

a ∂∂x = e

2πixL+a∂x+ 2πia

2L [x,∂x ] = λ−1/2eA+B

VU = λ1/2eA+B,

and yields

UV = e2πiN V U : Heisenberg commutation equation. (3.48)

Note Eq. 3.48 is the continuous version of the commutation equation derived forWeyl operators in Eq. 3.42.

In summary, the unitary operators

V = eia�p, U = e

2πixL (3.49)

provide a unitary representation of Heisenberg algebra.

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50 Operators

The Weyl matrices discussed in Section 3.5 provide a finite dimensional realiza-tion of the unitary representation of the algebra given in Eq. 3.49.

Noteworthy 3.2 Position and momentum incompatible

Position and momentum operators do not commute since [x, p] = i�I and hencethere is no state function that is a simultaneous eigenfunction of both x and p. Thenoncommutativity of x and p is an operator expression of the fact that if the positionof the quantum particle is known at instant t , its momentum is not known. Thisproperty of a quantum particle reflects the fact that the quantum particle does nothave a unique trajectory – leading to the lack of a unique derivative of the positionand consequently to an indeterminate momentum. The Heisenberg commutationequation is an operator expression of the fact that the path of the quantum particle isindeterminate.

3.9 Density matrix: pure and mixed states

The measurement of the expectation value of observable O can be expressed interms of the density matrix. From Eq. 2.6, the expectation value of an operator Ocan be expressed as

E[O] ≡ 〈χ |O|χ〉 = tr(O|χ〉〈χ |) = tr(OρP), (3.50)

where the density matrix of a pure state ρP is given by

⇒ ρP= |χ〉〈χ | : pure density matrix. (3.51)

Equation 2.8 for the expectation value of an operator O with eigenvectors O|ψi〉 =λi |ψi〉 can be re-written in terms of the mixed density matrix ρ

Mas

E[O] = 〈ψ |O|ψ〉 = tr(∑i

|ci |2O|ψi〉〈ψi |) = tr(OρM)

⇒ ρM=∑i

pi |ψi〉〈ψi |, pi = |ci |2 : mixed density matrix.

The mixed density matrix ρM

can be used for evaluating the expectation valueof any function of the operator O. However, there are uncontrollable errors if oneuses ρ

Mfor evaluating the expectation value of another operator Q that does not

commute with O, namely [O,Q] �= 0, and this is discussed in Baaquie (2013e).Consider a quantum mechanical system, specified by a Hamiltonian H , in ther-

mal equilibrium with a heat bath at temperature T . The system now has a quantummechanical uncertainty as well as classical uncertainty due to thermal randomness.

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3.10 Self-adjoint operators 51

Let H be the quantum mechanical Hamiltonian with the following resolution interms of the energy eigenfunctions |ψi〉 and eigenvalues Ei :

H =∑i

Ei |ψi〉〈ψi |.

The canonical ensemble yields a probability distribution of energy eigenstates forthe quantum system given by the Boltzmann distribution

1

Ze−H/kBT = 1

Z

∑i

e−Ei/kBT |ψi〉〈ψi |, (3.52)

where Z = tre−H/kBT , (3.53)

where kB is the Boltzmann constant.Hence the mixed state density matrix ρ

Mfor the canonical ensemble is given by

ρM=∑i

pi |ψi〉〈ψi |, (3.54)

pi = 1

Ze−Ei/kBT ,

∑i

pi = 1. (3.55)

Consider the expectation value of the operator O for which [O, H ] �= 0; then

E[O] = tr(OρM)

=∑i

piαi, αi = 〈ψi |O|ψi〉.

Note ρM

encodes both thermal and quantum uncertainty, with pi and αi reflectingthe thermal and quantum uncertainties, respectively.

3.10 Self-adjoint operators

In this section the expression 〈ψt |O|ψt〉 is carefully analyzed to determine pre-cisely how self-adjoint operators are defined.6 In particular, a point usually ignoredin physics textbooks is the question of the domain of the self-adjoint operator. Inthe following sections, the importance of the domain is discussed as an independentingredient in the definition of self-adjoint operators.

A self-adjoint operator is the generalization of the concept of a Hermitian matrixand takes into account the aspect of the domain of the Hermitian operator.

6 The term self-adjoint is synonymous with Hermitian and is used when the more mathematical aspect of theoperator is being discussed. Since in physics infinite dimensional self-adjoint operators on Hilbert space arecalled Hermitian, the two terms will be used interchangeably.

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52 Operators

A Hermitian matrix is only defined for complex square matrices N × N withMij with i, j = 1, 2, . . . N . A matrix is Hermitian if Mij satisfies

M† = M ⇒ M∗j i = Mij , i, j = 1, 2, . . . N. (3.56)

Note the crucial point that the equality in Eq. 3.56 can hold for all i, j only if i, jhave the same range. For finite square matrices there is no issue about the range ofthe indices i, j since they always have the same range.

The range over which the finite index i takes values has a generalization for statefunctions in Hilbert space V . The domain of the operator O, denoted by D(O) ⊂ Vis defined by all elements |ψ〉 in V such that O|ψ〉 ∈ V . Similarly, the vector |χ〉is in D(O†), the domain of O†, if O†|χ〉 ∈ V .

For an operator O acting on Hilbert space, its adjoint O† is defined, for an arbi-trary state vector |ψ〉 and a dual vector 〈χ |, as follows:

〈ψ |O†|χ〉 = 〈χ |O|ψ〉∗ for all |ψ〉 ∈ D(O), |χ〉 ∈ D(O†). (3.57)

Equation 3.57 is the definition of the adjoint operator O†.The analog of Hermitian conjugation being defined only for a square matrix is

that for operators on Hilbert space, the adjoint (Hermitian conjugate) operator canbe defined for only those operators O for which the domain of the operator and itsadjoint are isomorphic, or in other words, D(O) = D(O†).

Once the domains of the operator and its conjugate are isomorphic, the form ofthe operator has to be invariant under conjugation, that is, O = O†, for the operatorto be self-adjoint.

3.10.1 Momentum operator on finite interval

The example of the momentum operator p = −i�d/dx is analyzed to demonstratethe subtleties of self-adjoint operators; Planck’s constant � is set to 1 for notationalconvenience.

Consider the momentum operator being defined on a finite interval [a, b];naively one would expect p to be self-adjoint using the rule for conjugation,namely that (d/dx)† →−d/dx and hence under conjugation

p = −i ddx→ p† = −i d

dx: Is this self-conjugate?

The fact that the form of the operator is invariant under conjugation is not the wholestory since we need to show that the domains D(p) and D(p†) are equal; only thencan it be concluded that p† = p. Since p is a first order differential operator, itsdomain is fixed by specifying only one boundary condition. The domain of p isa linear vector space; the boundary condition is chosen as a linear function of the

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3.10 Self-adjoint operators 53

state vectors to preserve the linearity of the domain; the most general boundarycondition is given by

f (a)+ αf (b) = 0, f (x) ∈ D(p), x ∈ [a, b]. (3.58)

Integration by parts yields

〈f |p†|g〉 ≡ 〈g|p|f 〉∗ = [− i

∫ b

a

dxg∗(x)df

dx

]∗= 〈f |p|g〉 + i

[g∗(b)f (b)− g∗(a)f (a)

]⇒ 〈f |p†|g〉 − 〈f |p|g〉 = i

[g∗(b)f (b)− g∗(a)f (a)

].

Hence, from the above p† = p only if the boundary term is zero. Applying theboundary condition given in Eq. 3.58 to both g(a), f (a) yields

g∗(b)f (b)− g∗(a)f (a) = g∗(b)f (b)[1− |α|2] = 0.

Hence, the momentum operator is self-adjoint if and only if

α = eiθ .

In other words, for all functions f (x) in the domains D(p) and D(p†), the condi-tion of self-adjointness imposes an additional condition on the boundary values off (x), namely

f (a)+ eiθf (b) = 0, f (x) ∈ D(p); f (x) ∈ D(p†) ⇒ D(p) = D(p†).

(3.59)

Hence p = −id/dx is a self-adjoint on the interval [a, b] since

p = p†

and at the same time the requirement of Dθ (p) = Dθ (p†) is fulfilled. The equality

of the two domains results in the following boundary condition for the functions inthe domain of p:

f (a) = eiθf (b).

In conclusion, on finite open interval [a, b], there is a whole class of momentumoperators p that are Hermitian on domain Dθ indexed by an angular label θ ∈[0, 2π).

Note that for the special case of a particle in an infinitely deep potential well theboundary condition is f (a) = 0 = f (b), and hence the momentum operator p isself-adjoint for all values of θ .

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54 Operators

3.11 Self-adjoint domain

From the discussion in the previous sections, an operator O is self-adjoint if andonly if both the form of the operator and of its adjoint are equal, as well as thedomain of the operator and of its adjoint being equal. The domains being equal isthe analogy of only square matrices being Hermitian. Hence

O† = O, D(O) = D(O†) : self-adjoint domain. (3.60)

A (self-adjoint) Hermitian operator O satisfies

〈χ |O|ψ〉∗ = 〈ψ |O†|χ〉 = 〈ψ |O|χ〉|ψ〉 ∈ D(O), |χ〉 ∈ D(O†).

Note that only if the domain is self-adjoint, namely D(O) = D(O†), is the conceptof self-adjoint operator well defined. In particular, |χ〉 = |ψ〉 yields

〈ψ |O|ψ〉∗ = 〈ψ |O|ψ〉 : real. (3.61)

In other words, the diagonal elements of a self-adjoint operator are always real.Since the expected values of physical quantities are the diagonal elements of theself-adjoint operator that represents a physical quantity, the self-adjoint property ofthe operator ensures that the result of measurements is always a real number.

3.11.1 Real eigenvalues

The reality of the eigenvalues of an operator critically hinges on the operator beingself-adjoint. The condition that the eigenvalues of a Hermitian operator are real iscarefully re-examined below from the point of view of the domain of the operator.

Suppose a self-adjoint operator H has an eigenfunction ψ that satisfies H |ψ〉 =λ|ψ〉. Then, since 〈ψ |ψ〉 = 1, it follows that

λ = λ〈ψ |ψ〉 = 〈ψ |H |ψ〉 : eigenvalue condition

⇒ λ∗ = 〈ψ |H |ψ〉∗ : definition

= 〈ψ |H †|ψ〉 : since domain is self-adjoint

= 〈ψ |H |ψ〉 : since H is self-adjoint

= λ〈ψ |ψ〉 : eigenvalue condition

⇒ λ∗ = λ : Real

Note that it is essential that all the eigenvectors |ψ〉 belong in the domain of bothH and H †. There is a more general class of pseudo-Hermitian operators, studiedin Chapter 14, that obey H † = SHST , with SST =1. Many of the results that holdfor Hermitian operators carry over to the case of pseudo-Hermitian operators.

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3.12 Hamiltonian’s self-adjoint extension 55

3.12 Hamiltonian’s self-adjoint extension

The requirement that an operator be self-adjoint is a powerful constraint. A self-adjoint extension of an operator refers to fixing the boundary condition on the al-lowed state functions so that the operator is self-adjoint.

The concept of self-adjoint extension allows one to exactly solve a whole classof one-dimensional problems that have point-like interactions, and which are ex-pressed by potentials that are combinations of a Dirac delta function and its deriva-tives [Vanderbilt (1990)].

Consider the case of a Hamiltonian H that has a point-like interaction only atthe origin, of strength represented by V (x) = gδ(x). Let the kinetic operator bedenoted by T . The time independent Schrödinger equation is given by7

Hψ(x) = [T + V ]ψ(x) = Eψ(x), T = − d2

dx2, x ∈ R.

We need to ascertain D(H): the domain for which the Hamiltonian operator H isself-adjoint. Since the potential V is self-adjoint by inspection, it is the domain ofthe kinetic operator T that, in effect, determines the boundary conditions that needto be imposed on ψ to make H self-adjoint. We need to show that: a) D(H) =D(H †); and b) 〈ψ |H †|χ〉∗ = 〈χ |H |ψ〉 for all |ψ〉 ∈ D(H) and |χ〉 ∈ D(H †).

All integrations in the neighborhood of x = 0 need to be studied carefully dueto the presence of the delta function potential V (x) = gδ(x) located at x = 0.Define, for ε → 0, the following

ψ(ε) ≡ ψ+, ψ(−ε) ≡ ψ−dψ(ε)

dx≡ ψ ′+,

dψ(−ε)dx

≡ ψ ′−.

Consider the matrix element of T ; in the limit of ε → 0

〈ψ |T |χ〉 = −∫ +∞

−∞dxψ∗(x)

d2χ(x)

dx2

= −∫ −ε

−∞dxψ∗(x)

d2χ(x)

dx2−∫ +∞

ε

dxψ∗(x)d2χ(x)

dx2. (3.62)

Using the identity (dψ/dx = ψ ′)

ψ∗(x)d2χ(x)

dx2= d2ψ∗(x)

dx2χ(x)+ d

dx

(ψ∗χ ′

)− d

dx

(ψ ′∗χ

)(3.63)

and that all functions are zero at x = ±∞ yields, after an integration by parts,∫ +∞

ε

dxψ∗(x)d2χ(x)

dx2=∫ +∞

ε

dxd2ψ∗(x)dx2

χ(x)− ψ∗+χ′+ + ψ ′∗+χ+. (3.64)

7 Choose m so that 2m = 1.

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56 Operators

Hence, from Eqs.3.62 and 3.64

〈ψ |T |χ〉 = −∫ +∞

−∞dxψ∗(x)

d2χ(x)

dx2

= −∫ +∞

−∞dx

d2ψ∗(x)dx2

χ(x)+ R,

R = ψ∗+χ′+ − ψ ′∗+χ+ − ψ∗−χ

′− + ψ ′∗−χ−. (3.65)

Hence, the definition of the adjoint of an operator yields

〈χ |T †|ψ〉 − 〈χ |T |ψ〉 = 〈ψ |T |χ〉∗ − 〈χ |T |ψ〉= R∗, (3.66)

where the remainder R∗, in matrix notation, is given by Eq. 3.65 as

R∗ = (χ∗+ χ ′∗+

) (0 −11 0

)(ψ+ψ ′+

)− (χ∗− χ ′∗−

) (0 −11 0

)(ψ−ψ ′−

). (3.67)

Let us choose the (linear) boundary conditions to be the same for the domain ofboth the kinetic operator T and its adjoint T †, as is required for the operator to beself-adjoint. The boundary matrix U is hence postulated to be(

ψ+ψ ′+

)= U

(ψ−ψ ′−

),

(χ∗+ χ ′∗+

) = (χ∗− χ ′∗−

)U†, (3.68)

where

U = eiθ(γ β

α δ

)→

(γ β

α δ

), (3.69)

since it can be shown that θ = 0 for all stationary problems, such as scattering froma point-like potential located at x = 0. From Eqs. 3.67, 3.68, and 3.69 it followsthat

R∗ = 0

which yields

U†

(0 −11 0

)U =

(0 −11 0

). (3.70)

Taking the determinant of both sides of Eq. 3.70 shows that det2 U = 1 and hence

γ δ − αβ = 1. (3.71)

Consequently, only three independent real constants fully determine the boundarycondition matrix U .

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3.12 Hamiltonian’s self-adjoint extension 57

3.12.1 Delta function potential

Consider the special case of detU = 1 such that

U† =(

1 α

0 1

), U =

(1 0α 1

). (3.72)

It is straightforward to show that U satisfies Eq. 3.70 for all α. The boundary con-dition matrix yields the following boundary conditions:(

ψ+ψ ′+

)=(

1 0α 1

)(ψ−ψ ′−

)⇒ ψ+ = ψ−, ψ ′+ = ψ ′− + αψ− (3.73)

and similarly for the dual space vectors χ∗+ , χ ′∗+ .Note that the state function ψ(x) is normalizable with

∫dx |ψ(x)|2 = 1. The

boundary conditions of the state function ψ(x) given in Eq. 3.73 imply that it iscontinuous at x = 0 and its derivative ψ ′(x) has a jump (discontinuity) at x = 0,as shown in Figure 3.3(a). The discontinuity at x = 0 leads to the second derivativeof ψ(x) having a delta function singularity at x = 0.

Y– Y+

X

(a)

Y– Y+

X

(b)

Y–

Y+

X

(c)

Figure 3.3 Three distinct classes of discontinuities for the state function ψ(x) atthe origin x = 0. In (a) the state function is continuous but its derivative is discon-tinuous; in (b) the state function is discontinuous but its derivative is continuous;and in (c) both the value of the state function and its derivative are discontinuous.

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58 Operators

What is the potential in Schrödinger’s equation that yields a state function withthe discontinuity given in Eq. 3.73? The answer lies in ψ ′′(0) being singular atx = 0 and this indicates the presence of a delta function potential.

The simplest case of a point interaction is a delta function potential at the ori-gin of strength g, represented by V (x) = gδ(x). This yields a time independentSchrödinger equation given by (recall 2m = 1)

Eψ(x) = Hψ(x) = [T + V ]ψ(x)= −d

2ψ(x)

dx2+ gδ(x)ψ(x) (3.74)

and for which ψ ′′(0) has a delta function singularity.To make the connection between the potential gδ(x) and the boundary condition

matrix U , we integrate the Schrödinger equation from −ε to +ε. Since δ(x) is aneven function, one has

∫ L0 dxδ(x) = 1/2 = ∫ 0

−L dxδ(x) for any L > 0; this yields

∫ +ε

−εdxδ(x)ψ(x) =

∫ 0

−εdxδ(x)ψ(x)+

∫ +ε

0dxδ(x)ψ(x)

= 1

2(ψ− + ψ+). (3.75)

Using boundary conditions Eq. 3.73 to simplify the last two lines, Eq. 3.74 andthe above equation yields

−∫ +ε

−εdx

d2ψ(x)

dx2+ g

∫ +ε

−εdxδ(x)ψ(x) = E

∫ +ε

−εdxψ(x) = O(ε)

−(ψ ′+ − ψ ′−)+g

2(ψ− + ψ+) = 0

−αψ− + gψ− = 0 ⇒ α = g. (3.76)

From Eq. 3.76 it follows that the discontinuity of the state function at x = 0 is aconsequence of the delta function potential at the origin.

There are two more simple cases for U that satisfy the self-adjointness conditiongiven in Eq. 3.70, namely

UI =(

1 β

0 1

), UII =

(γ 00 1

γ

).

It is rather surprising that even these simple boundary matrices do not have anysimple explanation in terms of an underlying potential V (x) that is a combinationof delta functions. The three cases discussed so far are all special limits of a verygeneral potential called the Fermi pseudo-potential.

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3.13 Fermi pseudo-potential 59

3.13 Fermi pseudo-potential

The time-independent Schrödinger equation for the Fermi pseudo-potential is writ-ten in a nonlocal form as

−ψ ′′(x)+∫ +∞

−∞dx ′V (x, x ′)ψ)x ′) = Eψ(x), (3.77)

where the real Fermi pseudo-potential, as defined by Wu and Yu (2002), is given by

V (x, x ′) = g1δ(x)δ(x′)+ g2

[δ′P (x)δ(x

′)+ δ′(x)δP (x ′)]+ g3δ

′P (x)δ

′P (x

′)= V (x ′, x).

Note that V (x, x ′) is symmetric, as required by the conservation of probability. Themodified delta prime function is defined by

δ′P (x)ψ(x) ≡ δ′(x)ψ(x)

and the definition of ψ(x) is given by

ψ(x) ≡{ψ(x)− 1

2(ψ+ − ψ−) x > 0ψ(x)+ 1

2(ψ+ − ψ−) x < 0,

where we recall ψ+ = ψ(ε) and ψ− = ψ(−ε). Note that ψ(x) is continuous atx = 0 and in fact ψ(0) = (ψ+ + ψ−)/2; furthermore, even though δ′(x)ψ(x) isundefined for ψ(x) that is discontinuous at x = 0, the combination δ′P (x)ψ(x) iswell defined at x = 0 even when ψ(0) is discontinuous. In fact this is the mainmotivation for defining the (generalized) function δ′P (x).

For the Fermi pseudo-potential, the boundary condition matrix can be shown tobe given by [Wu and Yu (2002)]

UF = 1

�

((2− g2)

2 − g1g3 −4g3

4g1 (2+ g2)2 − g1g3

)� = (2+ g2)(2− g2)− g1g3. (3.78)

It can be directly verified that the matrix U given above satisfies the self-adjointnesscondition given in Eq. 3.70.

The special case of the delta function potential is treated by taking g1 = α andsetting g2 = g3 = 0; this yields the potential V (x, x ′) = g1δ(x)δ(x

′).To obtain UI one sets g3 = −β and g1 = g2 = 0, and finally to obtain UII one

sets g1 = g3 = 0 and γ = (2− g2)/(2+ g2).Since UF has three arbitrary parameters g1, g2, g3 it can be shown that UF is

the most general boundary matrix. Hence for a single degree of freedom the Fermipseudo-potential is the most general point-like interaction.

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60 Operators

3.14 Summary

Operators have a central role to play in quantum mathematics. In physics, op-erators represent observable quantities whereas in other applications of quantummathematics such as finance and statistical mechanics, operators have a differentinterpretation.

The structure of a Hermitian operator is realized by its spectral decomposition,in terms of all of its eigenfunctions and eigenvalues, and these also yield a repre-sentation of the completeness relation of the underlying state space. The positionand momentum operator have been discussed at length as these are the leading ex-emplars of Hermitian operators as well as being among the most important physicalobservables. The Weyl operators were shown to provide a finite dimensional uni-tary representation of the position and momentum operators as well as that of theHeisenberg algebra.

Physical quantities are represented in quantum mechanics by Hermitian oper-ators. The value of a physical quantity is obtained by the operator acting on theunderlying linear state space, elements of which represent the physical state of aquantum system. The set of all mutually commuting operators – which all com-mute with the Hamiltonian also – provide an exhaustive description of all the con-served quantities of a quantum system and are the quantum generalization of theconstants of motion of classical mechanics. The Heisenberg commutation equationis a reflection of the non-commuting nature of observables and results from theunderlying dynamics of the quantum system.

The concept of the domain of the adjoint (Hermitian conjugate) of an opera-tor was discussed; it was shown that the domain is an independent ingredient inthe definition of a self-adjoint (Hermitian) operator. The momentum operator ona finite interval exemplified the role of the domain. Self-adjoint extensions of anoperator were defined for generalized delta function potentials by choosing the ap-propriate domain for the Hamiltonian operator. The Fermi pseudo-potential wasshown to be the most general form of a generalized delta function potential thatsatisfies the extension of domain required for self-adjointness.

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The Feynman path integral

A path, in general, is defined by a determinate trajectory in time, from an initial toa final point. The classical trajectory is only one of the possible trajectories, and inquantum mechanics all the possible paths between the initial and final point comeinto play. Recall that the probability amplitude is a complex number that is as-signed to each determinate path. Indeterminate paths are defined as a collection ofdeterminate paths that are experimentally indistinguishable. In the Dirac–Feynmanapproach, the inherent indeterminacy of the quantum entity is realized by the de-gree of freedom – in undergoing time evolution – “taking” indeterminate paths[Baaquie (2013e)].

For a quantum degree of freedom evolving from an observed initial state to theobserved final state – and with no other observations made – the Feynman pathintegral is a mathematical construction that computes the probability amplitudesby summing over all the allowed determinate paths of the degree of freedom –discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005)and Baaquie (2013e).

4.1 Probability amplitude and time evolution

Recall that the description of a quantum system, at a particular instant, is given byits state vector, namely |ψ〉. To avoid confusion with the concept of a state vector,the term probability amplitude is used for describing a quantum entity undergoingtransitions in time.

Consider a quantum system making a transition from an arbitrary initial statefunction |ψ〉 at time ti = 0 to an arbitrary final state function |η〉 at final timetf = t . Note that in quantum theory both the initial state |ψ〉 and the final state |η〉can be independently specified.

The state vector |ψ〉 must be evolved for a duration of time t to reach the time atwhich the final state vector |η〉 is located, as shown in Figure 4.1. The initial state

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4.2 Evolution kernel 63

Hence, the fundamental expression that needs to be evaluated is the generalmatrix element

〈χi |e−itH/�|ψj 〉.All probability amplitudes can be evaluated from the general matrix element ofe−itH/� that is given above.

Consider the important special case of a quantum particle with the degree offreedom given by the coordinate x. From Eq. 4.2, the probability amplitude, usingthe completeness equation Eq. 3.6, is given by

〈η|e−itH/�|ψ〉 =∫

dxf dxiη∗(xf )〈xf |e−itH/�|xi〉ψ(xi).

The conditional probability for the particle that starts from initial state vector|ψ〉 and, after evolving for time t = tf − ti , ends up at the coordinate |xf 〉 is given– since 〈ψ |ψ〉 = 1 – by the following:

P(xf |ψ; t) = |〈xf |e−itH/�|ψ〉|2∫dx|〈x|e−itH/�|ψ〉|2 = |〈xf |e

−itH/�|ψ〉|2. (4.3)

The normalization is necessary since one is comparing the likelihood of the par-ticle’s degree of freedom making a transition from its initial state |ψ0〉 to xf withthe particle ending up at any other position. In particular,∫

dxf P(xf |ψ; t) = 1.

To obtain finite results, the initial state has to be normalizable.

4.2 Evolution kernel

The time evolution of the state vector is determined by the operator e−itH/�, withits matrix elements being given by

K(xf , tf ; xi, ti) = 〈xf |U(t)|xi〉 = 〈xf , tf |xi, ti〉; t = tf − ti

= 〈xf |e−i(tf−ti )H/�|xi〉. (4.4)

Simplifying the notation, the evolution kernel is written as

K(x, x ′; t) ≡ 〈x|e−itH/�|x ′〉. (4.5)

Using the time evolution of a state vector, from Eq. 2.12, and the completenessequation given in Eq. 3.6, we obtain

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�


0( )ψ x ′

t (t,x)ψ

x ′

Figure 4.2 The evolution kernel K(x, x′; t) propagates the initial state vectorψ0(x) through time t to state vector ψ(t, x).

ψ(t, x) = 〈x|e−itH/�|ψ0〉 =∫

dx ′〈x|e−itH/�|x ′〉〈x ′|ψ0〉

=∫

dx ′K(x, x ′; t)ψ0(x′). (4.6)

Equation 4.1 is illustrated in Figure 4.2 and graphically shows how the values ofthe initial state function ψ0(x) propagate in time to determine the value of ψ(t, x).

The solution of Schrödinger’s equation given in Eqs. 2.12 and 4.1 is formalsince one needs to evaluate the matrix elements of the evolution operator U(t),which in turn requires solving for all the eigenenergies and eigenfunctions of theHamiltonian H . Let the eigenfunctions of the Hamiltonian be given by

H |ψn〉 = En|ψn〉. (4.7)

The completeness equation, from Eq. 3.21, is then given by

I =∑n

|ψn〉〈ψn|. (4.8)

Using the completeness equation given in Eq. 4.8 yields the evolution kernel

K(x, x ′; t) = 〈x|e−itH/�|x ′〉 =∑n

e−itEn/�〈x|ψn〉〈ψn|x ′〉. (4.9)

Even though the expression for the transition amplitude K(xf , xi; t) has becomegreatly simplified, the sum over all eigenstates is still quite nontrivial to evaluate –even for the case of the harmonic oscillator.

All the eigenfunctions and eigenvalues of H are seldom known, and hence Eq.4.9 is in most cases only a formal expression for the transition amplitude. One

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4.3 Superposition: indeterminate paths 65

would like to have other avenues for (approximately) computing K(xf , xi; t) andthe path integral has been developed primarily to address this problem.

Noteworthy 4.1 Boundary conditions in classical and quantum mechanics

In classical mechanics, a classical particle is fully described by its position andvelocity. Since Newton’s equations of motion are based on acceleration that requiresthe second time derivative, one needs to specify two boundary conditions to uniquelyspecify a classical trajectory. Once the boundary conditions are specified Newton’sequations of motion yield a determinate and unique trajectory. In particular, if theposition and velocity of a particle are specified at some instant, its future trajectory isfully determined.

In quantum mechanics the situation is quite different. The quantum degree offreedom is described by a state vector that yields the likelihood for experimentallyobserving the expectation value of a particular projection operator. The Schrödingerequation involves only the first time derivative of the state vector; hence, one canspecify either the initial or the final state vector. Quantum mechanics, unlike classicalmechanics, is a theory of probabilities. The initial state vector |ψ〉 evolves into a state|ψt 〉 that has nonzero probability amplitude 〈χ |ψt 〉 with many different state vectors〈χ |. Hence, the time evolution of the quantum particle (degree of freedom) isindeterminate, with a likelihood of evolving from its initial state |ψ〉 to manydifferent possible final states 〈χ |.

4.3 Superposition: indeterminate paths

The probability amplitude of making a quantum transition from an initial state –that can go through many intermediate determinate paths – to a final state has twovery different and distinct cases:

• The intermediate paths are distinguishable and the path taken is determinate andexperimentally known.

• The intermediate paths are indistinguishable, namely the information on whichpath has been taken by the quantum particle is not experimentally determined.

When the path taken is not known the intermediate state of the quantum system isindeterminate, while it is determinate when the path is known.

The non-classical content of quantum mechanics comes out in a remarkablemanner for the case of the degree of freedom making a transition from an initialto a final quantum state via many indistinguishable intermediate paths. Hence thepath taken by the degree of freedom is indeterminate.

Consider the case of an initial state vector |xi, ti〉 making a transition to a finalstate vector |xf , tf 〉 via N intermediate slits specified by states |xn〉, n = 1, 2 . . . N

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Time

tf

ti

Space

x1t x2 x3 � xN

xi

xf

Figure 4.3 Probability amplitudes for transition from initial state vector |xi〉 attime ti to final state vector |xf 〉 at time tf for N different possible intermediatepaths.

and shown in Figure 4.3. In going from |xi, ti〉 to |xf , tf 〉, the particle can gothrough any of the N -slits. The probability amplitude, in the notation of Eq. 4.4, isgiven by K(xf , tf ; xi, ti) = 〈xf , tf |xi, ti〉.

The probability P of the transition from an initial state |xi, ti〉 to a final state|xf , tf 〉, as shown in Figure 4.3, has the following two different expressions:

• The path taken for the transition is known, and hence determinate, due to a mea-surement being made at time t that ascertains which intermediate position xi istaken by the particle; for this case, the probabilities for the different paths areadded and yield the probability of transition PD given by

PD =N∑n=1

|〈xf , tf |xn, t〉〈xn, t |xi, ti〉|2 =N∑n=1

Pxf ,xnPxn,xi , (4.10)

where

Pxf ,xn = |〈xf , tf |xn, t〉|2, Pxn,xi = |〈xn, , t |xi, ti〉|2.The result for PD follows from the classical composition of conditional prob-abilities, with the intermediate states being the allowed intermediate states. Inparticular there is no interference between the distinct paths as there is no cross-term for the different paths taken.

• For the case when the intermediate paths are indistinguishable , the probabilityamplitudes for the different determinate paths are added to yield the transitionprobability amplitude 〈xf , tf |xi, ti〉 given by

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4.4 The Dirac–Feynman formula 67

Time

Space(a)

ε

xi

x1

x2 x3

xN

xf

��

��

Time

Space

xi

x1

x2 x3

xN

xf

(b)

Figure 4.4 (a) A single determinate path, discretized by time steps ε, from initialto final position. (b) The ε → 0 continuum limit of the discretized path.

〈xf , tf |xi, ti〉 =N∑n=1

〈xf , tf |xn, t〉〈xn, , t |xi, ti〉. (4.11)

The probability amplitude yields the following observable transition probabil-ity PI

PI =∣∣∣〈xf , tf |xi, ti〉∣∣∣2 = ∣∣∣∣ N∑

n=1

〈xf , tf |xn, t〉〈xn, , t |xi, ti〉∣∣∣∣2

=N∑n=1

Pxf ,xnPxn,xi

+N∑

n �=m〈xf , tf |xn, t〉〈xn, , t |xi, ti〉〈xf , tf |xm, t〉∗〈xm, , t |xi, ti〉∗.

(4.12)

All the indeterminate paths interfere, as can be seen from their cross-terms givenin Eq. 4.12; the interference is a purely quantum effect and there is no analog ofthis result in classical probability theory, which is discussed in Baaquie (2013e).

4.4 The Dirac–Feynman formula

Consider the case of a determinate and discrete path, with infinitesimal steps ε, asshown in Figure 4.4(a), that goes from xi at tf to final position xf at time tf . Letthe points in the path, at intermediate times tn, be denoted by the following:

xi = x0, x1, x2, x3, xn, . . . , xN−1, xN = xf , tn = ti + εn, tN = tf .

The path is determinate since all the intermediate points xn are known (by hy-pothetical experimental observations). Hence the principle of quantum superposi-tion for successive steps tells us that the net amplitude φ[path] for the determinate

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path is equal to the product of the probability amplitude for each infinitesimal step[Feynman and Hibbs (1965)], and yields

φ[path] = 〈xN ; tN |xN−1; tN−1〉 . . . 〈xn+1; tn+1|xn; tn〉 . . . 〈x1; t1|x0; t0〉.Writing the probability amplitude in product notation yields

φ[path] =N−1∏n=0

〈xn+1; tn+1|xn; tn〉. (4.13)

For each infinitesimal step ε = tn+1 − tn, the probability amplitude is given bythe Dirac–Feynman formula

〈xn+1; tn+1|xn; tn〉 = N (ε) exp

{εi

�L(xn, xn+1; ε)

}, (4.14)

where N (ε) is a normalization constant and L is the Lagrangian of the particle, tobe defined more precisely in Section 4.5. The fact that for an infinitesimal step theprobability amplitude is an exponential that is proportional to the Lagrangian is adeep insight of Dirac that was further developed by Feynman.

The quantum particle makes a “quantum jump” from xn at time tn to xn+1 at timetn+1. Unlike classical mechanics, for which the position of the particle is known forevery instant t , the quantum particle can be said to be in a “trans-empirical” formof existence from time tn to time tn+1; this aspect of quantum mechanics has beendiscussed in Baaquie (2013e).

Using the Dirac–Feynman formula, given in Eq. 4.14, the probability amplitudefor the discretized determinate path given in Eq. 4.13 has the form

φ[path] =N−1∏n=0

〈xn+1; tn+1|xn; tn〉

= N exp{ε i�

N∑n=0

L(xn, xn+1; ε)} (4.15)

= N exp{ i�S[path]}, (4.16)

where the discrete and determinate path that appears in S[path] is shown in Figure4.4(a). N is a path independent normalization.

The term S is the action functional for the discrete paths, and is given fromEqs. 4.16 and 4.15 as follows:

S[path] = ε

N∑n=0

L(xn, xn+1; ε). (4.17)

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4.5 The Lagrangian 69

4.5 The Lagrangian

Let the total time interval tN − t0 = tf − ti be kept fixed, with N = (tf − ti)/ε. Inthe continuum limit ε → 0 and the paths become continuous. The discretized pathshown in Figure 4.4(a) converges to the continuous path shown in Figure 4.4(b).

The continuum limit ε → 0 (N →∞) yields

xn+1 − xn

ε→ dx

dt, t = nε

L(xn, xn+1; ε)→ L(x, dx/dt)

S[path] → S[x] =∫ tf

ti

dtL(x, dx/dt). (4.18)

The quantum particle’s Lagrangian for continuous time is L(x, dx/dt) and theaction functional is S[x]; the notation used for the action is to indicate that theaction depends on the entire path x(t) with t ∈ [ti, tf ].

The probability amplitude for the determinate continuous path x(t) – going fromxi at time ti to final position xf at time tf – is given by the continuum limit ofEq. 4.16, and the continuum action S[x] replaces the discretized action, namelyS[path]. Hence

φ[x] = N exp{ i�S[x]}. (4.19)

The action S[x] has the dimensions of � and dividing it by � is required since onlythe dimensionless quantity S/� can be exponentiated; it is an empirical result that� is given by Planck’s constant.

Equation 4.19 gives the probability amplitude for the quantum particle makinga transition from the initial to its final position via a specific possible path. In otherwords, the path x(t) is one possible determinate path from the initial to the finalposition, and not necessarily the classical path determined by classical mechanics.

The Hamiltonian given in Eq. 2.11 yields the Lagrangian

L(x, dx/dt) = 1

2m

(dx(t)

dt

)2

− V (x(t)). (4.20)

Although the action and Lagrangian given in Eqs. 4.18 and 4.20 look like classicalexpressions they are vastly different from the classical case. The reason is that inclassical mechanics, x(t) in the Lagrangian and action is restricted to one path,namely the classical path xc(t) that obeys Newton’s equation of motion, and forwhich the particle’s path is a numerical function of time t ; in contrast, for quantummechanics, the symbol x(t) that appears in the Lagrangian and action can be anypossible path from the initial to the final position.

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4.5.1 Infinite divisibility of quantum paths

The Dirac–Feynman formula given in Eq. 4.14 is the reason that the continuumlimit of ε → 0 exists for the probability amplitude. As one makes the step size εsmaller, the Dirac–Feynman formula for each infinitesimal transition, say from xn

to xn+1, yields the correspondingly smaller expression in the exponential, namelyexp{iεL(xn, xn+1; ε)/�}; this property of the paths leads, for N → ∞, to a welldefined limit of the infinite product for probability amplitude given in Eq. 4.13,namely φ[path] → φ[x] given in Eq. 4.19.

One can turn the above discussion around and argue that, for quantum mechan-ics to exist for continuous time, the probability amplitude for an infinitesimal stepin time, of necessity, needs to be an exponential of an infinitesimal – of the formgiven by the Dirac–Feynman formula. This is because any determinate path is in-finitely divisible in continuous time and hence requires a concomitant convergentprobability amplitude.

The requirement for a convergent probability amplitude for continuous pathsanswers a fundamental question as to why the action S that appears in classicalphysics needs to be exponentiated in quantum mechanics, as in exp{iS[x]/�} givenin Eq. 4.19. The classical to quantum transition is schematically given by

S[xc] → exp{ i�S[x]}.

One explanation provided by the probability amplitude is that the requirement ofquantum processes taking place in continuous time necessitates the exponentiationof the action. One may even state that the exponentiation of the action in quan-tum mechanics is also the reason why quantum mechanics is qualitatively differentfrom, and “exponentially” more complex, than classical mechanics.

4.6 The Feynman path integral

The result of the previous section provides an expression for the probability ampli-tude for the quantum entity to take a specific and determinate path in going from itsinitial to its final position. What is the probability amplitude if the quantum particleis only observed at its initial and final position? Due to the quantum indeterminacyof a quantum entity the paths of the entity’s degree of freedom are indeterminateand hence it “takes” all possible indeterminate paths simultaneously.

How many indeterminate paths are there between the initial and final positions?Clearly, there are many paths, and to develop a sense of these paths, considerputting barriers between the initial and final position to limit the number of pos-sible paths, as shown in Figure 4.5, so that we can enumerate the indeterminatepaths. Once the procedure for enumerating the indeterminate paths becomes clear,

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4.6 The Feynman path integral 71

Time

tf

tixi

xf

Space

�

�

�

�

Figure 4.5 Probability amplitudes for transition from initial state vector |xi, ti〉 tofinal state vector |xf , tf 〉 for many successive slits with indistinguishable trans-empirical paths.

the barriers will be removed and all the indeterminate paths will then be includedin our analysis.

Figure 4.5 shows a quantum particle going from initial state xi at time ti to finalposition xf at time tf , through barriers that restrict the number of paths available.Let the entire continuous path – going from initial state xi to final state xf throughthe successive slits as shown in Figure 4.5 – be denoted by path(n), with the prob-ability amplitude denoted by φ[path(n)]. One can take path(n) to be straight linesfrom xi, ti to the successive slit positions and another straight line from the last slitto xf , tf , as shown in Figure 4.5.

Let there be N total number of different paths going from xi to xf . All theN indeterminate paths from xi, ti to xf , tf are indistinguishable. From the super-position principle given in Eq. 4.11, the total probability amplitude is given byadding the probability amplitudes for all the indistinguishable determinate paths,and yields


φ[path(n)]. (4.21)

The probability amplitude φ[path(n)] for each determinate path is given by Eq.4.19 and yields

φ[path(n)] = N exp{iSn/�}, (4.22)

Sn = S[path(n)], n = 1, 2, . . . N, (4.23)

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t

tf

ti

xi xfx

Figure 4.6 All possible trans-empirical paths from the initial to the final statevector.

where S[path(n)] is the action for the continuous path(n) and N is a path indepen-dent normalization.

Hence, from Eqs. 4.21 and 4.22, the total spacetime probability amplitude thatthe initial state vector |xi, ti〉 makes a transition to the final state vector |xf , tf 〉 –via indistinguishable paths – is given by superposing the amplitude for all theindistinguishable paths and yields

〈xf , tf |xi, ti〉 = NN∑n=1

eiS[path(n)]/� = NN∑n=1

eiSn/�

= N{eiS1/� + eiS2/� + · · · }.

From Eq. 4.4, the evolution kernel has the representation

K(xf , tf ; xi, ti) = 〈xf , tf |xi, ti〉 = NN∑n=1

eiS[path(n)]/�. (4.24)

One can successively remove the barriers between the initial and final positionsof the quantum particle and, as shown in Figure 4.5, there will be a great prolif-eration of possible paths. When there are no longer any slits, one has the limit ofN → ∞ or what is the same thing, there are infinitely many indistinguishablepaths.

The transition amplitude is given by the sum over all possible trans-empiricalpaths, going from the initial position xi at time ti to the final state xf at time tf , asshown in Figure 4.6, and yields

K(xf , tf ; xi, ti) = N∑

all paths

eiS[path]/� : Feynman path integral. (4.25)

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4.7 Path integral for evolution kernel 73

The sum in Eq. 4.25 looks more figurative than a precise mathematical expression.After all, how are we supposed to actually perform a sum over infinitely manypaths? Eq. 4.25 is re-cast into a precise and mathematical expression in Section4.7.

In summary, from time ti to time tf , no measurement is performed on the parti-cle’s evolution and hence all the paths going xi to final state xf are indistinguish-able. The total probability amplitude to make a transition from initial state xi tofinal state xf is equal to the sum of probability amplitudes eiS/� over all the in-dividual (indistinguishable) determinate paths that go from initial state xi to finalstate xf .

4.7 Path integral for evolution kernel

To render the sum over all paths in continuous space, namely∑

all paths given in Eq.4.25, into a well-defined mathematical quantity, a derivation is given below of thepath integral starting from the Schrödinger equation. A corollary result is to showthat the definition of K(xi, xf ; t) given in Eq. 4.5 is equivalent to the one derivedin Eq. 4.25.

Consider the Hamiltonian

H = �

2mp2 + V (x), p = −i� ∂

∂x

and its evolution kernel, which from Eq. 4.5 is defined by

K(xi, xf ; t) = 〈xf |e−itH/�|xi〉.To simplify the notation, set � = 1.

Note that in general [p2, V ] �= 0; it is this noncommutativity that poses themain problem in evaluating the evolution kernel and makes quantum mechanicsnontrivial. Ignoring the noncommutativity yields

e−itH � e−itp2

2m e−itV

⇒ K(x, x ′; t) � 〈x|e−it p2

2m e−itV |x ′〉� e−itV (x

′)〈x|e−it p2

2m |x ′〉. (4.26)

To evaluate the evolution kernel K in this approximation requires only the evolu-tion kernel for the free particle Hamiltonian p2/2m, which is given in Eq. 5.72.

Note the remarkable fact that for noncommuting operators A and B

eAeB = eA+B+12 [A,B]+··· (4.27)

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and hence, for infinitesimal time, namely t = ε, we obtain

e−iεH = e−iε(p2

2m+V )

= e−iεp2

2m e−iεV +O(ε2). (4.28)

Hence for infinitesimal time ε, from Eq. 4.26 the transition amplitude K(x, x′ ; ε)

can be evaluated exactly to O(ε2).The path integral approach is employed fundamentally to build up the finite time

transition amplitude by composing the infinitesimal time transition amplitude byrepeatedly using the resolution of the identity operator given in Eq. 3.6.

The evolution kernel (transition amplitude) for a particle to go from initial posi-tion xi to final position xf in time t is written as

K(xf , xi; τ) = 〈xf |e−itH |xi〉 = 〈x|(e−i tNH )N |x ′〉,

where, for ε = tN

, we have

K = 〈xf | e−iεH e−iεH · · · e−iεH︸︷︷︸N−t imes

|xi〉. (4.29)

Inserting, N − 1 times, the completeness equation given in Eq. 3.6,

I =∫ ∞

−∞dx|x〉〈x|,

yields

K(xf , xi; τ) =∫

dx1dx2 . . . dxN−1〈xf |e−iεH |xN−1〉〈xN−1|e−iεH |xN−2〉· · · 〈xn+1|e−iεH |xn〉 · · · 〈x1|e−iεH |xi〉. (4.30)

Consider the matrix element

〈x|e−iεH |x ′〉 =∫ ∞

−∞dp

2π〈x|e−iεH |p〉〈p|x ′〉. (4.31)

Since 〈x|p〉 = eipx , one has from Eq. 4.28

〈x|e−iεH |x ′〉 = e−iεV (x′)∫

dp

2πe−

iεp2

2m eip(x−x′)

=√

m

2πiεei

m2ε (x−x′)2−iεV (x′). (4.32)

Recall from Eq. 4.14, the Dirac–Feynman formula, for each infinitesimal timestep ε, is given by

〈xn+1; tn+1|xn; tn〉 = N (iε) exp

{εi

�L(xn, xn+1; ε)

}.

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4.7 Path integral for evolution kernel 75

Recall from Eq. 4.4, the evolution kernel is defined by

K(xf , tf ; xi, ti) = 〈xf , tf |xi, ti〉 = 〈xf |e−i(tf−ti )H/�|xi〉.Hence, from Eqs. 4.4 and 4.14, and simplifying the notation, we obtain theLagrangian, defined for infinitesimal Euclidean time ε, given by

〈x|e−iεH |x ′〉 = N (ε)eεL(x,x′;ε) : Dirac-Feynman formula. (4.33)

Hence from Eq.4.32

N (ε)eiεL(x,x′;ε) = 〈x|e−iεH |x ′〉 =

√m

2πiεei

m2ε (x−x′)2−iεV (x′). (4.34)

In summary, the particle degree of freedom has the Hamiltonian given by

H = − �2

2m

∂2

∂x2+ V (x),

and its Lagrangian is given by Eq. 4.34, for discrete time t = nε, by

L = m

2

(x − x ′

ε

)2

− V (x). (4.35)

The Lagrangian is sometimes written more symmetrically as

L = m

2(x − x ′

ε)2 − 1

2[V (x)+ V (x ′)] (4.36)

and to O(ε) is the same as the one given in Eq. 4.35.Hence the transition amplitude, restoring � in the last equation below, is given by

K(xf , xi; τ) =∫ N−1∏

n=1

dxn

N−1∏n=0

〈xn+1|e−iεH |xn〉

=∫

Dxeiε∑N−1

n=0 L(xn+1,xn)

=∫

Dx exp{ i�S[x]}, (4.37)

Boundary conditions : x0 = xi, xN = xf . (4.38)

The lattice action and path integral integration measure is given by

S[x] = ε

N−1∑n=0

m

2

(xn+1 − xn

ε

)2

− ε

N−1∑n=0

V (xn),

∫Dx =

( m

2πiε

)N2

N−1∏n=1

∫ +∞

−∞dxn.

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In the continuum limit ε → 0 one obtains

S[x] =∫ τ

0Ldt, L = m

2

(dx

dt

)2

− V (x), (4.39)

∫Dx = N

τ∏t=0

∫dx(t).

The continuum path integral for the evolution kernel is given by

K(xf , xi; τ) = 〈xf |e−itH/�|xi〉 =∫B.C.

Dx exp{ i�S[x]}

: Feynman path integral, (4.40)

Boundary condition : x(0) = xi , x(τ ) = xf .

All paths between the initial and final position, figuratively shown in Figure 4.6,are summed over in the

∫DxeiS/� path integration given in Eq. 4.40. The figurative

summation over all paths∑

all paths eiS/� given in Eq. 4.25 is given a mathematical

realization in Eq. 4.40, which is a functional integration over all indistinguishablepaths from the initial to the final state.

At each instant, the position degree of freedom takes all its values; at instantt ∈ [ti , tf ], the degree of freedom is equal to the real line �t ; the total space ofall paths is given by a tensor product over all instants and yields the total spaceof all paths equal to ⊗t�t . In general, for degree of freedom space given by F , thepath space is given by ⊗tFt .

In summary, the Feynman path integral is an efficient mathematical instrumentfor evaluating the finite time matrix elements of the Euclidean continuation of theunitary operator U(t), namely of 〈xf |e−itH/�|xi〉.

4.8 Composition rule for probability amplitudes

Consider the case of a particle going through N -slits, as shown in Figure 4.3,with all the paths being indistinguishable. Equation 4.12 yields the probabilityamplitude


〈xf , tf |xn, t〉〈xn, t |xi, ti〉.

Suppose the slits have spacing a, so that xn = na, with n = 0,±1,±2, . . .±∞,that is, the slits extend over the entire x-axis. The probability amplitude, extending

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4.8 Composition rule for probability amplitudes 77

Eq. 4.12 to the entire x-axis, is given by

〈xf , tf |xi, ti〉 =+∞∑

n=−∞〈xf , tf |xn, t〉〈xn, t |xi, ti〉. (4.41)

To take the continuum limit of Eq. 4.41, the bra and ket vectors |xn, t〉, 〈xn, t |defined on a discrete set xn = na need to be written in continuum notation; fora → 0, let xn → z. The connection of the continuous and discrete state vector isgiven by Eq. 3.2,

lima→0

: |xn, t〉 →√a|z, t〉, xn = na, −∞ ≤ z ≤ ∞

〈xn, t | →√a|〈z, t |. (4.42)

Also, let ti = 0, t = τ and tf = τ + τ ′. The initial and final state vectors aredefined for continuous initial and final positions and hence have the limit

|xi, ti〉 → |x, 0〉, 〈xf , t | → 〈x ′, τ + τ ′|. (4.43)

As shown in Figure 4.7, taking the a → 0 limit, from Eqs. 4.41, 4.42, and 4.43,yields

〈x ′, τ + τ ′|x,0〉 = a

+∞∑n=−∞

〈x ′, τ + τ ′|xn, τ 〉〈xn, τ |x, 0〉, (4.44)

⇒ 〈x ′, τ + τ ′|x,0〉 →∫ +∞

−∞dz〈x ′, τ + τ ′|z, τ 〉〈z, τ |x, 0〉. (4.45)

Writing the transition amplitude in Eq. 4.43 in terms of the evolution kernelgiven in Eq. 4.4, for Euclidean time, yields

Time

Space(a)

zx

'x

τ

'x0

'τ

Time

Spacez1 z2 zN

…

zN–1

τ1

τ2

τN–1

τN

…

…

'xx

'x'τ

0(b)

Figure 4.7 (a) Probability amplitudes for transition from initial state vector |x〉 tofinal state vector |x′〉, summing over all indistinguishable paths passing throughposition z at time τ . (b) The probability amplitude with path going through manyintermediate positions z1, z2, . . . zn at times τ1, τ2, . . . , τN .

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K(x ′, x; τ + τ ′)∫ +∞

−∞dzK(x ′, z; τ ′)K(z, x; τ). (4.46)

Equation 4.46, illustrated in Figure 4.7, shows that the definition of the evolutionkernel is consistent with the rules for the composition of probability amplitudes bysumming over all intermediate indistinguishable paths.

In writing Eq. 4.46, only the property of the action was used. Consider a finitetime slice [0, τ+τ ′], as shown in Figure 4.7; due to the term (dx/dt)2 in the actiongiven in Eq. 4.39, one needs to specify the initial value x at t = 0 and a final valuez at t = τ , since two boundary conditions are required to specify paths going fromx to z. The state space appears in the path integral via the boundary conditionsimposed on the paths over which the path integration is defined.

The condition given in Eq. 4.46 is a fundamental property of probabilityamplitudes that allows one to define the state space – since the path integral canbe interpreted as the matrix element connecting the initial and final state vector.The fundamental reason that the action satisfies the composition law is because,writing the action in terms of its initial and final boundary variables as S[xf , xi],the action given in Eq. 4.39, in the notation of Figure 4.7, has the form

S[x ′, x] = S[x ′, z] + S[z, x]. (4.47)

Interestingly enough, the above equation holds only for state space expressed interms of coordinate state vectors |x〉. Unlike the Schrödinger equation that holdsequally in momentum space, the composition law given in Eq. 4.46 does not holdwhen expressed in terms of Fourier transformed variables, essentially because Eq.4.47 does not hold for Fourier transformed variables.

In many complicated cases such as quantum field theory on curved spacetime,the quantum theory is defined directly in terms of the action, and it may not bepossible to derive a Hamiltonian; in such cases, one can directly base the existenceof the state space on the properties of the Lagrangian and action.

For the case where there is a well defined Hamiltonian, Eq. 4.46 follows directlyfrom the definition of the evolution kernel in terms of the Hamiltonian given in Eq.4.40 and the completeness equation; more precisely, since e−i(t+t ′)H = e−itH e−it ′H ,we have

K(x ′, x; t + t ′) = 〈x ′|e−i(t+t ′)H |x〉=∫ +∞

−∞dz〈x|e−itH |z〉〈z|e−it ′H |x ′〉

=∫ +∞

−∞dzK(x ′, z; t)K(z, x; t ′).

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4.9 Summary 79

4.9 Summary

The path integral is an independent formulation of quantum mechanics. To showthe path integral’s connection to the underlying foundations of indeterminacy of thequantum degree of freedom, the path integral has been derived from Schrödinger’sstate vector formulation. The probability amplitude for a finite determinate pathwas evaluated by breaking up the path into a series of infinitesimal steps. TheDirac–Feynman formula yields the probability amplitude for an infinitesimal timestep; composing the infinitesimal paths yields that the probability amplitude for afinite determinate path is proportional to exp{iS/�}, where S is the action for thequantum degree of freedom.

The transition of a quantum entity from its initial to final state, without any obser-vations during the interregnum, is made by the degree of freedom simultaneouslytaking all the indeterminate paths, which is a collection of many indistinguishabledeterminate paths. The principle of quantum superposition yields the transitionamplitude as the sum of the probability amplitudes of all the indistinguishabledeterminate paths and leads to the summing of exp{iS/�} over all the indeterminatepaths, and yields the Feynman path integral.

The evolution kernel was defined starting from the Hamiltonian of the degreeof freedom. A path integral expression was obtained for the evolution kernel us-ing methods based on the state space, and the Lagrangian was derived from theHamiltonian.

In fields outside physics, the path integral is generalized and represents a randomsystem with a probability distribution function for the different possible outcomesgiven by exp{S/�}/Z, where S is the analog of the action. For these random sys-tems, the path integral is defined, similarly to quantum mechanics, by the sum overall possible outcomes.

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5

Hamiltonian mechanics

Path integrals are by and large defined directly in terms of the configurationspace representation of the quantum entity’s degrees of freedom and employ theLagrangian description of the quantum entity; most of the path integrals in thisbook follow this approach.

The Hamiltonian provides another independent approach for defining path in-tegrals and is discussed in this chapter. Two important path integrals, which aredirectly based on the Hamiltonian, are the following: (a) one that is defined onthe degree of freedom’s phase space, defined as the tensor product of the degree offreedom space and its canonical conjugate momentum space; and (b) path integralsusing the coherent state basis instead of the coordinate basis. Path integrals definedon phase space, or for coherent states, are both based on the Hamiltonian.

To put in the foreground the role of the Hamiltonian in quantum mechanics, thecanonical equations connecting the Lagrangian to the Hamiltonian are discussed.A brief review of Hamiltonian mechanics, also called the canonical equations, isgiven in Section 5.1 and the connection of symmetries with conservation laws isdiscussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Sec-tion 5.3, for both Minkowski and Euclidean time. Phase space path integrals aredefined in Section 5.4. Canonical quantization based on the Poisson brackets isdiscussed in Section 5.5, and Dirac brackets required for quantizing constrainedsystems are derived in Section 5.7. Coherent states and their path integrals are dis-cussed in Sections 5.10 to 5.14.

5.1 Canonical equations

Consider for now Minkowski time denoted by t ; a generic degree of freedom inconfiguration space is denoted by q and let q ≡ dq/dt ; the generalization to N

degrees of freedom qi, i = 1, 2, . . . N is straightforward and will be discussed

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5.1 Canonical equations 81

when necessary. The Minkowski Lagrangian L and action are defined, from initialtime ti to final time tf , by

L = 1

2mq2 − V (q), q = dq

dt, (5.1)

S[q] =∫ tf

ti

dtL, (5.2)

where S[q] denotes the dependence of the action on the entire path taken by q(t),from qi = q(ti) to qf = q(tf ).

The classical equations of motion are defined by requiring that the classical tra-jectory, with the given boundary conditions, must minimize the action S, and thisyields Lagrange’s equation of motion

0 = δS[q] =∫ tf

ti

dt

[∂L

∂qδq + ∂L

∂qδq

]=∫ tf

ti

dt

[∂L

∂q− d

dt

(∂L

∂q

)]δq

⇒ ∂L

∂q= d

dt

(∂L

∂q

). (5.3)

The Lagrangian is based on the spacetime description of the entity and, in par-ticular, depends on the degree of freedom q and its time derivative q and yields thefollowing variation of the Lagrangian:

dL = ∂L

∂qdq + ∂L

∂qdq. (5.4)

For many applications, such as studying a system that conserves energy, it ismore suitable to have a formulation that does not refer to time directly. Henceone would like to change the independent variables from q, q to another set ofindependent variables p, q and transform the Lagrangian to a new function, namelythe Hamiltonian H ; the variation of the Hamiltonian is given by

dH = ∂H

∂qdq + ∂H

∂pdp. (5.5)

The canonical momentum p is defined below and yields, from Eq. 5.3, the follow-ing expression for the equation of motion:

p = ∂L

∂q⇒ ∂L

∂q= p. (5.6)

The Hamiltonian is defined by the Legendre transformation of L and, fromEqs. 5.4 and 5.6, is given by

dL = ∂L

∂qdq + ∂L

∂qdq = pdq + pdq = pdq − qdp + d(pq)

⇒ dH ≡ d(pq − L) = −pdq + qdp. (5.7)

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82 Hamiltonian mechanics

The above equation yields the Hamiltonian H as well as the dynamical equationsfor q, p. From Eqs. 5.5 and 5.7, one obtains the canonical equations

H = pq − L ⇒ − p = ∂H

∂q, q = ∂H

∂p. (5.8)

The canonical equations given in Eq. 5.8 are two first order equations and are equiv-alent to the single second order equation of motion given by the Lagrangian in Eq.5.3.

5.2 Symmetries and conservation laws

One of the most fundamental symmetries of a Lagrangian is that it does not explic-itly depend on time parameter t ; what this means is that the system is homogeneousin time, having the same dynamics for each instant of time; in particular, there isno special value of time that affects the evolution of the system; all changes intime are effected via the dynamical variables p(t), q(t). For such a Lagrangianhomogeneous in time, energy conservation follows directly from the equations ofmotion.

Let the dynamical variables of the system be qi, pi with i = 1, 2, . . . N . Usingthe classical equations of motion given in Eq. 5.3 yields1

dL =∑i

[∂L

∂qidqi + ∂L

∂qidqi

]=∑i

[d

dt

(∂L

∂qi

)dqi + ∂L

∂qidqi

]

= d

dt

[∑i

(∂L

∂qi

)dqi

]⇒ dL

dt= d

dt

[∑i

(∂L

∂qi

)qi

]. (5.9)

Hence, it follows from Eq. 5.9 that energy E, defined below, is a conservedquantity,

E =∑i

(∂L

∂qi

)qi − L ⇒ dE

dt= 0. (5.10)

Note that although the definition of E looks very similar to the Hamiltonian H ,the two are quite different: E is defined only for the classical trajectory that fixes,for each instant, both the position and momentum; in contrast, the HamiltonianH = H(pi, qi) is defined for all values of the canonical momenta and coordinatespi, qi . The equation H(pi, qi) = E fixes a surface in phase space pi, qi ; for a givenfixed value of E = E0, the classical trajectory, as it evolves in time, moves in phasespace on the surface of constant E0.

1 An explicit dependence on time of L would yield an extra term dt (∂L/∂dt) in dL that would spoil theconservation law for energy.

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5.2 Symmetries and conservation laws 83

Suppose again that the Lagrangian L does not have any explicit dependence ontime t . From Eqs. 5.4 and 5.6

dL =∑i

[∂L

∂qidqi + ∂L

∂qidqi

]=∑i

[pidqi + pidqi]

= d

dt

[∑i

pidqi

]. (5.11)

Consider a general time independent transformation of the coordinates qi thatleaves the Lagrangian invariant, namely

δqi = fi(p, q)δ ⇒ δL = 0. (5.12)

Hence, from Eqs. 5.11 and 5.12

δL = 0 = d

dt

[∑i

piδqi

]= δ

d

dt

[∑i

pifi(p, q)

],

∑i

pifi(p, q) = constant. (5.13)

The canonical equations lead to a remarkable conclusion: for every symmetryof the Lagrangian there is a conserved quantity that remains constant over time.This result continues to hold in quantum mechanics, with every symmetry of theLagrangian resulting in a conserved quantity.

To illustrate the relation between symmetries and conservation laws, considerthe Lagrangian for two uncoupled harmonic oscillators, namely

L = 1

2m[x2 + y2] + 1

2mω2[x2 + y2]. (5.14)

It can be verified that under the following symmetry transformation, which is aninfinitesimal rotation of the coordinates about the z-axis,

δx = −yδ, δy = xδ ⇒ fx = −y, fy = x, (5.15)

the Lagrangian is invariant. Hence, the general result given in Eq. 5.13 yields, forthe transformation given in Eq. 5.15, the conserved quantity∑

i

pifi(p, q) = −ypx + xpy : constant.

The conserved quantity is seen to be the z-component of angular momentum.

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5.3 Euclidean Lagrangian and Hamiltonian

Euclidean time τ is defined as τ = it . The coordinate degree of freedom remainsunchanged and is q; Euclidean velocity q

Eis defined as follows:

τ = it, (5.16)

qE= dq

dτ= −i dq

dt= −iq. (5.17)

Hence, the Euclidean Lagrangian, from Eq. 5.1, is given by

LE(q, q

E) = −1

2m(q

E)2 − V (q) (5.18)

and Euclidean action, similar to Eq. 5.2, is given by

SE[q] =

∫ τf

τi

dτLE. (5.19)

The equations of motion for Euclidean time are obtained by minimizing the actionS and, similarly to Eq. 5.3, yield Lagrange’s equation of motion,

δSE[q] = 0 ⇒ ∂L

E

∂q= d

dτ

(∂L

E

∂qE

). (5.20)

Denoting Euclidean momentum by pE

, the Lagrangian yields

pE= ∂L

E

∂qE

= −mqE. (5.21)

Note the minus sign in the definition of pE

in terms of Euclidean velocity qE

,which is due to Euclidean time switching the sign of the kinetic in the EuclideanLagrangian as in Eq. 5.18. Since q

E= −iq, in terms of Minkowski momentum,

the Euclidean momentum is given by

pE= imq = ip,

p ∈ [−∞,+∞] ⇒ pE∈ [−i∞,+i∞]. (5.22)

Note the important fact that, since Minkowski momentum p is real, Euclideanmomentum p

Eis purely imaginary.

The Hamiltonian is defined by the Legendre transformation of LE

; fromEqs. 5.20 and 5.21

dLE= ∂L

E

∂qdq + ∂L

E

∂qE

dqE= p

Edq + p

Edq

E

⇒ dHE≡ d(p

EqE− L

E) = −p

Edq + q

Edp

E. (5.23)

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5.4 Phase space path integrals 85

The above equation yields the Hamiltonian HE

as well as the dynamical equationsfor q, p

E. In particular, similarly to Eq. 5.23, the Hamilton equations of motion are

given by

HE= p

EqE− L

E⇒ − p

E= ∂H

E

∂q, q

E= ∂H

E

∂pE

. (5.24)

The Euclidean Hamiltonian, from Eq. 5.29, is given by

HE(p

E, q) = p

EqE− L

E= − 1

2m(p

E)2 + V (q). (5.25)

The kinetic energy term of the EuclideanHE

has the opposite sign to the Minkowskikinetic term, reflecting the result obtained in Eq. 5.22.

Once the Euclidean Lagrangian and Hamiltonian are defined, all the results ob-tained earlier for Hamiltonian mechanics are valid for the Euclidean case.

5.4 Phase space path integrals

Path integral quantization discussed in Chapter 4 is based on the Lagrangian andthe configuration space representation of the degree of freedom, as expressed inEq. 4.40. In contrast, phase space path integration is based on the Hamiltonian H ,which is a function of pi, qi . The canonical coordinate qi and momentum pi , takentogether, define the phase space of the system, which for N particles (degrees offreedom) in three dimensional space is the 3N dimensional Euclidean space �3N .

The phase space path integrals for both Minkowski and Euclidean time are dis-cussed here. The phase space path is defined as the sum over all possible values ofp, q from initial time ti to final time tf , with the boundary condition qi , qf beingimposed on only coordinate q. For notational simplicity, unless required, the vectorindex is suppressed and phase space is represented by p, q.

Recall from Eq. 4.40 that the path integral quantization of the classical entityyields the transition amplitude given by

Z ≡ K(qi, qf ; tf , ti) =∫

Dq exp

{i

�S[q]

}.

Let us consider how this path integral is related to the phase space path integral. Todistinguish between the Minkowski and Euclidean cases, the subscripts M and E

are employed for the various quantities.

Minkowski path integral

The phase space action integral SM

for Minkowski time is given by

SM[p, q] =

∫ tf

ti

dt[pq −H

M(p, q)

]. (5.26)

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Similarly to Eq. 4.40, the path integral is given by integrating over all possibleconfigurations of the degrees of freedom in phase space and yields

ZM= N

∫DqDp exp

{i

�

∫ tf

ti

dt[pq −H

M(p, q)

]}. (5.27)

For the Lagrangian LM= 1

2mq2−V (q) given in Eq. 5.1, the canonical momen-

tum is given by

p = ∂LM

∂q= mq (5.28)

and hence the Hamiltonian is

HM= pq − L = 1

2mp2 + V (q). (5.29)

Hamiltonians that do not have velocity dependent potentials and that are freefrom any constraint have the generic form given in Eq. 5.29.

In general, the integration over canonical momentum cannot be done explicitly.For the Hamiltonian given in Eq.5.29 that is quadratic in the canonical momentum,one can exactly perform the Gaussian path integral over the momentum variablesin Eq. 5.27, which yields the Lagrangian

ZM= N ′

∫Dq exp

{i

�

∫ tf

ti

dtLM(q, q)

}, (5.30)

which is the result given earlier in Eq. 4.40.

Euclidean path integral

The Euclidean path integral can be obtained by an analytic continuation of theMinkowski path integral. From Eq. 5.26, define the action for Euclidean timeSE[p

E, q] by

SE[p

E, q] =

∫ τf

τi

dτ[p

EqE−H

E(p

E, q)

]. (5.31)

Continuing t = −iτ in Eq. 5.27 to imaginary Euclidean time yields

i

�SM = −i

2

�SE= 1

�SE[p

E, q]. (5.32)

Path integration is given by Eq. 4.40 and, from Eq. 5.22, because Euclideanmomentum is pure imaginary, yields the Euclidean path integral

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5.5 Poisson bracket 87

ZE= N

E

∫Dq

∫Dp

Eexp

{1

�SE[p

E, q]

}

= NE

∫ +∞

−∞Dq

∫ +i∞

−i∞Dp

Eexp

{1

�

∫ τf

τi

dτ[pEqE−H

E(p

E, q)

]}.

The Euclidean Hamiltonian HE

, in terms of real momentum p = −ipE

as in Eq.5.22, is given by

HE(p, q) = H

E(ip, q) = 1

2mp2 + V (q), S

E[p, q] ≡ S

E[−ip

E, q]. (5.33)

Hence

ZE= N

E

∫Dq

∫Dp exp{1

�SE[p, q]} (5.34)

= N ′E

∫ +∞

−∞Dq

∫ +∞

−∞Dp exp

{1

�

∫ τf

τi

dτ[ipq

E− H

E(p, q)

]}. (5.35)

Since the kinetic energy is quadratic in momentum, performing the Gaussian pathintegral over

∫Dp yields

ZE= N ′′

E

∫Dq exp

{1

�

∫ τf

τi

dτLE(q, q ′)

}, (5.36)

where the Euclidean Lagrangian LE

is given by

LE(q, q

E) = −1

2m(q

E)2 − V (q)

and agrees with the result obtained earlier in Eq. 5.18.

5.5 Poisson bracket

Consider arbitrary functions f (p, q; t) and g(p, q; t) of the canonical variablesp, q. The Poisson bracket {f, g}

Pis defined as

{f, g}P= ∂f

∂q

∂g

∂p− ∂f

∂p

∂g

∂q. (5.37)

The Poisson bracket for the canonical conjugate variables is given by

{q, p}P= ∂q

∂q

∂p

∂p− ∂q

∂p

∂p

∂q= 1. (5.38)

The time evolution of the function f , using the dynamical equations given in Eq.5.8, is given by

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df

dt= ∂f

∂qq + ∂f

∂pp + ∂f

∂t= ∂f

∂q

∂H

∂p− ∂f

∂p

∂H

∂q+ ∂f

∂t

⇒ df

dt≡ {f,H }

P+ ∂f

∂t. (5.39)

For the special case of the Hamiltonian, one has in general that

dH

dt= ∂H

∂t+ {H,H }

P= ∂H

∂t. (5.40)

The Poisson bracket {f, g}P

defined above can be generalized – for the case ofN dynamical variables q�, p�; � = 1, 2, . . . N – to

{f, g}P=

N∑�=1

(∂f

∂q�

∂g

∂p�− ∂f

∂p�

∂g

∂q�

). (5.41)

Poisson brackets have the following general properties:

{f, g}P= −{g, f }

P, {f, gh}

P= {f, g}

Ph+ g{f, h}

P(5.42)

and obey the Jacobi identity

{f, {g, h}P}P+ {h, {g, h}

P}P+ {g, {f, g}

P}P= 0. (5.43)

The Euclidean Hamiltonian and canonical variables obey the Poisson bracketrelations. Similarly to Eq. 5.38,

{q, pE}P= ∂q

∂q

∂pE

∂pE

− ∂q

∂pE

∂pE

∂q= 1. (5.44)

Furthermore, from Eq. 5.24 and similarly to Eq. 5.39,

df

dτ= {f,H

E}P+ ∂f

∂τ. (5.45)

5.6 Commutation equations

Path integral quantization, as discussed in Section 5.4, is based on functional inte-gration over the phase space of the degree of freedom. Another method for quan-tizing the degree of freedom is to introduce the Heisenberg commutation equationsfor the canonical dynamical variables q, p.

Minkowski case

Consider the single degree of freedom q; for Minkowski time t , the Minkowskimomentum is p = mdq/dt . The procedure for canonical quantization is to elevatethe dynamical variables q, p to Hermitian operators q, p and replace the Poisson

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5.6 Commutation equations 89

bracket of the classical dynamical variables q, p by the Heisenberg commutationequations for Hermitian operators q, p.

The degree of freedom q is quantized by postulating the following quantizationcondition [Weinberg (2013)]:

[q, p] = i�I{q, p}P. (5.46)

Note that the factor i on the right hand side of Eq. 5.46 is due to operators q, p andthe operator rendition of {q, p}

Pall being Hermitian.

The right hand side of Eq. 5.46 above is to be interpreted as follows: a) firstevaluate the Poisson bracket considering p, q as classical dynamical variables; andb) then generalize p, q to Hermitian operators, with a symmetrical ordering of theterms in {q, p}

Pchosen to ensure that the right hand side is a Hermitian operator.

The Poisson bracket for the canonical conjugate variables given in Eq. 5.38yields the well-known Heisenberg commutation equation2

[q, p] = i�I since {q, p}P= 1

⇒ p = −i ∂∂q

. (5.47)

Euclidean case

Euclidean momentum pE

for Euclidean time τ = it is given, from Eq. 5.21

pE= ip. (5.48)

Note the Hermitian operators are related in the following manner:

pE= ip ⇒ p†

E= −p

E: anti-Hermetian. (5.49)

Hence, for Euclidean time, the rule for canonical quantization is given by

[q, pE] = −�I{q, p

E}P.

The Euclidean commutation equation does not have a factor of i since the Eu-clidean momentum operator p

Eis anti-Hermitian. Since {q, p

E}P= 1,

[q, pE] = −�I (5.50)

⇒ pE= ∂

∂q. (5.51)

The result obtained in Eq. 5.50 above is verified by an independent path integralderivation given in Eq. 6.58.

2 The hat symbol is often dropped from the operators for simplicity of notation.

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5.7 Dirac bracket and constrained quantization

The transition from classical dynamical variables to quantum operators, as in thecommutation equations given in Eqs. 5.46 and 5.50, needs to be modified for thecase of when the dynamical variables are constrained.

Consider a dynamical system in Euclidean time with L constraints that restrictthe degree of freedom q = (q1, . . . qN) to lie in an N−L dimensional subspace of�N specified by

f1(q) = f2(q) . . . = fL(q) = 0. (5.52)

The question one needs to address is: what are the commutation equations of pE

and q (in boldface vector notation)? The commutation equations given in Eq. 5.50are invalid for the constrained system as they do not respect the constraint equationsgiven in Eq. 5.52.

Dirac (1964) has developed a procedure for quantizing constrained systems andin particular finding the commutation equations; the procedure is generalized to thecase of Euclidean time.

The primary constraints φi = fi(q), i = 1, 2 . . . L have to be supplemented bysecondary constraints φ(L+1), φ(L+2), . . . φK required for all the primary constraintsto be conserved in time. All the constraints are weakly zero, namely

φi(pE, q) ≈ 0, i = 1, 2, . . . K,

where the symbol ≈ means that the constraints φi are set to zero after performingthe differentiations required for evaluating all the Poisson brackets.

The modified Hamiltonian is defined, similarly to the procedure of defining con-straints using Lagrange multipliers in the Lagrangian, by

H(pE, q) = H

E(p

E, x)+

K∑j=1

uj (pE, q)φj (pE

, q), (5.53)

where HE

is the Euclidean Hamiltonian.The coefficient functions ui are determined by the requirement that the con-

straints are weakly conserved over time; this leads, from Eq. 5.45, to solving thefollowing equations:

∂φI

∂τ= {φI ,H }P = {φI ,HE

}P+

K∑j=1

({φI , uj }P φj + uj {φI , φj }P) ≈ 0. (5.54)

The constraints φ�, � = 1, 2 . . . , K define the anti-symmetric constraint matrix

C�m = {φ�, φm}P . (5.55)

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5.7 Dirac bracket and constrained quantization 91

The Dirac brackets are defined, for arbitrary function h(q, p), g(q, p), by

{h, g}D= {h, g}

P−

K∑�,m=1

{h, φ�}P C−1�m {φm, g}P .

The Dirac brackets satisfy all the relations fulfilled by the Poisson bracket givenin Eqs. 5.42, and in particular can be shown to satisfy the crucial Jacobi identitygiven in Eq. 5.43.

The rule for defining the Heisenberg commutation equations for the constraineddegree of freedom q is a generalization of Eq. 5.50, and for Euclidean time isgiven by

[qi , pEj ] = −�I{qi, pEj }D . (5.56)

The advantage of using the Dirac bracket for quantization is that the commutationequation holds for all values of the degree of freedom q and is not restricted by thecondition of weak equality.

5.7.1 Dirac bracket for two constraints

Consider the case of a single primary constraint [Weinberg (2013)] on the dynam-ical variable x given by

φ1 = f (x) = 0.

The Euclidean path integral provides a straightforward method for quantization byconstraining the degrees of freedom in the path integral and yields

Z =∫

Dx exp{∫

dtLE}∏t

δ(f (x(t))

)(5.57)

=∫

DxDλ exp{∫

dt[LE+ iλ(t)f (x(t))]}. (5.58)

The Euclidean Hamiltonian, from Eq. 5.25, is given by

HE(p

E, x) = p

E· dxdτ− L

E− iλf (x)

= − 1

2m(p

E)2 + V (x)− iλf (x). (5.59)

The primary constraint needs to be conserved over time and leads to the require-ment

∂φ1

∂τ= {φ1, HE

}P= − 1

mpE· ∇f ≈ 0. (5.60)

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Hence, the second constraint is chosen to be

φ2 = pE· ∇f ≈ 0, (5.61)

which yields, from Eqs. 5.60 and 5.61, that constraint φ1 is conserved over time.As required by Eq. 5.53, choose the modified Hamiltonian

H(pE, q) = H

E(p

E, x)+ iλφ1

= − 1

2m(p

E)2 + V (x). (5.62)

The time dependence of the constraint φ2 is given by

∂φ2

∂τ= {φ2, H }P = −

1

mp

Ei

∂

∂xi

(pE· ∇f ) ≈ 0. (5.63)

The last equality follows from the weak equality φ2 = pE· ∇f ≈ 0 given in Eq.

5.61; since it holds for all x, this implies that ∂(pE· ∇f )/∂xi ≈ 0.

Since both the constraints φ1, φ2 are (weakly) conserved in time, no new con-straints are required. The constraint matrix, from Eq. 5.55, is given by

C12 = {φ1, φ2}P =∑i

∂φ1

∂xi

∂φ2

∂pEi

= (∇f )2 = −C21. (5.64)

The inverse is given by

C−112 = −

1

(∇f )2= −C−1

21 . (5.65)

The Dirac bracket, from Eq. 5.56, is given by

{xi, pEj}D= {xi, pEj

}P− {xi, φ2}P C−1

21 {φ1, pEj}P

= δi−j − ∂f

∂xi

1

(∇f )2

∂f

∂xj. (5.66)

The other Dirac brackets have been shown by Weinberg (2013) to be zero, namely

{xi, xj }D = 0 = {pEi, p

Ej }D .Hence, for Euclidean time, the Heisenberg commutation equations for a con-strained degree of freedom, from Eqs. 5.56, 5.66, and 5.67, are given by

[xi , pEj ] = −�I{xi, pEj }D = −�I(δi−j − ∂f

∂xi

1

(∇f )2

∂f

∂xj

), (5.67)

[xi, xj ] = 0 = [pEi, p

Ej ], (5.68)

with the Hamiltonian given from Eq. 5.62 by

H(pE, q) = − 1

2m(p

E)2 + V (x).

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5.8 Free particle evolution kernel 93

Note an important general feature of quantum mechanics is that the state spacehas a structure that is specified independently of the Hamiltonian. This is whatcomes to the fore in the case of a constrained degree of freedom.

For the constrained system that is being considered, the Hamiltonian is thesame as the unconstrained case but the commutation equation given in Eq. 5.67is changed from the unconstrained case to account for the constraint. The commu-tation equation, in turn, is a reflection of the structure of the state space.

For a more complex constrained system such as the one for the accelerationHamiltonian considered in Section 13.4, both the Hamiltonian and the commuta-tion equation are modified to account for the constraints imposed on the degrees offreedom.

5.8 Free particle evolution kernel

Consider the Minkowski and Euclidean Hamiltonian of a free quantum particlegiven by

H = 1

2mp2 = − 1

2m(p

E)2 = − �

2

2m

∂2

∂x2. (5.69)

The Hamiltonian for both the Minkowski and Euclidean time yields the same dif-ferential operator in terms of the coordinate degree of freedom x ∈ �d . The eigen-states of H are given by the plane wave eigenstates | �p〉,

H |p〉 = p2

2m|p〉, (5.70)

that yields, from Eq. 3.34, the completeness equation∫ ∞

−∞dp

(2π�)d|p〉〈p| = I, 〈p|p′〉 = (2π�)dδ(p − p′). (5.71)

The free particle Minkowski evolution kernel, from Eq. 4.4, is given by

K(x, x′; t) = 〈x|e−it p2

2m� |x′〉 =∫

dp(2π�)d

e−itp2

2m� eip·(x−x′)/�

=(√

m

2πi�t

)d/2

exp

{i

�

m(x− x′)2

2t

}. (5.72)

The evolution kernel for Euclidean time τ is given by

K(x, x′; τ) = 〈x|e−τ(pE)2

2m� |x′〉

=(√

m

2π�τ

)d/2

exp

{−1

�

m(x− x′)2

2τ

}. (5.73)

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As can be seen by comparing Eqs. 5.72 and 5.73, the Euclidean evolution kernel isa well defined Gaussian distribution whereas the Minkowski case is an oscillatingexponential that needs to be defined using the theory of distributions.

5.9 Hamiltonian and path integral

The phase space path integral is defined by summing over all possible indeter-minate evolution of the degree of freedom and its canonical conjugate variable,and is one of the ways of defining a quantized Hamiltonian. Another approach,which leads to the Schrödiner equation, is to first quantize the classical Hamil-tonian based on the commutation equations, and then use state space methods todetermine the indeterminate evolution of the quantum degree of freedom encodedin its state vector.

In this section, the evolution kernel for a general Hermitian HamiltonianH(p, x)

is studied in Euclidean time. The fundamental reason that analytic continuation iswell defined is because the Hamiltonians for all physical systems are bounded frombelow. For all physical systems the lowest energy state, namely the ground state,has finite energy (and not infinite negative energy). Since the eigenspectrum of His bounded from below, the operator exp{−τH } is well defined with none of thematrix elements having an ill-defined value.

The Euclidean time evolution kernel is given by3

K(xN, x0; τ) = 〈xN | e−εH · · · e−εH︸︷︷︸N−factors

|x0〉,

where xi = x0, xf = xN for notational convenience. The general matrix elementsof H are given by

〈p|e−εH |x〉 = 〈x|e−εH † |p〉∗ = 〈x|e−εH |p〉∗ = e−εH (p,x)e−ipx.

Using the completeness equation∫dx|x〉〈x| = I = ∫

dp

2π |p〉〈p| yields

K(xN, x0; τ) =N−1∏i=1

∫dxi〈xN |e−εH |xN−1〉 · · · 〈xn+1|e−εH |xn〉 · · · 〈x1|e−εH |x0〉

=∫

Dx

N∏i=1

∫dpi

2π〈xN |pN 〉〈pN |e−εH |xN−1〉 · · ·

· · · 〈xn+1|pn+1〉〈pn+1|e−εH |xn〉〈xn|pn〉 · · · 〈x1|p1〉〈p1|e−εH |x0〉3 Set � = 1.

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5.10 Coherent states 95

=∫

DxDpei∑N

n=1 pnxne−ε∑N

n=1 H(pn,xn−1)e−i∑N

n=1 pnxn−1,

=∫

DxDpeS, (5.74)

where∫

DxDp =N−1∏n=1

∫ +∞

−∞dxn

N∏i=1

∫ +∞

−∞dpi

2π.

Hence, from Eq. 5.74, the limit of ε → 0 yields

S = i

N∑n=1

pn(xn − xn−1)− ε

N∑n=1

H(pn, xn−1)→ i

∫ τ

0dtpx −

∫ τ

0H(pt, xt )dt

⇒ S =∫ τ

0dt{ipx −H(p, x)}.

For the special case of a Euclidean Hamiltonian given by Eq. 5.25, namely

H = p2

2m+ V (x),

we obtain

K(xN, x0; τ) =∫

DxDpe−∫dtV e−

∫dt[ p2

2m−ipx] = N∫

Dxe− ∫ dt(m2 x2+V

)

⇒ K(xN, x0; τ) =∫

DxeS with Dx = (m

2πε)N2

N−1∏n=1

dxn.

Hence

S =∫ τ

0dtL, L = −m

2x2 − V

and the earlier result for the Euclidean Lagrangian given in Eq. 5.18 is recovered.

5.10 Coherent states

Coherent states are the natural basis states for studying systems that are defined interms of the creation and destruction operators and many computations can be donein a transparent manner in this basis. Coherent states also provide a mathematicalframework for the formulation of path integrals that, unlike many cases that requirea Lagrangian, requires only a Hamiltonian.

Consider a Hamiltonian H that is expressed in terms of the annihilation andcreation operators a, a† and has the form H = H(a, a†). The oscillator basis isdefined by

[a, a†] = 1, (5.75)

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a|0〉 = 0, |n〉 = (a†)n√n! |0〉, n = 1, 2, 3 . . .

〈n|m〉 = δm−n.

A coherent state is defined for complex number z by

|z〉 = eza† |0〉 ⇒ |z〉 =

∑n=0

zn√n! |n〉. (5.76)

Note that the |z〉 states are over-complete since z is complex and hence thereare many more states |z〉 than the integer valued oscillator basis {|n〉; n =0, 1, 2 · · ·∞}, which is complete.

The commutation equation given in Eq. 5.75 yields

a|n〉 = √n|n− 1〉, a†|n〉 = √n+ 1|n+ 1〉.Hence

a|z〉 =∑n

zn√n!a|n〉 =

∑ zn√n!√n|n− 1〉 = z|z〉.

Similarly,

〈z| = 〈0|ez∗a ⇒ 〈z|a† = 〈z|z∗.Consider the scalar product

〈z2|z1〉 = 〈0|ez∗2aez1a† |0〉.

Note that

ez∗2aez1a

† = ez1a†e−z1a

†ez∗2aez1a

† = ez1a†

exp(z∗2e

−z1a†aez1a

†)

= ez1a†ez∗2(a−z1[a†,a]) = ez1a

†ez∗2aez

∗2z1 .

Hence

〈z2|z1〉 = ez∗2z1〈0|ez∗2a†

ez1a|0〉 = ez∗2z1 . (5.77)

5.11 Coherent state vector

The normalized coherent state function is defined by

|ψz〉 = C|z〉,with the normalization given by

〈ψz|ψz〉 = 1 = C2e+|z|2 ⇒ C = e−

12 |z|

2

.

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5.11 Coherent state vector 97

One can use state functions |ψz〉 to define the completeness equation instead of |z〉;this just leads to a change of normalization and does not yield any changes in theresults obtained.

The expectation value of the number operator N = a†a is given by

N = 〈ψz|N |ψz〉 = 〈ψz|a†a|ψz〉 = |z|2.Since N = |z|2, the likelihood of finding the coherent state in the �th excited stateof the oscillator is given by

P�(N) = |〈�|ψz〉|2 =∣∣∣e−N

2|z|�√�!∣∣∣2 = e−N

N�

�! .

The result shows that the probability is given by the Poisson distribution. This is awell-known property of the photon field, in that the probability of a classical sourceemitting N photons of a given energy is given by the Poisson distribution.

Consider the harmonic oscillator Hamiltonian given by

H = ωaa†.

The evolution of the coherent state is given by

e−τωa†a|z〉 = e−τωa

†aeza† |0〉 = e−τωa

†aeza†eτωa

†ae−τωa†a|0〉

= exp(ze−τωa

†aa†eτωa†a)|0〉.

Note that

e−λa†aa†eλa

†a = e−λa†.

To prove this consider

w(λ) = e−λa†aa†eλa

†a

⇒ dw

dλ= e−λa

†a[a†, a†a]eλa†a = −w(λ),and integrating the above equation yields

w(λ) = e−λw(0) = e−λa†. (5.78)

Hence

e−τωa†a|z〉 = exp

(ze−ωτa†

)|0〉 = |e−ωτ z〉. (5.79)

For the simple harmonic oscillator, the coherent state’s time evolution results ina rescaling of its label z; this is the reason that coherent states are so useful inthe study of quantum optics, where the photons are equivalent to a collection ofharmonic oscillators.

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5.12 Completeness equation: over-complete

For z = x + iy, z∗ = x − iy, the completeness equation for the coherent states canbe written as

1 =∫ +∞

−∞dxdy

π|z〉e−z∗z〈z| ≡

∫dzdz∗

2πi|z〉e−z∗z〈z|. (5.80)

Note the new feature that there is a metric on the state space given by e−z∗z, whichis due to the fact that the basis |z〉 are over-complete; in other words, there arefar more basis states |z〉 than are required for spanning the state space. The over-completeness is to be expected since the oscillator state space is spanned by theoscillator basis |n〉, with n ≥ 0, whereas the |z〉 basis states are indexed by twocontinuous variables x, y.

To prove the completeness equation, note that consistency requires that the scalarproduct given in Eq. 5.77 be reproduced if the completeness equation is used forforming the scalar product. Hence consider

〈z2|z1〉 =∫

dxdy

π〈z2|z〉e−z∗z〈z|z1〉 =

∫dxdy

πez∗2ze−z

∗zez∗z1

=∫

dxdy

πeS. (5.81)

Since z = x + iy and z∗ = x − iy,

S = z∗2(x + iy)− (x2 + y2)+ (x − iy)z1

= −(x2 + y2)+ x(z1 + z∗2)+ iy(z∗2 − z1). (5.82)

Therefore

〈z2|z1〉 = 1

π

∫dxdye−x

2+x(z1+z∗2)e−y2+iy(z∗2−z1)

= 1

π· √πe 1

4 (z1+z∗2)2 · √πe− 14 (z

∗2−z1)

2

= ez∗2z1 (5.83)

as expected, hence verifying the completeness equation.

5.13 Operators; normal ordering

The matrix element of an operator Q(a†, a) is in general an arbitrary function ofa, a†. Since a and a† do not commute, the operator Q(a†, a) has to be brought intoa standard form, which is chosen to be all the destruction operators a to the rightof all the creation operators a†. The procedure for bringing an operator Q(a†, a)

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5.14 Path integral for coherent states 99

into the standard format is called normal ordering, and to emphasize the standardnormal ordered form of the operator, it is represented by : Q(a†, a) :.

One can then obtain the matrix elements of the operator to be given by

〈z| : Q(a†, a) : |z〉 =: Q(z∗, z) : 〈z|z〉 =: Q(z∗, z) : ez∗z, (5.84)

where, in the operator Q(a†, a), one has the replacement

a→ z, a† → z∗. (5.85)

An example of normal ordering is

H = a2a† ⇒ : H :=: a2a† :≡ a†a2.

The relation between H and : H : is

H = a(aa†) = a(a†a + 1) = (a†a + 1)a + a

=: H : +2a ⇒ : H := a†a2. (5.86)

One then obtains, from Eq. 5.86,

〈z2| : H(a†, a) : |z1〉 = 〈z2| : H(z∗2, z1) : |z1〉 = z∗2z21e

z∗2z1 . (5.87)

This is the general matrix element for the normal ordered operator.

5.14 Path integral for coherent states

In general, for an arbitrary Hamiltonian, using the completeness equation N − 1times yields the evolution kernel

K(zf , zi; τ) = 〈zf |e−τH(a†,a)|zi〉=∫

DzDz∗〈zN |e−εH |zN−1〉e−z∗N−1zN−1〈zN−1|· · · e−z∗n+1zn+1〈zn+1|e−εH |zn〉 · · · 〈z1|e−εH |z0〉,

where DzDz∗ =N−1∏i=1

dzidz∗i

2πi=

N−1∏i=1

dxidyi

π, zf = zN, zi = z0.

To find the matrix elements of 〈zn+1|e−H(a,a†)|zn〉 we first assume that theHamiltonian H(a, a†) has been normal ordered by moving all the destructive oper-ators to the right. Hence, from the general property of the normal ordered operatorgiven in Eq. 5.84,

〈zn+1|e−εH (a†,a)|zn〉 = e−εH(z∗n+1,zn)〈zn+1|zn〉 = e−εH(z∗n+1,zn)ez∗n+1zn .

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The evolution kernel is given by

K(zf , zi; τ) =∫

DzDz∗eS,

where S = −εN−1∑n=0

H(z∗n+1, zn)+N−1∑n=0

z∗n+1zn −N−1∑n=1

z∗nzn

= −εN−1∑n=0

H(z∗n+1, zn)+N−1∑n=0

(z∗n+1 − z∗n)zn + z∗0z0.

The action has a new feature: the appearance of a boundary term, namely z∗0z0; theterm z0 is fixed by the boundary condition whereas z∗0 is an integration variable.The continuum limit is given by

S = −∫ τ

0dtH(z∗(t), z(t))+

∫ τ

0dtdz∗

dtz+ z∗(0)z(0).

Since ∫ τ

0dtdz∗

dtz =

∫ τ

0dt

[d

dt(zz∗)− z∗

dz

dt

]

= z∗(τ )z(τ )− z∗(0)z(0)−∫ τ

0z∗dz

dt

one obtains

S = −∫ τ

0dt

[H(z∗, z)+ z ∗ dz

dt

]+ z∗(τ )z(τ ). (5.88)

A symmetric expression for the action yields

S = −∫ τ

0dtH(z∗(t), z(t))+ 1

2

∫ τ

0dt

(dz∗

dtz− z∗

dz

dt

)

+ 1

2

(z∗f z(τ )+ z∗(0)zi

),

Boundary conditions : zf = z(τ ), zi = z(0) : fixed.

Note both z(τ ) and z∗(0) are integration variables and couple the boundary termszi, zf via the time derivative terms in the action S to the other integration variablesof the path integral.

The discrete coherent state path integral is well defined for the linear as well asfor the nonlinear case. However, for the nonlinear case the continuum limit of thediscretized coherent state path integral apparently has anomalies and ambiguities,as is the case when studying a spin system [Zinn-Justin (2005)]. The ambiguitiescan be resolved by using the fact that the coherent basis states are over-complete.

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5.14 Path integral for coherent states 101

5.14.1 Simple harmonic oscillator

The Hamiltonian is given by

H = ωa†a,

and from Eq. 5.79

e−τωa†a|z〉 = |e−ωτ z〉.

Hence

K(zf , zi; τ) = 〈zf |e−τωa†a|zi〉 = 〈zf |e−ωτ zi〉= exp

(e−ωτ z∗f zi

). (5.89)

The evolution kernel for the simple harmonic oscillator can also be obtained byperforming the path integral. In the coherent state representation, from the defini-tion of the matrix elements of operators given in Eq. 5.84, the matrix elements ofthe Hamiltonian are given by

〈z|H(a†, a)|z〉 = ω〈z|a†a|z〉 = ωz∗zez∗z ⇒ H(z∗, z) = ωz∗z.

From Eq. 5.88, the action, for z ≡ dz/dt , is given by

S = −∫ τ

0(z∗z+ ω z∗z)+ z∗f z(τ ).

The classical equation of motion is given by

δS

δz∗= zc + ωzc = 0.

For z = x + iy, one has z∗ = x∗ − iy∗, and x, y yields the complex classicaltrajectory of the particle,

zc = e−ω tz(0) ⇒ z∗c (t) = e−(τ−t)ωz∗f .

Consider the change of variables

z = zc + ς, z∗ = z∗c + ς∗,where

ς(0) = 0 , ς ∗ (τ ) = 0.

Hence

S = −∫ τ

0(ς∗ς + ωςς)+ z∗f e

−ωτ zi (5.90)

and yields the evolution kernel

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K(z∗f , zi; τ) = A exp{z∗f e−ωτ zi},with A being a constant.

Let us perform a trace of K using the coherent basis completeness equation, andobtain

K = 〈z|e−TH |z〉⇒ trK =

∫dzdz∗

2πie−z

∗zK(z∗, z) = A

∫dzdz∗

2πie−(1−e

−wτ )|z|2

= A

1− e−wτ. (5.91)

Performing the trace directly on the Hamiltonian using the oscillator basis yields

trK = tr(e−τH ) = 1

1− e−wτ⇒ A = 1,

and hence, as given in Eq. 5.89, the evolution kernel is given by

K(z∗f , zi; τ) = exp{z∗f e−ωτ zi}.

5.15 Forced harmonic oscillator

Consider the Hamiltonian

H = ωa†a + λ(a† + a)

= ω

(a† + λ

ω

)(a + λ

ω

)− λ2

ω2.

Define the shifted operators by

b = a + λ

ω, b† = a† + λ

ω.

Then, for a|0〉 = 0 define the vacuum state for the shifted operators |υ〉 by

b|υ〉 = 0 = (a + λ

ω)|υ〉 ⇒ a|υ〉 = − λ

ω|υ〉. (5.92)

The state |υ〉 is given by

|υ〉 = eλω(a−a†)|0〉.

To prove this consider the expression

w(t) = e−t (a−a†)aet(a−a

†).

It can be readily shown, similarly to the derivation of Eq. 5.78, that

w(t) = −t + a.

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5.16 Summary 103

Hence one obtains the result given in Eq. 5.92, namely

a|υ〉 = aeλω(a−a†)|0〉 = e

λω(a−a†)[− λ

ω+ a]|0〉 = − λ

ω|υ〉.

Using the CBH formula eAeB = eA+B+ 12 [A,B] for [A,B] ∝ I yields

e−λωa†eλωa = e

λω(a−a†)− λ2

ω2 [a†,a] = e+ λ2

ω2 eλω(a−a†)

⇒ |υ〉 = e− λ2

ω2 e−λωa† |0〉.

The evolution kernel for the anharmonic oscillator, in the coherent basis states, isgiven by

K = 〈z2|e−τH |z1〉 = 〈0|ez2ae−τH ez1a† |0〉

= e2 λ2

ω2 〈υ|e+ λωaez

∗2ae−τH ez1a

†e+

λωa† |υ〉

= eλ2

ω2b〈υ|ea( λω+z∗2)e−τωb†bea

†(z1+ λω)|υ〉

= eλ2

ω2 e−λω( λω+z∗2) e−

λω(z1+ λ

ω)〈υ|eb(z∗2+ λ

ω) e−τωb

†be(z1+ λω)b† |υ〉. (5.93)

The definition of a coherent state given in Eq. 5.76, namely |z〉 = eza† |0〉, yields

for the b, b†-oscillator

|z〉b = ezb† |υ〉, b|υ〉 = 0.

Hence, from Eq. 5.93

K = 〈z2|e−τH |z1〉= e

− λ2

ω2− λω(z1+z∗2)

b〈z∗2 +λ

ω|e−τωb†b|z1 + λ

ω〉b

= exp

{− λ2

ω2− λ

ω(z1 + z∗2)

}exp

{(z∗2 +

λ

ω

)e−ωτ

(z1 + λ

ω

)}.

The evolution kernel of the simple harmonic oscillator, given in Eq. 5.89, has beenused to obtain the final result.

5.16 Summary

Hamiltonian mechanics shows the close connection of the Lagrangian and Hamil-tonian formulations of the dynamics of a system. The Hamiltonian is obtained bya Legendre transformation and yields the dynamics of the system in phase space,defined by the canonical coordinate and the conjugate momentum of the system.Conservation laws and symmetries are seen to emerge in a transparent manner inthe canonical formulation.

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The cardinal results of canonical equations, including the relation between theLagrangian and the Hamiltonian, are seen to hold for the quantum case, but with anappropriate generalization of all the underlying symbols. In particular, in makingthe transition to a quantum case, the Hamiltonian is elevated to a Hermitian oper-ator and the Lagrangian, via the action, determines the probability amplitude forinfinitesimal quantum paths. The Hamiltonian operator obtained from Hamiltonianmechanics leads to the formulation of the path integral in phase space as well as tothe formulation of coherent state quantization.

The Poisson and Dirac brackets provide a well-defined algorithm for making thetransition from the classical to the quantum case, with Dirac brackets playing a cen-tral role is providing a Hamiltonian formulation for constrained quantum systems.The case of constrained systems shows that the quantum system’s state space is anindependent ingredient that needs to be specified in addition to the Hamiltonian.The constraint can be formulated directly in terms of the path integral; it is seenthat the path integral has complete information about both the state space and theHamiltonian.

The oscillator basis provides another perspective to the study of the Hamiltonianand path integral. Coherent states extend the concepts of basis states and com-pleteness to over-complete states. A few calculations using the coherent basis wereperformed to illustrate some of the interesting and novel features of this approach.

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6

Path integral quantization

The Dirac–Feynman approach leads to the Feynman path integral, which is basedon the mathematical concept of functional integration. Besides its application inquantum mechanics, the mathematical formalism of functional integration is one ofthe pillars of quantum mathematics and can be applied to classical random systemsthat are outside the domain of quantum mechanics.

The starting point of path integral quantization is the degree of freedom. Onepostulates that the degree of freedom takes all possible determinate paths in mak-ing a quantum transition from its initial to its final state. The probability amplitudefor each determinate path is given by the properties of the degree of freedom en-coded in the action S, and this in turn entails knowledge of the degree of freedom’sLagrangian L.

In particular, the result obtained in Eq. 4.40 can be taken to be the starting pointof path integral quantization. Hence, instead of starting from a Hamiltonian, aswas done in Chapter 4, the quantum phenomena are specified by postulating aLagrangian.

In Section 6.1 the Hamiltonian is derived from the Lagrangian and path integral.In Section 6.2 it is shown how the classical limit emerges from the path integraland in Section 6.7 the Heisenberg commutation equation is derived from the pathintegral, both for Minkowski and Euclidean time.

For a continuous degree of freedom the Lagrangian consists of a kinetic term thatis usually the same for a wide class of systems; one needs to choose an appropriatepotential V (x) to fully describe the system. For the sake of rigor, consider theEuclidean Lagrangian and action given by (�= 1),

L = −1

2

(dx

dt

)2

− V (x), S =∫ tf

ti

dtL.

The evolution kernel is given by the superposition of all the indeterminate (in-distinguishable) paths and is equal to the sum of eS over all possible paths; hence

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106 Path integral quantization

〈xf |e−τH |xi〉 =∫B.C.

DXeS : Euclidean Feynman path integral, (6.1)

Boundary condition : x(0) = xi , x(τ ) = xf .

Path integral quantization is more general than starting from the Schrödingerequation for the following reasons:

• As discussed in Section 2.7, the Schrödinger approach is based on the propertiesof state space in addition to the Hamiltonian driving the Schrödinger equation.

• The spacetime symmetries of the quantum system are explicit in the Lagrangianbased path integral approach, whereas in the Schrödinger approach these areimplicit and need to be extracted using the properties of the Hamiltonian andstate space. In particular, one has to derive the symmetry operators that commutewith the Hamiltonian.

• Path integral quantization yields a transparent formulation of constrained sys-tems, as for example discussed in Section 5.7.1 and later in Chapter 13. In theSchrödinger formulation, one needs both the Hamiltonian and commutation re-lations, which for a constrained system are far from obvious and require a fairamount of derivation.

These considerations come to the forefront for complicated systems like non-Abelian gauge fields, where the starting point is the Lagrangian, and path integralquantization turns out to be more efficient than the Schrödinger approach.

6.1 Hamiltonian from Lagrangian

Recall that in Section 4.6 the Lagrangian was derived from the Hamiltonian usingthe Dirac–Feynman formula. The question naturally arises that if the Lagrangianis known, how would one derive its Hamiltonian H . As discussed in Section 5.1,Hamiltonian mechanics provides one procedure for obtaining H from L; the pur-pose of this section is to provide an alternate derivation using quantum mechanicaltechniques.

A Lagrangian that is more general than the one discussed in Section 4.6, andarises in the study of option theory in finance [Baaquie (2004)], is chosen to il-lustrate some new features. Option theory is based on classical random processesthat are similar to the diffusion equation and hence the time parameter t in the pathintegral appears as “Euclidean time” t , which for option theory is in fact calendartime.

Let the degree of freedom be the real variable φ. Consider the following La-grangian and action:

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6.1 Hamiltonian from Lagrangian 107

L(t) = −1

2

[me−2νφ

{dφ

dt+ α(φ, t)

}2

+ V (φ)

],

S =∫ τ

0dtL(t) = −1

2

∫ τ

0dt

[me−2νφ

{dφ

dt+ α(φ, t)

}2

+ V (φ)

]. (6.2)

For greater generality, a φ dependent mass equal to me−2νφ and a drift term α(φ, t)

have been included in L.The path integral is given by generalization of Eq. 4.39,

K(φi, φf ; τ) =∫

Dφe−νφeS, (6.3)∫Dφe−νφ ≡

τ∏t=0

∫ +∞

−∞dφ(t)e−νφ(t),

Boundary conditions φ(τ) = φf , φ(t = 0) = φi.

Note that the path integral integration measure∫Dφ has a factor of e−νφ needed

to obtain a well-defined Hamiltonian.Recall from the discussion of the evolution kernel in Section 4.6, that the path

integral is related to the Hamiltonian H by Eq. 4.40, namely

K(φi, φf ; T ) =∫

Dφe−νφeS = ⟨φf |e−τH |φi

⟩. (6.4)

One needs to extract the Hamiltonian H from the path integral on the left hand sideof Eq. 6.4.

The Hamiltonian propagates the system for infinitesimal time; the time index t isdiscretized into a lattice with spacing ε, where t = nε with N = T/ε and φ(x)→φn. The path integral reduces to a finite (N−1)-fold multiple integral, analogous towhat was obtained in Eq. 4.30. Discretizing the time derivative dφ/dt → (φn+1 −φn)/ε yields the following lattice action and Lagrangian:

〈φN |e−εNH |φ0〉 =N−1∏n=1

∫dφne

−νφneS(ε), (6.5)

S(ε) = ε

N−1∑n=0

L(n),

L(n) = −me−2νφn

2ε2[φn+1 − φn + εαn]2 − 1

2[V (φn+1)+ V (φn)] .

As in Section 4.6, the completeness equation∫dφn|φn〉〈φn| = I is used N−1 times

to write out the expression for e−εNH , and the Hamiltonian is identified as

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〈φn+1|e−εH |φn 〉 = N (ε)e−νφneεLn

= N (ε)e−νφn exp

{−me

−νφ

2ε[φn+1 − φn + εαn]2 − ε

2[V (φn+1)+ V (φn)]

}.

Since the Hamiltonian depends on the value of φ at two different instants, to sim-plify notation let

φn+1 = φ, φn = φ′, αn = α.

Ignoring terms that are of O(ε) in Eq. 6.5, the matrix elements of the Hamiltonianare given by

〈φ|e−εH |φ′〉 = N (ε)e−νφ exp

{−me

−2νφ

2ε

[φ − φ′ + εα

]2 − εV (φ)

}. (6.6)

Note that unlike Eq. 4.33, for which the Hamiltonian is known and from whichthe Lagrangian was derived, in Eq. 6.6 one needs to derive the Hamiltonian fromthe known Lagrangian. This derivation is the quantum mechanical analog of thederivation of H given by Hamiltonian mechanics in Section 5.1.

The key feature of the Lagrangian that in general allows one to derive itsHamiltonian is that the Lagrangian contains only first order time derivatives; henceon discretization the Lagrangian involves only φn that are nearest neighbours intime, thus allowing it to be represented as the matrix element of e−εH , as in Eq.6.6.

In contrast, for Lagrangians that contain second order or higher order timederivatives, as in Chapter 13, the derivation of the Hamiltonian from the Lagrangianand path integral is nontrivial since the entire framework of coordinate and canon-ical momentum, as discussed in Section 5.1, is no longer applicable. Instead, onehas to employ the method required for quantizing constrained systems, and in par-ticular, evaluate the Dirac brackets for the system in order to obtain the Hamiltonianand commutation relations.

In Eq. 6.6, the time derivatives appear in a quadratic form; hence one can useGaussian integration to re-write Eq. 6.61 as

〈φ|e−εH |φ′〉 = e−νφe−εV (φ)∫ +∞

−∞dp

2πexp

{− ε

2mp2 + ip

[φ − φ′ + εα

]e−νφ

}= e−εV (φ)

∫ +∞

−∞dp

2πexp

{−εe

2νφ

2mp2 + ip

(φ − φ′ + εα)

}, (6.7)

where the pre-factor of e−νφ has been canceled by re-scaling the integration vari-able p→ peνφ .

1 Henceforth N (ε) is ignored since it is an irrelevant constant contributing only to the definition of the zero ofenergy.

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6.2 Path integral’s classical limit �→ 0 109

The Hamiltonian H =H(φ, ∂/∂φ) is a differential operator and acts on thedual coordinate φ, as is required for all differential operators, as mentioned ear-lier after Eq. 3.31. Hence, for the state function |ψ〉, which is an element ofthe state space, the Hamiltonian acts on the dual basis state 〈φ|, and yields〈φ|H |ψ〉=H(φ, ∂/∂φ)ψ(φ), similarly to the result given in Eq. 2.10.

The Hamiltonian is hence given by2

〈φ|e−εH |φ′〉 = e−εH(φ,∂/∂φ)〈φ|φ′〉 = e−εH(φ,∂/∂φ)

∫ +∞

−∞dp

2πeip(φ−φ

′), (6.8)

since 〈φ|φ′〉 = δ(φ − φ′). Ignoring overall constants and using the property of theexponential function under differentiation, one can re-write Eq. 6.7 as

〈φ|e−εH |φ′〉 = exp

{1

2mεe2νφ ∂2

∂φ2+ εα

∂

∂φ− εV (φ)

}∫ +∞

−∞dp

2πeip(φ−φ). (6.9)

Comparing Eq. 6.9 with Eq. 6.8 yields the Hamiltonian

H = − 1

2me2νφ ∂2

∂φ2− α(φ)

∂

∂φ+ V (φ). (6.10)

The Hamiltonian is quite general since both V (φ) and α(φ) can be functionsof the degree of freedom φ. Note that the Hamiltonian H is non-Hermitian – andis Hermitian only for ν= 0 and a pure imaginary α. The path integral has a non-trivial integration measure exp{−νφ} that needs to be specified in addition to theHamiltonian.

6.2 Path integral’s classical limit � → 0

It is well known that classical physics is the limit of the quantum theory in thelimit that �→ 0. In this section it is shown how the classical limit emerges from acompletely quantum world; in particular it is shown how the classical limit emergesfrom the path integral. Hence, to consider the classical limit, the presence of � isrestored in the path integral, and the behavior of the path integral is studied inphysical Minkowski time and in the limit of �→ 0.

Recall that the evolution kernel is given in Euclidean time by

K(x, x ′; τ) =∫

DXeS/�,

S =∫ τ

0Ldt, L = −m

2x2 − V (x).

2 From Eq. 3.36, the convention for scalar product is 〈p|φn 〉= exp(−ipφn), and the sign of the exponential inEq. 6.8 reflects this choice. The definition of H requires it to act on the dual state vector 〈φ|; if one chose towrite the Hamiltonian as acting on the state vector |φ〉, H † would then have been obtained instead. Since His not Hermitian, this would lead to an incorrect result.

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To discuss the �→ 0 limit, analytically continue back to Minkowski time t = iτ ,

LM= m

2

(dx

dt

)2

− V (x), S → iSM,

K(x, x ′; t) =∫

DXeiSM /�.

In the �→ 0 limit, one expects only the classical path to contribute; to achievethe classical limit one expects

lim�→0

eiSM/� → N eiSc/�T∏t=0

δ(x(t)− xc(t)),

where the normalization N is required for dimensional consistency. The equationabove should be thought of only in a heuristic sense, since there are many paths“near” the classical path that also contribute, and in fact are responsible for theprecise determination of the normalization constant. One hence obtains

K → N∫

Dxδ[x − xc]eiSc/�

= N eiSc/�. (6.11)

A geometric proof of Eq. 6.11 is given here, following Feynman and Hibbs(1965). Recall that xc is the solution to the equation of motion given by

δS

δx(t)

∣∣∣x(t)=xc(t)

= 0 ⇒ δSc = 0. (6.12)

In other words, as one varies the path x(t), to first order Sc is stationary withrespect to this variation. Consider the discrete time path integral

K =∫

DxeiSM /� ∼=∑paths

eiSpath//� (6.13)

= eiS1/� + eiS2/� + · · · (6.14)

Note that, since |eiSpath//�| = 1, each term in the summation for K is a complexnumber of modulus unity. Hence K can be represented by a sum of vectors withunit length in the complex plane, one vector for each term eiSpath/�. Figure 6.1 showsthe addition of the eiSpath/� for different paths.

Consider two neighboring paths; these contribute eiSa/� + eiSb/� to K . Note thatif Sb � Sa + π� we have

eiSa/� + eiSb/� = eiSa/� − eiSa/� = 0.

In other words, if the action changes by about π� or more for neighboring paths,then these paths interfere destructively and give no contribution to K .

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6.2 Path integral’s classical limit �→ 0 111

Figure 6.1 Classical path, indicated by the heavy arrow, results from the interfer-ence of quantum paths

Since Sc is stationary, we expect the paths close to xc(t) to have a value for theaction S that is very close to Sc, and hence should interfere constructively, as shownin Figure 6.1. We expect that the paths in the neighborhood of the classical path allinterfere constructively. Hence

K � eiSc/� + eiS′/� + · · · + eiS

′′/� (6.15)

� N eiSc/�. (6.16)

The classical path is indicated by the heavy arrowed vector in Figure 6.1.Note that S ′ S ′′ . . . are contributions to K from paths close to xc, and for coher-

ence (constructive interference) we expect

S ′, S′′, · · · ≤ Sc + π�. (6.17)

6.2.1 Nonclassical paths and free particle

Consider a free particle of mass m traveling for a time interval T . The classicalpath is x = vt and consider a nonclassical path x= vt2/T . Let v= 10−2 km s−1,T = 1 s; this yields the following classical Sc and non-classical SNc actions:

Sc = 1

2mv2

∫ T

0dt = 1

2mv2T = 1

2m10−4 km2 s−1,

SNc = 2mv2

T 2

∫ T

0t2dt = 2

3mv2T = 2

3m 10−4 km2 s−1,

�S = SNc − Sc = 1

6m× 10−4 km2 s−1.

For a classical particle m= 10−3 kg; using �= 1.05× 10−34 kg km2 s−1 yields

�S = 1

6× 10−3 × 10−4

1.05× 10−34 kg km2 s−1�

� 1.6× 1026�� π�. (6.18)

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Hence, for a classical particle the nonclassical path is not coherent with the classi-cal path.

Consider the quantum case of the electron, with its mass given by

me = 9.1× 10−31kg

and

�S = 1

6× 9.1× 10−31 × 10−4

1.05× 10−34 � � �/6

⇒ �S/� < π. (6.19)

We have the important conclusion that for a nonrelativistic electron (moving inthe example at 1 cm/sec), in addition to the classical path, there are also manynonclassical paths that contribute to the transition amplitude.

Note that � sets the scale just right so that a nonrelativistic electron is required tobe treated in a completely quantum mechanical manner. In fact, in most of chem-istry and all of biology, the nuclei are treated as internally structure-less classicalpoint charges, and it is the quantum mechanical nature of the nonrelativistic elec-trons that determines most of chemistry and biology.

6.3 Fermat’s principle of least time

Consider light reflection off a mirror. In the limit of geometrical optics, where lightcan be considered as a ray,3 the well-known law of reflection states that

θi = θr . (6.20)

The light ray consists of photons. An approximate quantum action for the pho-tons that arrive at B from source A after reflecting off the mirror, as shown in Figure6.2, is given by

Sray/� � ωtp = θp, (6.21)

where ω is the frequency of light and tp is the time taken for light to travel the pathfrom A to B. A path in this section always means the path of a ray in going from Ato the mirror and then to B.

3 Geometrical optics is a description of photons (light particles) when the wavelength and energy of individualphotons are much smaller than the dimensions of the equipment and the experimental resolution of thephoton’s energy respectively.

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6.3 Fermat’s principle of least time 113

A B

C0

qi qr

Figure 6.2 Reflection of light ray off a mirror. The classical path is the trajectoryAC0B.

The quantum mechanics principle of superposition states that light simultane-ously takes all possible paths in going from A to B. The probability amplitude(A → B) is given by

KAB =∑

{all paths}eiS/�

= eiθ1 + eiθ2 + eiθ3 + eiθ4 + · · · (6.22)

Note path AC0B in Figure 6.3 is a classical path and consequently a minimumfor the action S, since for this path δS= 0; hence, paths that are infinitesimallyclose to the classical path AC0B all contribute coherently.

Each phase eiθ in Eq. 6.22 can be considered as a two-dimensional unit vectorin the two-dimensional plane and hence KAB , which is a sum of complex numbers,can be obtained by adding the phases like vectors.

If the path is far from AC0B the angle θ changes by large amounts, the reasonbeing that the wavelength of light is much smaller than the distance from the sourceA to the mirror; hence, if one is not near the classical path, two paths differ by manywavelengths, giving rise to a large change in θ in going from one path to the other.

Adding all the phase contributions to KAB yields the graphical representationgiven in Figure 6.1: one starts at the left end and starts to add paths starting frompath AC1B with phase eiθAC1B , the path AC2B contributes a phase eiθAC2B that addsdestructively to the phase from path AC1B and so on, and these are shown byarrows at the left end of Figure 6.1; similarly, phases from paths AC3B and AC4B

cancel out.What remains are phases that are close to the classical path AC0B, and they all

add coherently, as shown in Figure 6.1 by the heavy vector pointing straight from

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C1

BA

C2 C0 C3 C4

Figure 6.3 Many paths from source at A to detector at B reflected off the mirrorat points C0, C1, C2, . . . C3, C4. The classical path is AC0B.

left to right. The destructive and constructive interference yields the following thefinal result

KAB = N eiSc/�. (6.23)

In summary, all paths away from the classical path interfere destructively,whereas paths in the neighborhood of the classical paths interfere constructivelyyielding the final result that is dominated by the classical path. The fact that lighttaking the classical path contributes the most to light received at B can also be seenfrom the fact that if one removes most of the mirror except for a portion near thepoint C0, the light received at B does not significantly change, showing that theother paths for light do not contribute significantly.

The path of least distance, which minimizes the action, yields the law that angleof incidence is equal to the angle of reflection,

θi = θr . (6.24)

The path of least distance is also the path of least time and yields

KAB =N eiSc/�,

⇒ light takes path of least time in going from A to B: Fermat’s principle.

Fermat’s principle also yields Snell’s law of refraction for light going from onemedium to another having a different index of refraction.

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6.4 Functional differentiation 115

6.4 Functional differentiation

Consider variables fn, n= 0,±1,±2 . . .±N that satisfy

∂fn

∂fm= δn−m.

Let t = nε, with N →∞. The limit ε → 0 yields

∂fn

∂fm→ δf (t)

δf (t)′ ≡ lim

ε→0

1

ε

∂fn

∂fm

⇒ δf (t)

δf (t ′)= lim

ε→0

1

εδn−m → δ(t − t

′). (6.25)

In general, the functional derivative of �[f ] – an arbitrary functional of the pathf (t) – is denoted by δ/δf (t) and is defined by

δ�[f ]δf (t)

= limε→0

�[f (t ′)+ εδ(t − t ′)

]−�[f ]ε

. (6.26)

In the notation of state space one has⟨f

∣∣∣∣ δ

δf (t)

∣∣∣∣�⟩= δ

δf (t)〈f |�〉 = δ�[f ]

δf (t).

Note that ε has the dimensions of [f ] × [t].

Examples

• Consider the simplest function �[f ] = f (t0); then, from Eq. 6.26

δ�[f ]δf (t)

= δf (t0)

δf (t)= lim

ε→0

f (t0)+ εδ(t − t0)− f (t0)

ε= δ(t − t0).

• Let �[f ] = ∫dτf n(τ ); from above

δ�[f ]δf (t)

=∫

dτnf n−1(τ )δf (τ)

δf (t), (6.27)

=∫

dτnf n−1(τ )δ(t − τ) = nf n−1(t). (6.28)

6.4.1 Chain rule

The chain rule for the calculus of many variables has a generalization to functionalcalculus. Consider a change of variables from fn to gn; the chain rule of calculusyields

∂

∂fn=

N∑m=1

∂gm

∂fn

∂

∂gm.

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As before, let t = nε, t ′ =mε; we can re-write the above expression as

1

ε

∂

∂fn= ε

N∑m=1

[1

ε

∂gm

∂fn

] [1

ε

∂

∂gm

].

Taking the limit of N →∞ and ε → 0 yields

limε→0

1

ε

∂

∂fn→ δ

δf (t)=∫

dt ′δg(t ′)δf (t)

δ

δg(t ′): chain rule. (6.29)

6.5 Equations of motion

For T →∞, we have

e−TH = e−T E0 |�〉〈�| + · · · ,|�〉 = vacuum (ground) state. (6.30)

Hence

Z = limT→∞ tr(e−TH )

� 〈�|�〉e−T E0 . (6.31)

The path integral representation gives

Z =∫

DxeS, S =∫ +∞

−∞dtL(t).

The equations of motion follow from the fact that the path integration measureis invariant under the displacement, namely

x(t)→ x(t)+ ε(t)

⇒∫

Dx →∫

D(x + ε). (6.32)

On displacing the paths by ε(t) one obtains a functional Taylors expansion for theaction given by

S[x(t)+ ε(t)] = S[x(t)] +∫ +∞

−∞dtε(t)

δS[x(t)]δx(t)

+O(ε2). (6.33)

Hence, in abbreviated notation

Z =∫

DxeS[x] =∫

DxeS[x+ε]

=∫

DxeS[x][

1+∫

ε(t)δS[x]δx(t)

dt · · ·]. (6.34)

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6.6 Correlation functions 117

Note that the left hand side of Eq. 6.34 is independent of the ε parameter thatappears in the right hand side of the same equation. In particular, the first orderterm in the Taylors expansion in ε of the right hand side of Eq. 6.34 must be zero,and yields

⇒ 0 =∫

dtε(t)

∫Dx

δS[x]δx(t)

eS. (6.35)

Since ε(t) is arbitrary, Eq. 6.35 yields that for each t the coefficient multiplyingε(t) must be zero, and hence∫

DxδS[x]δx(t)

eS[x] = 0 : equations of motion. (6.36)

We have thus obtained the quantum mechanical generalization of the classicalequations of motion; namely it is only the average value of δS[x]/δx(t) that isequal to zero; denoting the functional average by

E[O] ≡ 1

Z

∫DxOeS, (6.37)

the equations of motion can be succinctly written as

E[δS] = 0. (6.38)

6.6 Correlation functions

Let τ1 < τ2 < τ3 . . . < τk; the k-point correlation is defined by

G(τ1, τ2 . . . τk) = 〈�|xH (τk) . . . xH (τ1)|�〉, (6.39)

where xH (τ ) are Heisenberg operators.In general, time ordering operator T is defined so as to place the operators at

the earliest time to the right of the operators at later time. More precisely, the timeordering operator is defined by

T (Ot Ot′ ) =

{Ot

′ Ot t′> t

OtOt′ t

′< t

. (6.40)

To simplify the notation, denote xH (τ ) by xτ ; time ordering yields

T (xτkxτ1xτ2 . . .) = xτk . . . xτ2xτ1

and the correlation function is

G(τ1, . . . τk) = 〈�|T (xτkxτ1xτ2 . . .)|�〉. (6.41)

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Since all correlation functions are the time ordered vacuum expectation values ofthe Heisenberg operators,

|�〉〈�| � limT→∞ e−T (H−E0),

G(τ1 . . . τk) = eE0T tr{T (xτkxτ1xτ2 . . .)e

−TH}= 1

Z

∫Dx xτk . . . xτ1e

S

= symmetric function of τ1, τ2 . . . τk.

The correlation functions are interrelated through differential equations, calledWard identities. Consider for example

G(τ0) = 1

Z

∫Dxxτ0e

S[x]

= 1

Z

∫Dx

[xτ0 + ε(τ0)

]eS[x+ε]

= 1

Z

∫Dx

[xτ0 + ε(τ0)

] [1+

∫dtε(t)

δS

δx(t)

]eS,

which yields

0 =∫

dtε(t)E

[xτ0

δS

δx(t)+ δ(t − τ0)

]

⇒ E

[xτ0

δS

δx(t)

]= −δ(t − τ0).

In general, for L= − 12mx

2−V (x) all the correlations are required for defining thetheory. The only special case is the Gaussian action with V =ω2x2/2 for which thetwo-point correlation functionE[xτxτ ′ ] is sufficient for obtaining all the correlationfunctions.

6.7 Heisenberg commutation equation

For a single degree of freedom, from Eq. 3.43, the Heisenberg commutation equa-tion is given by

[x, p] = i�I. (6.42)

Since the momentum operator p involves a time derivative, working in Euclideantime would yield an extra factor of i in the commutation equation. The calcu-lation to obtain the Heisenberg commutation equation is carried out in physicalMinkowski time so that the familiar expression is obtained.

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Recall in the Schrödinger canonical quantization, the coordinate was taken to bea degree of freedom and it was then postulated in Eq. 3.31 that momentum is anoperator conjugate to the coordinate degree of freedom and given by

p = −i� ∂

∂x: canonical quantization.

One postulates that the momentum of a degree of freedom is given by the “clas-sical” looking expression for momentum, namely that

p = mdx

dt: path integral quantization. (6.43)

However, Eq. 6.43 is not a classical expression since dx/dt is a degree of freedomand is indeterminate – having a determinate value only with a certain likelihood.

In the path integral quantization, only the matrix elements of the operators ap-pear. Hence, one would like to take the expectation value of both sides of Eq. 6.42to establish the Heisenberg commutation equation.

For action SM

in Minkowski time, one could try and evaluate the expectationvalue of the commutation equation, that simplistically is given by

E[([x, p])] = 1

Z

∫Dx

[x, p

]eiSM /� = 1

Z

∫Dx

(xt pt − pt xt

)eiSM /�. (6.44)

If one takes p = mx, then apparently∫Dx (xt xt − xt xt ) e

iSM/� = 0 (incorrect). (6.45)

The solution is to consider the product of operators as being time ordered,as is required when one goes from operators to correlation functions. Definept = (xt+ε − xt )/ε; time ordering then yields, for ε → 0+,

T (xtpt ) = xtpt−ε = m

εxt (xt − xt−ε) ,

T (ptxt ) = ptxt = m

ε(xt+ε − xt ) xt ,

from which it follows that the commutator has the time ordered form

E [([x, p])] → E [T ([x, p])] = E [T (xtpt − ptxt )] , (6.46)

and hence

T ([x, p]) = xtpt−ε − ptxt . (6.47)

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For discrete time t = nε, the (discretized) action is given by

SM=∫

dt(m

2x2 − V

)� m

2ε

∑n

(xn+1 − xn)2 − ε

∑n

V (xn), (6.48)

where it has been assumed that the potential V (x) does not depend on velocity.The commutator yields the discrete expression

T ([x, p]) = m

εxt (2xt − xt+ε − xt−ε) . (6.49)

Note that for any lattice variable x�= ±∞, the action SM[x�= ±∞]=∞ and

yields

x�eiS

M/�|x�=±∞ = 0.

An integration by parts for the specific coordinate x� yields∫Dx

∂

∂x�

(x�e

iSM/�) = ∫

D′X[x�e

iSM/�|x�=+∞ − x�eiSM/�|x�=−∞

] = 0,

(6.50)

where∫D′X is the path integral excluding the integration variable x�. Using the

chain rule for differentiation for the left hand side of Eq. 6.50 yields4

0 = E

[1+ i

�x�∂SM

∂x�

]. (6.51)

Hence

1 = − i

�E

[x�∂SM

∂x�

]

= − i

�E

[x�m

ε(2x� − x�+1 − x�−1)− εxl

∂V (x�)

∂x�

]

= − i

�E [T ([xl, pl])]+O(ε)

⇒ 1 = − i

�[xl, pl]+O(ε).

Taking the limit of ε → 0 yields

[x, p] = i�I : Heisenberg commutation equation. (6.52)

The commutation equation depends only on kinetic energy of action S, whichcontains the time derivative term. In particular, the commutation equation is inde-pendent of the potential energy V (x) and of the boundary conditions.4 Recall E[O(x)] ≡ ∫

DxO(x)eS/Z.

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6.7.1 Euclidean commutation equation

The commutation for Euclidean time has been derived in Section 5.6 using thePoisson bracket of the coordinate with its canonical momentum. Another deriva-tion of the same result is given here using the path integral approach to verify theconsistency of the two approaches.

Euclidean time τ and momentum pE

are given by

τ = it, pE= −mdx(τ)

dτ. (6.53)

The Euclidean action, for τ = nε, is given by

SE = −∫

dτ

{m

2(dx(τ )

dτ)2 + V

}⇒ SE � −m

2ε

∑n

(xn+1 − xn)2 − ε

∑n

V (xn)

= −m

2ε

∑n

xn (2xn − xn+1 − xn−1)− ε∑n

V (xn). (6.54)

Similarly to the Minkowski case, the Euclidean commutator is given by

T ([x, pE]) = −m

εxτ(2xτ − xτ+ε − xτ−ε

) = −mεxn(2xn − xn+1 − xn−1

). (6.55)

Note the difference of a minus sign between the Minkowski commutation equationgiven in Eq. 6.49 and the Euclidean case given in Eq. 6.55.

The discretized path integral, similarly to Eq. 6.44, yields the expectation value

E[T ([x, pE])] = −m

ε

1

Z

∫Dx xn

(2xn − xn+1 − xn−1

)eS/�. (6.56)

Similarly to Eq. 6.51, for the Euclidean case

0 = E

[1+ 1

�x�∂S

∂x�

]

⇒ 1 = −1

�E

[x�∂S

∂x�

]= −1

�E[T ([x, p

E])] , (6.57)

and the Euclidean commutator is given by

[x, pE] = −�I : Euclidean commutation equation. (6.58)

The result obtained above confirms the earlier results obtained in Eq. 5.50 basedon canonical quantization. Hence, the counterintuitive definition of Euclidean mo-mentum p

E= −mdx(τ)/dτ with an overall minus sign is confirmed by two inde-

pendent derivations.

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6.8 Summary

Path integral quantization is an independent formulation of quantum mathematicsbased on the concept of the Lagrangian and the action. The path integral measurehas to be defined in addition to the Lagrangian in order to formulate the path inte-gral. The Hamiltonian and state space are both incorporated in the path integral andthe path integral measure arises from the properties of the state space. The Hamil-tonian and the state space are independent structures of a quantum system that canboth be derived from the path integral.

In the limit of �→ 0, the classical limit emerges from the path integral due to thedestructive interference of paths far from the classical path. A concrete illustrationof this cancellation is displayed by light’s reflection off a mirror and yields that theangle of incidence and reflection are equal.

The correlation functions of a quantum system obey a set of identities that havea transparent derivation using the invariance properties of the path integral’s mea-sure. The Heisenberg commutation equations are shown to follow from the prop-erties of the correlation function of a large class of Lagrangians that depend onlyon the velocity squared of the degree of freedom. In particular, the commutationequation for Euclidean time was derived.

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Part two

Stochastic processes

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7

Stochastic systems

A quantum entity is intrinsically indeterminate and is not in a determinate(classical) state when it is not being observed. The quantum entity is found tobe in a determinate state only if an experiment is carried out to observe it, asdiscussed in Baaquie (2013e). The indeterminate behavior of the quantum entity isdescribed by the formalism of quantum mathematics discussed in Chapter 2. Themeasure of quantum indeterminacy and uncertainty is Plank’s constant �.

Randomness also exists in classical systems; however, a classical system’srandomness is in principle different from quantum indeterminacy, since a priori,a classical random system is intrinsically in a determinate state. Classical random-ness arises entirely due to our lack of complete knowledge of the determinate state.The theory of classical probability describes a classical system about which onehas incomplete knowledge.

There are many ways of introducing randomness in a classical system, depend-ing on its nature. A classical random system in thermal equilibrium is described bystatistical mechanics. A classical system undergoing random time evolution can bedescribed by taking the classical equations of motion and adding a random forceto it. The evolution of the classical particle is then described by a stochastic pro-cess. The random force is usually represented by Gaussian white noise.

Classical random processes driven by white noise, such as those that arise infinance and other areas, form a wide class of problems that can be described byclassical probability theory. It is shown in this chapter that the mathematical frame-work of quantum mechanics, and in particular, the Hamiltonian operator and pathintegrals, provide an appropriate mathematical formalism for describing processesthat are driven by Gaussian white noise.

Many classical random systems are described by probability distribution func-tions [also called probability density functions] that obey linear partial differentialequations, such as the Fokker–Planck equation [Risken (1988)]. These systems

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126 Stochastic systems

can be recast in terms of a Hamiltonian, which, in turn, leads to a path integralrepresentation for conditional probabilities.

A more fundamental reason for the applicability of quantum mathematics toclassical random systems is that random systems and processes have many pos-sible determinate states, and all of them need to be taken into account: quantummathematics is ideally suited for representing such systems.

Consider the classical equation of motion given by

mdv

dt+ γ v+�(v) = 0, (7.1)

where γ is the coefficient of friction and �(v) is an arbitrary potential. For largeobjects, the solutions of Eq. 7.1 give an adequate description of the smooth trajec-tory that the body follows under the influence of the potential �(v).

Consider a tiny grain of pollen moving in a fluid medium with which it hasmany collision per second. For example, a particle of air at room temperature andpressure has over 109 collisions per second. The trajectory of the grain or of theparticle of air does indeed exist, but it becomes near to impossible to describe itsmotion in complete detail. The classical equation of motion given in Eq. 7.1 is notenough for describing it. To encode this lack of knowledge of the trajectory of thegrain, classical randomness is introduced in the description of the motion of theparticle.

Langevin proposed that the medium that is responsible for the random motionof the particle be modeled by a random force, and replaced Eq. 7.1 by a stochasticdifferential equation given by

mdv

dt+ γ v+�(v) = √2AR(t). (7.2)

The random force√

2AR(t) is given by Gaussian white noise R(t).The introduction of a random force

√2AR(t) makes the trajectories of the par-

ticle random. For every choice of a random sample of values drawn for white noiseR(t), the trajectory changes. Hence, for a stochastic process one needs to specifythe likelihood of different trajectories rather than the unique trajectory that is givenin Eq. 7.1.

In Section 7.1 classical probability theory is briefly reviewed and in Section 7.3Gaussian white noise is discussed. Ito calculus is discussed in Section 7.4 and itsgeneralization given by the Wilson expansion is discussed in Section 7.5. In Section7.6 the Langevin equation and its random paths are studied; Sections 7.7 and 7.8examine the Langevin equation in an external potential and the nonlinear Langevinequation. In Section 7.9 stochastic quantization based on white noise is discussed.In Sections 7.10 and 7.11 the focus is on the Fokker–Planck Hamiltonian.

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7.1 Classical probability: objective reality 127

7.1 Classical probability: objective reality

Classical probability theory underpins the description and understanding of classi-cal random phenomena, and is briefly reviewed here.

Consider a large collection of classical particles in a container. Each particleobjectively exists in some determinate state and the uncertainty in the knowledgeof the state of the particle is attributed to our ignorance of the microscopic state ofa very large collection of molecules. In statistical mechanics, the large collectionof particles is described by assuming that the position and velocity of each particleare classical random variables.

Classical probability is based on the concept of a random variable, which takesa range of values, and that exists objectively regardless of whether it is measured(sampled) or not. A unique probability, called the joint probability distribution, isassigned to a collection of random variables and predicts how frequently a collec-tion of specific values will appear when the random variables are sampled.

Consider a random variable taking only two discrete values, say B (black) and W(white); each possible outcome has a probability, in this case, pB and pw. One canthink of the random variable as an infinite collection of black and white balls withthe relative frequency of B and W being given by their probability distribution.The probability of picking either B or W is given by pB and pw respectively, andis shown in Figure 7.1.

Following Kolomogorov, classical probability theory is defined by the followingpostulates:

• A collection of all possible allowed random sample values labeled by ω, formsa sample space �.

• A joint probability distribution function P(ω) determines the probability for thesimultaneous occurrence for these random events and provides an exhaustiveand complete description of the random system.

The events can be enumerated by random variables, say �X = (X, Y,Z, . . .),that map the random events ω of the sample space � to real numbers, namely

�X : � → �N,

X, Y,Z : ω→ �⊗�⊗�, ω ∈ �,

P (X, Y,Z) : joint probability distribution.

Recall that every element of the sample space � is assigned a likelihood ofoccurrence that is given by the joint probability distribution function P(ω); for themapping ofω by random variablesX, Y,Z to the real numbers, the joint probabilitydistribution function is P(X, Y,Z).

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BB

W

BBBBBB

W

B

W

Classical Probability

pB

pWpW

pB

Figure 7.1 Classical random events: the number of black and white balls inside aclosed box exist objectively, independent of being observed or not. Furthermore,there exists a probability pB, pW that is intrinsic to each possible outcome and isassigned to each black and white ball.

The assignment of a likelihood of occurrence P(ω) to each element of thesample space, namely to each ω ∈ �, is the defining property of classical prob-ability theory; this assignment implicitly assumes that each element ω of � existsobjectively – regardless of being observed or not – and an experiment finds it inits pre-existing state with probability specified by the probability distribution. Itis precisely on this point that quantum probability is fundamentally different fromclassical probability since, if a measurement is not made, the quantum degree offreedom in inherently indeterminate – having no objective existence, having nodeterminate position [Baaquie (2013e)]. This is the reason that the concept of thequantum degree of freedom replaces the classical concept of the random variablein the description of quantum phenomenon.

7.1.1 Joint, marginal and conditional probabilities

The joint probability distribution function obeys all the laws of classical proba-bility. Consider random variables X, Y,Z. Their joint probability distribution isgiven by

P(X, Y,Z) ≥ 0,∫ ∞

−∞dxdydzP (x, y, z) = 1.

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7.2 Review of Gaussian integration 129

In other words, P(X = x, Y = y,Z = z) yields the probability for the simultane-ous occurrence of the random values x, y, and z of the random variables X, Y,Z.Consider a function H that depends on the random variables X, Y,Z; its (average)classical expectation value is given by

Ec[H ] =∫

dxdydzH(x, y, z)P (x, y, z).

If the random variables are independent, the joint probability distribution func-tion factorizes and yields

P(x, y, z) = P1(x)P2(y)P3(z).

For many random variables one can form various marginal and conditional prob-ability distributions. The probability that random variables are observed havingrandom values X, Y , regardless of the value of Z, is given by the marginal distri-bution for two random variables, namely

P(X, Y ) =∫ ∞

−∞dzP (X, Y, z),

∫ ∞

−∞dxdyP (x, y) = 1. (7.3)

The conditional probability for events A,B is defined as follows. Let P(A,B) bethe joint probability distribution that events A and B both occur. The conditionalprobability P(A|B) that A occurs, given that B has definitely occurred, is given byconditional probability

P(A|B) = P(A,B)

P (B)⇒ P(A|B)P (B) = P(B|A)P (A).

For the case of a classical random particle such as a gas molecule in a room, theprobability of finding the classical particle at point x, y, given that it has beendefinitely observed at z, is given by the conditional probability

P(X, Y |Z) = P(X, Y,Z)

P (Z)= P(X, Y,Z)∫∞

−∞ dxdyP (x, y, Z),

∫ ∞

−∞dxdyP (x, y|Z) = 1.

7.2 Review of Gaussian integration

The mathematical framework of white noise and stochastic processes is based onthe normal random variable, which in turn is described using Gaussian integration.Gaussian integration also plays a key role in studying path integrals in quantummechanics.

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The main results of Gaussian integration that will be employed later are derivedbelow. For λ > 0 and complex parameter j , the basic Gaussian integral is given by

∫ +∞

−∞dxe−

12λx

2+jx =√

2π

λe

12λ j

2. (7.4)

Normal (Gaussian) random variable

The normal, or Gaussian, random variable – denoted by N(μ, σ) – is a variable xthat has a probability distribution given by

P(x) = 1√2πσ 2

exp

{− 1

2σ 2(x − μ)2

}. (7.5)

From Eq. 7.4

E[x] ≡∫ +∞

−∞xP (x) = μ : mean,

E[(x − μ)2] ≡∫ +∞

−∞(x − μ)2P(x) = σ 2 : variance.

Any normal random variable is equivalent to the N(0, 1) random variable via thefollowing linear transformation

X = N(μ, σ), Z = N(0, 1)⇒ X = μ+ σZ.

All the moments of the random variable Z = N(0, 1) can be determined by thegenerating function given in Eq. 7.4; namely

E[zn] = dn

dJ nZ[J ]|J=0.

The cumulative distribution for the normal random variable N(x) is defined by

Prob(−∞ ≤ z ≤ x) = N(x) = 1√2π

∫ x

−∞e−

12 z

2dz. (7.6)

A sum of normal random variables is also another normal random variable,

Z1 = N(μ1, σ1), Z2 = N(μ2, σ2), . . . Zn = N(μn, σn)

⇒ Z =n∑i=1

Zi = N(μ, σ) ⇒ μ =n∑i=1

μi, σ 2 =n∑i=1

σ 2i .

The result above can be proved using the generating function given in Eq. 7.4.

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7.2 Review of Gaussian integration 131

N -dimensional Gaussian integration

The moment generating function for the N -dimensional Gaussian random variableis given by

Z[j ] =∫ +∞

−∞dx1 · · · dxneS

with S = −1

2

N∑i,j=1

xiAijxj +∑i

Jixi .

Let Aij be a symmetric and positive definite matrix that has only positive eigen-values. Aij can be diagonalized by an orthogonal matrix M ,

A = MT

⎛⎜⎝λ1

. . .

λN

⎞⎟⎠M

MTM = I.

Define new variables

zi = Mijxj , xi = MTij zj

N∏i=1

dzi = detMN∏i=1

dxi =N∏i=1

dxi ≡ Dx.

Hence

Z[j ] =N∏i

∫dzie

− 12λiz

2i+(JMT )izi =

N∏i=1

[√2π

λie

12λi(JMT )i(JM

T )i

].

In matrix notation

∑i

1

λi

(JMT

)i

(JMT

)i= J

1

AJ = JA−1J,

N∏i=1

√2π

λi= (2π)N/2 1√

detA.

Hence

Z[J ] = (2π)N/2

√detA

e12 JA

−1J , (7.7)

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where Z[J ] is called the generating function since all the correlators can beevaluated from it. In particular, the correlator of two of the variables is given by

E[xixj ] ≡ 1

Z

∫Dx xixje

S, Z = Z[0]

E[xixj ] = ︷︸︸︷xixj = ∂2

∂Ji∂JjlnZ[J ]

∣∣∣J=0

= A−1ij : contraction.

The correlation of N variables is given by

〈xi1xi2xi3 . . . xiN−1xiN 〉 =︷︸︸︷xi1xi2 . . .

︷︸︸︷xiN−1xiN +all possible permutations.

Let t = nε, n = 0,±1,±2 . . .±N . Taking the limit of N →∞, ε → 0 yields

S0 = −1

2

∫ +∞

−∞dtdt ′xtAtt ′xt ′,

Z =∫

DxeS.

The generating functional for the simple harmonic oscillator is given by1

Z[j ] = 1

Z

∫DxeS0+

∫dtj (t)xt = exp

{1

2

∫ +∞

−∞dtdt ′jtA−1

t t ′ jt ′

}, (7.8)

where ∫A−1t t ′ At ′t ′′dt

′ = δ(t − t ′′). (7.9)

7.3 Gaussian white noise

The properties of white noise are analyzed in this section. The fundamental prop-erties of Gaussian white noise are that

E[R(t)] = 0, E[R(t)R(t ′)] = δ(t − t ′). (7.10)

Figure 7.2 shows how there is an independent (Gaussian) random variable R(t) foreach instant of time t .

Let us discretize time, namely t = nε, with n = 1, 2 . . . N , and withR(t)→ Rn.The probability distribution function of white noise is given by

P(Rn) =√

ε

2πe−

ε2R

2n . (7.11)

1 The term generating functional is used instead of generating function as in Eq. 7.7 to indicate that one isconsidering a system with infinitely many variables.

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7.3 Gaussian white noise 133

t*

t0

R(t)

t

Figure 7.2 One random variable R(t) for each instant of time.

Hence, Rn is a Gaussian random variable with zero mean and 1/√ε variance, and

is denoted by N(0, 1/√ε). The following result is essential in deriving the rules of

Ito calculus:

R2n =

1

ε+ random terms of 0(1). (7.12)

To prove the result stated in Eq.(7.12), it will be shown that, to leading order in 1/ε,the generating function of R2

n can be derived by considering R2n to be deterministic.

All the moments of R2n can be determined from its generating function, namely

E[(R2

n)k] = dk

dtkE[etR

2n

] ∣∣∣t=0

.

Note one needs to evaluate the generating function E[etR

2n

]only in the limit of

t → 0. Hence, for ε small but fixed

limt→0

E[etR

2n

]=∫ +∞

−∞dRne

tR2n

√ε

2πe−

ε2R

2n = 1√

1− 2tε

∼ exp

(t

ε

)+O(1).

The probability distribution function P(R2n) for R2

n, which gives the above gener-ating function, is given by

P(R2n

) = δ

(R2n −

1

ε

)⇒ E

[etR

2n

]=∫ +∞

−∞dRne

tR2nδ

(R2n −

1

ε

)= exp

(t

ε

).

In other words, although Rn is a random variable, the quantity R2n is not a random

variable, but is instead fixed at the value of 1/ε.To write the probability measure for R(t), with t1 ≤ t ≤ t2 discretize continu-

ous time, namely t → nε. White noise R(t) has the probability distribution given

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Eq. 7.11. The probability measure for the white noise random variables in the in-terval t1 ≤ t ≤ t2 is given by

P[R] =N∏n=1

P(Rn) =N∏n=1

e−ε2R

2n, (7.13)

∫dR =

N∏n=1

√ε

2π

∫ +∞

−∞dRn.

Taking the continuum limit of ε → 0 yields, for t1 < t < t2,

P[R, t1, t2] → eS0, S0 = −1

2

∫ t2

t1

dtR2(t), (7.14)

Z =∫

DReS0,

∫dR→

∫DR.

The action functional S0 is ultra-local with all the variables being decoupled. Gaus-sian integration, given in Eq. 7.7, yields

Z[j, t1, t2] = 1

Z

∫DRe

∫ t2t1dtj (t)R(t)

eS0[ = e12

∫ t2t1dtj2(t)

. (7.15)

The correlation functions are given by

E[R(t)] = 0, E[R(t)R(t ′)] = 1

Z

∫DR R(t)R(t ′)eS0 = δ(t − t ′),

and yield the result given in Eq. 7.10.The results given in Eqs. 7.14 and 7.15 show that white noise is represented by

a path integral with an ultra local action S0. Path integrals can be used to representa great variety of white noise, as discussed by Kleinert (1990).

7.3.1 Integrals of white noise

Consider the integral of white noise

I =∫ T

t

dt ′R(t ′) ∼ ε

M∑n=0

Rn, M =[T − t

ε

],

where ε is an infinitesimal. For Gaussian white noise

Rn = N

(0,

1√ε

)⇒ εRn = N

(0,√ε).

The integral of white noise is a sum of normal random variables and hence, fromEq.(7.7) and above, is also a Gaussian random variable given by

I ∼ N(

0,√εM

)→ N

(0,√T − t

). (7.16)

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7.3 Gaussian white noise 135

In general, for

Z =∫ T

t

dt ′a(t ′)R(t ′) ⇒ Z = N(0, σ 2

), σ 2 =

∫ T

t

dt ′a2(t ′).

Due to the Dirac delta function singularity in white noise R(t), there is an ambi-guity in discretizing white noise. Consider the integral of white noise

I =∫ τ

0dtf (R(t); t)R(t) =

∫ τ

0f (W(t); t)dW(t), dW(t) = R(t)dt,

E[W(t)W

(t ′)] = tθ

(t ′ − t

)+ t ′θ(t − t ′

) = min(t, t ′

), (7.17)

where the theta function is defined in Eq. 3.9.Let tn = nε and with τ = Nε;W(t)→ Wn; there are two different and inequiv-

alent ways of discretizing the integral, namely the Ito and Strantovich discretiza-tions, which are given by

II=

N−1∑n=0

f (Wn; tn) (Wn+1 −Wn), E[WnWn′ ] =[min(n, n′)

]ε, (7.18)

IS=

N−1∑n=0

f

(Wn+1 +Wn

2; tn+1 + tn

2

)(Wn+1 −Wn).

The expectation value of the stochastic integrals for the case of f = W(t) is usedbelow to illustrate the ambiguity in the discretization of the stochastic integral.From Eq. 7.18

E[II] =

N−1∑n=0

E [Wn(Wn+1 −Wn)] =N−1∑n=0

(nε − nε) = 0. (7.19)

In contrast

E[IS] = 1

2

N−1∑n=0

E [(Wn+1 +Wn)(Wn+1 −Wn)]

= 1

2

N−1∑n=0

E[W 2

n+1 −W 2n

] = 1

2

N−1∑n=0

{(n+ 1)ε − nε} = τ

2. (7.20)

Hence, Eqs. 7.19 and 7.20 show that the two methods for discretizing the stochasticintegral are inequivalent.

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Both the Ito and Strantovich discretizations are used extensively. It can be shownthat the Ito discretization is the correct one for mathematical finance since it isconsistent with the requirement that the market is free from arbitrage opportunities[Baaquie (2004)]; the Strantovich discretization requires a future value of f (W),which in turn is in conflict with the requirement of no arbitrage. The Strantovichdiscretization is used in the study of the Fokker–Planck equation [Risken (1988)].

7.4 Ito calculus

A brief discussion is given below of Ito calculus and its relation to stochastic differ-ential equations. Due to the singular nature of white noise R(t), functions of whitenoise have new features. In particular, the infinitesimal behavior of such functions,as seen in their Taylor’s expansions, acquire new terms.

Consider the stochastic differential equation that occurs in finance; the timederivative of an underlying security S(t) is generically expressed as

dS(t)

dt= φ(t)S(t)+ σ(t)S(t)R(t). (7.21)

Let f be some arbitrary function of S(t). Ito’s definition of a derivative yields

df

dt= lim

ε→0

f (t + ε, S(t + ε))− f (t, S(t))

ε

or, using Taylors expansion

df

dt= ∂f

∂t+ ∂f

∂S

dS

dt+ ε

2

∂2f

∂S2

[dSdt

]2 + 0(ε1/2

). (7.22)

The last term in Taylor’s expansion is order ε for smooth functions, and goes tozero. However, due to the singular nature of white noise, Eq. 7.21 yields[

dS

dt

]2

= σ 2S2R2 + 0(1) = 1

εσ 2S2 + 0(1). (7.23)

Hence, from Eqs. 7.21, 7.22, and 7.23, for ε → 0

df

dt= ∂f

∂t+ ∂f

∂S

dS

dt+ σ 2

2

∂2f

∂S2= ∂f

∂t+ 1

2σ 2S2 ∂

2f

∂S2+ φS

∂f

∂S+ σS

∂f

∂SR.

(7.24)

Since Eq. 7.24 is of central importance for the theory of security derivatives,a derivation is given based on Ito-calculus. Let us rewrite Eq. 7.21 in terms ofdifferentials as

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7.4 Ito calculus 137

dS = φSdt + σSdz, dz = Rdt, (7.25)

where dz is a Wiener process. Equation 7.12 yields R2(t) = 1/ε = 1/dt and hence

(dz)2 = R(t)2(dt)2 = dt + 0(dt3/2)

and

(dS)2 = σ 2S2dt + 0(dt3/2

).

From the equations for dS and (dS)2 given above,

df = ∂f

∂tdt + ∂f

∂SdS + 1

2

∂2f

∂S2(dS)2 + 0

(dt3/2

)=(∂f

∂t+ 1

2σ 2S2 ∂

2f

∂S2

)dt + σS

∂f

∂Sdz,

and Eq. 7.24 is recovered using dz/dt = R.Suppose g(t, S(t)) ≡ gt is another function of the white noise S(t). The abbre-

viated notation δgt ≡ gt+ε − gt yields

d(fg)

dt= lim

ε→0

1

ε

[ft+εgt+ε − ftgt

]= lim

ε→0

1

ε

[δftgt + ftδgt + δftδgt

].

Usually the last term δftδgt is of order ε2 and goes to zero. However, due to thesingular nature of white noise

d(fg)

dt= df

dtg + f

dg

dt+ df√

dt

dg√dt

: Ito’s chain rule. (7.26)

Similarly to Eq. 7.26, in terms of infinitesimals, the Ito chain rule is given by

d(fg) = dfg + f dg + df dg.

7.4.1 Stock price

To illustrate stochastic calculus, the stochastic differential equation, Eq. 7.21, isintegrated. Consider the change of variable and the subsequent integration

x(t) = ln[S(t)], ⇒ dx

dt= φ − σ 2

2+ σR(t), (7.27)

⇒ x(T ) = x(t)+ (φ − σ 2

2)(T − t)+ σ

∫ T

t

dt ′R(t ′). (7.28)

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The random variable∫ Ttdt ′R(t ′) is a sum of normal random variables and is shown

in Eq.7.16 to be equal to a normal N(0,√T − t) random variable. Hence

S(T ) = S(t)e

(φ− σ2

2

)(T−t)+(σ√T−t)Z with Z = N(0, 1). (7.29)

The stock price evolves randomly from its given value of S(t) at time t to a wholerange of possible values S(T ) at time T . Since the random variable x(T ) is a nor-mal (Gaussian) random variable, the security S(T ) is a lognormal random variable.

Geometric mean of stock price

The probability distribution of the (path dependent) geometric mean of the stockprice can be exactly evaluated. For τ = T − t and m = x(t)+ (φ− σ 2

2 )τ , Eq. 7.28yields

SGM = eG, τ = T − t,

G ≡ 1

τ

∫ T

t

dt ′x(t ′) = m+ σ

τ

∫ T

t

dt ′∫ t ′

t

dt ′′R(t ′′)

= m+ σ

τ

∫ T

t

dt ′(T − t ′

)R(t ′).

From Eq.7.16 the integral of white noise is a Gaussian random variable, whichis completely specified by its means and variance. Hence, using E[G] = m andEq. 7.10 for E[R(t)R(t ′)] = δ(t − t ′) yields

E[(G−m)2

] = (στ

)2∫ T

t

dt ′(T − t ′

) ∫ T

t

dt ′′(T − t ′′

)E[R(t ′)R(t ′′)]

=(στ

)2∫ T

t

dt ′(T − t ′

)2 = σ 2τ

3.

Hence

G = N

(m,

σ 2τ

3

). (7.30)

The geometric mean of the stock price is lognormal with the same mean as thestock price, but with its volatility being one-third of the stock price’s volatility.

7.5 Wilson expansion

The product of nonlinear (non-Gaussian) quantum fields is the subject matter ofwhat is called the “short distance” Wilson expansion, discussed by Wilson (1969),Wilson and Zimmermann (1972), and Zinn-Justin (1993). The Wilson expansion ofquantum fields is a very general technique that allows one to isolate the singularities

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7.5 Wilson expansion 139

in the product of quantum fields. In the context of stochastic systems, the Wilsonexpansion provides a generalization of Ito calculus to the case where the stochasticphenomenon is driven by the two dimensional Gaussian quantum field A(t, x), asdiscussed by Baaquie (2009).

The time derivative of various quantities like the underlying security S(t) orstochastic volatility are generically expressed as

dS(t)

dt= μ(t)+ σ(t)R(t).

Ito’s stochastic calculus, for discrete time t = nε, is a result of the identity [Baaquie(2004)]

E [R(t)R(t)] = δ(t − t ′) ⇒ R2(t) = 1

ε+O(1). (7.31)

The singular term for R2(t) in Eq. 7.31 is deterministic, namely, equal to 1/ε; allthe random terms that occur for R2(t) are finite as ε → 0.

The two dimensional quantum field A(t, x) is an independent degree of freedomfor each t and each x. For Gaussian quantum fields such as A(t, x) that have aquadratic action, the full content of a Gaussian (free) quantum field, as discussedby Zinn-Justin (1993), is encoded in its propagator. Consider a propagator given by

E[A(t, x)A(t ′, x ′

] = δ(t − t ′

)D(t, x; t ′, x ′). (7.32)

Similarly to white noise correlator E[R(t)R(t)] = δ(t − t ′), the correlationfunction E[A(t, x)A(t ′, x ′)] given in Eq. 7.32 is infinite for t = t ′. The singularproduct of two Gaussian quantum fields is the simplest case of the Wilson expan-sion. The singularity of the correlation function in the product of the quantum fieldA(t, x), similarly to Eq. 7.31, can be expressed as

A (t, x)A(t, x ′

) = 1

εD(x, x ′; t)+O(1). (7.33)

All the fluctuating components, which are contained in A(t, x)A(t, x ′), are regularand finite as ε → 0. The correlation of A(t, x) is singular for t = t ′, very muchlike the singularity of white noise R(t).

Since A(t, x) is an indeterminate quantity, one may ask how one can assignit a deterministic numerical value as in Eq. 7.33? What Eq. 7.33 means is thatin any correlation function, wherever a product of fields is at equal time, namelyA(t, x)A(t, x ′), then – to leading order in ε – the product can be replaced by thedeterministic quantity D(x, x ′; t)/ε. In terms of symbols, Eq. 7.33 states

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E [A(t1, x1)A(t2, x2) . . .A(t, xn)A(t, xn+1) . . .A(tN , xN)]

= 1

εE [A(t1, x1)A(t2, x2) . . .A(tn−1, xn−1)D(xn, xn+1; t)×A(tn+2, xn+2) . . .A(tN , xN)]+O(1).

Equations 7.31 and 7.33 have a similar singularity structure, and Gaussian quan-tum fields are a natural generalization of Ito calculus. Ito calculus is a special caseof the Wilson expansion, and is given by taking the limit

A(t, x)→ R(t), D(x, x ′; t)→ 1, (7.34)

⇒ E[A(t, x)A(t ′, x ′)

]→ E [R(t)R(t)] = δ(t − t ′).

7.6 Linear Langevin equation

Consider the case with no external potential, namely �(v) = 0. One can rewritethe Langevin equation given in Eq. 7.2, for β ≡ γ /m, as

dv

dt= −γ v

m+√

2A

mR ≡ −βv+

√2A

mR. (7.35)

The term R is white noise, with probability density given in Eq. 7.14 by

P [R] = 1

Zexp

(−1

2

∫R2(t)dt

).

Integrating the first order stochastic differential yields for v(t)

dv

dt+ γ

mv =

√2A

mR = e−

γmt d

dt

(eγmtv)=√

2A

mR.

Hence

v = v0e−γ t/m +√2A

e−γ tm

m

∫ t

0dt ′e

γmt ′R(t ′).

Define

u = v− v0e−γ t/m, (7.36)

such that E[u(t)] = 0; hence

E[u2(t)

] = √2A

m2

∫ t

0dηdτe−

γm(t−η)e−

rm(t−τ)E [R(η)R(τ)]

= 2A

m2

∫ t

0dηe−2β(t−η) = A

mγ

(1− e−2γ t/m

)⇒ lim

t→∞E[u2(t)

] = E[v2(t)] = A

mγ.

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7.6 Linear Langevin equation 141

The particle reaches equilibrium with its medium after a long time, namely fort = ∞. For the particle in a medium at temperature T , the Maxwell velocitydistribution is given by

P [v] =√

m

2πkTe−

1kT· 12mv2 ⇒ E

[v2] = kT

m. (7.37)

Hence, in equilibrium

A

γm= kT

m⇒ A = kT γ. (7.38)

The velocity v(t) is a random variable, and hence is determined by a probabilitydistribution function P [v; t]. Solving a stochastic differential equation, unlike anordinary differential, consists of determining the probability distribution P [v; t; v0]for the occurrence of different values for v(t), given the initial velocity v0.

Recall from Eq. 7.2 that the Langevin equation yields

v = v0e− γ t

m +∫ t

0dt ′ρ(t ′)R(t ′). (7.39)

It can be shown that the linear sum of Gaussian random variables is also a Gaussianrandom variable. Hence all we need to do is to determine the mean and variance ofv(t) to determine P [v; t]. Recall from Eq. 7.36 that u = v− v0e

−γ t/m and hence

E[v] = v0e− γ

mt ⇒ σ 2

v = σ 2u = E[u2(t)] = kT

m(1− e−2 γ

mt ). (7.40)

Hence

P [v; t; v0] = 1√2πσ 2

v

e− 1

2σ2v

(v−v0e

− γm t)2

=√√√√ m

2πkT(

1− e−2 γmt) exp

⎧⎪⎨⎪⎩−

m

2kT

(v− v0e

− γmt)2

(1− e−

γmt)⎫⎪⎬⎪⎭ . (7.41)

Note, as expected

limt→0

P [v; t; v0] → δ(v− v0)

limt→∞P [v; t; v0] →

√m

2πkTe−

m2kT v2

. (7.42)

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7.6.1 Random paths

From Eq. 7.35, the Langevin equation is given by

dv

dt= −βv+

√2A

mR.

The random paths x(t), as determined by the Langevin equation, are given by

dx(t)

dt= v(t), (7.43)

x(t) = x0 +∫ t

0dξv(ξ)⇒ E[x(t)] = x0 +

∫ t

0dξE[v(ξ)],

or, E[x(t)] = x0 + v0

∫ t

0dξe−βξ = x0 + v0

β(1− e−βt ). (7.44)

Let x0 = −v0/β, then

E[x(t)] = −v0

βe−βt → 0 as t →∞.

Let y(t) = x(t) + v0βe−βt , y(0) = 0. The change from x(t) to y(t) is made to

remove unnecessary terms arising from the boundary conditions on x(t) and v(t).Defining y = dy/dt yields

dy

dt= v− e−βtv0 ⇒ y = √2A

e−βt

m

∫ t

0dξeβξR(ξ).

Hence

E[y(t)y(t ′)

] = 2A

m2

∫ t

0dξ

∫ t ′

0dηe−β(t−ξ)e−β(t

′−η)E [R(ξ)R(η)]

= e−β(t+t′) 2A

m2

∫ t

0dξ

∫ t ′

0dηeβξ eβηδ(ξ − η).

But ∫ t ′

0dηδ(ξ − η) =

{0 ξ > t ′

1 ξ � t ′= θ(t ′ − ξ).

Let t ≥ t ′, then

E[y(t)y

(t ′)] = A

m2e−β(t+t

′)∫ t ′

0dξe2βξ = A

2βm2e−β(t+t

′)[−1+ e2βt ′

].

Due to time ordering expressed in t ≥ t ′, the correlator is given by

E[y(t)y

(t ′)] = A

2βm2

∫ t

0dτ

∫ t ′

0dτ ′

[−e−β(τ+τ ′) + e−β|τ−τ ′|

]. (7.45)

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7.7 Langevin equation with potential 143

For t = t ′,

E[y2(t)

] = A

2βm2

∫ t

0dτdτ ′

[−e−β(τ+τ ′) + e−β|τ−τ ′|

]. (7.46)

Integrating the equation above yields

E[y2(t)

] = 2A

β2t + Am

β3

[−3+ 4e−

γmt − e−

2γmt]. (7.47)

In the limit of t →∞ we have that y(t)→ x(t), and this yields

√E[x2(t)

] =√

2A

γ 2· √t, (7.48)

where√t is the characteristic signal of a random walk. Equivalently

E[x2(t)

] = 2A

γ 2t ≡ 2Dt, (7.49)

where D is the diffusion constant, hence

D = A

γ 2= kT

γ: Einstein relation. (7.50)

Note E[x(t)] = 0 and hence√E[x2(t)] measures the degree of dispersal of the

randomly moving particle.For a uniformly moving particle x ∼ vt . In contrast, for a randomly walking

particle x ∼ √t , which is much slower than a freely moving particle, due to thefrequent random collisions that the particle has with the medium it is in.

7.7 Langevin equation with potential

In the presence of an external force due to a potential field B, the Langevin equa-tion is

d�vdt= −β�v+ 1

m�K + 1

m�F, (7.51)

where �K = −�∇B.For the one-dimensional simple harmonic oscillator, B = 1

2mω2x2 and

dv

dt= −βv− ω2x + 1

mF, (7.52)

or

d2x

dt2+ β

dx

dt+ ω2x = 1

mF. (7.53)

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Let D = d/dt , then

(D2 + βD + ω2

)x = 1

mF, (7.54)

or

(D + r1)(D + r2)x = σR

(σ =

√2A

m

), (7.55)

r1,2 = β

2±√β2

4− ω2.

Let xH be the homogeneous solution of Eq.7.53 with

(D + r1)(D + r2)xH = 0. (7.56)

Note the boundary values that specify a unique solution for Eq.7.53 are carried byxH . The complete solution of Eq.7.53 is then given by

x = xH + 1

(D + r1)(D + r2)σR

= xH + σ

r2 − r1

[1

D + r1− 1

D + r2

]R

= xH + σ

∫ t

0B(t, t ′)R(t ′)dt ′. (7.57)

Using

1

D + r1R =

∫ t

0e−r1(t−ξ)R(ξ)dξ, xH = Ae−r1t + Be−r2t

yields

B(t, t ′) = 1√β2 − 4ω2

{e−r2(t−t ′) − e−r1(t−t ′)

}. (7.58)

The coefficients A,B are determined from the initial conditions using x(0) = x0,x(0) = v0, or any other specifications of the boundary conditions.

7.7.1 Correlation functions

Consider the unequal time correlator given by

Ct,t ′ =E[x(t)x(t ′)] = xH (t)xH (t′)+ 2A

m2

∫ t

0dξ

∫ t ′

0dξ ′B(t, ξ)B(t ′, ξ ′)δ(ξ − ξ ′).

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7.8 Nonlinear Langevin equation 145

Choose time ordering t > t ′, and for CH = xH (t)xH (t′), one obtains

Ct,t ′ = CH + 2A

m2

∫ t ′

0dξ ′B

(t, ξ ′

)B(t ′, ξ ′

). (7.59)

For the under damped case

λ2 = ω2 − β2

4> 0 (7.60)

and hence

r1 = −β2+ iλ, r2 = −β

2− iλ (7.61)

⇒ Ctt ′ = CH + 2A

m2

1

2βω2e−

β2 (t−t ′)

{cos λ

(t − t ′

)+ β

2λsin λ

(t − t ′

)}.

(7.62)

Similar expressions can be obtained for the over damped case for which λ2 > 0.

7.8 Nonlinear Langevin equation

Consider the nonlinear Langevin equation given by

dv

dt= βv− gv3 + σR, (7.63)

with β, g > 0 and v(0) = v0. Note that the sign of β is positive, and is the oppositeof the sign of the viscosity term γ in the Langevin equation as given in Eq. 7.2.

There are two competing effects in the nonlinear Langevin equation, namely:

• the β term acts to exponentially increase v;• the dissipative term g damps and reduces the value of v.

It is not known how to exactly solve Eq. 7.63. Instead, a self-consistent meanfield approximation is used to find a solution. The accuracy of this approximatesolution is discussed later.

Let f (t) = E[v2(t)] and f (0) = v20. The mean field approximation consists of

(self-consistently) linearizing the equations of motion, namely

dv

dt→ βv− gvE[v2] + σR

⇒ dv

dt= βv− gvf + σR, mean field approximation. (7.64)

Note that

df

dt= 2E

[v(t)

dv

dt(t)

]= 2E

[v(t)

{βv(t)− gv3(t)+ σR(t)

}]. (7.65)

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Using the mean field approximation for the equations of motion given in Eq. 7.64yields

E[v(t)4

] � E[v(t)2

]2 ≡ f 2(t).

Hence, in the mean field approximation

df

dt= 2βf − 2gf 2 + 2σE [v(t)R(t)] .

Defining h(t) ≡ g∫ t

0 dτf (τ)− βt , Eq. 7.64 yields

v(t) = e−h(t)v0 + eh(t)σ

∫ t

0e−h(τ)R(τ)dτ

and hence

E [v(t)R(t)] = e−h(t)σ∫ t

0eh(τ)δ(t − τ)dτ = 1

2σ,

where for consistency∫ t

0 δ(t − τ)dτ = 12 . Hence, from Eq. 7.65

df

dt= 2βf − 2gf 2 + σ 2 : Riccati equation. (7.66)

Let

f = 1

2g

u

u⇒ f = 1

2g

(u

u−(u

u

)2).

Substituting the expression for f, f on both sides of the Riccati equation given inEq. 7.66 yields

1

2g

(u

u− u2

u2

)= 2β · 1

2g

u

u− 1

2g

(u

u

)2

+ σ 2.

Note that the nonlinear term (u/u)2 cancels in the above equation, and yields alinear equation for u(t), namely

u− 2βu− 2gσ 2u = 0.

Using the ansatz u(t) � eαt yields

α2 − 2βα − 2gσ 2 = 0 ⇒ α = β ±√β2 + 2σ 2g = β ± λ.

Hence the solution is given by

u(t) = ceβt(eλt + ke−λt

),

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7.8 Nonlinear Langevin equation 147

where c, k are two integration constants. The superfluous constant c cancels out, asis required, for the “physical” quantity f (t), in the following manner:

E[v2(t)

] = f = 1

2g

u

u

⇒ E[v2(t)

] = 1

2g

(β + λ)eλt + k(β − λ)e−λt

eλt + ke−λt

v20 =

1

2g

β + λ+ k(β − λ)

1+ k, (7.67)

or, equivalently

k = 2gv20 − β − λ

β − λ− 2gv20

.

Note that

λ ≡ β

(1+ 2σ 2g

β2

)1/2

� β

(1+ σ 2g

β2

)� β.

Consider 1 � βt � ∞, that is 1β� t � ∞. Let v0 � 1, then, for infinite time,

the average fluctuation for v2(t) is given by

⇒ limt→∞E

[v2(t)

] ∼= 1

2g

[2βeλt

eλt + ke−λt

]= β

g

1ke2λt

1+ 1ke2λt

→ β

g.

The cross-over of the velocity v(t) from its initial value v0 to its equilibrium valueβ/g takes place at t0 such that E[v2(t)] � β/g, as shown in Figure 7.3.

Hence, 1+ e2λt0/k � e2λt0/k � 1, and this yields

1

ke2λt0 � 1, t0 ∼ ln(k)/2λ.

At time t0 dissipative structures far from equilibrium are formed due to the syn-ergy between nonlinear dissipation driven by g and random fluctuations due toσ 2/β. As time flows the dissipative structures evolve into the final equilibriumstate. Figure 7.3 shows the evolution of the particle from its initial state at t = 0,via the formation of dissipative structures t0, to equilibrium at tE .

To check whether the mean field result is consistent, note that on reachingequilibrium

dv

dt= 0 = βv− gv3 + R,

⇒ βE[v] = gE[v3] � gE[v]E[v2] : mean field,

⇒ E[v2] � β

g: consistent.

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<V2(t)>

V20

t=0 t0 tE t

β/g

Figure 7.3 The value of E[v2(t)] as a function of time, which starts at the initialvalue v2

0 and follows a nonlinear trajectory to its final value, with cross-over att0 � ln(k)/(2λ).

How good is the mean field approximation? Can one systematically improveit? There is no small parameter in terms of which the mean field approximationis defined, and hence one cannot go to higher and higher order in the mean fieldapproximation.

The mean field approximation can correctly ascertain whether, for example,a system undergoes a phase transition, but cannot give an accurate estimate of thephysical parameters that describe the transition. Rather, mean field approximationis a guide to the qualitative behaviour of a system and can capture many featureslike the system having instabilities or other global properties.

7.9 Stochastic quantization

The stochastic Langevin differential equation expresses the evolution of the fun-damental random degree of freedom, say the random velocity v(t) itself, and isanalogous to the Heisenberg operator equations. The solution of a stochastic dif-ferential equation entails determining the probability distribution function for ev-ery time P(v; t; v0), given the initial condition at t = 0 by v(0) = v0. P(v; v0; t)is the evolution kernel for a nonequilibrium and dissipative system, driven by theFokker–Planck Hamiltonian.

As a warm-up to a more detailed analysis of the Fokker–Planck equation, a quickand heuristic derivation is given for P(v; t; v0) for the (linear) Langevin equa-tion. Recall

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7.9 Stochastic quantization 149

dv

dt= −βv+�(v)+ σR, v(0) = v0.

Since both vt and Rt are random variables, stochastic quantization defines a pathintegral by integrating over all possible values for vt and Rt .

The conditional probability P(v; v0; T ) is the likelihood of the final value ofthe velocity v(T ) having a value equal to v, given that its initial value was v0.The Langevin equation is a first order differential and needs only one boundarycondition, which has been taken to be the initial condition; the second conditionv(T ) = v is a constraint on the allowed random paths and is put in as a delta-function constraint in the path integral. The path integral is given by

P(v; v0; T )

=∫

DvDR

[∏t

δ

(vt + βtvt −�(v)

σ− R(t)

)]δ(v(T )− v)e−

12

∫R2(t)dt

=∫

Dvδ (v(T )− v) e−1

2σ2

∫dt(v+βv−�(v))2

, B.C. : v(0) = v0.

Once the white noise path integral has been performed, one can write the pathintegral for finite time by directly incorporating the two boundary conditions onthe velocity, namely v(0) = v0, v(T ) = v; the path integral can be written morecompactly as

P(v; v0; T ) =∫

Dve−1

2σ2

∫dt(v+βv−�(v))2

. (7.68)

For the nonlinear Langevin equation given in Eq. 7.63 one obtains the action

S = − 1

2σ 2

∫ T

0dt(v− βv+ gv3

)2. (7.69)

The path integral for the nonlinear Langevin equation in principle is exact andcan be used for systematically generating a power series in any small parameterby expanding the action and doing the path integral term by term. In particular,one could for example compute the expectation values E[v3], E[v2], E[v] as aperturbation expansion in g << 1, and then compare E[v3] with E[v2]E[v] todetermine the accuracy of the mean field approximation.

7.9.1 Linear Langevin path integral

Consider the linear Langevin equation for which �(v) = 0. The stochastic differ-ential equation is given by

dv

dt= −βv+ σR

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with volatility σ=constant. The action is given by

S = − 1

2σ 2

∫ T

0dt (v+ βv)2 .

Consider change of variables

w = v+ βv = e−βtd

dt

(eβtv

).

For incorporating the initial condition v(0) = v0, consider

v(T ) = e−βT v0 + e−βT∫ T

0dt ′eβt

′wt′ . (7.70)

As discussed in deriving Eq. 7.68, the second condition v(T ) = v is put in as adelta-function constraint in the path integral and yields the following path integralfor the conditional probability:

P (v; v0; T ) = N

∫DWδ (v− v(T )) e−

12σ2

∫w2

.

Using the Fourier representation for the δ-function and using Eq. 7.70 for v(T )yields

P(v; v0; T ) = N

∫dξ

2πeiξ(v−v0e

−βT )∫

DWe−iξeβT∫dt ′eβt ′wt ′ e

− 12σ2

∫w2

=∫

dξ

2πeiξ(v−v0e

−βT ) exp

{−σ

2

2e−2βT ξ 2

∫ T

0dt ′dt ′′eβt

′δ(t ′ − t ′′)eβt

′′}

=∫

dξ

2πeiξ(v−v0e

−βT ) exp

{−σ

2v

2ξ 2

},

where

σ 2v = σ 2e−2βT

∫ T

0dte2βt = σ 2

2β

(1− e−2βT

)and, as expected from Eq. 7.40,

σ 2

2β= 2A

2m2· mγ= A

mγ= kT

m.

Performing the ξ integration yields the final result

P (v; v0; T ) = 1√2πσ 2

v

e− 1

2σ2v(v−e−βT v0)

2

, (7.71)

as was obtained earlier in Eq. 7.41.

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7.10 Fokker–Planck Hamiltonian 151

7.10 Fokker–Planck Hamiltonian

The Fokker–Plank equation is a partial differential equation for evolving the con-ditional probability P(v; v0; t) [Risken (1988)].

Consider the general Langevin equation

dv(t)

dt= a(v, t)+ σ(v, t)R(t). (7.72)

There are two possible boundary conditions, namely

• Initial condition v(t0) = v0;or

• Final condition v(T ) = vT .

The Langevin equation has the forward Euler discretization given by t = nε and

v(t + ε) = v(t)+ ε [a(v, t)+ σ(v, t)R(t)] , (7.73)

v = v′ + ε{a(v′ + σ(v′)Rn

)}. (7.74)

For discrete time, white noise has the correlators

E[Rn] = 0, E[R2n

] = 1

ε.

The forward conditional probability is given by

PF (v, v0, t + ε) =∫

dv′E[δ(v− v′ − ε[a (v′)+ σ

(v′)Rt ])]PF

(v′, v0, t

)=∫

dv′E[δ(v− v′

)+ ε(a(v′)+ σ

(v′)R) ∂

∂v′δ(v− v′

)+ε

2

2(a + σR)2 ∂2

∂v′2δ(v− v′

)]PF

(v′; v0, t

)= PF (v, v0, t)− ε

∂

∂v(aPF )+ ε

2

∂2

∂v2

(σ 2PF

)+O(ε2).

Hence

∂PF

∂t≡ 1

ε

[PF (v, v0, t + ε)− PF (v, v0, t)

]=[

1

2

∂2

∂v2σ 2 − ∂

∂va

]PF

≡ −HFPF ,

where the forward Fokker–Planck Hamiltonian is given by

HF = −1

2

∂2

∂v2σ 2(v)+ ∂

∂va = −1

2

∂2

∂v2σ 2(v)+ a

∂

∂v+ ∂a

∂v(7.75)

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and

PF (v, v0; t) =⟨v∣∣e−(t−t0)H ∣∣ v0

⟩.

Note that in the discretization of the Langevin equation

v = v′ + ε(a(v′)+ σ

(v′)R) �= v′ + ε(a(v)+ σ(v)R),

due to singular nature of the white noise.Consider for example

a(v′) = a

(v+ εσ

(v′)R + 0(ε)

) = a(v)+ εσ(v′)Ra′(v)+ 0(ε)

� a(v)+ ε1/2σ(v)+ · · ·The 0(ε1/2) term changes the Langevin equation since(

v− v′

ε

)= (

a(v′)+ σ

(v′)R + εσRa′(v)

)2

⇒(dv

dt

)2

= (a + σR)2 + 2εσ 2a′R2 + 0(ε2)

�= (a + σR)2.

The backward probability distribution is given by the backward Fokker–Planckequation results from the same discretization of the Langevin equation as the for-ward case. The only difference being that for the backward case one integrates overthe later velocity v(t + ε) ≡ v, which yields

PB(vT , v′; t) =∫

dvE[δ[v− v′ − ε(a(v′)+ σ(v′)Rt)

]]P(vT , v, ; t + ε)

= P(vT , v′, t + ε

)+ ε

[a(v)

∂P

∂v+ 1

2σ 2(v)

∂2P

∂v2

].

The backward Fokker–Planck Hamiltonian is hence

∂PB

∂t= HBPB, HB = −σ

2

2

∂2

∂v2− a

∂

∂v.

Note that for the backward FP time is given by −t , that is, time is running back-ward, and consequently

1

ε[PB(vT , v; t)− PB(vT , v; t + ε)] = − ∂PB

∂(−t) = −HBPB

⇒ ∂PB

∂t= HBPB.

To write PB as an evolution kernel, note that in Dirac’s notation |initial〉 alwaysrepresents the |starting state〉 and the ending state is given by 〈final| = 〈endingstate|. Hence

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7.11 Pseudo-Hermitian Fokker–Planck Hamiltonian 153

PB(vT , v; t) =⟨v∣∣∣e−(T−t)H ∣∣∣ vT

⟩, PB(vT , v; T ) = δ(v− vT ),

where

H = HB = −σ2

2

∂2

∂v2− a

∂

∂v.

7.11 Pseudo-Hermitian Fokker–Planck Hamiltonian

For non-Hermitian operators, the rule for the action of operators is that all operatorsact on the dual vectors; that is⟨

v|H |ψ⟩= H

(v,

∂

∂v

)〈v|ψ〉 = H(v, ∂v)ψ(v).

The definition of Hermitian conjugation is given by⟨ψ |H |v

⟩=(⟨

v|H †|ψ⟩)∗ = (H †(v, ∂v)ψ(v))

∗.

The forward Fokker–Planck Hamiltonian is given by the Hermitian conjugationof the backward Fokker–Planck Hamiltonian as follows:

HF = H†B = −

1

2

∂2

∂v2σ 2 + ∂

∂va �= HF (7.76)

: non-Hermitian.

HF has two sources of non-Hermiticity, namely [Risken (1988)]:

1. The volatility depends on velocity, that is σ = σ(v).

2. The first partial derivative a ∂/∂v term is not Hermitian.

An effective Hermitian Fokker–Planck Hamiltonian can be obtained for one di-mension in which volatility σ(v) is made into a constant by a change of variablesand the first derivative term is removed by a similarity transformation.

Consider the backward Fokker–Planck Hamiltonian given by

H = −1

2σ 2(v)∂2

v − a(v)∂v, ∂v ≡ ∂

∂v, (7.77)

where both σ and a depend on the velocity v. Change variables from v to w suchthat

H = −1

2σ 2

0 ∂2w + α(w)∂w,

where σ 20 is chosen to be a constant, independent of v.

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Using the chain rule for differentiation yields

∂2

∂v2= ∂w

∂v

∂

∂w

(∂w

∂v

∂

∂w

)=(∂w

∂v

)2∂2

∂w2+ ∂w

∂v

[∂

∂w

(∂w

∂v

)]∂

∂w. (7.78)

Choose constant σ 20 ,

σ 20 =

(∂w

∂v

)2

σ 2,

which yields the definition for the change of variables from v to w as

dw

dv= σ0

σ(v)⇒ w = σ0

∫ v

0

dv′

σ(v′). (7.79)

Hence, the Hamiltonian in Eq. 7.77 yields from Eqs. 7.78 and 7.79

H = −1

2σ 2(v)∂2

v − a(v)∂v

= −1

2σ 2

0

∂2

∂w2− 1

2σ 2 · σ0

σ(v)·[∂

∂w

(σ0

σ(v)

)]∂

∂w− a(v)

∂w

∂v

∂

∂w

= −σ20

2∂2w + α(w)∂w, (7.80)

where

α(w) = σ0

σ

[1

2σ0∂σ

∂w− a

].

The volatility σ0 is constant and hence the first term in the Hamiltonian given inEq. 7.80 is Hermitian. A similarity transformation is now defined that removes thefirst order derivative term in Eq. 7.80.

For an abitrary function � = �(w), note that

e−�(−1

2σ 2

0 ∂2w

)e� = −σ

20

2

(�′′ +�′2 + 2�′∂w + ∂2

w

),

−σ20

2∂2w −

σ 20

2(2�′∂w) = e−�

(−1

2σ 2

0 ∂2w +

σ 20

2�′′ + σ 2

0

2�′2)e�, (7.81)

where �′ = d�/dw. Choose � such that

−σ20

2

(2�′∂w

) = α(w)∂w

⇒ �′ = − 1

σ 20

α ⇒ � = − 1

σ 20

∫ w

0α(w)dw. (7.82)

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7.11 Pseudo-Hermitian Fokker–Planck Hamiltonian 155

Hence from Eqs. 7.81 and 7.82

−σ20

2∂2w + α(w)∂w = e−�

(−1

2σ 2

0 ∂2w +

σ 20

2�′′ + σ 2

0

2�′2)e�

= e−�Heffe�, (7.83)

where the effective Hermitian Hamiltonian is given by

Heff = H†eff : Hermitian

= −1

2σ 2

0 ∂2w +

σ 20

2�′′ + σ 2

0

2�′2. (7.84)

In summary, the backward and forward Fokker–Planck Hamiltonians are given by

H = HB = −σ20

2∂2w + α(w)∂w = e−�Heffe

�,

HF = H † = e�Heffe−� �= H.

Hamiltonian H is pseudo-Hermitian since

H † == e2�He−2�.

Pseudo-Hermitian Hamiltonians are a special class of operators with many inter-esting properties, and are studied in great detail in Chapter 14.

The effective Hamiltonian given in Eq. 7.84 can be further simplified usingEq. 7.82 as follows:

Heff = 1

2σ 2

0

(−∂2

w −1

σ 20

α′ + 1

σ 40

α2

)

= 1

2σ 2

0

(−∂w + 1

σ 20

α

)(∂w + 1

σ 20

α

)

= 1

2σ 2

0Q†Q, (7.85)

where

Q = ∂w + 1

σ 20

α, Q† = −∂w + 1

σ 20

α.

The eigenfunction equation for Heff is given by

Heff|ψn〉 = En|ψn〉.

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The eigenvalues En of Heff are nonnegative since, from Eq. 7.85

En = 〈ψn|Heff|ψn〉 = 1

2σ 2

0 〈ψn|Q†Q|ψn〉

= 1

2σ 2

0 ||Q|ψn〉||2 ≥ 0. (7.86)

The ground state can be obtained exactly. Note, for |ψ0〉 ≡ |�〉Heff|�〉 = 0, E0 = 0

⇒ Q� =(∂w + 1

σ 20

α

)� = 0, (7.87)

�(w) = exp

(− 1

σ 20

∫ w

0α(w′)dw′

), (7.88)

: stationary (time independent) solution.

Since Heff|ψn〉 = En|ψn〉, the eigenfunctions of H are given by

H[e−�|ψn〉

] = e−�Heff|ψn〉 = En

[e−�|ψn〉

].

Hence the eigenfunctions and eigenvalues of H are given by

e−�|ψn〉, En.

In particular, the conditional transition probability is given by

PB(vT , v; t) = ⟨v|e−(T−t)H |vT

⟩ = e�(v)−�(vT )∑n

cne−En(T−t), (7.89)

where cn = 〈v|ψn〉〈ψn|vT 〉.Note that, sinceEn≥ 0, the conditional probability PB(vT , v; t) given in Eq. 7.89

is convergent. This is a reflection, in the Hamiltonian framework, of the fact thatthe path integral expression for PB(vT , v; t) is convergent – in spite of the Fokker–Planck Hamiltonian not being Hermitian – since the Lagrangian is a squared quan-tity due to Gaussian white noise driving the random system.

For T →∞, since E0 = 0,

PB(vT ; v) � �∗v(v)�(vT )e�(v)−�(vT ). (7.90)

7.12 Fokker–Planck path integral

Consider the case σ = σ0 = constant; the path integral, from Eq. 7.68, is given by

ZF =∫

DveSF. (7.91)

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7.12 Fokker–Planck path integral 157

The forward Fokker–Planck Hamiltonian, from Eq. 7.75, is given by

HF = −1

2

∂2

∂v2σ 2(v)+ a

∂

∂v+ ∂a

∂v.

Hence⟨v|e−εHF |v⟩ = ∫

dp

2π

⟨v∣∣e−εHF

∣∣p⟩ 〈p|v〉= N

∫dp

2πe−ε(σ22 p2+iap+a′

)eip(v−v′) = e

− ε

2σ2 (v−v−εa)e−εa

′

= NeεL ⇒ L = −1

2(v− a)2 − ∂a

∂v.

Hence, the forward action is given by

SF = −1

2

∫dt

1

σ 2

(dv

dt− a

)2

−∫

dt∂a

∂v. (7.92)

The backward path integral ZB has the intuitive stochastic quantization dis-cussed in Section 7.9,

ZB =∫

DrDR∏t

δ

[dv

dt− a − σR

]e−

12

∫R2(t).

Rescaling the white noise variable R → R/σ and doing the path integral overwhite noise yields

ZB =∫

DvDRδ [v− a − R] e−12

∫ 1σ2 R

2 =∫

Dve−12

∫ 1σ2 (v−a)2 . (7.93)

Recall that

HB = −1

2σ∂2

∂v2− a

∂

∂v, HF = H

†B

and yields

ZB =∫

Dve− 1

2σ20

∫(v−a)2dt = ⟨

v|e−(T−t)HB |vF⟩, (7.94)

which is the expected result.Similarly to the arbitrariness in choosing complete basis states, as discussed in

Section 3.2, the choice of the coordinates for the integration variables in the pathintegral is arbitrary. Just as one can make a change in the basis states of Hilbertspace by a unitary transformation, one can make a change of the integration vari-ables; the analog of the requirement of unitarity in changing the basis states isthat the change of the path integration variables needs to be invertible, and this in

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turn yields a positive Jacobian of the transformation. More precisely, let the newintegration variables be defined by

y = y(x) ⇒ dy(t) =∫

dt ′C(t, t ′)dx(t ′) ⇒ DY = J [x]DY,where J = det[C] is the Jacobian of the transformation.

The forward and backward Fokker–Planck Lagrangians differ by the term∂a(r)/∂r . The difference lies in the manner in which the path integral

∫DR is

carried out. For the forward Lagrangian, if one repeats the calculation expressedin Eq. 7.93, the integration yields an extra Jacobian term that precisely gives theextra term in Eq. 7.92, namely ∂a(r)/∂r , that is present in the forward Lagrangian[Zinn-Justin (1993)].

7.13 Summary

A classical random process driven by white noise was addressed and shown to bedescribed by the formalism of path integrals and Hamiltonians.

Gaussian white noise, that forms the backbone of the study of stochastic pro-cesses, is based on Gaussian integration, and this was briefly reviewed. Ito calculusand the Wilson expansion were seen to follow from the properties of Gaussianstochastic processes, with the Wilson expansion discussed in this chapter being theproperty of Gaussian quantum fields.

The linear and nonlinear Langevin equation was analyzed as a stochastic dif-ferential equation and a number of exact and approximate results were obtained.In particular, the mean field approximation led to many interesting results forthe nonlinear Langevin equation, including the formation of structures far fromequilibrium.

Stochastic quantization was defined based on Gaussian white noise and the pathintegral for the Langevin equation was derived. The rest of the chapter was focusedon the Fokker–Planck equation, and the Fokker–Planck Hamiltonian was shown tobe a pseudo-Hermitian Hamiltonian.

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Part three

Discrete degrees of freedom

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8

Ising model

The simplest possible quantum degree of freedom has only two discrete possiblevalues and is called an Ising spin. This could be a spin system having only twopossible values, namely “up” or “down,” or this could be a quantum particle thatcan occupy only two possible positions.

To simplify the problem to its bare essentials, a two-state system is studied inzero-dimension space and in Euclidean time; the two-state system occupies a singlepoint and, as it evolves in time, it can point either up or down. Furthermore, timeevolution is simplified by discretizing time into a finite time lattice.

The two-state degree of freedom in zero-dimension space propagating on a timelattice is called the Ising model. The one dimensional Ising model is a toy modelthat is studied to develop the essential ideas of quantum mechanics. The conceptsof state space, Hamiltonian, evolution kernel, and path integration have a discreteand simple representation in the Ising model.

The Ising model can be viewed as a statistical mechanical system in thermalequilibrium, with the Ising spins occupying a one-dimensional space lattice, andhence is called the one-dimensional Ising model. The Ising model can be defined onhigher dimensional lattices and forms one of the bedrocks of theoretical statisticalmechanics.

In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path inte-gral and correlator for the Ising model are discussed. In Section 8.6, the techniqueof spin decimation is illustrated to compute an Ising model with coupling depend-ing on the lattice site. The last Section 8.7 discusses how the one-dimensional canbe generalized by consider the Ising model on a 2×N lattice.

8.1 Ising degree of freedom and state space

The Ising degree of freedom is studied in some detail to motivate the results formore complicated degrees of freedom.

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162 Ising model

The electron has an intrinsic spin 1/2 angular momentum; the electron’s spinin three dimensions can point in any direction, which is specified by two angularvariables; the (z-component of the) spin of the electron can either point up or down,and hence forms a double-valued system. If one observes only the z-component ofthe spin, then the spin 1/2 degree of freedom can be represented by the Ising spindegree of freedom. Another example of a two-state system is a particle that canhave only two possible positions. In quantum information theory, the Ising degreeof freedom is called a qubit since it is the quantum generalization of the classical“bit” of information science.

The quantum state of the Ising degree of freedom is mathematically describedby specifying the state vector |ψ〉, and belongs to the state space V . The generalstate vector of the Ising spin can be described by basis states |μ〉 with μ = ±1 andare sometimes called “up” =+1 or “down” =−1 states. The two distinct statescorresponding to the two possible values of the Ising spin (degree of freedom)should be “orthogonal” to each other, as in Eq. 2.3, so that that being in one basisstate is completely different from being in the other basis state,

|ψup〉 = |u〉 = | + 1〉, |ψdown〉 = |d〉 = | − 1〉, 〈u|d〉 = 0.

A general state vector of the Ising spin can be represented by a collection of com-plex valued two-dimensional vectors. One should note that the state space V hasno relation with physical space, but rather is a mathematical construction for de-scribing the state vector of the Ising spin.

One possible representation of the Ising basis states is to assign |u〉 and |d〉,the basis vectors of a two-dimensional vector space, as shown in Figure 8.1. Thedistinct basis state vectors and their dual basis vectors have the representation

|u〉 =[

10

], |d〉 =

[01

]〈u| = [

1 0], 〈d| = [

0 1]. (8.1)

The inner product of two vectors is defined as

〈d|u〉 = [0 1

] · [10

]= 0. (8.2)

Spin up Spin down

Figure 8.1 The Ising spin is represented by up and down vectors.

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8.1 Ising degree of freedom and state space 163

The completeness equation expresses the fact that a collection of basis vectorsforms a complete basis of a linear vector space and is given in Eq. 3.2 for a degreeof freedom having infinitely many discrete values. For the Ising spin, the com-pleteness equation for the two dimensional state space is realized, as follows, bythe outer product of two vectors

|u〉〈u| + |d〉〈d| =[

10

]⊗ [1 0

]+ [01

]⊗ [0 1

]=[

1 00 0

]+[

0 00 1

]=[

1 00 1

]= I. (8.3)

8.1.1 Ising spin’s state space VA spin pointing purely up |u〉 or down |d〉 is the closest that, in quantum mechanics,one can get to a classical object since, on being observed, it is certain to be ineither the up or down. The general state vector of the Ising spin is more subtle:the two basis states can be superposed, as discussed after Eq. 2.13, yielding anindeterminate state that simultaneously points up and down, but with only a certainlikelihood.1 Superposing a quantum spin pointing up with one pointing down yieldsa state vector given by

|ψ〉 = α|u〉 + β|d〉, 〈ψ | = α∗〈u| + β∗〈d|〈ψ |ψ〉 = |ψ |2 = |α|2 + |β|2, (8.4)

where α and β are complex numbers.For a probabilistic interpretation of |ψ〉, the total probability has to be unity, and

one obtains the normalization of the state vector given by

〈ψ |ψ〉 = |ψ |2 = |α|2 + |β|2 = 1. (8.5)

The coefficients now have the physical interpretation

|α|2 = probability that the spin is pointing up,

|β|2 = probability that the spin is pointing down.

The coefficients α, β parameterize the state space V2 of the two-state system.Note that state vector |ψ〉 given in Eq. 8.4 cannot be in general represented by atwo-dimensional unit vector in two-dimensional Euclidean space �2, since onlyvectors with real coefficients can be drawn in �2. In contrast, α, β are complexnumbers and, from Eq. 8.5, it can be seen that their possible values constitute athree-dimensional sphere S3.

1 The concept of quantum superposition is discussed in-depth in Section 4.3.

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164 Ising model

θ

ϕ

x

y

zu

d

ψ

Figure 8.2 Bloch sphere for the two-dimensional state space.

There is, however, a redundancy in this description since the coefficients α, βcan both be re-scaled by the same constant phase and yield

|ψ〉 = eiφ|ψ〉 ⇒ 〈ψ |ψ〉 = 〈ψ |ψ〉. (8.6)

Hence, both state vectors |ψ〉 and |ψ〉 provide the same description; in other wordsstates linked by a global pure phase, namely |ψ〉 → eiφ|ψ〉, are equivalent.2

The phase φ forms a space that is isomorphic to a circle S1. Hence, one needsto “divide out” S3 by S1 to form equivalence classes of state vectors, yieldingS3/S1 = S2, which is a two-dimensional surface.3

In summary, all unique state vectors of the Ising spin are parameterized by theangles of the Bloch sphere S2; each point on the Bloch sphere corresponds to astate vector of the two-state system state space V .

8.1.2 Bloch sphere

The state space V of the Ising spin is isomorphic to the Bloch sphere S2, shownin Figure 8.2. The Bloch sphere allows for a specific representation of the mostgeneral state vector for the two-state system.

Consider the spherical polar coordinates θ, φ shown in Figure 8.2. An arbitrarynormalized two-state ket vector |ψ〉, shown in Figure 8.2, is given by the followingmapping of the Bloch sphere into two-dimensional state space:

2 φ being a pure phase means that it is real.3 To prove this result, one needs to construct S3 by a Hopf fibration, using the mathematics of fiber bundles,

by fibrating the base manifold S2 with fibers given by S1.

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8.2 Transfer matrix 165

|ψ(θ, φ)〉 = cos

(θ

2

)[10

]+ eiφ sin

(θ

2

)[01

], (8.7)

0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.

The basis states have been chosen to reflect the fact that θ = 0 points in the “up”direction and θ = π points in the “down” direction, as shown in Figure 8.2; moreprecisely, Eq. 8.7 yields

|ψ(0, φ)〉 =[

10

]= |u〉,

|ψ(π, φ)〉 = eiφ[

01

]≡[

01

]= |d〉. (8.8)

Note that the ambiguity in the definition of the state vector |ψ(θ, φ)〉 for θ = π

is automatically removed for quantum states since states differing by a pure phaseare equivalent, as shown in Eq. 8.6.

The presence of the phase eiφ in Eq. 8.7 shows that vector |ψ(θ, φ)〉 does not cor-respond to a two-dimensional unit vector in Euclidean space, which has only real-valued components. In fact, every state vector |ψ(θ, φ)〉 corresponds to a (unique)unit three-dimensional Euclidean vector, as shown in Figure 8.2, and is uniquelyparameterized by the coordinates θ ∈ [0, π] and φ ∈ [0, 2π ]. In summary, everypoint on the surface of the Bloch sphere S2 corresponds to a unique state vectorin V2.4

8.2 Transfer matrix

The state function |ψt〉 is evolved in time by the Hamiltonian operator H and isgiven by

|ψt〉 = e−itH/�|ψ0〉.It is generally more efficient to analytically continue Minkowski time t to imagi-nary Euclidean time τ by the mapping t = −i�τ , which yields

|ψτ 〉 = e−τH |ψ0〉.To reduce the system to its bare essential ingredients consider discrete Euclidean

time τ = nε, with n = 0, 1, 2, . . .M , shown in Figure 8.3. There is an Ising spinμn for lattice site n; the spin variable takes two values ±1, indicated by an arrowpointing up or down. There are in total 2M+1 possible spin configurations for anM + 1 lattice, and Figure 8.4 shows some of the possible Ising configurations forfive lattice sites.4 The ambiguity for the special value of θ = π is removed by Eq. 8.8.

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166 Ising model

Figure 8.3 Ising lattice.

Figure 8.4 A few Ising spin configurations for five sites.

The system evolves in time through jumps over a discrete time interval ε. Thetime evolution of the state function |ψτ 〉 → |ψnε〉 ≡ |ψn〉 is hence given by

|ψn+1〉 = e−εH |ψn〉 = L|ψn〉⇒ L ≡ e−εH : transfer matrix.

In statistical mechanics L is called the transfer matrix.The |μ〉 basis yields

〈μ|ψn+1〉 = 〈μ|L|ψn〉=∑μ′=±

〈μ|L|μ′〉〈μ′|ψn〉.

To define the dynamics of the system 〈μ|L|μ′〉 needs to be specified. Sinceμ2= 1 one has the most general expansion for a Hermitian L given by

L(μ,μ′

) = 〈μ|L|μ′〉= a + b(μ+ μ′)+ cμμ′.

A more convenient representation is given by (discarding an overall constant)

L(μ,μ′

) = eKμμ′+ h2 (μ+μ′).

The constant K quantifies the strength of interaction of nearest neighboring spins,and the constant h is the strength of the external magnetic field that the spins aresubjected to. Note all the quantities in the Ising model are dimensionless, and thesize of the lattice spacing (or time step ε) is encoded in the coupling constant K;this aspect of the lattice model is discussed later in Section 9.6.

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8.3 Correlators 167

The Heisenberg operators XH(t) satisfy the evolution equation

∂XH (t)

∂t= i

�

[H, XH (t)

]. (8.9)

From Eq. 8.9, the Heisenberg operators have the evolution

XH (t) = eitH/�Xe−itH/�, (8.10)

where X is the Schrödinger operator. Euclidean time τ = it/� yields

XH (τ ) = eτH Xe−τH . (8.11)

For discrete time, τ → kε and L = e−εH . The Heisenberg spin operator μk obeysthe lattice version of Eq. 8.11 given by

μk = L−kμLk. (8.12)

The operator μ is the co-ordinate Schrödinger operator for the Ising discrete degreeof freedom; its explicit representation, using the completeness equation given inEq. 8.3, is given by

μ = μI =∑μ

μ|μ〉〈μ|. (8.13)

Since μ|μ〉 = μ|μ〉, one has

μ =∑μ

μ|μ〉〈μ| = |+〉〈+| − |−〉〈−|

=(

1 00 0

)−(

0 00 1

)=(

1 00 −1

). (8.14)

Note that as expected μ2 = I.

8.3 Correlators

A typical quantity of interest is the correlation function, called correlator forbrevity. If a disturbance is created at time k, one would like to know for how longdoes the disturbance propagate. In general, for a d-dimensional system, one wouldlike to know what is the system’s correlation length, which quantifies the extent towhich any disturbance propagates in the system.

For the open chain the normalized correlator is defined by

Cr = 1

Z〈μ|LN/2μk+r μkL

N/2|μ′〉, (8.15)

where Z is a normalization constant and μk is the Heisenberg operator at time k.The correlator is given in Figure 8.5.

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168 Ising model

Figure 8.5 Ising correlation function with open boundary conditions.

μN

μ2

μ1

μK

Figure 8.6 Periodic lattice with μN+1 = μ1.

For r = 0, since μ2k = 1 one expects C0 = 1 to be the maximum value of the

self-correlation. Hence

C0 = 1 ⇒ Z = 〈μ|LN |μ′ 〉.From Eq. 8.12, the Heisenberg spin operators yield

Cr = 1

Z〈μ|LN/2−(k+r)μLrμLN/2+k|μ′〉, (8.16)

: open chain with μ,μ′ end-points.

8.3.1 Periodic lattice

To simplify the computation consider the case of periodic boundary conditionssuch that μ = μ′ and a sum is carried out over the states μ with the periodicboundary condition that μ1 = μN+1. The periodic lattice is shown in Figure 8.6.The correlator simplifies since the trace is cyclic, that is tr(ABC) = tr(CAB).

The correlator is defined by

Cr = 1

Z

∑μ=±1

〈μ|LN/2μk+r μkLN/2|μ〉

= 1

Ztr(μk+r μkL

N). (8.17)

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8.4 Correlator for periodic boundary conditions 169

Using the cyclicity of the trace, the correlator is given by

Cr = 1

Ztr(L−(k+r)μLrμLN+k)

= 1

Ztr(LN−r μLrμ

). (8.18)

The normalization is given by C0 = 1. Hence

Z = tr(LN) : partition function, (8.19)

and the normalized correlator for the periodic lattice is

Cr = 1

Ztr(LNμk+r μk

). (8.20)

8.4 Correlator for periodic boundary conditions

The correlator is explicitly evaluated for the case of h = 0. One has that

L(μ,μ′

) = eKμμ′, (8.21)

with

L =(L11 L1−1

L−11 L−1−1

)

=(eK e−K

e−K eK

). (8.22)

The eigenfunction equation is given by

L|λ〉 = λ|λ〉, (8.23)

det|L− λ| = 0, (8.24)

and yields

λ1 = eK + e−K, λ2 = eK − e−K. (8.25)

Hence the partition function, from Eq. 8.19, is given by

Z = tr(LN) = λN1 + λN2 . (8.26)

The eigenvectors are

|λ1〉 = 1√2

(11

)≡ |1〉, (8.27)

|λ2〉 = 1√2

(1−1

)≡ |2〉, (8.28)

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170 Ising model

I =2∑

i=1

|i〉〈i| : completeness equation. (8.29)

Hence the transfer matrix is given by

L =2∑

i=1

λi |i〉〈i|. (8.30)

The correlator is given by

Cr = 1

Ztr(LN−r μLrμ

)= 1

Z

2∑i,j=1

λN−ri λrj tr(|i〉〈i|μ|j〉〈j |μ)

= 1

Z

2∑i,j=1

λN−ri λrj 〈i|μ|j〉〈j |μ|i〉, (8.31)

since tr(M|a〉〈b|) = 〈b|M|a〉.The coordinate operator is Hermitian; hence

〈j |μ|i〉 = 〈jμ|i〉∗ = 〈i|μ†|j〉= 〈i|μ|j〉. (8.32)

Since

〈1|μ|1〉 = 0 = 〈2|μ|2〉, (8.33)

〈1|μ|2〉 = 1 = 〈2|μ|1〉, (8.34)

the correlator is given by

Cr = 1

Z

∑ij

(〈i|μ|j〉)2λN−ri λrj . (8.35)

Hence

Cr = 1

Z

{λN−r1 λr2 + λN−r2 λr1

}= λN1

λN1 + λN2

[tanhr K + tanhN−r K

]= tanhr K + tanhN−r K

1+ tanhN K. (8.36)

Note that C0 = 1 as expected. For N →∞, tanhNK → 0; hence

Cr = tanhr K = e−r/ξ , (8.37)

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8.5 Ising model’s path integral 171

where the correlation length(time) ξ is given by

ξ = − 1

ln(tanhK), (8.38)

∼{ 1

2e2K K � 1

1− ln(K)

K ∼ 0. (8.39)

From the above it can be seen that for K � 1 the system becomes strongly corre-lated, whereas for K → 0 the system has short-time correlation.

8.4.1 Correlator as vacuum expectation values

Since λ1 > λ2, the state |1〉 has the lower energy. Hence, for N →∞, and labeling|1〉 = |�〉 = the vacuum state, yields

LN = λN1

(|�〉〈�| +O

((λ2

λ1

)N)). (8.40)

Hence, for N →∞, the periodic correlator given in Eq. 8.17 reduces to

Cr = 1

ZλN1 tr(|�〉〈�〈|μr μ0)

= 〈�|μr μ0|�〉. (8.41)

Recall from Eq. 6.40 that time ordering of operators is given by

T (Ot Ot′ ) =

{Ot

′ Ot t′> t

OtOt′ t

′< t

.

Hence, it follows, in general, that

Cr = 〈�|T (μk+r μk)|�〉. (8.42)

8.5 Ising model’s path integral

The derivation given in Section 8.4 is for the partition function and correlator ofthe periodic chain L. Another derivation is now given (for a periodic lattice ofsize N ) of the partition function and the correlator by summing over all the possi-ble 2N configurations. This derivation is an example of evaluating a discrete pathintegral.

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172 Ising model

8.5.1 Ising partition function

The partition function is given by Z = tr(LN); one can evaluate the trace for Z byinserting, N times, the complete set of states given in Eq. 8.1, yielding∑

μ=±1

|μ〉〈μ| = I, |μ〉 , μ = −1,+1.

Introducing an index μi on μ to distinguish the different completeness equationsthat are inserted into the trace to evaluate Z, one can write LN as an N -fold matrixproduct and the trace is achieved by having periodic boundary conditions on thespin variables, namely μi = μi+N ; in particular μ1 = μN+1. Hence the partitionfunction is given by

Z = tr(LN)

=∑μN

〈μN |LLL · · ·LLL · · ·LLL︸︷︷︸N−t imes

|μN 〉

=[ N∏i=1

∑μi=±1

][ N∏i=1

〈μi+1|L|μi〉]=[ N∏i=1

∑μi=±1

] N∏i=1

eKμi+1μi

=∑{μ}

eS : periodic boundary conditions μ1 = μN.

The action S is given by

S = K

N∑i=1

μi+1μi. (8.43)

Note μ1, μ2, . . . μN are discrete random variables taking values of ±1. There are2N possible spin configurations, and eS[μ1,...μN ]/Z is the (normalized) probabilitydistribution for the occurrence of a particular spin configuration (also called a fluc-tuation).

The action can be re-written using the special property of the spin variables that(μnμn+1)

2 = 1; this fact yields

eS = eK∑N

n=1 μn+1μn =N∏n=1

[eKμnμn+1

]

=N∏n=1

[coshK + sinhKμnμn+1]

= coshN K

N∏n=1

[1+ tanhKμnμn+1] . (8.44)

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8.5 Ising model’s path integral 173

Representing μnμn+1 by a bond connecting the lattice site n with n+ 1, one cansee that eS has an expansion that consists of the sum of all possible combinationsof bonds, with the term 1 arising from no bond to the term

∏Nn=1 μn that has one

contribution from every bond.Consider the simple product

(1+ a)(1+ b) = 1+ a + b + ab

(1+ a)(1+ b)(1+ c) = 1+ a + b + ab + c + ac + bc + abc

...... = ........ (8.45)

Since∑

μ μ = 0, we conclude that, at each lattice site n, only even or no spinvariables can contribute to the sum forZ. There are only two terms in the expansionfor S that satisfy this condition, namely the term with no bonds and the term withall the bonds. Every sum contributes a factor of 2, since

∑ui=±1 = 2, and hence∑

{μ} = 2N . The partition function is

Z = tr(LN) = 2N coshN K[1+ tanhN K]

= λN1 + λN2

and we obtain the expected result given in Eq. 8.26.

8.5.2 Path integral calculation of Cr

The correlator, given by Eq. 8.17, can be written as

Cr = 1

Ztr(μk+r μkL

N)

= 1

Ztr(LN−r μLrμ

)= 1

Z

∑μk=±1

〈μk|LN−r μLrμ|μk〉.

Inserting a complete set of states yields

Cr = 1

Z

∑μk

∑μk+r〈μk|LN−r μ|μk+r〉〈μk+r |Lr |μk〉μk

= 1

Z

∑μk

∑μk+r

μk+rμk〈μk|LN−r |μk+r〉〈μk+r |Lr |μk〉

= 1

Z

∑{μ}

eK∑N

n=1 μnμn+1μk+rμk,

where∑{μ} is a sum over all possible configurations. Hence

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174 Ising model

(tanh K)r

(tanh K)N-r

μk

μk

μk+rμk+r

+

Figure 8.7 The two bond configurations that contribute to the correlation function∑{μ} eSμk+rμk/Z. Note that the correlation function is independent of k.

Cr = 1

Z

∑{μ}

eSμrμ0 : path integral,

where the action is given in Eq. 8.43. Since the lattice is periodic, the correlator Cr

does not depend on the index k and hence

Cr = 1

Z

∑{μ}

eSμrμ0.

Using the expression Eq. 8.44 for the Ising action yields

Cr = coshN K

Z

∑{μ}

∏n

(1+ tanhKμnμn+1)μrμ0. (8.46)

The μr and μ0 terms must be canceled by bonds. Only two terms survive from theproduct, namely a product of bonds clockwise going from μr and μ0 and anotherterm going counter-clockwise, as shown in Figure 8.7. Hence performing the sumyields, as expected

Cr = 2N coshN K

Z


]= 1

1+ tanhN K


]. (8.47)

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8.6 Spin decimation 175

Figure 8.8 The spins are decimated (summed over) from the boundary.

8.6 Spin decimation

Consider an open chain one dimensional Ising model with coupling constants Kn

and the action given by

S =N−1∑n=1

Knμnμn+1, (8.48)

Z =∑{μ}

eS. (8.49)

The partition function Z can be evaluated by summing over the spin μN at theboundary, called spin decimation. Then decimate spin μN−1, μN−2,. . . all the wayto spin μ1. In symbols one has

S =N−1∑n=1

Knμnμn+1, (8.50)

Z =N∏n=1

∑μn=±1

eKnμnμn+1

=∑μN

eKN−1μN−1μN∑μN−1

eKN−2μN−2μN−1 . . .

= (eKN−1μN−1 + e−KN−1μN−1

) ∑μN−1

eKN−2μN−2μN−1 . . .

= 2 cosh(KN−1)∑μN−1

eKN−2μN−2μN−1 . . .

Recursively decimating the spins from the boundary inwards, as shown inFigure 8.8, yields

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176 Ising model

Z =[N−1∏n=1

2 cosh(Kn)

]∑μ1

= 2

[N−1∏n=1

2 cosh(Kn)

].

To evaluate 〈μnμn+r〉 note that the correlation function can be re-written bydifferentiating the action with respect to the coupling constants Km; one starts withthe coupling constants Kn corresponding to the term in the action Knμnμn+1 andkeeps differentiating on the next lattice site until one reaches site n + r − 1. Insymbols one has

E[μnμn+r] = 1

Z

∑μnμn+reS

= 1

Z

∂

∂Kn

∂

∂Kn+1· · · ∂

∂Kn+r−1

∑eS, (8.51)

=n+r−1∏�=n

sinh(K�)

cosh(K�)=

n+r−1∏�=n

tanh(K�). (8.52)

As expected, the limit K� = K = constant yields

E[μnμn+r ] → tanhr K for K� = K = const.

8.7 Ising model on 2 × N lattice

Consider the Ising model on a two site by N steps in the time direction. At eachstep there are two spin variables μn and μ′n. Suppose the time lattice is a periodiclattice, that is μN+1 = μ1 and μ′N+1 = μ′1. The action is given by

S = K

N∑n=1

μnμ′n +K

N∑n=1

μnμn+1 +K

N∑n=1

μ′nμ′n+1 =

N∑n=1

L(n).

The action is defined on the Ising “ladder”, as shown in Figure 8.9, and yieldsthe Lagrangian

L(n) = K

2

(μnμ

′n + μn+1μ

′n+1

)+Kμnμn+1 +Kμ′nμ′n+1. (8.53)

To simplify the notation, let us make the simplification

μn+1 = μ, μ′n+1 = μ′

μn = λ, μ′n = λ′.

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8.7 Ising model on 2×N lattice 177

Figure 8.9 Ising “ladder” for 2×N lattice.

This yields

L(μ,μ′; λλ′) = K

2

(λ′λ+ μμ′

)+K(λμ+ λ′μ′

). (8.54)

The partition function is given by

Z =∑{μ;μ′}

eS[μ;μ′] =

∑{μ,λ}

eS[μ,λ].

One has to choose a set of basis vectors to write out the matrix elements of thetransfer matrix L. Since there are two spins at each time n, the natural basis statesfor each n are given by the tensor product of the basis states of the two lattice sites,namely |μn;μ′n〉 ≡ |μn〉 ⊗ |μ′n〉. In the simplified notation introduced above, thematrix elements of L are given by 〈μ;μ′|L|λ; λ′〉.

The explicit construction of the basis vectors |λ〉⊗ |λ′〉, using the rules of tensorproduct of state vectors [Baaquie (2013e)], yields

|1〉 = |+〉 ⊗ |+〉 =(

10

)⊗(

10

)=

⎛⎜⎜⎝

1000

⎞⎟⎟⎠,

|2〉 = |+〉 ⊗ |−〉 =(

10

)⊗(

01

)=

⎛⎜⎜⎝

0100

⎞⎟⎟⎠,

|3〉 = |−〉 ⊗ |+〉 =(

01

)⊗(

10

)=

⎛⎜⎜⎝

0010

⎞⎟⎟⎠,

|4〉 = |−〉 ⊗ |−〉 =(

01

)⊗(

01

)=

⎛⎜⎜⎝

0001

⎞⎟⎟⎠.

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178 Ising model

The matrix elements of L are defined with respect to the four basis vectors, andthe matrix can be written as Lij ≡ 〈i|L|j〉, where |i〉, i = 1, 2, 3, 4 as given aboveand 〈j | is the transpose of the |i〉 vectors. Hence

L =4∑

i,j=1

Lij |i〉〈j | =

⎛⎜⎜⎝L11 L12 L13 L14

.. L22 L23 L24

.. .. L33 L34

.. .. .. L44

⎞⎟⎟⎠, Lij = Lji, (8.55)

with the other elements being fixed by the fact that L is a symmetric matrix, forexample L11 = 〈1|L|1〉 and so forth. Thus the transfer matrix is given by

Lij = eL(μ,μ′;λλ′),

where the matrix elements are given by

L11 = eL(1,1;1,1) = e3K; L12 = L13 = eL(1,1;−1,1) = 1,

L14 = eL(1,1;−1,−1) = e−K,L22 = eL(1,−1;1,−1) = eK,

L23 = eL(1,−1;−1,1) = e−3K,L24 = eL(1,−1;−1,−1) = 1,

L33 = eL(−1,1;−1,1) = eK, L34 = eL(−1,1;−1,−1) = 1,

L44 = eL(−1,−1;−1,−1) = e3K.

Collecting all the results yields the symmetric transfer matrix L given by

L =

⎛⎜⎜⎝e3K 1 1 e−K

1 eK e−3K 11 e−3K eK 1

e−K 1 1 e3K

⎞⎟⎟⎠,

with eigenvalues λ1 > λ2 > λ3 > λ4 (decreasing in magnitude) given by

λ1 = e−3k(−1+ e4k), λ2 = e−k(−1+ e4k),

λ3 = 1

2e−4k

(ek + e3k + e5k + e7k − ek(1+ 22k)

√1− 4e2k + 10e4k − 4e6k + e8k

),

λ4 = 1

2e−4k

(ek + e3k + e5k + e7k + ek(1+ 22k)

√1− 4e2k + 10e4k − 4e6k + e8k

).

The partition function is given by

Z ==∑{μ,λ}

eS[μ,λ] = tr(LN) =4∑

i=1

λNi .

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8.8 Summary 179

The Ising model can be extended to a square lattice by adding lattice sites toextend the Ising “ladder” and create an N×N two dimensional lattice. The transfermatrix is 2N × 2N and has 2N eigenvalues λI with the partition function given by

Z =2N∑I=1

[λI ]N.

The limit of N → ∞ of the two dimensional Ising model was exactly solvedby Onsager in a landmark derivation and exhibits a second order phase transition[Papon et al. (2002)].

Similarly, one can go to higher dimensions by defining Ising spins on an Nd

lattice given by N × N × N . . . × N . The transfer matrix rapidly becomes moreand more complicated.

8.8 Summary

The Ising model is based on the simplest possible degree of freedom, namely onehaving only two possible values. The one-dimensional Ising model is a toy modelthat has all the mathematical structures of quantum mechanics, from the state spaceto operators and onto path integrals and correlation functions.

The periodic lattice was studied for explicitly deriving the partition function andthe correlation function, and these derivations illustrate the general features of thecomputations that are carried out for more complex systems.

One significant feature of the Ising model is the expansion of the action in apower series. This is possible since the Ising spin is a compact variable, taking val-ues in a finite range. This property does not hold for Gaussian degrees of freedomand hence is a special feature of the Ising model.

The generalization of the one-dimensional Ising model to the 2×N Ising laddercan be extended to defining the Ising model on an N × N lattice and shows theprocedure that is required for defining the model in higher dimensions.

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9

Ising model: magnetic field

The Ising model in a magnetic field is a continuation of the discussions ofChapter 8. The equations have to be generalized to include the presence of amagnetic field. A number of new features are studied, in particular, the effect ofthe Ising model’s boundary conditions on the partition and correlation functions.

The Ising model in a magnetic field is introduced in Section 9.1 and the evolutionfor this system is obtained. The magnetization of the Ising models is an importantmanifestation of the magnetic field and is discussed for a periodic lattice in Section9.3. The concept of correlation function is discussed in Section 9.4 using the con-cept of linear regression. Magnetization for an open chain is discussed in Section9.5, and block spin renormalization is discussed in Section 9.6.

9.1 Periodic Ising model in a magnetic field

The earlier calculation of the Ising model for a periodic lattice is generalized tothe case with a nonzero magnetic field. The calculation is more complex than theearlier case showing new features of the transfer matrix.

The partition function for an N size periodic lattice is defined by

ZN =∑{μ}

eS,∑{μ}≡

N∏n=1

∑μn=±1

, μN+1 = μ1,

where

S ≡N∑j=1

(Kμjμj+1 + hμj

). (9.1)

Writing the partition function as

ZN = tr(LN)

(9.2)

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9.1 Periodic Ising model in a magnetic field 181

yields the Hermitian (symmetric) transfer matrix

L(μ,μ′) = exp

{Kμμ′ + h

1

2

(μ+ μ′

)}. (9.3)

The eigenvalues and eigenfunctions of the transfer matrix are

L =(eK+h e−K

e−K eK−h

), L|φi〉 = λi |φi〉, i = 1, 2, (9.4)

λ1 = eK coshh+ (e2K sinh2(h)+ e−2K)1/2

,

λ2 = eK coshh− (e2K sinh2(h)+ e−2K)1/2

, (9.5)

|φ1〉 = 1√1+ a2

(1a

), (9.6)

|φ2〉 = 1√1+ a2

(a

−1

), (9.7)

a = eK(λ1 − eK+h

).

The completeness equation is given by∑i=1,2

|φi〉〈φi | = 1.

From Eqs. 9.6 and 9.7

〈φ1|μ|φ1〉 = −〈φ2|μ|φ2〉 = 1− a2

1+ a2, 〈φ1|μ|φ2〉 = 〈φ2|μ|φ1〉 = 2a

1+ a2. (9.8)

The eigenvalues yield

L = λ1|φ1〉〈φ1| + λ2|φ2〉〈φ2| ⇒ LN = λN1 |φ1〉〈φ1| + λN2 |φ2〉〈φ2|. (9.9)

Hence, the partition function is given by

ZN = tr(LN) = λN1 + λN2 . (9.10)

The periodic Ising model with negative coupling, namely −K , is given by

S ′ ≡N∑j=1

(−Kμjμj+1 + hμj

).

By a change of variables for only the odd spins, that is, μ2n+1 →−μ2n+1, one canchange the sign of K for all the bonds. This yields

S ′ ≡N∑j=1

(Kμjμj+1 + h(−1)nμj

).

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182 Ising model: magnetic field

Hence, one can see that −K corresponds to anti-ferromagnetic coupling, with thenearest neighbor spins tending to be oppositely aligned. The alternating sign ofthe magnetic field drives this anti-alignment. All the formulas derived for +K canbe applied to the case for −K by a simple change of sign.

9.2 Ising model’s evolution kernel

From Eq. 9.4, the transfer matrix of the Ising model in a magnetic field is given by

L =(eK+h e−K

e−K eK−h

).

The eigenvalues and eigenvectors of L, given in Eqs. 9.5, 9.6, and 9.7, yield

LN = 1

1+ a2

(1 a

a −1

)(λN1 00 λN2

)(1 a

a −1

)

= 1

1+ a2

(λN1 + a2λN2 a

(λN1 − λN2

)a(λN1 − λN2

)a2λN1 + λN2

). (9.11)

The evolution kernel for the Ising model is given by

K(μ′, μ;N) = 〈μ′|LN |μ〉 = exp

{β + αμμ′ + γ

2

(μ+ μ′

)}.

In matrix notation

K = eβ(eα+γ e−α

e−α eα−γ

). (9.12)

From Eqs. 9.11 and 9.12, after some algebra

e2γ = λN1 + a2λN2

a2λN1 + λN2, e4α =

(λN1 + a2λN2

) (a2λN1 + λN2

)a2(λN1 − λN2

)2 ,

eβ =√a

1+ a2

(λN1 − λN2

)1/2 (λN1 + a2λN2

)1/4 (a2λN1 + λN2

)1/4.

In the limit of h→ 0

a→ 1, λ1 → κ1, λ2 → κ2,

with

κN1 = 2N coshN(K), κN2 = 2N sinhN(K).

The evolution kernel for zero magnetic field h → 0 is hence given by theparameters

α→ 1

2ln

(κN1 + κN2

κN1 − κN2

); β → 1

2ln(κ2N

1 − κ2N2

)− ln 2; γ → 0.

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9.3 Magnetization 183

9.3 Magnetization

The magnetization is defined by

MN = E[μn] = 1

Z

∑{μ}

μneS.

Due to the periodicity of the lattice, 〈μn〉 = 〈μn+N 〉 and hence MN is independentof n. Hence

MN = 1

N

N∑n=1

E[μn] = 1

N· 1

Z

∂

∂h

∑{μ}

eS

= 1

N

∂

∂hln(ZN)

= ∂

∂hln(λ1)+ 1

N

∂

∂hln

(1+

(λ2

λ1

)N).

Consider the limit of N →∞, called the thermodynamic limit. Since λ1 > λ2, thepartition function is given by

ZN = λN1[1+O

(e−N

)].

Hence, the magnetization is given by

M = limN→∞MN = ∂

∂hln(λ1)+O(e−N)

= 1

λ1

[eK sinhh+ e2K sinhh coshh(

e2K sinh2(h)+ e−2K)1/2

]. (9.13)

The magnetization can also be derived using the transfer matrix. For a periodiclattice

MN = 〈μn〉 = 1

ZN

∑{μ}

μneS

= 1

ZN

tr(LNμ

). (9.14)

From Eqs. 9.8, 9.9, and 9.10

MN = 1

ZN

2∑i=1

〈φi |LNμ|φi〉

= 1

ZN

[λN1 〈φ1|μ|φ1〉 + λN2 〈φ2|μ|φ2〉

]

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Figure 9.1 Ising magnetization for the case of +K and −K .

= 1

ZN

· (1− a2)

(1+ a2)· (λN1 − λN2 )

= 1− a2

1+ a2· λ

N1 − λN2

λN1 + λN2. (9.15)

Taking the limit of N →∞ yields the magnetization

M = limN→∞MN = 1− a2

1+ a2, (9.16)

and it can be shown to be equal to the result given in Eq. 9.13.Figure 9.1 plots the magnetization given in Eq. 9.15 as a function of h with

K = ±1 and for two cases, namely N = 2 and N = 100. It can be seen that themagnetization converges very fast to its large lattice value. For −K , the magneti-zation has a smaller value since, due to the anti-ferromagnetic coupling, the spinstend to anti-align.

9.3.1 Correlator

The periodic chain correlation function is given by

E[μkμk+r ] = 1

ZN

∑{μ}

μkμk+reS,

where the partition function is given by Eq. 9.10. From Eq. 8.17, in terms of thetransfer matrix, the correlator is given by

Cr = 1

ZN

tr(LN−r μLrμ

).

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9.4 Linear regression 185

The expression given above for the nonzero magnetic field looks the same as forthe case of zero magnetic field, the only difference between the two cases being thevalues of the eigenvalues and eigenvectors of the transfer matrix L. The correlationfunction can then be written, similarly to Eq. 8.31, as

Cr = 1

ZN

2∑i,j=1

λN−ri λrj(〈φi |μ|φj 〉)2

.

A straightforward but long derivation, using the result above and Eqs. 9.5 and 9.10,yields

Cr = 1

(1+ a2)2

⎡⎢⎣(1− a2

)2 + 4a2

(λ2λ1

)r + (λ2λ1

)N−r1+

(λ2λ1

)N⎤⎥⎦ . (9.17)

As expected, C0 = 1. Recall from Eq. 9.16

M = 1− a2

1+ a2

and hence

Cr =M2 + 4a2(1+ a2

)2

(λ2λ1

)r + (λ2λ1

)N−r1+

(λ2λ1

)N . (9.18)

Note that since λ1 > λ2, we can define correlation length ξ by(λ2

λ1

)r

≡ exp(−r/ξ), limN→∞

(λ2

λ1

)N−r�(λ2

λ1

)N

→ 0.

Hence, for an infinitely large lattice, N →∞, from Eq. 9.18

Cr =M2 + 4a2(1+ a2

)2 e−r/ξ +O

(e−N

). (9.19)

The limit of h = 0 leads to a → 0 and results in M = 0; the value of Cr

converges to the value of the correlator for zero magnetic field given in Eq. 8.36.

9.4 Linear regression

For two random variables to be correlated means that the variables take their ran-dom values in tandem, that is, the value of one of them takes a predictable range ofvalues if the other takes certain values and vice versa.

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The concept of correlation is very different from causation, in which the causedetermines the effect: heat and fire are correlated as well as causally linked, withheat causing fires.

Consider the height and weight of a person in a given age group; although bothare random variables they are nevertheless related (correlated) since the taller theperson is, the greater is the likelihood that the person is heavier and vice versa.However, although they are correlated, height is not taken to be the cause of weightand vice versa. Another example is intelligence and say height – which have nocorrelation – again without one being the cause of the other.

Consider random variables X and Y . Suppose they are linearly related and let

Y = p + qX. (9.20)

The equation above is written so that the random variable X is considered as theindependent random variable and Y the dependent random variable. For the case ofheight and weight, the height can be taken to be the independent random variablewith the weight being the dependent random variable. One uses the average valueand the correlation of X, Y and variance of X to fix the parameters p, q in thefollowing manner. Let

E[XY ]c = E[XY ] − E[X]E[Y ],σ 2(X) = E[X2] − E[X]2.

Then

q = E[XY ]cσ 2(X)

, p = E[Y ] − qE[X]. (9.21)

The estimate of the variance of Y from above is given by

σ 2Est(Y ) = q2σ 2(X) = 〈XY 〉

2c

σ 2(X). (9.22)

The exact variance of Y is σ 2(Y ). Hence, one measure of the accuracy of the linearregression between X and Y is given by the fractional error

FE =√σ 2(Y )− σ 2

Est(Y )

σ 2(Y )

=√

1− E[XY ]2cσ 2(X)σ 2(Y )

. (9.23)

From the point of view of probability theory, the spins μ0 and μr are two randomvariables; let

X = μr, Y = μ0.

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9.4 Linear regression 187

The average value of the random variable is given by the magnetization, namely

E[μr ] = 1

ZN

∑{μ}

eSμr. (9.24)

For the periodic lattice, the magnetization is given by

E[μr ] = E[μ0] =MN. (9.25)

The expectation value of the product of two spin variables, namely μ0 and μr , isgiven by

E[μrμ0] = 1

ZN

∑{μ}

eSμrμ0. (9.26)

The connected two spin correlation function is given by

E[XY ]c = Gr = E[μrμ0] − E[μr ]E[μ0]. (9.27)

For the case of the Ising spin on a finite periodic lattice, from Eqs. 9.15 and 9.18,the correlator is

Gr = Cr −M2N

= (1− a2)2

(1+ a2)2

[2λN2

λN1 + λN2

]+ 4a2

(1+ a2)2

⎡⎣(λ2

λ1

)r + (λ2λ1

)N−r1+ (λ2

λ1

)N⎤⎦ . (9.28)

Furthermore, the variance of μr is given by

σ 2 (μr) = E[μ2r

]− (E [μr ])2 = 1−M2

N = σ 2(μ0), (9.29)

where the last equation follows due to the periodic lattice.From the point of view of probability theory all the spins in the lattice, namely

μn with n = 1, 2, . . . N are random variables. The behavior of all the spins can befully described by specifying how the different random variables are correlated.

One can start with the simplest relation between the random variables’ as givenby the linear regression. From Eqs. 9.20 and 9.21

μr � p + qμ0,

q = Gr

σ 2(μr)= Gr

1−M2N

,

p = 〈μr〉 − q〈μ0〉 = [1− q]MN.

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From Eq. 9.23 the fractional error for the Ising spins is given by

FE =√

1−[ Gr

1−M2N

]2

={

0 r = 01 r >> 1

. (9.30)

For simplicity, consider an infinitely large lattice N →∞ that yields, from Eq.9.19,

μr � p + e−r/ξμ0, p = [1− e−r/ξ

]M,

FE =√

1− e−2r/ξ .

For r � ξ we have to leading order that the random variables μr and μ0 are de-correlated and take values independently of each other; that is

r � ξ ⇒ μr �M : independent of μ0 .

The fractional error FE is small when r , the distance of the two spins μ0 and μr ,is less than ξ and hence the linear regression is a good fit; when r � ξ the errorFE ∼ 1, which is equal to the magnitude of the spin, and hence the linear regressionis no longer a good fit.

In summary, the correlation length ξ is the crucial quantity in the validity oflinear regression. The correlation length ξ specifies the distance within which thespins are correlated. In fact, in statistical mechanics and quantum field theory, thecorrelation length ξ plays a central role in describing physical phenomena.

To fully specify the theory one needs to have a generalized nonlinear regressionthat expresses all the product spins, namely all possible combination of bonds,in terms of the other remaining spins. For instance, one needs to specify μ0μn

is terms of products of other spins excluding the spins μ0 and μn. Note that onecannot arbitrarily assign values to the regression of various products of spins sincethese must satisfy consistency conditions. At most, one can try and define a theoryperturbatively by specifying the regression coefficients for the product of a fewrandom variables.

The assignment of the probability distribution – in statistical mechanics andquantum theory – is given by the expression eS[μ0,...μN−1]/Z for the (normalized)probability distribution for a particular spin configuration. This completely anduniquely defines the theory, as well as determining the regression and correlationof all the random spin variables in a self-consistent manner.

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9.5 Open chain Ising model in a magnetic field 189

μ1 μN

Figure 9.2 Open lattice with boundary spins given by μ1, μN .

9.5 Open chain Ising model in a magnetic field

Consider the one dimensional Ising model for an open chain of length N with arbi-trary boundary values for μ1, μN as shown in Figure 9.2. The open chain partitionfunction is defined by

ZN =∑{μ}

eS

and the action is given by

S ≡ K

N−1∑j=1

μjμj+1 + h

N∑j=1

μj,

∑{μ}≡

N∏n=1

∑μn=±1

.

The partition function can be written in terms of the transfer matrix as

ZN =∑{μ}

exp

⎡⎣K N−1∑

j=1

μjμj+1 + h

2

N−1∑j=1

(μj + μj+1)

⎤⎦ exp

[h

2(μN + μ1)

].

(9.31)

The sum for the magnetic field runs from 1 to N − 1 since it extends to the bound-aries. The term in the action

exp

{h

2(μN + μ1)

}is left over due to the open chain condition. The transfer matrix is given by

L(μ,μ′) = exp

{Kμμ′ + h

1

2

(μ+ μ′

)}, (9.32)

which is the same as the one given in Eq. 9.3 for the periodic lattice.From Eq. 9.31, the partition function ZN can be written as

ZN =∑μ1

. . .∑μN

⎡⎣N−1∏

j=1

〈μj |L|μj+1〉⎤⎦ exp

[h

2(μN + μ1)

]

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=∑μ1

. . .∑μN

〈μ1|⎡⎣N−2∏

j=2

L|μj 〉〈μj |⎤⎦L|μN 〉e h

2 (μN+μ1)

=∑μ1

∑μN

〈μ1|LN−2 · L|μN 〉e h2 (μN+μ1)

= tr

(∑μ1

∑μN

LN−1|μN 〉〈μ1|e h2 (μN+μ1)

).

In matrix notation, the partition function is given by

ZN = tr(LN−1B), (9.33)

where the boundary condition B is given by

B =∑μ1

∑μN

|μN 〉〈μ1|e h2 (μN+μ1).

In components

〈μN |B|μ1〉 = eh2μN e

h2μ1 ⇒ B =

(eh 11 e−h

). (9.34)


⇒ ZN = λN−11 〈φ1|B|φ1〉 + λN−1

2 〈φ2|B|φ2〉

= λN−11

(eh2 + ae− h

2

)2

1+ a2+ λN−1

2

(e− h

2 − aeh2

)2

1+ a2. (9.35)

The case of h = 0 yields the partition function

ZN = 2(2 coshK)N−1. (9.36)

9.5.1 Open chain magnetization

Consider the expectation value 〈μk〉 of single spin variable μk; note that for anopen chain 〈μk〉 depends on how far the lattice site k is from the boundaries. Inparticular

〈μk〉 = 1

ZN

∑μ1

· · ·∑μN

⎛⎝N−1∏

j=1

〈μj |L|μj+1〉⎞⎠μke

h2 (μ1+μN)

= 1

ZN

tr(Lk−1μLN−kM

), (9.37)

where, from Eq. 8.14

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9.6 Block spin renormalization 191

μ =(

1 00 −1

).

A long calculation yields

〈μk〉 = 1

ZN

1

(1+ a2)2

[λN−1

1

(eh2 + ae−

h2

)2 (1− a2

)− λN−1

2

(ae

h2 − e−

h2

)2 (1− a2

)+ 2a

{λN−k1 λk−1

2 + λk−11 λN−k2

} (aeh − 1+ a2 − ae−h

)].

(9.38)Note that for h→ 0, one recovers the expected result

E[μk] = 0.

9.6 Block spin renormalization

Wilson’s (1983) concept of renormalization plays a central role in quantum fieldtheory. The essential idea is that at each length scale, there is an effective theory thatcompletely describes the physics at that length scale. The one-dimensional Isingmodel provides a toy model to examine the essential features of renormalizationtheory; the concept of length scale in the case of the Ising model is the distancebetween two spins in a correlation function.

The large distance action is related to the short distance action by the proce-dure of renormalization. In Wilson’s (1983) approach, the short distance degreesof freedom are summed over (integrated out) to generate the action appropriate fordescribing the longer distance physics.

Let the lattice spacing for the Ising model be denoted by a, described by actionS0. Consider the odd and even spins of the one-dimensional Ising model. The dis-tance between the even spins is twice the distance between adjacent spins. Henceif all the odd spins are summed over, the new lattice will have a spacing of a1 = 2aand describe a new effective action S1; the transformation relating the action forthe original lattice S0 to the action of the new lattice S0 is the renormalizationtransformation R and S1 = R[S0] [Kardar (2007)].

Successive transformations form a group, and hence the name renormalizationgroup for this procedure for studying the distinct scales of a system. If one repeatsthe renormalization transformation l times, then the lattice spacing of the final lat-tice is al = 2la and the action Sl is given by Sl = Rl[S0].

The correlation on the original lattice is say ξ0 = La; hence, the correlation onthe lattice with spacing al is given by ξl = 2−lξ0, where ξl is the correlation lengthmeasured by the scale of the final lattice, namely al = 2la.

Divide the lattice into odd and even sites and we sum over all the spins residingat the even sites, thus generating a lattice of double the original lattice. The partitionfunction remains invariant. The renormalization transformation is given by

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x

a

2a

4a

x x x

μ λ μ’ λ’ μ’’ λ’’

μ μ‘ μ’’

μ μ’

Figure 9.3 Block-spin renormalization.

Z =∑{μ:all}

eS0 =∑{μ:odd}

eS1, (9.39)

S0 = g0 +K0

∑n

μnμn+1. (9.40)

Let the even spins be labeled by λn as shown in Figure 9.3; the renormalizationtransformation is given by

eS1 =∑{μ:even}

eS0 =∑{λ}

eS0 .

One can think of integrating over the even spins as combining two spins together,as shown in Figure 9.3, to generate a new block-spin that is defined on the newlattice and described by its own effective action. For a lattice of infinite size, thisprocess can be repeated indefinitely.

Each λn spin is coupled only to its nearest neighbor; hence the sum over the λnspins can be done separately over each λn spin. As shown in Figure 9.3, the Isingspin at the even site labeled by λ = ±1 couples to its nearest neighbor spins μ,μ′,the two bonds λμ and μ′λ that contain the spin variable λ; summing over λ yieldsthe fundamental renormalization group transformation

e2g0∑λ=±1

eK0λ(μ+μ′) = eg1+K1μμ′, (9.41)

where it is postulated that the effective bonds coupling the lattice at double thelattice spacing are given by the right hand side of Eq. 9.41. The new couplingconstants g1,K1 are now determined in terms of the original coupling constantsg0,K0. Consider the following cases for Eq. 9.41:

• μ = 1 = μ′ and μ = −1 = μ′ both yield the same equation, namely

2e2g0[e2K0 + e−2K0

] = eg1+K1 . (9.42)

• μ = 1 = −μ′ and μ = −1 = −μ′ both yield the same equation

2e2g0 = eg1−K1 . (9.43)

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Equations. 9.42 and 9.43 yield

eg1 = 2e2g0 cosh1/2 2K0, eK1 = cosh1/2 2K0,

and we obtain the renormalization group transformation

K1 = 1

2ln cosh 2K0, g1 = 2g0 + 1

2ln cosh 2K0 + ln 2. (9.44)

The lattice spacing for S1 is twice the lattice spacing of S0; hence, the correlationlengths ξ as computed from the two actions should scale accordingly, namely

ξ1 = 1

2ξ0 ⇒ ξ(K1) = 1

2ξ(K0). (9.45)

To prove Eq. 9.45, recall that Eq. 8.38 yields the correlation lengths

ξ(K0) = − 1

ln(tanhK0)and ξ(K1) = − 1

ln(tanhK1).

The expression for K1 in Eq. 9.44 yields

2(e2K1 ± 1

) = [eK0 ± e−K0

]2 ⇒ tanh(K1) = [tanh(K0)]2.Hence, the result given in Eq. 9.45 is realized as

ξ(K1) = − 1

ln(tanhK1)= −1

2

1

ln(tanhK0)= 1

2ξ(K0). (9.46)

Note the remarkable result that the new coupling constant K1 on the larger lattice“remembers” that it is the result of summing over the smaller lattice with couplingconstant K0. One of the fundamental insights that the renormalization group pro-vides about the different length scales in a problem is the following: that each scale2� of the system has its own corresponding coupling constant K�.

The recursion equation connecting coupling constants gl,Kl defined for a latticeof size 2la, with coupling constants gl+1,Kl+1 defined for a lattice of size 2l+1a, isgiven by

gl+1 = 2gl + 1

2ln cosh 2Kl + ln 2 = R(gl), Kl+1 = 1

2ln cosh(2Kl) = R(Kl).

One can write the renormalization group recursion equations more generally as

(Kl, gl) = R(Kl−1, gl−1) = R�(K0, g0), (9.47)

where R is a 2× 2 matrix.The coupling constant Kl is the renormalized coupling constant, describing the

physics at length scale 2la and is obtained from the initial (bare) coupling constant

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K0 by the repeated application of the renormalization group transformation R. Thecorrelation length is given by

ξ� = 1

2ξ�−1 = 1

2�ξ0.

The fixed point of the renormalization group transformation is given by

K∗ = R(K∗)⇔ ξ ∗ = 1

2ξ ∗

and hence leads to the result

ξ ∗ = 1

2ξ ∗ ⇒ ξ ∗ = 0 or ξ∗ = ∞.

The case of ξ ∗ = 0 corresponds to a decoupled system. A system is said to becritical – undergoing a phase transition – if its correlation length is infinite, namleyξ ∗ = ∞; hence the fixed point of the renormalization transformation correspondsto the system being critical.

Note the remarkable fact that the fixed point of the renormalization group trans-formation R depends only on R and not on the details of the (Ising) system westart from; this feature of phase transitions leads to the property of universality inthat many different types of phase transitions are described by the same criticalsystem defined by R.

The fixed point is analyzed as follows:

• To find the fixed point of Kl , consider the limit liml→∞Kl → K∗.• Fixed point of K0 = ∞ = K∗ result from the fact that Kl = K∗ = ∞. The ini-

tial Ising system (fixed by K0) is strongly correlated with the correlation lengthbeing ξ ∗ = ∞. Hence, under renormalization, the correlation length remainsinfinite no matter how large the effective lattice spacing.

• For any Kl � 0,

Kl+1 � 1

2ln

(1+ 4K2

l

2

)≈ K2

l < Kl.

Hence liml→∞Kl → K∗ = 0. The system is decoupled at large enough dis-tances for any initial value K0 < 0. In fact, for any K0 < ∞, liml→∞Kl →0 = K∗. For any initial finite correlation length, the large distance correlation isalways zero, namely ξ ∗ = 0.

The flow of the coupling constant Kl is shown in Figure 9.4; if one starts withany finite value for K0 < ∞, then the effective coupling Kl for larger and largerlattice size flows towards zero, with the one-dimensional Ising model becomingdecoupled at large distances.

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0=tanh K* K tanh K*=1

Figure 9.4 Flow of coupling constant under renormalization.

Near K∗ � ∞,

Kl+1 � 1

2ln

(e2Kl

2

)� Kl − 1

2ln 2 < Kl.

Hence, for any Kl <∞Kl+1 < Kl, lim

l→∞Kl → 0.

9.6.1 Block spin renormalization: magnetic field

One can introduce a magnetic field h, as in Eq. 9.1, and repeat the block spincalculation starting with the initial Ising model given by

S0 ≡+∞∑

j=−∞

(Kμjμj+1 + hμj

).

The renormalization group transformation is given by

exp

[K ′μμ′ + h′

2(μ+ μ′)+ g′

]

=∑λ=±1

exp

[Kλ

(μ+ μ′

)+ h

2

(μ+ μ′

)+ hλ+ 2g

]. (9.48)

To solve for the renormalized interaction it is convenient to set⎧⎨⎩x = eK, y = eh, z = eg

x ′ = eK′, y ′ = eh

′, z′ = eg

′.

(9.49)

Similarly to the case for zero magnetic field, the four possible configurations of thebond this time yield only three equations in three unknowns. The solution is givenby Kardar (2007)⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

z′4 = z8{x2y + x−2y−1[x−2y + x2y−1(y + y−1)2]}

y ′2 = y2 x2y+x−2y−1

x−2y+x2y−1

x ′4 = (x2y+x−2y−1)(x−2y+x2y−1)

(y+y−1)2 .

(9.50)

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tanh K*=0 tanh(K) tanh K*=1

tanh(h)

+1

-1

Figure 9.5 Flow of coupling constant K and magnetic field h under renormaliza-tion, from an initial value of K0, h0.

The renormalization of coupling constants with three equations is a generaliza-tion of the renormalization group equations given in Eq. 9.47; taking the logarithmof Eq. 9.50 yields

(Kl+1, gl+1, hl+1) = R(Kl, gl, hl), (9.51)

where R is now a 3× 3 matrix.Similarly to the case of hl = 0, the flow of gl is irrelevant for the critical proper-

ties of the Ising system in the presence of a magnetic field. The coupling constantsKl, hl have flows as shown in Figure 9.5. The system is critical for K∗ = ∞ andfor any value of the magnetic field [Kardar (2007)].

For K0 < ∞, as one recurses and goes to long distance, the system decoupleswith the coupling constant going to zero, namely Kl → 0; the magnetic field flowsto hl → h∗, namely a system with a fixed magnetic field h∗, the value of h∗ beingfixed by the initial values K0, h0. Figure 9.5 shows the renormalization flow forK0 = K∗ = ∞ and h0 = 0.

9.7 Summary

The Ising model with a magnetic field has many new features. An exact solution ofthe partition function, of the magnetization, and of the correlation function can

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9.7 Summary 197

be obtained using the transfer matrix. Unlike the zero magnetic field case, theexpansion eS in a power series does not lead to a tractable method for calculationdue to the difficulty in doing the combinatorics in the presence of a magnetic field.

Linear regression in the presence of a magnetic field provides an intuitive andphysical interpretation of the concept of the correlation function.

An important feature of all lattice systems is that the lattice spacing a does notappear explicitly. The consequence of this plays a central role in the concept ofrenormalization and of the renormalization group.

Let the physical correlation length of a system described by the Ising model beL,in units of say meters; let the lattice spacing be a meters; then on the lattice, sinceall quantities are dimensionless, the dimensionless correlation length is given byξ(K0) = L/a. The correlation for the lattice of spacing 2a with coupling constantK1 is ξ(K1) and hence the dimensionless correlation is given by ξ(K1) = L/(2a).

The dimensionless numerical value of the correlation ξ(K1) is seen to be halfthe value of the correlation for lattice a with coupling constant K0 given by ξ(K0),namely ξ(K1) = L/(2a) = ξ(K0)/2. The result of the renormalization transfor-mation leads to the nontrivial result that the effective lattice spacing, instead ofappearing explicitly, appears through the value of the coupling constant Kl whichcorresponds to a lattice of spacing 2la.

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10

Fermions

The degrees of freedom studied so far have been either real or complex variables.These variables commute under multiplication, in the sense that two numbers a, bsatisfy ab = ba; commuting variables are generically called bosonic variables, orbosonic degrees of freedom. Typical of the bosonic case are the degrees of freedomfor a collection of quantum mechanical particles.

Interactions of fundamental particles are generally mediated by bosonic fieldssuch as the Maxwell electromagnetic field, whereas mass is usually carried by par-ticles that are fermions, the most familiar being the electron.

Unlike bosonic variables, fermionic variables, also called fermionic degrees offreedom, anticommute; namely, if ζ, η are two fermionic variables, then ζη =−ηζ . Two key features distinguish fermionic from bosonic variables:

• Fermions obey the Pauli exclusion principle, which states that no two fermionscan occupy the same quantum state. This is the reason the concept of intensitydoes not apply to a fermion. A high intensity electric field is a reflection of thepresence of a large number of photons, which are bosons, in the same quantumstate; for photons, any number of photons can be in the same quantum state.In contrast, an electron is either in a quantum state or it is not; in particular, afermion exists at a point or there is no electron there.

• The state function of a multi-bosonic system is totally symmetric in that theexchange of any two bosonic degrees of freedom yields the same state function.In contrast, a multi-fermion system is totally anti-symmetric: the exchange ofany two fermion degrees of freedom gives the same state – but with a negativesign.

The Pauli exclusion principle implies a discrete nature for fermions since afermion degree of freedom has only two possibilities, either occupying a state ornot occupying a state. In discussing the fermion Hilbert space in Section 10.3,

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10.1 Fermionic variables 199

it will be seen that the Hilbert space of a single fermion is identical (isomorphic)to the Hilbert space of a discrete degree of freedom that takes only two values.

Both the key features of fermions, namely obeying the Pauli exclusion principleand the state function being anti-symmetric, can be mathematically realized byintroducing a new type of variable, namely fermionic variables; similarly to thebosonic case, fermionic degrees of freedom can be described by either real or com-plex fermionic variables.

Fermionic variables are defined in Section 10.1 and fermion integration is dis-cussed in Section 10.2. The fermionic Hilbert space as well as its dual space aredefined in Section 10.3. The concept of the antifermionic Hilbert space is discussedin Section 10.4.

Gaussian integration for real and complex fermions is discussed in Section 10.6.The fermionic path integral is defined for the particle and anti-particle system inSection 10.8, and the transition probability amplitude is obtained in Section 10.11.A simple one-dimensional toy model is constructed in Section 10.12 to show howquark confinement can arise by coupling the fermions to a gauge field.

10.1 Fermionic variables

The defining property of fermions is the Pauli exclusion principle, namely that atmost only one fermion can occupy a quantum state. In other words, for fermions,a state either has no fermions, or at most one fermion.

Let|0〉F be the fermion vacuum state, and let a†F be the fermion creation operator.

Then

|0〉F : ground state; no fermions,

a†F |0〉F : one fermion,

(a†)2F |0〉 = 0 : null state. (10.1)

The second defining property of fermions is that two different fermions mustgive an anti-symmetric state function on being exchanged; hence two distinctfermionic creation operators, represented by say a

†1,a†

2 , must satisfy the anti-symmetry

a†1a

†2 |0〉 = −a†

2a†1 |0〉,

which is realized by imposing the anti-commutation relation

a†1a

†2 = −a†

2a†1 ⇒ {a†

1, a†2} = 0,

where the anti-commutator is defined for any two quantities A,B by

{A,B} ≡ AB + BA. (10.2)

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200 Fermions

Instead of working with fermionic creation and annihilation operators acting on theground state |0〉, one can instead describe fermions using a calculus distinct fromthe calculus based on real numbers that is used for describing bosons.

An independent and self-contained formalism for realizing all the defining prop-erties of fermions is provided by a set of anti-commuting fermionic variablesψ1, ψ2, . . . ψN and its conjugate ψ1, ψ2, . . . ψN , defined by the properties

{ψi, ψj } = −{ψi, ψj },{ψi, ψj } = −{ψj, ψi},{ψi, ψj } = −{ψj , ψi}.

Hence, it follows that

ψ2i = 0 = ψ2

i .

Fermionic differentiation is defined by

δ

δψi

ψj = δi−j ,δ

δψi

ψj = 0

and

δ2

δψiδψj

= − δ2

δψjδψi

⇒ δ2

δψ2i

= 0 = δ2

δψ2i

.

Similarly, all the fermionic derivative operators δ/δψi ,δ/δψi anti-commute.

10.2 Fermion integration

Similarly to the case of∫ +∞−∞ dxf (x) which is invariant under x → x + a, that is∫ +∞

−∞ dxf (x) = ∫ +∞−∞ dxf (x + a), we define fermion integration by∫

dψf (ψ) =∫

dψf (ψ + η). (10.3)

Since ψ2 = 0, Taylors expansion shows that the most general function of thevariable ψ is given by

f = a + bψ.

It follows that rules of fermion integration that yield Eq. 10.3 are given by∫dψ = 0 =

∫dψ,∫

dψψ = 1 =∫

dψψ,

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10.3 Fermion Hilbert space 201∫dψdψψψ = 1 = −

∫dψdψψψ. (10.4)

For N fermionic variables ψi , with i = 1, 2, . . . N , one has the generalization[N∏n=1

∫dψn

]ψi1ψi2 . . . ψin = εi1,i2,...in , (10.5)

where εi1,i2,...in is the completely anti-symmetric epsilon tensor.Consider a change variable for a single variable, namely

ψ = aχ + ζ,

where a is a constant and ζ is a constant fermion. From Eq. 10.4, the non-zerofermion integral yields

1 =∫

dψψ =∫

dψ(aχ + ζ ) =∫

dψaχ ⇒ dψ = 1

adχ. (10.6)

Note that this is the inverse for the case of real variables, since x = ay yieldsdx = ady.

For the case of N fermions, the anti-symmetric matrix Mij ) = −Mji yields thechange of variables

ψi =N∑j=1

Mijχj ⇒ ψ = Mχ.

Similarly to Eq. 10.6, it follows that

N∏i=1

dψi = 1

detM

N∏j=1

dχj ⇒ Dψ = 1

detMDχ, (10.7)

where Dψ =∏Ni=1 dψi and so on.

10.3 Fermion Hilbert space

The discussion in Section 2.4 on state space and its dual is valid for fermions aswell. There are two distinct fermionic variables, namely the variable ψ and its dualψ . Comparing with the bosonic case, if one takes the fermionic variable ψ to bethe analog of the coordinate x, then ψ is the analog of the conjugate momentumvariable p. The two different state spaces Vψ and Vψ are based on the coordinatevariable ψ and its conjugate ψ , respectively. There are consequently two Hilbertspaces, namely Vψ and Vψ , that are dual to each other, as shown in Figure 2.3.

In analogy with bosonic variables x for which 〈φ|φ〉 = ∫dxφ∗(x)φ(x) ≥ 0, the

norm for the fermionic Hilbert space needs to be defined so as to yield a positivenorm fermionic Hilbert space.

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202 Fermions

Choose Vψ to be the state space of the fermionic degree of freedom ψ . A fermionstate function is then given by the scalar product of the dual coordinate state vector〈ψ | ∈ Vψ with |f 〉 ∈ Vψ and yields

f (ψ) = 〈ψ |f 〉 = a + bψ.

The dual state of |f 〉, denoted by 〈f |, is defined such that

〈f |f 〉 = |a|2 + |b|2 > 0. (10.8)

Physical (normalizable) state functions have 〈f |f 〉 = 1 and yield the interpreta-tion

〈f |f 〉 = 1 ⇒ |a|2 + |b|2 = 1,

|a|2 = probability, there is no fermion,

|b|2 = probability, there is one fermion.

Since a, b are complex numbers, the space of all physical state functions is equalto a three-dimensional sphere S3. Note that a fermion state function, similarly tothe boson case, is equivalent to all state functions related to it by a global phaseeiφ . Factoring out the phase from the physically distinct state functions yields, asdiscussed in Section 8.1.1, the Hilbert space

Vψ ≡ S3/S1 = S2 : Bloch sphere.

Hence, similarly to a spin 1/2 system, the distinct physical states of a singlefermion Hilbert space are parameterized by the points of a two-dimensional sphere.Or more formally, each state vector of the single fermion space of states corre-sponds to one point on a two-dimensional sphere.

The single fermion state space is seen to be isomorphic to the state space ofthe Ising spin discussed in Section 8.1.1, and shows that in essence a fermion is adiscrete degree of freedom.

Define the dual state by

ψ → ψ ⇒ f D(ψ) = 〈f |ψ〉 = a∗ + b∗ψ. (10.9)

To achieve the required scalar product, note that

〈f |f 〉 =∫

dψdψf D(ψ)e−ψψf (ψ)

=∫

dψdψ(a∗ + b∗ψ)e−ψψ(a + bψ)

=∫

dψdψ(|a|2 + |b|2ψψ + a∗bψ + ab∗ψ

)(1− ψψ). (10.10)

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10.3 Fermion Hilbert space 203

Using the rules for fermion integration given in Eq. 10.4, the positive definite scalarproduct is given by

〈f |f 〉 = |a|2 + |b|2.

10.3.1 Fermionic completeness equation

The fermionic variables have a phase space representation with (ψ, ψ) being anal-ogous to the coherent state representation of the creation and destruction operators(a†, a). Let |ψ〉 be the fermionic eigenstate and 〈ψ | be the dual fermionic eigen-state. The completeness equation for the fermion degree of freedom is given by

I =∫

dψdψ |ψ〉e−ψψ〈ψ |. (10.11)

The fermion completeness equation is similar to the completeness equation forcoherent states discussed in Section 5.12, and to Eq. 5.80 in particular. The fermionbasis states are over-complete – as is the case for the bosonic coherent basis states– and the metric on the fermion state space that accounts for the over-completenessis given by exp{−ψψ}.

The inner product of the basis with its dual being is given by the self-consistencyequation that follows from the completeness equation and yields

〈ψ |ψ〉 = eψψ . (10.12)

To verify that Eq. 10.12 is indeed consistent with the completeness equation givenin Eq. 10.11, let ζ , ζ be fermionic variables; then

〈ζ |ζ 〉 =∫

dψdψ〈ζ |ψ〉e−ψψ〈ψ |ζ 〉 =∫

dψdψ(1+ ζψ

) (1− ψψ

) (1+ ψζ

)=∫

dψdψ(ζψψζ − ψψ

) = ∫dψdψ

(1+ ζ ζ

)ψψ = eζ ζ

as expected.The inner product of the basis states and completeness is self-consistently de-

termined since one requires the other: to prove completeness one needs the innerproduct. In particular, a proof of the resolution of the identity as given in Eq. 10.11is the following:

I2 =

∫dψdψdζdζ |ψ〉e−ζψ〈ζ |ζ 〉e−ψζ 〈ψ |

=∫

dψdψdζdζ |ψ〉e−ζψeζ ζ e−ψζ 〈ψ |

=∫

dψdψ |ψ〉e−ψψ〈ψ | = I,

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204 Fermions

where the result follows from the inner product given in Eq. 10.12 and performingthe ζ , ζ integrations.

10.3.2 Fermionic momentum operator

The state space depends on ψ , so we need to determine the representation of thedual coordinate ψ on the state space. The fermion coordinate operator ψ has thefermionic coordinate eigenstate given by

ψ |ψ〉 = ψ |ψ〉,where the coordinate eigenvalue ψ is fermionic. The scalar product yields

〈ψ |ψ |ψ〉 = ψ〈ψ |ψ〉 = ψeψψ = δ

δψ(eψψ),

⇒ ψ〈ψ |ψ〉 = δ

δψeψψ ⇒ ψ = δ

δψ.

Hence, on state space Vψ the dual coordinate ψ yields

ψf (ψ) = δ

δψf (ψ). (10.13)

Hence, as mentioned in Section 10.3, the variable ψ is the analog of the momentumoperator p and is evidenced by its action on state space, as given in Eq. 10.13.

In terms of the fermionic variables the creation and annihilation operators havethe realization

a† = ψ, a = δ

δψwith {a†, a} = 1.

The identification of the fermion variables with the fermion creation destructionoperators is consistent with the identification made earlier, in the discssion on Eq.10.11, of the fermion completeness equation with the completeness equation forthe coherent state space.

10.4 Antifermion state space

Let χ , χ be a set of fermionic variables; the dual coordinate eigenstate is definedby 〈χ | ∈ Vχ ; hence

〈χ |f 〉 = a + bχ.

This change of definition for the coordinate of the state space will lead to the con-clusion that Vχ is the state space for antifermions. The completeness equation con-tinues to be

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10.4 Antifermion state space 205

I =∫

dχdχ |χ〉e−χχ 〈χ |. (10.14)

Consistency with the completeness equation for the antifermionic variables re-quires that the inner product continues to be the same as the particle case, namely

〈χ |χ〉 = exp{χχ}. (10.15)

To following derivation provides a consistency check for the above completenessequation:

〈η|η〉 =∫

dχdχeχηe−χχ eηχ

=∫

dχdχ (1+ χη) (1− χχ) (1+ ηχ)

=∫

dχdχ (−χχ + χηηχ) =∫

dχdχ (−χχ) (1+ ηη)

= exp{ηη}.In the completeness equation given in Eq. 10.14, the variables χ, χ have beeninterchanged for the state space vectors – as compared to the completeness equationfor variables ψ, ψ , as in Eq. 10.11 – but with the metric being unchanged, namelyexp{−χχ} and exp{−ψψ}.

To compensate for the difference in these two cases, an extra minus sign needsto introduce the conjugation of the state space vector f (χ). Conjugation is definedby χ →−χ . 1 Hence

f D(χ) = f ∗(−χ ) ⇒ a∗ − b∗χ .

It is verified that the rule for conjugation yields a positive definite norm for thestate space Vχ ,

〈f |f 〉 =∫

dχdχf D(χ)e−χχf (χ)

=∫

dχdχ(a∗ − b∗χ)(a + bχ)(1− χχ)

=∫

dχdχ(|a|2 + |b|2)χχ = |a|2 + |b|2.The fermion momentum operator is defined by

〈χ |χ |χ〉 = χeχχ = − δ

δχeχχ

and yields1 This is because the order of integration in the scalar product is reversed compared to the fermion case, for

which, under conjugation ψ → ψ .

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206 Fermions

χf (χ) = − δ

δχf (χ).

The fact that anticommuting variables χ, χ represent antifermions becomes clearwhen the antifermions are combined with fermions, as is done in the followingsection.

10.5 Fermion and antifermion Hilbert space

One can choose either of the Hilbert spaces Vψ or Vχ to be fermionic state space;once a choice is made for the fermion state space, it automatically allows for theintroduction of the concept of antifermions. The representation of antifermions isfixed by the choice that is made for the fermion state space. The normal conventionis to choose Vψ to be the fermionic state space and ψ the fermionic degree offreedom and Vχ to be the antifermionic state space and χ to be the anitfermionsdegree of freedom.

The state space and path integral for the fermion and antifermion system is dis-cussed below. A system containing both particle and anti-particle has a state spacegiven by the tensor product of the fermion and antifermion state spaces, namelyVψ ⊗ Vχ .

The most general state vector is given by

f (ψ, χ) = 〈ψ, χ |f 〉 = a + bψ + cχ + dψχ,

|a|2 + |b|2 + |c|2 + |d|2 = 1. (10.16)

The interpretation of the state vector is the following:

• |a|2 = probability of the system having no fermion or antifermion,

• |b|2 = probability of the system having one fermion,

• |c|2 = probability of the system having one antifermion,

• |d|2 = probability of the system having one fermion and one antifermion.

Hermitian conjugation for the fermion and antifermion state space is defined bythe operations:

1. Complex conjugate all the coefficients;

2. Reverse the order of all the fermion variables;

3. Make the substitution (ψ

χ

)→

(1 00 −1

)(ψ

χ

). (10.17)

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10.6 Real and complex fermions: Gaussian integration 207

The completeness for the fermion–antifermion degrees of freedom is

I =∫

dψdψdχdχ |ψ, χ〉e−ψψ−χχ 〈ψ, χ |.

Consider a state vector |f 〉 given in Eq. 10.16; its dual state vector, using therules for fermion and antifermion conjugation, is given by

f D(ψ, χ) = 〈f |ψ, χ〉 = a∗ + b∗ψ − c∗χ − d∗χψ.

The rule for conjugation yields the positive definite inner product

〈f |f 〉 =∫

dψdψdχdχ 〈f |ψ, χ〉 e−ψψ−χχ ⟨ψ, χ |f ⟩=∫

dψdψdχdχ(a∗ + b∗ − c∗χ − d∗χψ

) (1− ψψ − χχ + ψψχχ

)× (a + b + cχ + dψχ

)=∫

dψdψdχdχ(|a|2 + |b|2 + |c|2 + |d|2) ψψχχ

= |a|2 + |b|2 + |c|2 + |d|2 .

10.6 Real and complex fermions: Gaussian integration

The rules of fermion integration are used for evaluating real and complex fermionGaussian integration. Consider N real fermion variables χn with n = 1, 2, . . . , N .Define the partition function

Z [J ] =N∏n=1

∫dχn exp

{−1

2χnMnmχm + Jnχn

}. (10.18)

The external source Jn is fermionic.For N odd, Z [J ] is always zero. To see this, consider the case of J = 0; on ex-

panding the exponential, one always has powers of the even product of fermions;hence, term by term, the partition function is given by terms that are all zero,namely

N∏i=1

∫dχi

(even products of χj

) = 0.

The term M is a real anti-symmetric matrix. One way of evaluating the partitionfunction is to try and diagonalize M . However, one can only use a real transfor-mation since all the fermions are real. Matrix algebra yields the result that every

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208 Fermions

real anti-symmetric matrix M can be brought into a block-diagonal form by anorthogonal transformation O in the following manner:

M = OT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 λ1

−λ1 00 λ2

−λ2 0. . .

0 λN2−λN

20

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠O = OT�O, (10.19)

where

OTO = 1 : orthogonal.

Let us perform a change of variables, and from Eq. 10.7

Oχ = ξ : real

⇒N∏i=1

dχi = det(O)

N∏i=1

dξi =N∏i=1

dξi.

The partition function factors into a product of N/2 terms and yields

Z [0] =N∏i=1

∫dξi exp

{−1

2ξn�nmξm

}

=N/2∏i=1

[∫dξ1dξ2 exp

{−1

2

(ξ1 ξ2

) ( 0 λi

−λi 0

)(χ1

χ2

)}]

=N/2∏i=1

∫dξ1dξ2e

+λiξ1ξ2 =N/2∏i=1

λi. (10.20)

From Eq. 10.19

detM =N/2∏i=1

det

(0 λi

−λi 0

)=

N/2∏i=1

λ2i .

Hence, from Eq. 10.20, the partition function is given by

Z [0] = √detM (10.21)

The square root of an antisymmetric matrix is known as a Pfaffian and has manyremarkable mathematical properties.

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10.6 Real and complex fermions: Gaussian integration 209

The general case of the partition function in the presence of an external fermionicsource Ji can be derived from the results obtained. Rewriting the action given inEq. 10.18 yields

Z [J ] =∫

Dχ exp

{−1

2

(χ + JM−1

)M(χ +M−1J

)+ 1

2J−1MJ

}. (10.22)

Using the fundamental invariance of fermion integration under the shift of thefermion integration variable given in Eq. 10.3 allows us to shift the integrationvariable,

χ → χ − JM−1,

and yields, from Eqs. 10.22 and 10.21

Z [J ] =∫

Dχe−12χMχe

12 J

T M−1J = Z [0] e12 J

T M−1J

⇒ Z [J ] = √detMe12 J

T M−1J .

The propagator can be obtained by fermionic differentiations on Ji ,

⟨χiχj

⟩ = δ2

δJiδJj· 1

Z

∫DχeS+Jχ

∣∣∣∣J=0

= δ2

δJiδJjexp

{1

2

N∑mn=1

J TmM

−1mnJn

}∣∣∣∣∣J=0

= δ

δJi

[(1

2M−1

jk Jk −1

2JkM

−1kj

)eS]∣∣∣∣

J=0

= 1

2M−1

j i −1

2M−1

ij

⇒ ⟨χiχj

⟩ = M−1j i .

10.6.1 Complex Gaussian fermion

Consider the N -dimensional Gaussian integral for fermions ψn and ψn,

Z[J ] =N∏n=1

∫dψndψn exp

{−ψnMnmψm + Jnψn + ψnJn},

where Mnm = −Mmn is an antisymmetric matrix. For real fermions ψ = ψ∗. Forcomplex fermions ψ = ψ1 + iψ2.

An antisymmetric matrix M = −MT can be diagonalized by a unitary transfor-mation

M = U †

⎛⎜⎝λ1

. . .

λN

⎞⎟⎠U,U †U = 1.

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210 Fermions

In matrix notation

M = U †�U, UU † = 1, det(UU †) = 1, (10.23)

where � = diag(λ1, . . . λN). Since the fermions ψ, ψ are complex, let us definethe change of variables using the unitary matrix U , and from Eq. 10.72

ψU † = η, η = Uψ,

DηDη = 1

det(U)Dψ × 1

det(U †)Dψ = DψDψ. (10.24)

Hence, the fermion integrations completely factorize and yield

Z[0] =∏n

∫dηndηn exp{−

N∑n=1

λnηnηn}

=∏n

[∫dηndηne

−λnηnηn]=∏n

λn = detM. (10.25)

The partition function with an external source is given as before by a shift offermion integration variables. Write the partition function as

Z[J ] =∫

DψDψ exp{− (ψ − JM−1

)M(ψ −M−1J

)+ JM−1J}.

Using the fundamental property of fermion integration that it is invariant, a constantshift of fermion variables as given in Eq. 10.3 yields

ψ → ψ + JM−1, ψ → ψ +M−1J

and hence

Z[J ] =∫

DψDψ exp{−ψMψ + JM−1J

} = (detM) exp{JM−1J

}.

The fermion Gaussian integration obtained in Eq. 10.25 can be directly done usingthe rules of fermion integration. On expanding the exponential term exp{ψMψ},only one term –

(ψMψ

)N/N ! containing the product of all the fermion variables –

is nonzero inside the integrand. Using the notation of summing over repeatedindexes, Eq. 10.5 yields∫

DψDψeψMψ =∫

DψDψ

[1

N !(ψMψ

)N]

= 1

N !Mi1j1Mi2j2 · · ·MiNjN

∫DψDψψi1ψj1ψi2ψj2 · · · ψiNψjN

2 Note that for real fermions one could not use a unitary transformation for a change of variables as this wouldlead to the transformed fermions being complex.

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10.8 Fermionic path integral 211

= 1

N !Mi1j1Mi2j2 · · ·MiNjN εi1i2···iN εj1j2···jN

= detM.

10.7 Fermionic operators

Operators for fermions, such as the Hamiltonian and momentum have a represen-tation in terms of fermionic variables. Given that the dual of the fermionic coor-dinate ψ is given by ψ , all operators for fermions are expressed in terms of boththe fermion coordinate and its dual. In this sense, the operators of the fermion de-gree of freedom are defined analogously to bosonic operators that are defined onphase space, as in Section 5.4.

Consider a fermionic operator O; the operator is a mapping from Vψ to itself andhence, similar to the bosonic case given in Eq. 2.5, O is an element of the tensorproduct space Vψ ⊗ Vψ ,

O ∈ V ⊗ VD ≡ Vψ ⊗ Vψ.

The matrix elements of O are given by

〈ψ |O|ψ〉 = O(ψ, ψ)〈ψ |ψ〉 = O(ψ, ψ)eψψ .

In particular, the Hamiltonian operator H is defined by

〈ψ |H |ψ〉 = H(ψ, ψ)〈ψ |ψ〉 = H(ψ, ψ)eψψ (10.26)

and yields, from Eq. 10.26,

〈ψ |e−εH |ψ〉 = 〈ψ |[1− εH ]ψ〉 = [1− εH(ψ, ψ)]eψψ +O(ε2)

� e−εH(ψ,ψ)eψψ +O(ε2). (10.27)

10.8 Fermionic path integral

One way of understanding the difference between a fermion and an antifermionis to examine the evolution of fermions in (Euclidean) time. For clarity, considera time lattice with spacing ε and let lattice time be denoted by nε. The fermiondegrees of freedom are defined on the lattice and denoted by ψn;ψn. Consider atypical action for the lattice fermions given by

SP = −∑n

ψnψn + 2K∑n

ψn+1ψn

=∑n

L(n),

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212 Fermions

where the fermion Lagrangian is given by

L(n) = −ψnψn + 2Kψn+1ψn

= 2K[ψn+1 − ψn

]ψn + (2K − 1)ψnψn. (10.28)

The path integral is given by

ZP =∏n

∫dψndψne

S. (10.29)

The path integral is written in the fermion coherent state basis, similarly to thecoherent basis path integral for bosons (real and complex degrees of freedom) asdiscussed in Section 5.14.

Note that the fermion variable ψn at time n propagates to variable ψn+1 at timen+ 1. The partition function is written by repeatedly using the completeness equa-tion given in Eq. 10.11, and yields

ZP =∫

DψDψ〈ψn+1|e−εH |ψn〉e−ψnψn〈ψn|e−εH |ψn−1〉e−ψn−1ψn−1 . . .

Hence, from the definition of fermionic operators given in Eq. 10.27

e2Kψn+1ψn = 〈ψn+1|e−εH |ψn〉� e−εH(ψn+1,ψn)〈ψn+1|ψn〉 = e−εH(ψn+1,ψn)eψn+1ψn. (10.30)

Dropping the index of time on the fermion variables yields the particle Hamiltonian

−εHP (ψ, ψ) � 2Kψψ − ψψ

⇒ HP (ψ, ψ) �(

1− 2K

ε

)ψψ = 〈ψ |H |ψ〉.

Note that, as is the case for coherent states (Section 5.13), the Hamiltonian is auto-matically normal ordered.

The limit of continuous time is taken by defining the following continuumfermions ψ(t), ψ(t) and Hamiltonian:

ψt =√

2Kψn, ψt =√

2Kψn, t = nε,

HP

(ψ, ψ

) = (1− 2K

2Kε

)ψtψt = ωψt

δ

δψt

, ω = 1− 2K

2Kε. (10.31)

The continuum Lagrangian L(t) and action, from Eqs. 10.28 and 10.31, aregiven by

L(n) = −ε[∂ψt

∂tψt + 2K − 1

2εKψtψt

]≡ εL(t),

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10.8 Fermionic path integral 213

SP =∫

dtL(t), L(t) = −[∂ψt

∂tψt + ωψtψt

].

The term SP is the action for a particle propagating forward in time. Consideranother action that describes the time evolution of anti-particles, namely

SA = −∑n

χnχn + 2K∑n

χnχn+1

with

ZA =∫

DχDχeSA. (10.32)

The antifermion variable χn propagates from time n to the variable χn+1 at timen + 1. If one thinks of the variable χn+1 as representing a particle, then one canthink of the anti-particle as being equivalent to a particle propagating backwardsin time; although this way of thinking is not required, it does help to develop somephysical intuition about the peculiarities of anti-particles.

Given the nature of the state space of the anti-particles,ZA can be decomposed as

ZA =∫

DχDχ . . . 〈χn+1|e−εHA |χn〉e−χnχn〈χn|e−εHA |χn−1〉 . . . (10.33)

and by inspection

〈χn+1|e−εHA |χn〉 = e2Kχnχn+1

= e−εHA(χnχn+1)eχnχn+1

⇒ HA =(

1− 2K

ε

)χχ = 〈χ |H |χ〉.

Note that the order of the matrix element of the antifermions, namely 〈χ |H |χ〉, isthe reverse of the fermion case. We define continuum fermion variables by

χt =√

2Kχn, χt =√

2Kχn, ω =(

1− 2K

2Kε

), t = nε.

Anti-commuting the fermionic variables and ignoring an irrelevant constant yields,similarly to Eq. 10.31, the continuum antifermion Hamiltonian

HA = −ωχt χt . (10.34)

After normal ordering, interpreting HA as an operator yields, using χ = −δ/δχ ,

HA = ωχtδ

δχt.

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214 Fermions

10.9 Fermion–antifermion Hamiltonian

From Eq. 10.31, a Hamiltonian for fermions is given by

HP = ωψψ = ωψδ

δψ. (10.35)

The eigenstates and eigenenergies of the Hamiltonian are given by

HP�n = En�n

�0 = 1 E0 = 0

�1 = ψ E1 = ω.

From Eq. 10.34, a typical antifermion Hamiltonian is given by

HA = ωχδ

δχ= −ωχχ (10.36)

with eigenstates and eigenenergies

�A0 = 1 E0 = 0

�A1 = χ E1 = ω.

For a single fermion degree of freedom, the Hamiltonian H has the form ωψψ

and this is all that one can construct. Hence, it is necessary to look at more compli-cated systems, such as considering systems coupling fermion and antifermion aswell as coupling of the fermions to bosons.

Consider a fermion and antifermion system. The Hilbert state space is four di-mensional since there are four possible states for the system, as enumerated in Eq.10.16. A simple Hamiltonian for the fermion and antifermion system is a sum ofthe fermion and antifermion system with a coupling term, namely

H = ωψδ

δψ+ ω′χ

δ

δχ+ λψχ

≡ ωψψ − ω′χχ + λψχ (10.37)

with real values for ω,ω′, λ.The normalized eigenfunctions and eigenvalues are given by

H�n = En�n.

One can directly verify that

�1 = χ E1 = ω′

�2 = ψ E2 = ω

�3 = ψχ E3 = ω + ω′.

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10.9 Fermion–antifermion Hamiltonian 215

Note that the eigenstates do not form a complete set since the Hilbert space is fourdimensional, the reason being that this Hamiltonian is not Hermitian. Using therules of conjugation given in Eq. 10.17,

ψ → ψ, χ →−χψ → ψ, χ →−χ

and reversing the order of the fermion variables shows that

H † = ωψψ − ω′χχ − λχψ �= H. (10.38)

A Hermitian Hamiltonian for the fermion–antifermion system is given by

H = ωψδ

δψ+ ω′χ

δ

δχ+ λψχ

δ2

δψδχ

= ωψψ + ω′χχ − λψχψχ.

Hermitian conjugation, explicitly shown below, shows that the Hamiltonian is Her-mitian,

H † = ωψψ + ω′χχ − λχcψcχcψc

= ωψψ + ω′χχ − λ (−χ) ψ (−χ ) ψ= ωψψ + ω′χχ − λχψχψ

= ωψψ + ω′χχ − λψχψχ

= H.

The eigenfunctions can be read off by inspection and are

�0 = 1 E0 = 0

�1 = ψ E1 = ω

�2 = χ E2 = ω′

�3 = ψχ E3 = ω + ω′ − λ.

The Hamiltonian given in Eq. 10.37 can be made Hermitian by adding the termrequired, and yields

H = ωψδ

δψ+ ω′χ

δ

δχ+ iλ(ψχ + χψ). (10.39)

Two of the four orthogonal eigenstates can be obtained by inspection, and thisyields

�1 = χ E1 = ω′

�2 = ψ E2 = ω.

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216 Fermions

For the other two eigenfunctions consider the ansatz

� = ψχ + ic.

Applying the Hamiltonian given in Eq. 10.39 on � yields

H� = [ω + ω′ − cλ

] (ψχ − i

λ

ω + ω′ − cλ

).

One obtains the eigenvalue condition

c = − λ

ω + ω′ − cλ,

which has two solutions, namely

c± = 1

2λ

(ω + ω′ ±

√(ω + ω′)2 + 4λ2

), c+c− = −1. (10.40)

Hence, the remaining two eigenfunctions and eigenvalues are given by

�3 = 1√1+ c2+

[ψχ + ic+

], E3 = ω + ω′ − λc+

�4 = 1√1+ c2−

[ψχ + ic−

], E4 = ω + ω′ − λc−. (10.41)

The interpretation of the state �3 is that |c+|2/√

1+ |c+|2 is the likelihood that thesystem has no particles and 1/

√1+ |c+|2 is the likelihood of having a fermion–

antifermion pair, with a similar interpretation for �4.

10.9.1 Orthogonality and completeness

To illustrate the workings of fermion calculus, the orthogonality of the states�3, �4 is explicitly computed. Using the rules for forming the conjugate statefunction yields

〈�3|�4〉 =∫

dψdψdχdχ〈�3|ψ; χ〉e−ψψ−χχ 〈ψ;χ |�4〉

=∫

dψdψdχdχ�†3[ψ; χ ]�4[ψ;χ ]e−ψψ−χχ

=∫

dψdψdχdχ(− χψ − ic+

)(ψχ + ic−

)e−ψψ−χχ

= (1+ c+c−)∫

dψdψdχdχ ψψχχ

= (1+ c+c−) = 0,

since c+c− = −1.

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10.10 Fermion–antifermion Lagrangian 217

The completeness equation can be expressed in terms of the eigenfunctions by

I =4∑

i=1

|�i〉〈�i |. (10.42)

To verify Eq. 10.42, one needs to prove that

4∑i=1

〈ψ, χ |�i〉〈�i |ψ, χ〉 = exp{ψψ + χχ}.

Using the explicit form of the state functions derived above yields

4∑i=1

〈ψ, χ |�i〉〈�i |ψ, χ〉 = χχ + ψψ + N+[ψχ + ic+

][−χψ − ic+]

+N−[ψχ + ic−

][−χψ − ic−]

= χχ + ψψ + [c2+N+ + c2

−N−]+ [N+ +N−] ψψχχ

− i[c+N+ + c−N−][ψχ + χψ]where N+ = 1

1+ c2+, N− = 1

1+ c2−.

From Eq. 10.40 it follows that

c2+N+ + c2

−N− = 1 = N+ +N−,c+N+ + c−N− = 0.

Hence4∑

i=1

〈ψ, χ |�i〉〈�i |ψ, χ〉 = 1+ χχ + ψψ + ψψχχ = exp{ψψ + χχ

},

thus verifying Eq. 10.42.

10.10 Fermion–antifermion Lagrangian

Consider the propagation of a fermion–antifermion system given by the Lagrangian

εLn = −ψnψn − χnχn + 2K(ψn+1ψn + χnχn+1

). (10.43)

The term Ln consists of a fermion propagating forward in time and its antifermion(since K is the same for both) propagating “backward” in time.

Define a two-component spinor

ψ =(ψu

ψ�

)=(ψ

χ

)(10.44)

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218 Fermions

and

ψ = (ψ χ) (10.45)

and let

γ0 =(

1 00 −1

). (10.46)

Then

Ln = −ψnψn +K{ψn+1(1+ γ0)ψn + ψn(1− γ0)ψn+1

}. (10.47)

The fermion action is

S = ε∑n

Ln. (10.48)

Consider the continuum limit by defining t = nε and take the limit of ε → 0,yielding

Kψn+1ψ = K(ψn+1 − ψn)ψn +Kψnψn

∼= Kε∂0ψnψn +Kψnψn. (10.49)

Similarly

Kχnχn+1 = εKχn∂0χ +Kχnχn. (10.50)

Hence

εLn = (−1+ 2K)[ψnψn + χnχn

]+ 2Kε[∂0ψnψn + χn∂0χn

]. (10.51)

Let us define continuum fermionic variables by

ψt = (2K)1/2ψn, ψt = (2K)1/2ψn,

χt = (2K)1/2χn, χt = (2K)1/2χn.

The continuum Lagrangian is given by

εL(t) =(

2K − 1

2K

) [ψtψt + χtχt

]+ ε[ψt (−∂0)ψt + χt ∂0χt

].

We define

mε = −[

2K − 1

2K

]⇒ 2K = 1

1+mε

� =(ψ

χ

)� = (

ψ χ)

γ0 =(

1 00 −1

). (10.52)

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10.11 Fermionic transition probability amplitude 219

Hence, the continuum action and Lagrangian are given by

S =∫ +∞

−∞dtL,

L(t) = −�(γ0∂0 +m)�

: one-dimensional Dirac Lagrangian.

10.11 Fermionic transition probability amplitude

The derivation done earlier for finding the eigenstates and eigenenergies of thefermionic Hamiltonian can be obtained directly working with the fermion transi-tion probability amplitude. Recall for a fermion particle

〈ψ |e−εH |ψ〉 = e2Kψψ . (10.53)

The eigenvalue and eigenstates that are given by �0 ∼ 1 and �1 ∼ ψ can bedirectly obtained by performing fermion integration in the following manner:

〈ψ |e−εH |�0〉 =∫

dςdς〈ψ |e−εH |ς〉e−ςς 〈ς |�0〉

=∫

dςdςe2Kψςe−ςς

=∫

dςdς(1+ 2Kψς)(1− ςς)

= 1, (10.54)

⇒ �0(ψ) = 〈ψ |�0〉 = 1, E0 = 0.

For the eigenstate �1 ∼ ψ consider the calculation

〈ψ |e−εH |�1〉 =∫

dςdςe2Kψςe−ςς ς

= 2Kψ

∫dςdςςς = 2Kψ = e−εE1ψ,

⇒ 〈ψ |�1〉 = ψ,

which yields the expected answer. The eigenenergy is given by

⇒ E1 = −1

εln(2K)

= −1

εln

(1

1+mε

)� m+O(ε). (10.55)

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220 Fermions

Similarly for the antifermion, recall

〈χ |e−εH |χ〉 = e2Kχχ . (10.56)

Hence, for the antifermion ground state, e−εH |�0〉 = |�0〉 since �0 ∼ 1.For 〈χ |�1〉 = χ one has

〈χ |e−εH |�1〉 =∫〈χ |e−εH |ς〉e−ςς 〈ς |�1〉

=∫ςς

e2Kςχe−ςςς

=∫ςς

(1+ 2Kςχ)(1− ςς)ς

= 2Kχ

∫(−ςς) = 2Kχ. (10.57)

Note that the transition amplitude automatically yields a normal ordered Hamil-tonian with the energy given by

E1 = −1

εln(2K)

� m. (10.58)

Until now, fermions and antifermions have equal mass but have not been cou-pled, and hence their contrasting properties have not come into play. One can cou-ple them by nonlinear terms in the Lagrangian such as λψψχχ , as well as by theircoupling to gauge fields, and this is briefly explored below.

10.12 Quark confinement

Consider a one-dimensional toy model of quarks (fermions) and antiquarks (an-tifermions) given by

Ln = −ψnψn + 2K(ψn+1ψn + χnχn+1

)− χnχn.

A gauge transformation on the fermions is defined by

ψn → eiφnψn ψn → ψne−iφn

χn → eiφnχn χn → χne−iφn .

To leave Ln invariant we need fix the nearest neighbor term. Let us introduce gaugefield eiBn and modify Ln to

Ln = −ψnψn − χnχn + 2K(ψn+1e

−iBnψn + χneiBnχn+1

).

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10.12 Quark confinement 221

Under a gauge transformation, let Bn have the transformation

eiBn → eiφneiBne−iφn+1,

and hence the combined transformations on the fermions and gauge field leave theLagrangian L invariant.

Note that Bn is an angular variable taking values in [−π,+π]. The partitionfunction is

Z =∏n

∫ π

−πdBn

2πdψndψndχndχne

S.

Since eiBn couples nearest neighboring instants of time, one can derive as beforethe transition amplitude for quark–antiquark system, namely

〈ψn+1, χn+1|e−εH |ψn, χn〉,by generalizing Eq. 10.30. One can choose to include the integration over the gaugefield in the definition of the transition probability amplitude for the fermions, andthis yields

〈ψn+1, χn+1|e−εH |ψn, χn〉 =∫ π

−πdBn

2πeLn .

Dropping the subscripts from Ln yields

〈ψ, χ |e−εH |ψ, χ〉 =∫ π

−πdB

2π

[1+ 2Kψe−iBψ

] [1+ 2KχeiBχ

]= 1+ (2K)2ψψχχ.

The transition probability amplitude acting on a general state function |�〉 yields

〈ψ, χ |�〉 =∫

dζ dςdξdξ〈ψ, χ |e−εH |ζ, ξ〉e−ζ ζ−ξ ξ 〈ζ , ξ |�〉

=∫

dζ dζdξdξ[1+ (2K)2ψζ ξχ

]e−ζ ζ−ξ ξ�(ζ , ξ). (10.59)

The only nonzero integral is∫dζ dζdξdξ ζ ζ ξ ξ = 1.

Solving the eigenvalue equation of the quark–antiquark Hamiltonian

e−εH |�n〉 = e−εEn |�n〉using Eq. 10.59 yields the following eigenfunctions and eigenvalues.

Note that single quark ψ and antiquark χ states are confined since they haveinfinite energy and cannot propagate in time; they are fixed at whatever moment in

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222 Fermions

�n(ζ , ξ) En State

1 0 vacuumψ ∞ one quarkχ ∞ one antiquark

ψχ 2[− ln(2K)

ε

]quark+antiquark

time they are created. Only the paired states of quark–antiquark, namely ψχ , arethe finite energy eigenstates and hence can propagate in time.

10.13 Summary

The fermion variable takes only two values and is fundamentally different from areal variable. The fermion degree of freedom describes a system that is essentiallydiscrete and at the same time quite distinct from the Ising variable, that belongs tothe category of real variables.

A differential and integral calculus was developed for the fermion variable andthe concepts applicable to a real variable were generalized to the fermion case.Gaussian integration was defined for both real and complex fermions and the re-sults are similar to the real variable case, but with a few significant differences.

Fermion and antifermion variables emerge naturally, based on the manner inwhich conjugation of the state vector is defined. The state space and Hamiltonianfor fermions and antifermions were derived and the state space was shown to be-have in the manner that one intuitively expects for a discrete system. A few sim-ple models of the fermion and antifermion path integral, Hamiltonian, and statespace were discussed. A one-dimensional toy model based on fermion and an-tifermion degrees of freedom coupled to a gauge field was shown to exhibit quarkconfinement.

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Part four

Quadratic path integrals

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11

Simple harmonic oscillator

A simple harmonic oscillator, or oscillator in short, is one of the most impor-tant systems in physics as well as in quantum mechanics. The path integral andHamiltonian of a simple harmonic oscillator are leading exemplars for the descrip-tion of a wide class of physical systems. An oscillator is also a theoretical modelof great utility, primarily due to its simplicity that allows for the exact derivation ofmany results.

The simple harmonic oscillator can be generalized to the case of infinitely manyoscillators, and is also the starting point for analysis of many nonlinear quantumsystems.

Path integrals for the simple harmonic oscillator are all based on Gaussian in-tegration, which has been discussed in Section 7.2. The harmonic oscillator is de-scribed by Gaussian path integrals, and is also a bedrock of the general theory ofpath integration. Moreover, all perturbation expansions of nonlinear path integralsabout the oscillator path integral, which includes the semi-classical expansion, arebased on results of Gaussian integration

The properties of the harmonic oscillator are analyzed here in coordinate spacerepresentation, and its path integral as well as its correlation functions are analyzed.All formulas and derivations are for Euclidean time.

In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator arestudied and the correlator is derived using state space methods. In Section 11.2 theinfinite time oscillator is introduced and in Sections 11.3–11.5 the path integralfor the oscillator is evaluated. In Sections 11.6–11.11 the oscillator path integral isstudied on a finite lattice and the path integral is reduced to a finite dimensionalordinary multiple integral. In Section 11.12 the finite lattice is derived using thetechnique of the transfer matrix.

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226 Simple harmonic oscillator

11.1 Oscillator Hamiltonian

The oscillator Hamiltonian H has a kinetic term equal to the free particle case –as in Eq. 5.69 – and which has an operator form valid both in Minkowski andEuclidean time; the oscillator Hamiltonian has a quadratic potential and hence isgiven by

H = − 1

2m

∂2

∂x2+ 1

2mω2x2.

Let us define the creation and annihilation operators as follows:

x = 1√2mω

(a + a†

),

∂

∂x=√mω

2

(a − a†

),

a = 1√2mω

(mωx + ∂

∂x

), a† = 1√

2mω

(mωx − ∂

∂x

),

which yield the commutation equations[x,

∂

∂x

]= −I ⇒ [

a, a†] = I.

We define the oscillator ground state |0〉 by a|0〉 = 0. The Hamiltonian and itseigenfunctions and eigenvectors are given by

H = ωa†a + 1

2ω,

|n〉 = (a†)n√n! |0〉, 〈m|n〉 = δn−m, a|0〉 = 0,

H |n〉 = En|n〉, En = nω + 1

2ω.

Furthermore, the oscillator algebra [a, a†] = I yields the basis states

a† |n〉 = √n+ 1 |n+ 1〉, a |n〉 = √n |n− 1〉,which are complete, namely ∑

n

|n〉〈n| = I. (11.1)

11.2 The propagator

The Heisenberg operator is given by

xH (t) = etH xe−tH ≡ xt .

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11.2 The propagator 227

Time

xt^ xt¢^

<O| |O>

Figure 11.1 The correlator of two coordinate operators acting on the vacuumstate.

Consider the correlation function, shown in Figure 11.1,

D(t, t ′) = 〈0|T [xH (t)xH (t ′)]|0〉, (11.2)

which is a measure of the time interval over which disturbances on the vacuumare correlated. Given the special role of D(t, t ′) – the correlator of two coordinateoperators – it is called the propagator.

Taking t ′ = 0 and t > 0, the correlator is given by

D(t) = eωτ2 〈0|xe−tH x|0〉, (11.3)

= 1

2mω〈0|(a + a†)e−tωa

+a(a + a†)|0〉

= 1

2mω〈1|e−tωa+a|1〉

= 1

2mωe−ωt . (11.4)

For the general case given in Eq. 11.2, it can readily be shown that

D(t, t ′) = 1

2mωe−ω|t−t

′|. (11.5)

11.2.1 Finite time propagator

The propagator for finite time can be evaluated exactly for the case of the harmonicoscillator. The oscillator basis is used for the derivation and another derivation willbe given later using the path integral. Figure 11.2 shows two coordinate operatorsacting at two different times, with a finite time interval with periodic time givenby τ .

To simplify the notation let xH (t) ≡ xt ; the finite time correlator is

D(t, t ′, τ

) = 1

Z(τ)tr(T[xt xt ′

]e−τH

), Z(τ) = tr

(e−τH

). (11.6)

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x0^

xt^

τ

Figure 11.2 The correlator of two coordinate operators for periodic time.

Taking t ′ = 0 and t > 0 yields, for D (t, 0, τ ) = D (t, τ ),

D (t, τ ) ≡ 1

Z(τ)tr(xt x0e

−τH)= 1

Z(τ)tr(e−(τ−t)H xe−tH x

)= N

Z(τ),

where N = tr(e−(τ−t)H xe−tH x

).

The numerator N is given by using the complete oscillator basis states |n〉 given byEq. 11.1; hence

N =∞∑n=0

〈n∣∣∣e−(τ−t)H xe−tH x∣∣∣ n〉

= 1

2mω

∞∑n=0

e−(τ−t)(n+12)ω〈n

∣∣∣(a + a†)e−tH

(a + a†

)∣∣∣ n〉.Using the oscillator algebra yields

N = 1

2mω

∞∑n=0

e−(τ−t)(n+12)ω{(√

n+ 1 〈n+ 1| + √n 〈n− 1|)e−tH

×(√

n+ 1 |n+ 1〉 + √n |n− 1〉)}

= 1

2mω

∞∑n=0

e−τ(n+12)ω

{(n+ 1) e−ωt + neωt

}

= e−ωτ2

2mω

∞∑n=0

{ne−nωτ

(e−ωt + eωt

)+ e−nωτ e−ωt}.

Note the two summations∞∑n=0

e−nωτ = 1

1− e−ωτ,

∞∑n=0

ne−nωτ = − ∂

∂(ωτ)

∞∑n=0

e−nωτ = e−ωτ

(1− e−ωτ )2 .

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11.2 The propagator 229

Hence, the partition function is given by

Z(τ) = tr(e−τH ) = e−ωτ2

∞∑n=0

e−nωτ = 1

2 sinh(ωτ2

) ,and the numerator simplifies to

N = e−ωτ2

2mω

[(e−ωt + eωt

)e−ωτ

(1− e−ωτ )2 + e−ωt

1− e−ωτ

]

= e−ωτ2

2mω

[e−ωte−ωτ + eωte−ωτ + e−ωt − e−ωτ e−ωt

(1− e−ωτ )2

]

= 1

2mω

[eωte−

ωτ2 + e−ωte

ωτ2(

eωτ2 − e− ωτ

2)2

]

= 1

2mω

cosh(ωτ2 − ωt

)2 sinh2

(ωτ2

) .

Hence, the propagator is given by

D(t, τ ) = N

Z(τ)= 1

2mω

cosh(ωt − ωτ

2

)sinh

(ωτ2

) . (11.7)

In general, for two arbitrary times t and t ′, the correlator is given by time orderingthe operators and yields

D(t, t ′; τ) = 1

2mω

cosh(ω|t − t ′| − ωτ

2

)sinh

(ωτ2

) . (11.8)

Consider the limiting case of τ →∞,

limτ→∞ e−τH � e−

ωτ2 |0〉〈0|

and hence, from Eq. 11.6,

limτ→∞D(t, t ′; τ)→ tr

(T[xt xt ′

]e− ωτ

2 |0〉〈0|)tre− ωτ

2 |0〉〈0|= 〈0|T [xt xt ′]|0〉.

Equation 11.2 shows that the finite time correlator reduces to the one for infinitetime. Directly taking the limit of τ →∞ in Eq. 11.8 yields the limiting value

limτ→∞D(t, t ′; τ)→ 1

2mωe−ω|t−t

′|,

as was obtained earlier in Eq. 11.5.

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11.3 Infinite time oscillator

The action for the infinite time oscillator is given by

S = −m2

∫ +∞

−∞dt(x2 + ω2x2

)= −m

2

∫dtx

(− d2

dt2+ ω2

)x

= −m2

∫dtdt ′xt

(− d2

dt2+ ω2

)δ(t − t ′)xt ′,

and yields

Att ′ =(− d2

dt2+ ω2

)δ(t − t ′

).

The inverse of Att ′ is given by

A−1t t ′ =

(1

− d2

dt2+ ω2

)δ(t − t ′) = 1(

− d2

dt2+ ω2

) ∫ dp

2πeip(t−t

′)

=∫

dp

2π

eip(t−t ′)

p2 + ω2= 1

2ωe−ω|t−t

′|. (11.9)

Hence

E[xτx0] = 1

Z

∫Dx xτx0e

S

= 1

mA−1τ0 =

1

2mωe−ω|τ |, (11.10)

as expected from Eq. 11.4.To interpret the meaning of the correlator one can repeat the earlier linear regres-

sion analysis discussed in Section 9.4. Since E[xτ ] = 0 and E[x20 ] = 1/(2mω),

the random variables are related by the linear regression

xτ � A−1τ0

E[x20 ]x0 � e−ω|τ |x0.

11.4 Harmonic oscillator’s evolution kernel

The simple harmonic oscillator’s action and Lagrangian is given by

L = −m2x2 − 1

2mω2x2.

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11.4 Harmonic oscillator’s evolution kernel 231

The Euclidean time path integral is given by

K(xf , xi; τ) = 〈x ′|e−τH |x〉 =∫

Dx eS.

The continuum action for finite time is given by

S =∫ τ

0Ldt = −m

2

∫ τ

0dt(x2 + ω2x2

).

Consider the classical field equation

δS

δx(t)= mxc(t)−mω2xc(t) = 0, (11.11)

⇒ xc(t) = Ae−ωt + Beωt , (11.12)

boundary conditions: xc(0) = x, xc(τ ) = x ′. (11.13)

Define new quantum variables ξ(t) by

x(t) = ξ(t)+ xc(t)

that obey the boundary conditions

ξ(0) = 0 = ξ(τ ).

Hence, the path integral yields

∫Dx eS =

∫DξeS[ξ+xc],

where Dx = Dξ . The action, for ξ = dξ/dt , is

S[ξ + xc] = −m2

∫ τ

0dt{(ξ + xc

)2 + ω2 (ξ + xc)2}

= −m2

∫ τ

0dt{ξ 2 + 2ξ xc + x2

c + ω2(ξ 2 + 2ξxc + x2

c

)}= −m

2

∫dt(x2c + ω2x2

c

)−m

∫ (ξ xc + ω2ξxc

)− m

2

∫dt(ξ 2 + ω2ξ 2

).

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Note, since ξ(0) = ξ(τ ) = 0,∫ τ

0ξ xc =

∫ τ

0

{d

dt(ξ xc)− ξ xc

}= ξ(0)xc(0)− ξ(τ )xc(τ )−

∫ξ xcdt

= −∫

ξ xcdt.

The cross-terms of xc and ξ cancel using the classical equation, Eq. 11.11, and

S[xc + ξ ] = S[xc] + S[ξ ]. (11.14)

Hence, the evolution kernel factorizes and yields

K(x ′, x; τ) = N eSc ,

where using Dx = Dξ , we have

N =∫

DξeS[ξ ] : independent of x, x ′. (11.15)

It is shown in Eq. 11.24 that∫DξeS[ξ ] = N =

√mω

2π sinhωτ,

yielding the result

K(x ′, x; τ) =√

mω

2π sinhωτeSc . (11.16)

The classical action Sc is given by

Sc = −m2

∫ τ

0dt(x2c + ω2x2

c

)= −m

2(xcxc) |τ0 −

m

2

∫ τ

0dtxc

(−xc + ω2xc)

= −m2

[xc(τ )xc(τ )− xc(0)xc(0)] .

Equation 11.12 yields

xc(t) = ω(−Ae−ωt + Beωt

).

The classical action is hence

Sc[xc] = −mω2

[B2(e2ωτ − 1

)+ A2(1− e−2ωτ

)]. (11.17)

The boundary conditions, from Eq. 11.13, are

xc(0) = A+ B = x, xc(τ ) = Ae−ωτ + Beωτ = x ′,

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11.5 Normalization 233

and yield

A = 1

2 sinhωτ

(xeωτ − x ′

), B = 1

2 sinhωτ

(x ′ − xe−ωτ

), (11.18)

and hence the classical solution for S0, from Eqs. 11.12 and 11.18, is given by

xc(t) = 1

2 sinhωτ

[(xeωτ − x ′

)e−ωt + (x ′ − xe−ωτ

)eωt]. (11.19)

From Eq. 11.17, the classical action is

Sc = − mω

2 sinhωτ

[(x2 + x

′2)

coshωτ − 2xx ′], (11.20)

yielding the final result for the evolution kernel

K(x ′, x; τ) =√

mω

2π sinhωτexp

{− mω

2 sinhωτ

[(x2 + x

′2)

coshωτ − 2xx ′]}

.

(11.21)

11.5 Normalization

Recall K(x ′, x; τ) = 〈x ′|e−τH |x〉. Consider the periodic trace of K , namely

trK =∫

dx〈x|e−τH |x〉 =∫

dxK(x, x; τ).Using K = N exp{Sc} yields

trK = N∫ +∞

−∞dxeSc(x,x;τ)

= N∫

dx exp{− mω

sinhωτ[cosh τ − 1]x2

}

= N√

2π

mω

sinhωτ

2(coshωτ − 1). (11.22)

In the oscillator basis

H = ωa†a + ω

2

the trace yields, using this basis,

tr(K) = tr(e−τH

)= e−

τω2 tr

(e−τωa

†a)= e−

τω2

∑n

〈n|e−τωa†a|n〉

= e−τω2

∞∑n=0

(e−nτω

) = e−τω2

1

1− e−τω. (11.23)

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N =√

mω

2π sinh(ωτ). (11.24)

11.6 Generating functional for the oscillator

Consider the simple harmonic oscillator in the presence of an external source j (t)given by

L = −1

2mx2(t)− 1

2mω2x2(t)+ j (t)x(t)

and with the finite time action

S = −1

2m

∫ τ

0

(x2 + w2x2

)+ ∫ τ

0j (t)x(t) = S0 +

∫ τ

0j (t)x(t). (11.25)

The path integral for the normalized generating functional is given by

Z[j ] = 1

Z

∫DxeS = 1

Z

∫DxeS0+

∫j (t)x(t),

Z =∫

DxeS.

The generating functional contains the full content of the quantum system; all thecorrelation functions for the action S0, in the presence of the boundary conditions,can be evaluated by functional differentiation. In particular, the first few correlatorsare given by

1

Z

δZ[j ]δj (t)

∣∣∣j=0

= 1

Z

∫Dx x(t)eS0,

1

Z

δ2Z[j ]δj (t)δj (t ′)

∣∣∣j=0

= 1

Z

∫Dx x(t)x(t ′)eS0,

1

Z

δ3Z[j ]δj (t)δj (t ′)δj (t ′′)

∣∣∣j=0

= 1

Z

∫Dx x(t), x(t ′)x(t ′′)eS0,

and so on.

11.6.1 Classical solution with source

Similarly to the derivation in Section 11.4, one can use the classical equationsof motion to evaluate the Gaussian path integral. The classical solution obeys theequation

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11.6 Generating functional for the oscillator 235(− d2

dt2+ w2

)xc = J (t)

m,

xc = xH + xI , (11.26)

where xH , xI are the homogeneous and inhomogeneous solutions respectively.Note that the homogeneous solution depends on the boundary conditions. In par-ticular,

(− d2

dt2+ w2

)xH = 0 ⇒ xH = A′e−wt + B ′ewt .

The inhomogeneous solution is given by

xI (t) = 1

m〈t | 1(

− d2

dt2+ w2

) |J 〉 = 1

2mw

∫ τ

0dt ′e−w|t−t

′|J (t ′)

= 1

2mw

[∫ t

0dt ′e−w(t−t

′)J (t ′)+∫ τ

t

dt ′e−w(t′−t)J (t ′)

], (11.27)

where

(− d2

dt2+ w2

)−1

δ(t − t ′) = e−w|t−t ′|

2w. (11.28)

We fix boundary conditions by

xc(0) = x = xH (0)+ xI (0), xc(τ ) = x ′ = xH (τ)+ xI (τ ).

Hence

x(0) = x = A′ + B ′ − 1

2mω

∫ τ

0dte−ωtJ (t),

x(τ ) = x ′ = A′e−ωτ + B ′eωτ − 1

2mω

∫ τ

0dte−ω(τ−t)J (t),

which yields

A′ = 1

2 sinh(ωτ)

[x ′ − e−ωτx − 1

m

∫ τ

0dt sinhω(τ − t)J (t)

],

B ′ = 1

2 sinh(ωτ)

[e−ωτx − x ′ − e−ωτ

mω

∫ τ

0dt sinh(ωt)J (t)

].

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After considerable simplifications,

S[xc; j ] = − mω

2 sinhωτ

[(x2 + x

′2) coshωτ − 2xx ′]

+ x

sinhwτ

∫ τ

0dtJ (t) sinhw (τ − t)+ x ′

sinhwτ

∫ τ

0dtJ (t) sinh(wt)

+ 1

mw sinhwτ

∫ τ

0

∫ t

0dt ′J (t) sinhw (τ − t) sinh

(wt ′

)J(t ′).

(11.29)

The generating functional is hence given by the decomposition

x(t) = xc(t)+ ξ(t), ξ(0) = 0 = ξ(τ ),

that yields ∫DxeS = AeS[xc;j ]

∫DξeS0[ξ ],

⇒ A

∫DξeS0[ξ ] = N : independent of x, x ′, and j (t),

with normalization constant N given by Eq. 11.24. The generating functional, fromEq. 11.29, is given by

Z[j ] = 1

Z

∫DxeS = eS[ξ ;j ],

with the normalization constant N canceling out.

11.6.2 Source free classical solution

The classical solution yields the inhomogeneous term xI , as in Eq. 11.26, arisingfrom the source j (t). To obtain the final answer given in Eq. 11.29 requires a fairamount of algebra.

Given the importance of the simple harmonic oscillator, another derivation ofthe same result is given here. This derivation turns out to be useful in situationsthat are more complicated than the simple one that we are considering.

For this derivation, consider the classical solution for the source free action,that is only S0[x] – the external source j (t) is not included in S0[x] – and hencethe classical solution does not have an inhomogeneous term xI . The change ofvariables x = xc + ξ now yields a path integral for which the quantum variableis coupled to the source j (t) – and one needs to perform a path integral over thequantum degree of freedom ξ(t) to obtain the result.

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11.6 Generating functional for the oscillator 237

Recall, from Eq. 11.25, that the simple harmonic action for the generating func-tional is given by

S = S0 +∫ τ

0j (t)x(t), S0 = −1

2m

∫ τ

0

(x2 + w2x2

).

The classical solution for S0, from Eq. 11.19, is given by

xc(t) = 1

2 sinhωτ

[(xeωτ − x ′

)e−ωt + (x ′ − xe−ωτ

)eωt].

Define the quantum variables ξ by

x(t) = xc(t)+ ξ(t), ξ(0) = 0 = ξ(τ ).

The classical solution yields, from Eq. 11.14, the factorization

S0[xc + ξ ] = S0[xc] + S0[ξ ].The path integral for the generating functional, up to an irrelevant normalization,is hence given by ∫

DxeS = eF0[xc;j ]∫

DξeF [ξ ;j ],

F0[xc; j ] = S0[xc] +∫ τ

0j (t)xc(t),

F [ξ ; j ] = S0[ξ ] +∫ τ

0dtj (t)ξ(t).

The following path integral over the quantum variables ξ(t) needs to be performed:∫Dξ exp{F [ξ ; j ]}, ξ(0) = 0 = ξ(τ ).

The boundary conditions ξ(0) = 0 = ξ(τ ) are satisfied by the Fourier sine expan-sion for ξ(t),

ξ(t) =∞∑n=1

sin

[nπt

τ

]ξn,

where, for each n, ξn is an independent and real (indeterminate ) integration vari-able. Hence, using the orthogonality of the sine functions,∫ τ

0dt sin

[nπt

τ

]sin

[mπt

τ

]= τ

2δn−m

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yields

F [ξ ; j ] = −1

2

∞∑n=1

λnξ2n +

∞∑n=1

jnξn,

λn = 1

2mτ

{(nπτ

)2 + w2

}, jn =

∫ τ

0j (t) sin

[nπt

τ

].

All the ξn variables have decoupled and we obtain, up to a normalization,∫Dξ exp {F [ξ ; j ]} =

+∞∏n=1

∫ +∞

−∞dξn exp

{−1

2

∞∑n=1

λnξ2n +

∞∑n=1

jnξn

}

=√

1

2πλnexp

{1

2

∞∑n=1

jn1

λnjn

}.

The result further simplifies,

∞∑n=1

jn1

λnjn =

∫ τ

0dt

∫ τ

0dt ′j (t)D(t, t ′)j (t ′),

D(t, t ′) =∞∑n=1

1

λnsin

[nπt

τ

]sin

[nπt ′

τ

].

Using the identity

+∞∑n=1

cos(θ)

a2 + n2= π

2a

cosh[a(π − |θ |)]sinhπa

− 1

2a2(11.30)

yields

D(t, t ′

) = 1

mw sinhwτsinhw (τ − t) sinh

(wt ′

), t > t ′ (11.31)

and hence

1

2

∫ τ

0dt

∫ τ

0dt ′j (t)D

(t, t ′

)j(t ′)

= 1

mw sinhwτ

∫ τ

0dt

∫ t

0dt ′J (t) sinhw(τ − t) sinh

(wt ′

)J(t ′). (11.32)

Note, from Eq. 11.19, that∫ τ

0j (t)xc(t) = x

sinhwτ

∫ τ

0dtJ (t) sinhw(τ − t)

+ x ′

sinhwτ

∫ τ

0dtJ (t) sinh(wt). (11.33)

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11.7 Harmonic oscillator’s conditional probability 239

The term S0[xc] is given in Eq. 11.20; hence, from Eqs. 11.32 and 11.33∫DxeS = exp {S[xc; j ]} ,

S[xc; j ] = S0[xc] +∫ τ

0j (t)xc(t)+ 1

2

∫ τ

0dt

∫ τ

0dt ′j (t)D(t, t ′)j (t ′),

and, as expected, S[xc; j ] is given by Eq. 11.29.Furthermore, the generating functional is given by

Z[j ] = 1

Z

∫DxeS

= exp{1

2

∫ τ

0dt

∫ τ

0dt ′j (t)D(t, t ′)j (t ′)+

∫ τ

0j (t)xc(t)

},

with the source free classical solution S0[xc] canceling out due to the normaliza-tion by Z.

11.7 Harmonic oscillator’s conditional probability

Consider the Lagrangian given by

L = −1

2m

[(dx

dt)2 + ω2x2

].

The evolution kernel is the probability amplitude that the harmonic oscillator’sdegree of freedom will reach a final point xf from initial point xi in time τ . It isgiven, up to a normalization constant that cancels out, by

K(xf , xi; τ) = 〈xf |e−τH |xi〉= N exp

{− mω

2 sinhωτ

[(x2

i + xf2) coshωτ − 2xixf

]}.

The conditional probability quantifies the likelihood that the outcome of an exper-iment will yield the value of the coordinate to be xi , given that xi has occurred; itis given by1

P(xf |xi; τ) = |K(xf , xi; τ)|2∫dxf |K(xf , xi; τ)|2 .

The denominator can be simplified as follows:∫dxf |K(xf , xi; τ)|2 =

∫dxf e

− mω2 sinhωτ

[(x2i +x2

f

)coshωτ−2xixf

]

=√

2π sinhωτ

2mωexp

{−mωx2i tanhωτ

},

1 The normalization constant N cancels out and is set to unity in this section.

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yielding the conditional probability given by

P(xf |xi; τ) =√

mω

π sinhωτexp

{mω tanhωτx2

i

} |K(xf , xi; τ)|2.Note that the normalization of the evolution kernel drops out of P(xi, xf ; τ).

As expected, one obtains the normalization∫ +∞

−∞dxf P (xf |xi; τ) = 1.

The conditional probability is different for Euclidean time compared to Minkowskitime.

11.8 Free particle path integral

A path integral derivation is given of the evolution kernel for a free particle degreeof freedom moving in one dimension (d = 1), which was obtained earlier in Eq.5.72 using the eigenfunctions of the free particle Hamiltonian.

Let ε = t/N ; from Eq. 4.30 the path integral for finite ε is given by a multipleintegral. Using the infinitesimal form of Eq. 5.72, which also directly follows fromthe Dirac–Feynman formula, yields (set � = 1)

K(x, x ′; τ) =N−1∏n=0

K(xn+1, xn; ε)

=(√

m

2πε

) N−1∏n=1

√m

2πε

∫ +∞

−∞dxne

− m2ε

∑N−1n=0 (xn+1−xn)2

≡∫

Dx e−m2ε (x−xN−1)

2 · · · e− m2ε (x2−x1)

2e−

m2ε (x1−x′)2,

boundary conditions : x ′ = xN, x = x0, (11.34)

where ∫Dx ≡

(√m

2πε

) N−1∏n=1

√m

2πε

∫ +∞

−∞dxn. (11.35)

Note the identity√m

2πε

∫ +∞

−∞dx1e

− m2ε (x2−x1)

2− m2ε (x1−x′)2 =

√1

2e−

m2 · 1

2ε (x2−x′)2 .

One can evaluate the path integral exactly by performing theDx-integrations recur-sively, starting from the end with x1. The successive integrations over the variablesx1 → x2 → x3 · · · → xN−1 yield

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11.9 Finite lattice path integral 241

K(x, x ′; τ) =√

m

2πNεe−

m2 · 1

Nε(x−x′)2 =

√m

2πτe−

m2τ (x−x′)2, (11.36)

which is the result obtained in Eq. 5.72.The case for a free particle in arbitrary d-dimensional space follows from

Eq. 11.36, since the d-dimensional transition amplitude factorizes into separateone-dimensional components.

To obtain the evolution kernel in Minkowski time, recall from Eq. 5.16 that

τ = it.

Hence, from Eq. 11.36, the Minkowski time evolution kernel, denoted by subscriptM , and restoring the � in the formula, yields

KM(x, x′; t) =

√m

2πτe−

m2τ� (x−x′)2 =

√m

2πi�tei

m2t� (x−x′)2, (11.37)

which was obtained in Eq. 5.72 using the free particle Hamiltonian.

11.9 Finite lattice path integral

A rigorous definition of the path integral can be given by approximating it by afinite dimensional multiple integral. One can then take the continuum limit and ob-tain the functional integration required for defining a path integral. There is anotherreason for considering the finite approximation of this integral. For numerical sim-ulation one always approximates the path integral by a finite dimensional multipleintegral.

Hence it is important to have some exact analytical results for a finite system soas to compare the numerical simulations with it. One of the few exact path integralresults that one can derive is for the finite approximation of the harmonic oscillatorpath integral, and in this section the oscillator is analyzed in some detail.

Consider an open lattice with boundary values of xN and x0; the action isgiven by

S = −m

2ε

N−1∑n=0

(xn+1 − xn)2 − 1

2ε mω2

N−1∑n=1

x2n,

K(xN, x0; τ) = 〈xN |e−τH |x0〉 = N∫ N−1∏

i=1

dxneS.

To simplify the analysis, consider time to be periodic with boundary conditionxN = x0. Furthermore, for a symmetric labeling of the lattice sites, consider aperiodic lattice of size 2N + 1 as shown in Figure 11.3.

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–10 1

–N N

Figure 11.3 Finite periodic lattice, with symmetric numbering of the lattice sites.

The degrees of freedom are periodic in the discrete time index n, where

t = nε, ε = τ/(2N + 1),

xn = xn+2N+1 = 1

2N + 1

+N∑p=−N

e2πipn/(2N+1)xp.

For simplicity, use the notation 2πp/(2N + 1) = k; in this simplified notation onewrites

xn =∑k

eiknxk ≡ 1

2N + 1

+N∑p=−N

e2πipn/(2N+1)xp,

and one has the identity

N∑n=−N

ei(k+k′)n =

+N∑n=−N

e2πi

2N+1n(p+p′) = (2N + 1)δ(p + p′) ≡ δk+k′ .

The action is hence given by

S = −m

2ε

+N∑n=−N

(xn+1 − xn)2 − mω2

2ε

N∑n=−N

x2n

= −m

2ε

N∑n=−N

∑k,k′

{(eik − 1

)xk(e

ik′ − 1)xk′ + ε2ω2xkxk′}ei(k+k

′)n

= −m

2ε

∑k

{|eik − 1|2 + ε2ω2}xkx−k,

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11.10 Lattice free energy 243

where note that, due to the simplified notation being used, one has

|1− eik|2 + ε2ω2 = 4 sin2

(πp

2N + 1

)+ ε2ω2.

In the momentum basis, both the gradient and quadratic term are diagonalized,and hence the path integral can be performed exactly.

11.9.1 Coordinate and momentum basis

The change of integration variables from real space variables xn to momentumspace variables xk is analyzed below.

There are 2N + 1 real integration variables xn, and these are transformed to2N + 1 variables xk that are complex, and hence there seem to be too many xk

variables. However, note that the xn coordinates are all real and hence in the Fourierexpansion given by xn =∑

k eiknxk, Eq. 11.38 yields

x∗n = xn ⇒ x∗k = x−k, with k = 0,±1,±2, . . .±N.

Hence, although there are 2N + 1 complex variables xk, they are not all inde-pendent, and one can choose only the set of complex variables xk – with k =0, 1, 2, . . . N – to be the independent complex variables.

Note the special case of the zero momentum mode, which is real since x0 =xk=0 = x∗k=0: real.

In summary, there are 2N + 1 independent momentum space variables; writ-ing the momentum modes in real and imaginary components yields the 2N + 1independent momentum space variables to be

xk = xRk + ixIk : k = 1, 2, . . . N, x0 ≡ xk=0. (11.38)

The change of 2N + 1 variables from xn to xk has a Jacobian equal to one, andyields

+N∏n=−N

∫ +∞

−∞dxn =

∫ +∞

−∞dx0

N∏k=1

∫ +∞

−∞dxRk

∫ +∞

−∞dxIk ≡

∫Dx.

11.10 Lattice free energy

To illustrate the workings of the momentum space variables, the partition functionis evaluated as follows. Let the lattice simple harmonic Hamiltonian be denoted byHLat; the partition function for the periodic lattice is given by

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Z = tr(exp {−ε(2N + 1)HLat}

=( m

2πε

)(2N+1)/2 +N∏n=−N

∫dxne

S

= N∫

dx0

N∏k=1

∫dxRk dx

Ik e− m

2ε

∑Nk=−N{|1−eik |2+ε2ω2}x−kxk ,

where HLat is the lattice Hamiltonian. Note, since k ≡ 2πp/(2N + 1),

|1− eik|2 + ε2ω2 = 4 sin2

(πp

2N + 1

)+ ε2ω2 ≡ dk.

From Eq. 11.38 one has the simplification

x−kxk =(xRk − ixIk

) (xRk + ixIk

) = (xRk)2 + (xIk )2

.

Hence the action completely factorizes into independent Gaussian integrations foreach variable xk, k = 0, 1, 2, . . . N . This factorization is the reason in the firstplace why the variables were transformed from real space to momentum space,since in the momentum basis both the quadratic potential and the kinetic energyare diagonal. The partition function is given by

Z =( m

2πε

) 2N+12

N∏k=1

∫dxRk dx

Ik e− m

2ε dk[(xRk )2+(xIk )2]∫ ∞

−∞dx0e

− m2ε ·ε2ω2x2

0

=( m

2πε

)N (2πε

m

)N[

N∏k=1

(1

dk

)]( m

2πε

)1/2√

2πε

m· 1

ε2ω2

=√

1

d0

N∏k=1

(1

dk

)= exp

(−1

2

∑k

ln (dk)

)

= e−(2N+1)F , (11.39)

where d0 = ε2ω2 has been used in obtaining Eq. 11.39.The free energy for the simple harmonic partition function is given by

F = free energy per lattice site

= 1

2

∑k

ln(dk) = 1

2

∑k

ln

{4 sin2

(k

2

)+ ε2ω2

}

= 1

2

1

2N + 1

N∑p=−N

ln

{4 sin2

(πp

2N + 1

)+ ε2ω2

}.

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11.11 Lattice propagator 245

Taking the limit of N →∞ yields

F = 1

2

∫ −π

−πdk

2πln

{4 sin2

(k

2

)+ ω2

0

}, ω2

0 = limε→0

ε2ω2.

11.11 Lattice propagator

The propagator (correlator) for the lattice simple harmonic oscillator on a periodiclattice is given by

Dn = E[xn+mxm] = 1

Z

∫Dx xn+mxmeS.

The propagator does not depend on m due to periodicity of the lattice. Transform-ing to the momentum space basis yields

Dn =∑k,k′

eik(n+m)eik′m 1

Z

∫DXkxpxke

S

=∑k,k′

eikneim(k+k′) ︷︸︸︷xkxk′

= ε

m

∑k,k′

eikneim(k+k′)δk+k′

1

|1− eik|2 + ε2ω2

= ε

m

∑k

eikn

|1− eik|2 + ε2ω2

≡ ε

8m

1

2N + 1

+N∑p=−N

cos(2πpn/(2N + 1))

sin2(πpn

2N+1

)+ ε2ω2

4

. (11.40)

It can be shown by a direct and tedious calculation that the Fourier sum given inEq. 11.40 yields

Dn = rn + r2N+1−n

2ω′(1− r2N+1), (11.41)

where

ω′ = mω

(1+ ε2ω2

4

)1/2

,

r = 1+ ε2ω2

2− εω

(1+ ε2ω2

4

)1/2

=(εω

2−√(εω

2

)2 + 1

)2

. (11.42)

Note the important fact that Eq. 11.41 is convergent for large N because r obeysthe bounds

0 < r < 1.

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To define the continuum limit, the lattice parameters are defined in terms of con-tinuum time t and total time τ as

n = t

τ(2N + 1), ε = τ

2N + 1,

r = 1− ωτ

2N + 1+O

(1

(2N + 1)2

),

ω′ = mω +O

(1

(2N + 1)2

).

For limit N →∞, the propagator is given by

Dn = rn−2N+1

2 + r2N+1

2 −n

2mω(r− 2N+1

2 − r2N+1

2

) ,r2N+1 =

(1− ωτ

2N + 1

)2N+1

→ e−ωτ ,

rn−2N+1

2 = r(tτ− 1

2)(2N+1) → e−ωτ(tτ− 1

2) = e−ωt+ωτ2 ,

∴ Dn → 1

2mω

e−ωt+ ωτ2 + eωt− ωτ

2

eωτ2 − e− ωτ

2.

Hence, one obtains the continuum limit given by

Dn = 1

2mω

cosh(ωt − ωτ

2

)sinh

(ωτ2

) ,

which agrees with the result obtained earlier in Eq. 11.7.

11.12 Lattice transfer matrix and propagator

The propagator can be found exactly for the lattice simple harmonic oscillator usingthe exact transfer matrix [Creutz and Freedman (1981)]. The path integral can beexpressed as a trace due to the periodic boundary conditions. Hence

Z =∫(periodic)

DXeS

= tr(T 2N+1).

To obtain the transfer matrix T, consider two adjacent sites; then the integrandof the lattice path integral for Z is given by the product

· · · 〈xn+2|T |xn+1〉〈xn+1|T |xn〉〈xn|T |xn−1〉 · · ·

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11.12 Lattice transfer matrix and propagator 247

The terms linking variables xn and xn+1 are entirely due to matrix element〈xn+1|T |xn〉, whereas terms like x2

n in S can be symmetrically divided betweenadjacent matrix elements of T . A Hermitian transfer matrix is given by

〈x ′|T |x〉 = e−mεω2

4 x′2e−

m2ε (x−x′)2e−

mεω24 x2

.

Recall, for [x, p] = i, one has

〈x ′|e− ε2m p

2 |x〉 =√

m

2πεe−

m2ε (x

′−x)2,

⇒ T =√

2πε

me−ε

mω24 x2

e−ε

2m p2e−ε

mω22 x2

.

If one is working to only lowest order in ε, then using the CBH-formula one caneasily show that T ∼ e−εH , where H = p2/2m + mω2x2/2. For the finite sizelattice, we need the operator expression for T that is exact in ε, that is, correct toall powers of ε.

Note, for α = ε/m and u2 = m2ω2, it can be shown that [Creutz and Freedman(1981)]

[x, T ] = α2u2

2T x − iαTp,

[p, T ] = α2u2

2Tp + iαu2

(1+ α2u2

4

)T x.

Using the commutators given above it can be shown that

[T ,H ] = 0,

where

H = 1

2p2 + 1

2β2x2,

β = u

√1+ α2u2

4= mω

√1+ ε2ω2

4= ω′,

where the last relation follows from Eq. 11.42.The eigenstates of H also diagonalize T . Let the creation and annihilation oper-

ators be defined as

a = 1√2ω′

(ω′x + ip

),

a† = 1√2ω′

(ω′x − ip

),

[a, a†] = 1.

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One can show that

[a, T ] = (r − 1)T a

with

r = 1+ α2u2

2− αu

(1+ α2u2

4

)1/2

.

Since |n〉 are eigenstates of T ,

T |n〉 = λn|n〉,and the commutator [a, T ] yields

(r − 1)T a|n〉 = (aT − T a)|n〉or (r − 1)λn−1|n− 1〉 = (λn − λn−1)|n− 1〉,

(r − 1)λn−1 = λn − λn−1,

λn

λn−1= r ⇒ λn = Krn (K = constant).

Hence

T = AKr(H/W).

To determine the constant K , consider the trace of T

1

AtrT = K

∞∑n=0

rn+1/2 = Kr1/2

1− r.

Note that from Eq. 11.43 it follows that

r =[√

1+ α2u2

4− αu

2

]2

.

Hence

r1/2

1− r=

√1+ α2u2

4 − αu2

αu

[√1+ α2u2

4 − αu2

] = 1

αu.

The definition of T yields

1

AtrT =

∫dx〈x|e− αu2

4 x2e−

α2 p

2e−

αu24 x2 |x〉

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11.13 Eigenfunctions from evolution kernel 249

=∫ +∞

−∞dxe−

αu22 x2〈x|e− α

2 p2 |x〉

=√

2π

αu2·√

1

2πα= 1

αu,

which yields K = 1.Consequently

T =√

2πε

mrH/w.

The path integral is hence given by

Z = tr(T 2N+1)

=(

2πεr

m

) 2N+12 1

1− r2N+1.

Furthermore, the propagator is given by

Dn = E [xn+mxm] = 1

Ztr(xT nxT 2N+1−n)

= 1

2ω′(1− r2N+1)

(rn + r2N+1−n) ,

which is seen to agree with the earlier result given by Eq. 11.41

11.13 Eigenfunctions from evolution kernel

Recall

K(x, x ′; τ) = 〈x|e−τH |x ′〉=∑n

e−τEn〈x|ψn〉〈ψn|x ′〉

=∑n

e−τEnψn(x)ψ∗n (x

′),

where |ψn〉, En are the eigenfunctions and eigenvalues of H . Expanding the sumyields

K(x, x ′; τ) = eτE0ψ0(x)ψ

∗0

(x ′)+ e−τE1ψ1(x)ψ

∗1

(x ′)+ · · ·

The evolution kernel can be expanded in a power series of e−τEn term by term,which yields the eigenfunctions.

Consider the harmonic oscillator with

K(x, x ′; τ) = √

mω

2π sinh(ωτ)exp

{− mω

2 sinhωτ

[(x2 + x

′2) coshωτ − 2xx ′]}

.

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For τ →∞, one has to leading order

K(x, x ′; τ) � e−ωτ2

√mω

2πe−mω

2

(x2+x′2

),

⇒ E0 = ω

2ψ0(x) =

(mω2π

)1/4e−

mω2 x2

.

To next leading order

K(x, x ′; τ) � ψ0(x)ψ0(x

′)[e−

ωτ2 + 2mωxx ′e−

32ωτ

+{

1

2+ 2m2ω2x2x

′2 −mω(x2 + x′2)

}e−

52ωτ + · · ·

]and hence by inspection

E1 = 3

2ω , ψ1(x) =

(mω2π

)1/4 · √2mω · xe−mω2 x2

,

E2 = 5

2ω , ψ2(x) =

(mω2π

)1/4 1√2

(2mωx2 − 1

)e−

mω2 x2

.

11.14 Summary

The simple harmonic oscillator is a leading exemplar in the study of path integralssince many of the key ideas can be fully worked out for the oscillator. Startingfrom the Hamiltonian, the propagator of the oscillator was derived using state spacemethods. The oscillator path integral was then defined and the evolution kernel wasevaluated, both in the presence and absence of an external current.

The path integral for discrete time and for a finite interval was shown to reduceto an ordinary multiple integral. The propagator for the finite and discrete timewas exactly evaluated using both the multiple integral formulation and the transfermatrix formulation, showing the consistency of the integral and differential formu-lations.

The evolution kernel was lastly used to derive the ground state and the first fewexcited states of the simple harmonic oscillator.

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12

Gaussian path integrals

Several path integrals are exactly evaluated here using Gaussian path integration.A few general ideas are illustrated using the advantage of being able to exactlyevaluate Gaussian path integrals.

Path integrals defined over a particular collection of allowed indeterminate pathscan sometimes be represented by a Fourier expansion of the paths. This leads totwo important techniques for performing path integrations:

1. Expanding the action about the classical solution of the Lagrangian;2. Expanding the degree of freedom in a Fourier expansion of the allowed paths.

Various cases are considered to illustrate the usage of classical solutions andFourier expansions, and these also provide a set of relatively simple examples tofamiliarize oneself with the nuts and bolts of the path integral. The Lagrangian ofthe simple harmonic oscillator is used for all of the following examples; all thecomputations are carried out explicitly and exactly.

The following different cases are considered:

• Correlators of exponential functions of the degree of freedom are discussed inSection 12.1.

• The generating functional for periodic paths is evaluated in Section 12.2.• The path integral required for evaluating the normalization constant for the os-

cillator evolution kernel is discussed in Section 12.3. The path integral entailssumming over all paths that start from and return to the same fixed position.

• Section 12.4 discusses the evolution kernel for a particle starting at an initialposition xi and, after time τ , having a final position that is indeterminate.

• The path integral of a free particle in the presence of a fixed external current j isdiscussed in Section 12.5.

• Section 12.6 discusses the evolution kernel for a system with indeterminate ini-tial and indeterminate final positions.

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252 Gaussian path integrals

• The evolution kernel of the harmonic oscillator was derived using the classicalsolutions. Given its importance, the evolution kernel of the oscillator is derivedusing the Fourier expansion in Section 12.7.

In Section 12.8 the path integral for a free particle in the presence of a magneticfield is exactly evaluated; this example has a new feature in that the magnetic fieldcouples to the velocity of the particle. The calculation is carried out in Minkowskitime.

12.1 Exponential operators

Gaussian integration has the important property, as seen in Eq. 7.8, that the gen-erating function can be exactly evaluated. This result allows the exact study ofexponential operators of the coordinate operators, namely

eix(t)/a,

where a is an arbitrary length scale required for making the exponent dimensional.For the case of the simple harmonic oscillator, the degree of freedom x(t) is some-times called log normal since, in the path integral formulation, x(t) can be consid-ered a Gaussian (normal) random variable.

The correlators of the exponential operator can be exactly evaluated. FromEqs. 7.8 and 11.9 the infinite time case yields

Z[j ] = 1

Z

∫DXei

∫dtj (t)x(t)eS

= exp

{− 1

2mω

∫dtdt ′j (t)e−ω|t−t

′|j (t ′)}. (12.1)

Consider the correlator of two operators

G(t, t ′) = 1

Z

∫DXeSeix(t)/ae−ix(t

′)/a = 1

Z

∫DXei

∫dτj (τ )x(τ )eS,

⇒ j (τ ) = 1

a

[δ(τ − t)− δ(τ − t ′)

].

Hence

G(t, t ′) = exp

{− 1

2mωa2

∫dτdτ ′e−ω|τ−τ

′|

× [δ(τ − t)− δ(τ − t ′)

] [δ(τ ′ − t

)− δ(τ ′ − t ′)]}

= exp

{− 1

mωa2

(1− e−ω|t−t

′|)}

.

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12.2 Periodic path integral 253

Taking the limit of ω→ 0 yields

G(t, t ′) � exp

{− 1

ma2|t − t ′|

}= e

− |t−t ′|ξ ,

where correlation time is given by ξ = ma2.Note that for ω = 0, namely, the case of a free particle, it can be shown that the

correlation of a product of operators

eix1(t)/a1eix2(t)/a2 . . . eix1(t)/an

is nonzero only if

1

a1+ 1

a1. . .+ 1

an= 0.

12.2 Periodic path integral

The finite time action for a simple harmonic oscillator is

S = −m2

∫ τ

0dt (x2 + ω2x2). (12.2)

The correlator for finite time is given in Eq. 11.6,

D(t, t ′, τ

) = 1

Z(τ)tr(T[xt xt ′

]e−τH

), Z(τ) = tr(e−τH ).

Due to the trace, the propagator is given by the Feynman path integral over allperiodic paths, namely for x(t + τ) = x(t). More precisely, the propagator isgiven by

D(t, t ′, τ

) ≡ Dt−t ′ = 1

Z

∫DXx(t)x(t ′)eS,

x(t + τ) = x(t).

To calculate the propagator, consider the generating function given by

Z[j ] = 1

Z

∫DXeS+

∫j (t)x(t) = exp

{1

2

∫j (t)Dt−t ′j (t ′)

}. (12.3)

All the periodic path x(t) are given by the Fourier expansion

x(t) =+∞∑

n=−∞e2πint/τ xn. (12.4)

Since ∫ τ

0dte2πint/τ e2πimt/τ = τδn−m, (12.5)

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the action is given by

S = −mτ2

∑n

{(2πn

τ

)2

+ ω2

}xnx−n. (12.6)

The Fourier modes xn have all decoupled and one can exactly evaluate the pathintegral. From Eq. 12.3, using Gaussian integration yields

Z[j ] = exp

{1

2

∫j (t)Dt−t ′j (t ′)

},

where

Dt−t ′ = 1

mτ

+∞∑n=−∞

e2πin(t−t ′)/τ 1(2πnτ

)2 + ω2}

= 1

mτ

( τ

2π

)2 +∞∑n=−∞

e2πin(t−t ′)/τ

n2 + (ωτ2π )2. (12.7)

Using the identity given in Eq. 11.30, namely

+∞∑n=−∞

einθ

a2 + n2= π

a

cosh[(π − |θ |)a]sinhπa

,

yields

Dt−t ′ = 1

mτ

( τ

2π

)2(

2π

ωτ

)π

cosh(ωτ2 − ωτ |t − t ′|)sinh(ωτ2 )

= 1

2ωm

cosh(ωτ2 − ωτ |t − t ′|)sinh(ωτ2 )

and reproduces the result obtained earlier in Eq. 11.8 using the oscillator algebra.

12.3 Oscillator normalization

The normalization of the simple harmonic oscillator is given by the path integral∫DzeS[z] =

√mω

2π sinhωτ,

boundary conditions : z(0) = 0 = z(τ ).

As can be read from the boundary conditions, the path integral corresponds to thedegree of freedom starting and ending at the same point after time τ .

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12.3 Oscillator normalization 255

A Fourier expansion for z(t) yields

z(t) =∞∑n=1

sin

(nπt

τ

)zn,

z(t)

dt=

∞∑n=1

(nπτ

)cos

(nπt

τ

)zn.

Note that all the Fourier coefficients z∗n = zn are real since z(t) is real.Using the orthogonality relation,∫ τ

0dt sin

(nπt

τ

)sin

(mπt

τ

)=∫ τ

0dt cos

(nπt

τ

)cos

(mπt

τ

)= τ

2δn−m

yields the simplification for the action

S[z] = −τ2

m

2

∞∑n=1

{(nπτ

)2 + ω2

}z2n.

The action has completely factorized into a sum over the Fourier modes and the∫DZ path integral factorizes into infinitely many discrete integrations, one for

each zn. We obtain (N is a normalization constant)∫Dz eS[z] = N ′

∞∏n=1

∫ +∞

−∞dzne

− τ2 λnz

2n

= N ′∞∏n=1

√2π

τλn= N

∞∏n=1

(1(

nπτ

)2 + ω2

)1/2

= N∞∏n=1

1[1+ (ωτ)2

n2π2

]1/2 .

Since∞∏n=1

[1+ θ2

n2π2

]= sinh θ

θ,

the path integral yields ∫Dz eS[z] = N

√ωτ

sinhωτ.

Recall from Eq. 11.16, for the simple harmonic oscillator,

K(x ′, x; τ) = AN ′√

ωτ

sinhωτeSc .

For ω → 0, the simple harmonic oscillator reduces to the free particle. Theclassical action is given by limω→0 Sc(ω) → − m

2τ (x − x′)2. The evolution kernel

for a free particle, from Eq. 11.36, is given by

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limω→0

K(x, x ′; τ) =√

m

2πτe−

m2τ (x−x′)2 .

Hence

AN ′ =√

m

2πτ,

yielding the result

K(x′, x; τ) =

√mω

2π sinhωτeSc.

12.4 Evolution kernel for indeterminate final position

The simple harmonic oscillator’s action is given by

S = −m2

∫ τ

0(x2 + ω2x2)dt.

The transition amplitude of the degree of freedom from the initial to the final po-sition is denoted by K(x, x ′; τ). The transition amplitude K(x; τ) from x to allpossible positions at the final time is

K(x; τ) =∫

dx ′〈x|e−τH |x ′〉 =∫ +∞

−∞dx ′K(x, x ′; τ).

There are three ways of obtaining K(x; τ):• One can first evaluate the full evolution kernel K(x, x ′; τ) and then integrate

over x ′.• Since the boundary condition [dx(t)/dt]|t=τ = 0 is equivalent to integrating

over all possible values of the final position xf , one finds the classical solutionfor x(t) with boundary conditions given by

x(0) = x,dx(t)

dt

∣∣∣t=τ= 0. (12.8)

The classical solution provides the classical action that contains the dependenceon the final position x.

• Another method for evaluating K(xi; τ) is by directly expanding the paths usinga Fourier expansion that respects the boundary conditions given in Eq. 12.8,namely

x(t) = x +∞∑n=0

xn sin

[(2n+ 1

2

)πt

τ

].

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12.4 Evolution kernel for indeterminate final position 257

Evolution kernel of harmonic oscillator

The simple harmonic oscillator is given by Eq. 11.21. This yields

K(x, x ′; τ) =√

mω

2π sinhωτexp

{− mω

2 sinhωτ

[(x2 + x ′2

)coshωτ − 2xx ′

]}.

Hence, the kernel for reaching an arbitrary point after time τ is given by

K(x; τ) =∫

dx ′K(x, x ′; τ)

= N exp

{−mω coshωτ

2 sinhωτx2

}√2π sinhωτ

mω coshωτ

× exp

{1

2

1

mω

sinhωτ

coshωτ

( mω

sinhωτx)2}=√

1

coshωτeF ,

where

F = − mω

2 sinhωτx2

[coshωτ − 1

coshωτ

]= − mωx2

2 sinhωτ

sinh2 ωτ

coshωτ

= −mω2x2 tanhωτ.

Classical solution

The evolution kernel is determined by using the classical solution with appropriateboundary conditions,

δS

δx(t)= 0 = x − ω2x, xc(t) = Ae−ωt + Beωt .

We impose the following boundary conditions on the classical solution:

xc(0) = x; dxc(τ )

dt= 0,

which yields

x = A+ B,dx(τ)

dt= 0 = −ω (Ae−ωτ − Beωτ

)and hence

A = Be2ωτ = x

1+ e2ωτe2ωτ .

The classical solution is given by

xc(t) = x

1+ e2ωτ

[e2ωτ e−ωt + eωt

],

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with the classical action

Sc = −m2

[xc(τ )xc(τ )− xc(0)xc(0)]

= mx

2

xω

1+ e2ωτ

[−e2ωτ + 1]

= −mωx2

2

sinhωτ

coshωτ= −mω

2x2 tanhωτ.

The evolution kernel is hence

K(x; τ) = N eSc .

Finding the classical solution does not allow for determination of the normalizationN , since one cannot use the composition law to generate the equation for N .

Fourier expansion

The boundary conditions

x(0) = x; dx(τ)

dt= 0

are realized by the Fourier expansion

x(t) = x +∞∑n=0

xn sin

[(2n+ 1

2

)πt

τ

].

All possible paths obeying the given boundary condition are obtained by varyingthe real coefficients xn. The orthogonality equations∫ tf

ti

dt cos

[nπ

(t − ti)

τ

]cos

[mπ

(t − ti)

τ

]

=∫ tf

ti

dt sin

[nπ

(t − ti)

τ

]sin

[mπ

(t − ti)

τ

]= τ

2δm−n, m, n ≥ 1 (12.9)

yield the action

S = −m2

τ

2

∞∑n=0

[(2n+ 1)2π2

4τ 2+ ω2

]x2n

− mω2

2

∫ τ

0dt

(x2 + 2x

∞∑n=0

sin[(

2n+ 1

2

)πt

τ]xn).

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12.4 Evolution kernel for indeterminate final position 259

Note ∫ τ

0dt sin

[(2n+ 1

2

)πt

τ

]= − 2τ

(2n+ 1)π

{cos

[(2n+ 1

2

)π

]− 1

}

= 2τ

(2n+ 1)π.

Thus

S = −m2

π2

8τ

∞∑n=0

[(2n+ 1)2 +

(2ωτ

π

)2]x2n −

mτω2

2x2

− 2τmxω2

π

∞∑n=0

1

(2n+ 1)xn.

Performing the path integral yields

K(x; τ) = const∏n

[2π

(2n+ 1)2 + ( 2ωτn)2

]1/2

eF = N eF .

Using the numerical identity

∞∑n=0

1

(2n+ 1)4 + ( 2ωτπ)2(2n+ 1)2

= π4

26ω2τ 2

(1− tanhωτ

ωτ

)

yields

F = −mτω2

2x2 +

(2τmxω2

π

)2

× 8τ

mπ2×

∞∑n=0

1

(2n+ 1)2

[1

(2n+ 1)2 + ( 2ωτπ)2

]

= −mτω2

2x2 + 32mτ 3ω4

π4x2 π4

64ω2τ 2

(1− tanhωτ

ωτ

)

= −mτω2

2x2 + mω2τ

2

(1− tanhωτ

ωτ

)x2

= −mωx2

2tanhωτ,

and thence the expected solution

K(x; τ) = N eSc .

A careful treatment of the normalization of the path integral, as discussed in detailin Section 12.3, can be used to evaluate N .

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12.5 Free degree of freedom: constant external source

Consider a free degree of freedom with action with an external current j given by

S = −1

2m

∫ τ

0dt

(dx

dt

)2

+ j

∫ τ

0dtx

and the path integral

K(xf , xi; τ) =∫

DxeS,

boundary conditions : x(0) = xi; x(τ) = xf .

The general solution for an arbitrary time dependent j (t) given in Eq. 11.29 canbe used to perform the path integral. A different approach is to employ a changeof variables to simplify the source free quadratic action and then evaluate the pathintegral.

Consider change of variables,

dx(t)

dt= ξ(t) ⇒ x(t) = xi +

∫ t

0dt ′ξ(t ′),

x(τ ) = xi +∫ τ

0dt ′ξ(t ′).

The requirement that x(τ) = xf needs to be put in as a constraint in the pathintegral. Hence

K =∫

Dξδ(xi +∫ τ

0dtξ(t)− xf )e

S

=∫

dη

2π

∫Dξeiη(xi−xf )eiη

∫ τ0 dtξ(t)eS

=∫

dη

2π

∫Dξeiη(xi−xf )eS .

The action is given by

S = iη

∫ τ

0dtξ(t)− 1

2m

∫ τ

0dtξ 2 − j

∫ τ

0dt

∫ t

0dt ′ξ(t ′)− jxiτ

= −1

2m

∫ τ

0dtξ 2 +

∫ τ

0dt[iη − j (τ − t)]ξ − jxiτ.

Performing the ξ path integral yields1

K =∫

dη

2πeiη(xi−xf )eF−jxiτ , (12.10)

1 Using the results given in Section 11.8 for the free particle path integral one can obtain the normalizationgiven in Eq. 12.10.

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12.6 Evolution kernel for indeterminate positions 261

where

F = 1

2m

∫ τ

0dt[iη − j (τ − t)]2 = − τ

2mη2 − i

2mjητ 2 + 1

6mj 2τ 3.

Doing the η-integration yields

K(xf , xi; τ) = N eF ,

where2

F = − m

2τ(xi − xf − τ 2

2mj)2 + 1

6mj 2τ 3 − jτxi

= − m

2τ(xi − xf )

2 − mτ

2(xi + xf )j + 1

24mj 2τ 3,

N =√

m

2πτ.

Hence, the evolution kernel is given by

K(xf , xi, τ ) =√

m

2πτeF .

The normalization is the same as the case of j = 0.

12.6 Evolution kernel for indeterminate positions

A particle with paths that have indeterminate initial and final positions can be stud-ied using a Fourier expansion of its possible paths. Let the particle have mass m andspring constant ω, and be subject to an external force j ; the particle’s Lagrangianand action, from initial time and position ti , xi to final time and position tf , xf , aregiven by

S =∫ tf

ti

dtL, L = −1

2m

(dx

dt

)2

− 1

2mω2x2 + jx. (12.11)

The transition amplitude is given by

K(x; ti; xf , tf ) = 〈xf |e−(tf−ti )H |xi〉 =∫

DxeS.

For the case when the initial and final positions xi, xf are indeterminate the transi-tion amplitude is given by

K(ti, tf ; j) =∫ +∞

−∞dxidxfK(xi, ti; xf , tf ). (12.12)

2 There are some typographical errors in the result given for this computation in Feynman and Hibbs (1965).

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The integration over the initial and final positions that results in Z(ti, tf ; j) has asimple expression in terms of the boundary conditions imposed on the path integra-tion measure

∫DX. Instead of the initial and final positions being fixed, the paths

x(t) now have

dx(ti)

dt= 0 = dx(tf )

dt: Neumann BCs. (12.13)

The Neumann boundary conditions (BCs) allow one to do an integration by partsof the action given in Eq.(12.11), yielding the action

S = −1

2m

∫ tf

ti

dtx(t)

[− d2

dt2+ ω2

]x(t)+

∫ tf

ti

dtj (t)x(t). (12.14)

The generating functional is given by the path integral

K(ti, tf ; j) =∫

DxeS. (12.15)

The path integral∫DxeS is performed over all paths (functions) x(t) that satisfy

the Neumann boundary conditions given in Eq. (12.13). All such functions can beexpanded in a Fourier cosine series as follows:

x(t) = a0 +∞∑n=1

an cos

[nπ

(t − ti)

τ

], τ ≡ tf − ti ,

∫DX = N

∞∏n=0

∫ +∞

−∞dan : infinite multiple integral,

(N is a normalization constant). The orthogonality equations∫ tf

ti

dt cos

[nπ

(t − ti)

τ

]cos

[mπ

(t − ti)

τ

]

=∫ tf

ti

dt sin

[nπ

(t − ti)

τ

]sin

[mπ

(t − ti)

τ

]= τ

2δm−n, m, n ≥ 1 (12.16)

yield, for the action given in Eq. 12.14,

S = −1

2mω2τ

{a2

0 +1

2

∞∑n=1

[1+ (

nπ

ωτ)2]a2n

}

+∫ tf

ti

dtj (t)

{a0 +

∞∑i=1

an cos

[nπ

(t − ti)

τ

]}

= −1

2

∞∑n=0

κna2n +

∞∑n=0

jnan, (12.17)

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12.6 Evolution kernel for indeterminate positions 263

where

κ0 = mω2τ, κn = 1

2mω2τ

[1+

(nπωτ

)2], n ≥ 1,

jn =∫ tf

ti

dtj (t) cos

[nπ

(t − ti)

τ

], n = 0, 1, . . .∞.

All the Gaussian integrations over the variables an have decoupled in the actionS given in Eq. 12.17. The path integral has been reduced to an infinite product ofsingle Gaussian integrations, each of which can be performed using Eq. 7.7. Hence,from Eqs. 12.15 and 12.17

K(ti, tf ; j) =∫

DXeS = N∞∏n=0

[ ∫ +∞

−∞dane

− 12 κna

2n+jnan

]

= exp

{1

2

∞∑n=0

jn1

κnjn

}, (12.18)

yielding

K(ti, tf ; j) = e12

∫ tfti

dtdt ′j (t)D(t,t ′;ti ,tf )j (t ′), (12.19)

where the function D(t, t ′; ti , tf ) is the propagator for the simple harmonic oscil-lator. Using Eq. 12.19 to factor out the j (t)s from Eq. 12.18 yields

D(t, t ′; ti , tf )

= 1

mω2τ

{1+ 2

∞∑n=1

cos

[nπ

(t − ti)

τ

] [1

1+ (nπωτ)2

]cos

[nπ

(t ′ − ti)

τ

]}.

(12.20)

Let θ = t − ti > 0 and θ ′ = t ′ − ti > 0; then

2∞∑n=1

cos(nπθ/τ) cos(nπθ ′/τ)1+ (nπ

ωτ)2

=(ωτπ

)2 ∞∑n=1

cos(nπ(θ + θ ′)/τ )+ cos(nπ(θ − θ ′)/τ )(ωτπ)2 + n2

. (12.21)

The summation over integer n is performed using the identity3

∞∑n=1

cos(nθ)

a2 + n2= π

2a

cosh(π − |θ |)asinhπa

− 1

2a2(12.22)

3 The formula given in Eq. 11.30 is valid for any complex number a, and will be applied in later discussionsfor a case where a is indeed a complex number. The branch of the square root of a2 that is taken on the righthand side need not be specified since the rhs is a function of a2.

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and yields the result

D(t, t ′; ti, tf ) = coshω{τ − |θ − θ ′|}+ coshω

{τ − (θ + θ ′)

}2mω sinhωτ

. (12.23)

Hence, from Eq. 12.23, and since τ = tf − ti , the propagator is given by

D(t, t ′; ti , tf )

= coshω{(tf − ti)− |t − t ′|}+ coshω

{(tf − ti)− (t + t ′ − 2ti)

}2mω sinhω(tf − ti)

. (12.24)

Note that the propagator (also called a Green’s function) satisfies the differentialequation

m

[− d2

dt2+ ω2

]D(t, t ′; ti , tf

) = δ(t − t ′

) : Neumann BCs.

The case of infinite time for Eq. 12.24 is obtained by taking the limit of ti →−∞ , tf →+∞, and yields

D(t, t ′) = 1

2mωe−ω|t−t

′|, (12.25)

which has been derived earlier using different methods.

12.7 Simple harmonic oscillator: Fourier expansion

Given the importance of the simple harmonic oscillator’s evolution kernel, a deriva-tion is given below using the technique of Fourier expansions.

Consider a quantum particle with an initial position xi and, after time τ , with afinal position xf . The evolution kernel for this case has been obtained earlier in Sec-tion 11.6 based on evaluating the solution of the classical equation of motion. Thelimitation of finding the classical solution is that for many nonlinear Lagrangians,it is usually not possible to obtain the classical trajectory. The Fourier expansionavoids this problem by directly enumerating all the paths with a given initial andfinal position and hence can be used for a great variety of problems.

Consider the Fourier expansion for the possible paths of the quantum particle,

x(t) = xf +∞∑n=0

cos

[(2n+ 1)πt

2τ

]xn. (12.26)

The boundary conditions are

x(0) = xf +∞∑n=0

xn,

x(τ ) = xf .

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12.7 Simple harmonic oscillator: Fourier expansion 265

At time t = τ , the final condition is automatically satisfied. Note the velocity ofthe particle at t = 0 is zero, since

dx(0)

dt=

∞∑n=0

(2n+ 1)π

2τsin[0]xn = 0.

As discussed for the two previous cases, zero velocity at t = 0 means that theposition x(0) is indeterminate. Hence, to fix the initial position of the particle to bexi , one needs to put a delta-function constraint into the path integral, namely

xi = xf +∞∑n=0

xn : constraint.

Hence, the evolution kernel is given by

K(xi, xf ; τ) = 〈xf |e−τH |xi〉 (12.27)

= N ′∞∏n=0

∫ +∞

−∞dxn δ(

∞∑n=0

xn + xf − xi)eS

=∫

Dx

∫ +∞

−∞dη

2πeiη(

∑∞n=0 xn+xf−xi ) eS, (12.28)

where N is a normalization constant.Using the orthogonality relation∫ τ

0dt sin

[(2n+ 1)πt

2τ

)sin

[(2m+ 1)πt

2τ

]

=∫ τ

0dt cos

[(2n+ 1)πt

2τ

]cos

[(2m+ 1)πt

2τ

]= τ

2δn−m

yields a simplification for the action,

S = −m2

∫ τ

0dt

[(dx(t)dt

)2 + ω2x2(t)

]

= −m2

π2

8τ

∞∑n=0

[(2n+ 1)2 +

(2ωτ

π

)2]x2n −

m

2ω2τx2

f

+∞∑n=0

2mω2τ

(2n+ 1)π(−1)nxf xn,

since ∫ τ

0dt cos

[(2n+ 1)πt

2τ

]= − 2τ

(2n+ 1)π(−1)n.

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All the Fourier modes xn have completely factorized in the action. Performingthe Gaussian path integration (over the Fourier modes xn) given in Eq. 12.28 yields

K(xi, xf ; τ) = N∫ +∞

−∞dη

2πeF(η),

F (η) = 1

2m

8τ

π2

∞∑n=0

1

(2n+ 1)2 + ( 2ωτπ

)2

[iη + 2mω2τ

(2n+ 1)π(−1)nxf

]2

.

Using the numerical identities

∞∑n=0

1

(2n+ 1)2 + ( 2ωτπ

)2 =π2

8ωτtanh (ωτ),

∞∑n=0

(−1)n

(2n+ 1)3 + ( 2ωτπ

)2(2n+ 1)

= π3

8(ωτ)2· sinh2

(ωτ2

)cosh(ωτ)

,

∞∑n=0

1

(2n+ 1)4 + ( 2ωτπ

)2(2n+ 1)2

= π4

25ω2τ 2

(1− tanhωτ

ωτ

),

yields the result

F = − tanh(ωτ)

2mωη2 − 2ixf η

sinh2(ωτ2 )

cosh(ωτ)+ 1

2x2fmω

2τ

(1− tanh(ωτ)

ωτ

)

− 1

2x2fmω

2τ − iη(xi − xf )

= − tanh(ωτ)

2mωη2 − iη

[xi − xf

cosh(ωτ)

]− 1

2x2fmω tanh(ωτ).

Performing the η integration gives the final result

K(xi, xf ; τ) = N√

2πmω

tanh(ωτ)eG,

with

G = − mω

2 tanh(ωτ)

[xi − xf

cosh(ωτ)

]2

− 1

2x2fmω tanh(ωτ)

= − mω

2 tanh(ωτ)

[x2i + x2

f − 2xixf

cosh(ωτ)

]+ ξ.

The remainder ξ is zero since

ξ = −1

2mωx2

f

[1

tanh(ωτ) cosh2(ωτ)− 1

tanh(ωτ)+ tanh(ωτ)

]= 0.

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12.8 Evolution kernel for a magnetic field 267

Once G has been determined, the overall normalization of the evolution kernel canbe fixed by using the composition law, as in Section 11.5. Hence, the evolutionkernel is given by

K(xi, xf ; τ) =√

mω

2π sinh(ωτ)eG,

which agrees with the result obtained earlier in Eq. 11.21.

12.8 Evolution kernel for a magnetic field

A free charged particle moving in an electromagnetic field couples with the vec-tor potential which, in vector notation, is given by A. The Lagrangian, for x =(x, y, z), is given in Minkowski time by

L = 1

2mx2 + eA · x, x ≡ dx

dt. (12.29)

The calculation in this section is carried out for Minkowski time to get a flavor forsuch calculations; in particular, it will be seen that the computations in Minkowskitime have a plethora of i factors which are quite unnecessary in obtaining the resultfor Euclidean time.

Consider a free charged particle interacting with a constant external magneticfield of strength B acting along the z-direction. The vector potential A is chosen tobe in the symmetric gauge,

A = 1

2B(−y, x, 0). (12.30)

The Lagrangian is then given by

L(t) = 1

2m(x2 + y2

)2 −mσ (xy − xy) , σ = eB

2m, (12.31)

where σ is the cyclotron frequency. The particle moves only in the xy-plane andhence the z-axis can be completely dropped from the problem. The evolution kernelis given by the path integral

K(xb, yb, tb; xa, ya, ta) = 〈xb, yb, |e−i(tb−ta)H |xa, ya〉 =∫

DxDyeiS, (12.32)

where H is the Hamiltonian of the system; the action S is given by

S =∫ tb

ta

dtL(t).

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The action is quadratic in the degrees of freedom and hence the evolution kernelcan be obtained using the classical solution. The equations of motion for the La-grangian in Eq. 12.29 yield the classical solution xc, yc given by

xc = 2σ yc, yc − 2σ xc. (12.33)

Shifting the degree of freedom x, y by xc, yc

x → x + xc, y → y + yc

and keeping the same notation for the sake of simplicity yields the action

S [x + xc, y + yc; T ] = Sc [xc, yc; T ]+ S0 [x, y; T ]+�S, T = tb − ta, (12.34)

�S =∫ tb

ta

dt {m(xxc + yyc)−mσ (xyc + yxc − xyc − yxc)} .

The shifted quantum degrees of freedom x, y have new boundary conditionsgiven by

x(ta) = 0 = x(tb), y(ta) = 0 = y(tb) (12.35)

and lead to the result

�S = 0.

The decoupling of the classical solution – since �S = 0 – from the quantumvariables is true for any quadratic Lagrangian.

From Eqs. 12.32, 12.34, and 12.35, the Minkowski time evolution kernel isgiven by

K(xb, yb; xa, ya; T ) = eiSc[xc,yc;T ]∫

DxDyeiS0[x,y;T ]

= N (T )eiSc[xc,yc;T ]. (12.36)

Classical action

The classical action is evaluated using the Lagrangian given in Eq. 12.29,

Sc[xc, yc; T ] =∫ tb

ta

dt

{1

2m(x2c + y2

c

)−mσ (xcyc − xcyc)

}. (12.37)

Integrating by parts, Eq. 12.37 becomes4

Sc[xc, yc; T ] = 1

2m(xbxb + ybyb − xaxa − yaya) .

4 Since∫ tbtadt (xcxc + ycyc + 2σ xcyc − 2σxcyc) = 0.

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12.8 Evolution kernel for a magnetic field 269

Furthermore, integrating Eq. 12.33 gives

xc = 2σ(yc + C), yc = −2σ(xc −D), (12.38)

where C and D are constants of the integration. Equation 12.38 is the equation fora two-dimensional harmonic oscillator,

xc = −ω2(xc −D), yc = −ω2(yc + C), ω = 2σ. (12.39)

The solutions of Eq. 12.39 are given by

xc = A cosωt + B sinωt +D, yc = −A sinωt + B cosωt − C.

The constants C and D are derived from the boundary conditions,

xa = A cosωta + B sinωta +D, ya = −A sinωta + B cosωta − C,

xb = A cosωtb + B sinωtb +D, yb = −A sinωtb + B cosωtb − C. (12.40)

Solving Eq. 12.40 yields

B = (xb − xa) cos σT ′ − (yb − ya) sin σT ′

2 sin σT,

A = −(xb − xa) sin σT ′ − (yb − ya) cos σT ′

2 sin σT,

D = 1

2(xb + xa)+ (yb − ya) cos σT

2 sin σT,

C = 1

2(yb + ya)+ (xb − xa) cos σT

2 sin σT, (12.41)

where T = tb − ta , T ′ = ta + tb. Furthermore, Eq. 12.38 yields

xcxc + ycyc = 2σ(Cxc +Dyc).

Equation 12.37, together with Eqs. 12.38 and 12.41, yields

Sc[xc, yc; T ] = mσ {C(xb − xa)−D(yb − ya)},

and we obtain the final result

Sc[xc, yc; T ] = mσ[cos σT {(xb − xa)

2 + (yb − ya)2}

2 sin σT+ ybxa − xbya

]. (12.42)

The normalization of the evolution kernel N (T ) is given by the consistency equa-tion as discussed in Section 4.8,

K(xb, yb; xa, ya; 2T ) =∫

dξdηK(xb, yb; ξ, η; T )K(ξ, η; xa, ya; T ), (12.43)

⇒ N (2T ) = N 2(T )�(T ). (12.44)

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The evolution kernel is given by Eq. 12.36 and 12.42; performing the Gaussianintegration given on the right hand side of Eq. 12.43 yields, from Eq. 12.44, thefollowing recursion equation:

N (2T ) = N 2(T )iπ sin(σT )

mσ cos(σT )⇒ N (T ) = mσ

2πi sin σT. (12.45)

The final result for the Minkowski time path integral, from Eqs. 12.36, 12.42,and 12.45, is thus

K(xb, yb; xa, ya; T ) = N (T )eiSc[xc,yc;T ] = mσ

2πi sin σT

× exp[ i

2

mσ cos σT

sin σT

{(xb − xa)

2 + (yb − ya)2}+ imσ

(ybxa − xbya

)].

(12.46)

The evolution kernel for Euclidean time τ > 0 is given by τ = iT and

KE(xb, yb; xa, ya; τ) = mσ

2π sinh στ

× exp[− 1

2

mσ cosh στ

sinh στ

{(xb − xa)

2 + (yb − ya)2}+ imσ

(ybxa − xbya

)].

12.9 Summary

The techniques of using classical solutions and Fourier expansions for analyzingpath integrals were illustrated using Gaussian path integration. The exact solutionsobtained show the flexibility and power of these techniques, which have gener-alizations to many nonlinear systems. The evolution kernel for various quadraticLagrangians, in particular of the simple harmonic oscillator, reveal the rich mathe-matical structure of these relatively simple theories.

The evolution kernel of a charged particle in a magnetic field, evaluated for bothMinkowski and Euclidean time, shows the equivalence of these approaches as wellas illustrating the relative simplicity of the results in Euclidean time.

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Part five

Action with acceleration

Introduction to part five

The most widely used actions in quantum mechanics have a kinetic term that is thevelocity squared of the degree of freedom and a potential term that depends on thedegree of freedom.1 The kinetic term fixes the equal time commutation equation ofthe degree of freedom, as shown in Section 6.7. Furthermore, the velocity term inthe Lagrangian entails that all the possible indeterminate paths obey two boundaryconditions, which in turn yields a state space that depends on the degree of free-dom only. In Section 11.13, this property of the indeterminate paths was used forderiving the state function of the harmonic oscillator.

The action with acceleration has a kinetic term that is given by the accelera-tion squared of the degree of freedom, in addition to the usual velocity and poten-tial terms. It is an example of higher derivative Lagrangians, discussed in Simon(1990). The higher derivative quantum systems have many remarkable propertiesnot present for the usual cases studied so far.

The action with acceleration arises in many diverse fields and has been widelystudied; it describes the behaviour of “stiff” polymers, of cell walls, of the for-mation of microemulsions, the properties of chromoelectric flux lines in quan-tum chromodynamics, as well as the Big Bang singularity in cosmology. TheEuclidean action and path integral were studied by Kleinert (1986), where a listof the applications of the model is given. Bender and Mannheim (2008b) haveextensively studied the Lagrangian and Hamiltonian of the model, both in Eu-clidean and Minkowski time; Mannheim and Davidson (2005) showed that themodel in Minkowski time was free of ghost states using the concept of a PT-symmetric Hamiltonian; in Bender and Mannheim (2010) they demonstrated thereality of the model’s eigenspectrum. The acceleration action has been applied to

1 Velocity dependent terms can be included in the potential.

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272 Action with acceleration

the study of cosmology, and quantum and conformal gravity by Mannheim (2011a)and Mannheim (2011b).

The Euclidean model was studied by Hawking and Hertog (2002) for its rolein quantum gravity and in string theory’s D-brane dynamics; the Euclidean pathintegral was used by Fontanini and Trodden (2011) for analyzing the ghost statesfor Minkowski time.

The acceleration action appears in the study of mathematical finance in describ-ing the time dependence of financial derivative instruments and in the study ofinterest rates and equity [Baaquie (2010)]. The Euclidean path integral has beenstudied extensively by Baaquie et al. (2012) and Yang (2012) for describing thecorrelation function of equities. The model has been applied in the study of op-tions by Baaquie and Yang (2013) and to the study of microeconomics by Baaquie(2013b).

The acceleration Lagrangian yields a pseudo-Hermitian Hamiltonian that hasbeen studied in Mostafazadeh (2002) and Wang et al. (2010). The analysis of theEuclidean Hamiltonian in this book is a continuation of the study carried out ofthe Minkowski time Hamiltonian by Bender and Mannheim (2008a), (2008b). TheEuclidean case has some new features that are absent for Minkowski time, themost important being a transparent positivity of the Euclidean state space. Thepath integral for the pseudo-Hermitian Hamiltonian has been studied, starting fromthe propagator, by Jones and Rivers (2009) and Rivers (2011).

For a special value of the parameters, the acceleration Hamiltonian becomesessentially non-Hermitian. As shown by Mannheim and Davidson (2000), (2005),and elaborated further by Mannheim (2011b), at the special value of the parametersthe Hamiltonian is described by a Jordan block matrix; the Jordan block Hamilto-nian was shown to be pseudo-Hermitian by Bender and Mannheim (2011).

The following three chapters analyze the acceleration Lagrangian and Hamilto-nian and are largely based on papers by Baaquie (2013c), (2013d). The quantummechanical system for the acceleration Lagrangian is defined in Euclidean timeand using the path integral. Many of the unnecessary complications that appear inMinkowski time formulation – and which obscure key features of the system – areabsent in the Euclidean time formulation. Furthermore, applications in biophysicsand finance are directly based on the Euclidean formulation and consequently thissystem is interesting in its own right, regardless of its connection with the theoryin Minkowski time.

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13

Acceleration Lagrangian

The Lagrangian and path integral for the action with acceleration are defined inSection 13.1 and the quadratic potential is chosen in Section 13.2 so that the pathintegral can be performed exactly using the classical solution. The propagator isderived from the infinite time action in Section 13.3.

The Hamiltonian is obtained for the Euclidean time Lagrangian using the tech-nique of Dirac brackets in Section 13.4, and it is shown in Section 13.5 that theHamiltonian yields the expected path integral. The evolution kernel is carefullystudied in Section 13.6 to prove that the Lagrangian based path integral is equiva-lent to the one derived from the quantum Hamiltonian. It is shown in Section 13.7how the boundary conditions on the indeterminate paths (integration variables) ofthe path integral are transformed in going from the Hamiltonian to the action basedformulation of the path integral. In Section 14.14, the acceleration Hamiltonian isextended to many degrees of freedom.

13.1 Lagrangian

Euclidean time is defined by the analytic continuation of Minkowski time tM = −it ,where t is Euclidean time. The degree of freedom x does not change in going fromMinkowski to Euclidean time but Minkowski velocity vM picks up an extra factorof i since it is related to Euclidean velocity v, due to Eq. 5.17, by

v = −dxdτ= i

dx

dtM= ivM. (13.1)

The minus in the definition of Euclidean velocity, v = ivM , is consistent witht = itM .

The non-Hermitian Euclidean Hamiltonian and path integral are well behavedand a complexification of the degree of freedom is required.

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274 Acceleration Lagrangian

The Euclidean time Lagrangian with acceleration is given by

L = −γ2

(d2x

dt2

)2

− α

2

(dx

dt

)2

−�(x), (13.2)

with the acceleration action for finite Euclidean time τ given by

S[x] =∫ τ

0dtL. (13.3)

The Feynman path integral for finite Euclidean time is given by

K(xf , xf ; xi, xi) =∫

DX eS[x]∣∣∣(xi ,xi ;xf ,xf )

, (13.4)∫DX = N

τ∏t=0

∫ +∞

−∞dx(t), (13.5)

where N is a normalization constant. The paths have four boundary conditions,

x(0) = xi; dx(0)

dt= xi initial position and velocity, (13.6)

x(τ) = xf ; dx(τ)

dt= xf final position and velocity. (13.7)

To evaluate the path integral exactly using the classical solution, consider thequadratic potential

�(x) = β

2x2, (13.8)

which yields the Lagrangian

L = −γ2

(d2x

dt2

)2

− α

2

(dx

dt

)2

− β

2x2. (13.9)

The path integral given Eq. 13.4 is now quadratic and can be evaluated exactlyusing the classical solution. Let xc(t) be the classical solution given by

δS[xc]δx(t)

= 0 (13.10)

that satisfies the boundary conditions,

xc(0) = xi; dxc(0)

dt= xi initial position and velocity,

xc(τ ) = xf ; dxc(τ )

dt= xf final position and velocity. (13.11)

Consider the change of integration variables, from x(t) to ξ(t),

x(t) = xc(t)+ ξ(t), (13.12)

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13.2 Quadratic potential: the classical solution 275

with boundary conditions given by

ξ(0) = 0; dξ(0)

dt= 0 initial position and velocity,

ξ(τ ) = 0; dξ(τ )

dt= 0 final position and velocity. (13.13)

The change of variables yields

S[x] = S[xc + ξ ] = S[xc] + S[ξ ],K(xf , xf ; xi, xi) =

∫DX eS = N (τ )eS[xc], (13.14)

N (τ ) =∫

Dξ eS[ξ ].

The evolution kernel given in Eq. 13.14 has been evaluated explicitly in Hawkingand Hertog (2002) by solving for the classical solution xc(t) and then obtainingS[xc] and N (τ ).

As can be directly verified from the classical action S[xc], the classical solutionxc(t) yields another equally valid classical solution xc(t) given by

xc(t) = xc(τ − t), (13.15)

with boundary conditions, from Eq. 13.11, given by

xc(0) = xc(τ ) = xf ; dxc(0)

dt= −dxc(τ )

dt= −xf , (13.16)

xc(τ ) = xc(0) = xi; dxc(τ )

dt= −dxc(0)

dt= −xi , (13.17)

⇒ S[xc] = S[xc],⇒ Sc[xf , xf ; xi, xi] = S[xi,−xi; xf ,−xf ]. (13.18)

The evolution kernel, from Eqs. 13.14 and 13.18, hence has the symmetry

K(xf , xf ; xi, xi) = K(xi,−xi; xf ,−xf ). (13.19)

13.2 Quadratic potential: the classical solution

The acceleration Lagrangian given by Eq. 13.9 is re-written with a slight change ofnotation, more suitable for the classical solution, in the following manner:

L = −1

2

(ax2 + 2b(x)2 + cx2

), S =

∫ τ

0dtL. (13.20)

The parameterization chosen in Eq. 13.20 is more suitable for studying the classicalsolutions than the one given in Eq. 13.9

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The Euler–Lagrangian equation in Eq. 13.10 yields the equation of motion; theclassical solution xc(t) satisfies the equation of motion

a....x c(t)− 2bxc(t)+ cxc(t) = 0. (13.21)

Let us choose the boundary conditions,

Initial values : x(0) = xf = x1, x(0) = xf = x2 = −vf ,

Final values : x(τ) = xi = x4, x(τ ) = xi = x3 = −vi . (13.22)

We define the parameters r and ω by

r + iω ≡√b + i

√ac − b2

a. (13.23)

From Eq. 13.21 the classical solution of equations of motion is given by

xc(t) = ert (a1 sinωt + a2 cosωt)+ e−rt (a3 sinωt + a4 cosωt). (13.24)

The parameters a1, . . . , a4 are obtained from the boundary conditions and are givenby Yang (2012) as

a1 = �[r2xf e

2rτ sin(2τω)+ ωvf e2rτ − rvf e

2rτ sin(2τω)+ rωxf e2rτ cos(2τω)

− rωxf − ωvf − 2r2xierτ sin(τω)− 2rvie

rτ sin(τω)

− ωerτ(e2rτ − 1

)cos(τω) (vi + rxi)− ω2xie

rτ sin(τω)+ ω2xie3rτ sin(τω)

],

a2 = �[r2xf

(−e2rτ)+ rvf e

2rτ + re2rτ cos(2τω)(rxf − vf

)− ω2xf e2rτ

− rωxf e2rτ sin(2τω)+ ω2xf − ωvie

rτ sin(τω)+ ωvie3rτ sin(τω)

+ ω2xierτ(e2rτ − 1

)cos(τω)+ rωxie

rτ sin(τω)+ rωxie3rτ sin(τω)

],

a3 = �erτ[r2xf e

rτ sin(2τω)+ ωvf erτ − ωvf e

3rτ + rvf erτ sin(2τω)

+ rωxf e3rτ − rωxf e

rτ cos(2τω)− 2r2xie2rτ sin(τω)− 2rvie

2rτ sin(τω)

− ω(e2rτ − 1

)cos(τω) (rxi − vi)− ω2xie

2rτ sin(τω)+ ω2xi sin(τω)],

a4 = �erτ[r2xf

(−erτ )− rvf erτ + rerτ cos(2τω)

(rxf + vf

)− ω2xf erτ

+ ω2xf e3rτ + rωxf e

rτ sin(2τω)− ωvie2rτ sin(τω)

− ω2xi(e2rτ − 1

)cos(τω)− rωxie

2rτ sin(τω)− rωxi sin(τω)+ ωvi sin(τω)].

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13.3 Propagator: path integral 277

The term � is defined by

� = 1

ω2 + ω2e4rτ + 2r2e2rτ cos(2τω)− 2e2rτ(r2 + ω2

) . (13.25)

The boundary condition given in Eq.13.22 yields the classical action

Sc = Sc(xf , vf , xi, vi) = −1

2

4∑I,J=1

xIMIJ xJ . (13.26)

From Eq. 13.18, the classical action has the symmetry

Sc[xf , vf ; xi, vi] = S[xi,−vi; xf ,−vf ] (13.27)

and hence the matrices MIJ given in Eq. 13.26 satisfy symmetry,

M11 = M44, M22 = M33,

M12 = −M34, M13 = −M24.

The action can be simplified to

Sc(xf , vf , vi , xi) = −1

2M11(x

2i + x2

f )−1

2M22(v

2i + v2

f )+M12(xivi − xf vf )

+M13(xivf − xf vi)−M14xixf −M23vivf . (13.28)

The solutions for MIJ are given by Yang (2012) and listed below:

M11 = �[2arω

(r2 + ω2

) (ω(e4rτ − 1

)+ 2re2rτ sin(2τω)) ],

M12 = �[ω2e4rτ

(2ar2 − b

)− 2r2e2rτ(2aω2 + b

)cos(2τω)

− ω2(b − 2ar2

)+ 2be2rτ(r2 + ω2

) ],

M13 = �[4arωerτ

(e2rτ − 1

) (r2 + ω2

)sin(τω)

],

M14 = �[− 4arωerτ

(r2 + ω2

) (r(e2rτ + 1

)sin(τω)+ ω

(e2rτ − 1

)cos(τω)

) ],

M22 = �[− 2arω

(ω(−e4rτ

)+ 2re2rτ sin(2τω)+ ω) ],

M23 = �[4arωerτ

(r(e2rτ + 1

)sin(τω)− ω

(e2rτ − 1

)cos(τω)

) ].

13.3 Propagator: path integral

The quadratic potential given in Eq. 13.8 yields the Lagrangian given in Eq. 13.9,namely

L = −1

2

[γ x2 + αx2 + βx2

].

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A parameterization that is more suitable for studying the Hamiltonian and statespace is given by Kleinert (1986),

L = −γ2

[x2 + (ω2

1 + ω22)x

2 + ω21ω

22x

2]. (13.29)

The Lagrangian is completely symmetric in parameters ω1 and ω2. For the case ofreal ω1 and ω2, the entire parameter space is covered by choosing say ω1 > ω2; theroots are chosen accordingly and are

ω1 = 1

2√γ

(√α + 2

√γβ +

√α − 2

√γβ

),

ω2 = 1

2√γ

(√α + 2

√γβ −

√α − 2

√γβ

), (13.30)

ω1 > ω2 for ω1, ω2 real.

Note that ω1 > ω2 real ω1 and ω2.We define the critical value αc = 2

√βγ . The parameters have three branches,

namely the real and the complex, plus the critical branch separating them, and theseare shown in Figure 13.1:

• Complex branch α < αc.Frequencies ω1, ω2 are complex,

ω1 = ω∗2 = Reiφ : R > 0, φ ∈ [−π/2, π/2]. (13.31)

• Real branch α > αc.

α

Real BranchComplex Branch

2 βγa = 0

w1 = Reif w1 = Reb

w2 = Re−bw2 = Re−if = w1*

Figure 13.1 Parameter branches for the Euclidean Lagrangian. The critical valueof αc = 2

√βγ is equivalent to ω1 = ω2.

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13.3 Propagator: path integral 279

Frequencies ω1, ω2 are real and ω1 > ω2 is chosen without any loss of gener-ality,

ω1 = Reb, ω2 = Re−b : R > 0, b ∈ [0,+∞].• Equal frequency α = αc.

This is a special case that is treated in detail in Chapter 15,

ω1 = ω2, b = 0 = φ.

Note that φ ∈ [−π/2, π/2] for all α, β > 0, and this is also the range for whichthe path integral is convergent.

The infinite time path integral is given by

Z = limT→∞

tr(e−TH

) = ∫DxeS ,

S = −1

2γ

∫ +∞

−∞dt[x2 + (ω2

1 + ω22)x

2 + ω21ω

22x

2].

The propagator is given by the path integral

G(τ) = 1

Z

∫DxeSx(t)x(t ′). (13.32)

The acceleration action is a quadratic functional of the paths x(t) and hence thepropagator can be evaluated exactly. We define the Fourier transformed variablesthat diagonalize the action, namely

x(t) =∫ +∞

−∞dk

2πeikxxk, (13.33)

⇒ S = −1

2γ

∫ +∞

−∞dk

2π

[k4 + (ω2

1 + ω22)k

2 + ω21ω

22

]x−kxk. (13.34)

Using Gaussian path integration yields

G(τ) = 1

γ

∫ +∞

−∞dk

2π

eik(t−t ′)

(k2 + ω21)(k

2 + ω22)

= 1

2γ

1

ω21 − ω2

2

[e−ω2τ

ω2− e−ω1τ

ω1

], τ = |t − t ′|, (13.35)

where the last equation has been obtained using counter integration. Figure 13.2ashows a single exponential and Figure 13.2b shows how the acceleration term inthe Lagrangian smooths out the “kink” of the exponential at x = 0; this smoothingout holds for all branches, including the real and complex branches for ω1, ω2.

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(a) (b)

Figure 13.2 (a) Single exponential. (b) Propagator for equal frequencyexp{−ω|τ |}[1+ ω|τ |].

13.4 Dirac constraints and acceleration Hamiltonian

The acceleration Lagrangian belongs to the class of higher derivative Lagrangiansfor which the canonical derivation of the Hamiltonian from the Lagrangian doesnot hold. The reason is that the term (d2x/dt2)2 in the Lagrangian does not yielda pair consisting of a canonical coordinate and momenta, as is required by thecanonical framework.

This problem of higher derivative Lagrangians was addressed almost two cen-turies ago by Ostogradski and discussed by Simon (1990); but instead of follow-ing his approach, a more direct route is taken here, based on Dirac’s analysis ofconstrained systems. In this approach, the number of independent degrees of free-dom is increased by imposing constraints that re-cast the acceleration Lagrangianinto another equivalent form in which the term (dx2/dt2)2 is re-written as a firstderivative expression, namely as (dv/dt)2, with a constraint that v = −dx/dt ; theLagrangian can now be analyzed using the canonical method.

Dirac’s analysis shows how to impose a set of consistent constraints on theenlarged phase space on which the Hamiltonian is defined so that only the physicalphase space is obtained for the higher derivative system.

The Minkowski Hamiltonian for the action acceleration has been obtained byMannheim and Davidson (2000) using the Dirac bracket approach and this analysisis now carried out for Euclidean space. The canonical transformation that takesthe Lagrangian to the Hamiltonian makes no explicit reference to time and henceone needs to write the constraint equation for the Euclidean Lagrangian to obtainthe correct Euclidean Hamiltonian.

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13.4 Dirac constraints and acceleration Hamiltonian 281

The Euclidean Lagrangian is given by Eq. 13.29 as follows (· = d/dt):

L = −γ2

[x2 + (ω2

1 + ω22)x

2 + ω21ω

22x

2]. (13.36)

The first step is to re-write the path integral; from Eqs. 5.17 and 13.1, Euclideanvelocity v is made equal to −dx/dt by imposing a delta function constraint in thepath integral; ignoring the boundary condition for now, one obtains

Z =∫

Dx exp

{∫dtL

}=∫

DxDv exp

{∫dtL

}∏t

δ[v(t)+ x(t)

],

(13.37)

=∫

DxDvDλ exp

{∫dtLD

}, (13.38)

with

LD = −γ2

[v2 + (ω2

1 + ω22)v

2 + ω21ω

22x

2]+ iλ[x + v]. (13.39)

The equivalent Euclidean Lagrangian LD has only first order derivatives andhence can be treated by the canonical method for obtaining the Hamiltonian fromthe Lagrangian, as discussed in Section 5.1. Compared to the original LagrangianL, which has one degree of freedom x(t), the equivalent Lagrangian LD has threedegrees of freedom x(t), v(t), and λ(t). Dirac’s formalism of constraints removesthe redundant degree(s) of freedom.

For simplicity, the notation for Euclidean momenta px, E and pv, E has beenabbreviated to px, pv. Let us define the canonical momenta for Euclidean time asgiven in Eq. 5.21, which yields

px = ∂LD

∂x= iλ, pv = ∂LD

∂ v= −γ v, pλ = ∂LD

∂λ= 0. (13.40)

The canonical Euclidean Hamiltonian is given by expressing all time derivativesin terms of the canonical momenta; hence, using Eqs. 13.39 and 13.40 yields

Hc = xpx + vpv + λpλ − LD

= −γ2p2

v +γ

2

[(ω2

1 + ω22)v

2 + ω21ω

22x

2]− ivλ. (13.41)

Note the canonical momenta px and pλ do not appear in the Hamiltonian Hc;hence these two momenta have to be imposed, following the notation and termi-nology of Dirac, as constraints

φ1 = px − iλ, φ2 = pλ. (13.42)

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Similarly to including a Lagrange multiplier λ to impose a constraint on theLagrangian, as given in Eq. 13.39, the most general Euclidean Hamiltonianconsistent with the constraints is given by

H1 = Hc + u1φ1 + u2φ2

= Hc + u1(px − iλ)+ u2pλ, (13.43)

where u1, u2 are arbitrary functions of the degrees of freedom.Hamiltonian dynamics yields that the time dependence of any function F of

the dynamical variables is given by its Poisson bracket with the Hamiltonian H1,namely

dF

dt= {F,H1}P + ∂F

∂t, (13.44)

where the Poisson bracket for arbitrary functions F,G of the dynamical variablesx, v, λ is given by

{F,G}P ≡ ∂F

∂x

∂G

∂px− ∂G

∂x

∂F

∂px+ ∂F

∂v

∂G

∂pv− ∂G

∂v

∂F

∂pv+ ∂F

∂λ

∂G

∂pλ− ∂G

∂λ

∂F

∂pλ

⇒ {F,GH }P = {F,G}PH +G{F,H }P. (13.45)

In light of the derivation of Mannheim and Davidson (2000), consider the choiceof functions u1 = −v, u2 = γω2

1ω2 that, for the Hamiltonian given in Eq. 13.43,yields

H1 = Hc − v(px − iλ)+ γω21ω2xpλ

= −γ2p2

v +γ

2

[ω2

1 + ω22)v

2 + ω21ω

22x

2]− vpx + γω2

1ω2xpλ. (13.46)

The constraints φ1, φ2 must be conserved over time; hence, we need to evaluatethe Poisson brackets of the constraints withH1. A straightforward calculation usingEqs. 13.45 and 13.46 yields

{φ1, H1}P = −γω21ω2pλ, {φ2, H1}P = 0. (13.47)

The dynamical variable λ is cyclic since it does not appear in the HamiltonianH1. Hence for conjugate variables λ and pλ

dpλ

dt= {pλ,H1}P = ∂pλ

∂λ

∂H1

∂pλ− ∂H1

∂λ

∂pλ

∂pλ= −∂H1

∂λ= 0

⇒ pλ = constant.

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13.5 Phase space path integral and Hamiltonian operator 283

To decouple the degree of freedom λ from the degrees of freedom x and v,choose pλ = 0; the resulting physical Hamiltonian, from Eq. 13.46, is

H(v, pv, x, px) = −γ2p2

v +α

2v2 + β

2x2 − vpx

= −γ2p2

v +γ

2

[ω2

1 + ω22)v

2 + ω21ω

22x

2]− vpx. (13.48)

It can be shown that the Poisson brackets amongst x, px, v, pv, andH form a closedsubalgebra; on this subspace the constraint equations are all conserved. Hence, His the requisite classical Hamiltonian for the acceleration Lagrangian.

13.5 Phase space path integral and Hamiltonian operator

The classical Hamiltonian yields two pathways for quantization, namely one via theformalism of phase space path integration and the other by imposing commutationequations on the pairs of canonical coordinate and its canonical momentum. Thesetwo routes for quantization are discussed below.

The Euclidean Hamiltonian obtained in Eq. 13.48 yields the Lagrangian L forEuclidean time, which from Eq. 5.31, is given by

L[v, pv, x, px] = vpv + xpx −H(v, pv, x, px). (13.49)

As discussed in Eq. 5.22, the Euclidean momentum is pure imaginary and equalto ip, where p is real. Making the change of variables pv → ipv, px → ipx andkeeping the same notation for simplicity yields, from Eqs. 13.48 and 13.49, theEuclidean Lagrangian1

L[v, ipv, x, ipx] = ivpv + ixpx −H(v, ipv, x, ipx) ≡ L[v, pv, x, px]⇒ L[v, pv, x, px] = ivpv + ixpx − γ

2p2

v −α

2v2 − β

2x2 + ivpx, (13.50)

and the finite time Euclidean action is given by

S[v, pv, x, px] =∫ τ

0dtL[v, pv, x, px]. (13.51)

The Euclidean path integral, from Eq. 5.34, is hence defined by

Z =∫

Dx exp{S}

=∫

DxDpxDvDpv exp{∫ τ

0dtL[v, pv, x, px]}, (13.52)

1 Using the notation for the Lagrangian given in Eq. 13.9.

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with boundary conditions x ′ = x(0), v′ = v(0) and x = x(τ), v = v(τ ). Equation13.50 yields

Z =∫

DxDpxDvDpv exp

{∫ τ

0dt (ivpv + ixpx − γ

2p2

v −α

2v2 − β

2x2 + ivpx)

}

=∫

DxDpxDv exp

{∫ τ

0dt (ixpx − 1

2v2 − α

2v2 − β

2x2 + ivpx)

}

=∫

DxDv exp

{∫ τ

0dt (−1

2v2 − α

2v2 − β

2x2)

}∏t

δ[v+ x], (13.53)

and we have recovered the path given earlier in Eq. 13.37, thus verifying thatthe Hamiltonian is indeed correct. Note the sign of the constraint in the equation,namely that v = −x, above what expected, and the fact that Euclidean momentumis pure imaginary consistently yields the required result.

To obtain the quantum Hamiltonian, since the system has constraints, the Diracbrackets are required to obtain the Heisenberg commutators. In particular, theequal time Euclidean commutation equations have to be obtained from the Diracbrackets.

A set of constraints, given by φ�, � = 1, 2 . . .M , defines as per Eq. 5.55, theconstraint matrix

C�,�′ = {φ�, φ�′ }P . (13.54)

The Dirac brackets are given by Eq. 5.56 for arbitrary function f (q, p), g(q, p), by

{f, g}D = {f, g}P −M∑

�,�′=1

{f, φ�}PC−1�,�′ {φ�′, g}P .

For degrees of freedom x, px, v, pv, from Eq. 13.42 the constraints are given by

φ1 = px − iλ, φ2 = pλ, (13.55)

and the constraint matrix is given by

C12 = {φ1, φ2}P = −i{λ, pλ}P = −i = −C21, (13.56)

⇒ C−112 = i = −C−1

21 . (13.57)

Evaluating the Dirac bracket for only the x, v sector yields

{x, φ1}P = 1, {px, φ1}P = {x, φ2}P = 0 = {px, φ2}P, (13.58)

{v, φ1}P = 0 = {pv, φ1}P = {v, φ2}P = 0 = {pv, φ2}P . (13.59)

From the result above, the Dirac brackets become equal to the Poisson bracket forall the conjugate variables, namely

{x, px}D = {x, px}P, . . .

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13.5 Phase space path integral and Hamiltonian operator 285

The canonical quantization for equal Euclidean time is given by

[x, px] = −�I{x, px}D = −�I, (13.60)

[v, pv] = −�I{v, pv}D = −�I, (13.61)

[x, v] = [x, pv] = [v, px] = 0.

From the above, the nonzero commutation equations for the degrees of freedomx, v are given by

[x, px] = −�I = [v, pv]. (13.62)

Hence, from Eqs. 5.50 and 5.51, the Euclidean momentum operators are given by

px = �∂

∂x, pv = �

∂

∂v.

In summary, the degrees of freedom for the acceleration Lagrangian are given by(setting � = 1)

x, px = ∂

∂x, v, pv = ∂

∂v. (13.63)

Hence, from Eqs. 13.48 and 13.63, the quantum non-Hermitian Hamiltonian forthe acceleration Lagrangian is given by

H = −γ2

∂2

∂v2+ γ

2

[(ω2

1 + ω22

)v2 + ω2

1ω22x

2]− v

∂

∂x, (13.64)

and its Hermitian conjugate is

H † = −γ2

∂2

∂v2+ γ

2

[(ω2

1 + ω22

)v2 + ω2

1ω22x

2]+ v

∂

∂x�= H,

since (v∂/∂x)† = −v∂/∂x.

Noteworthy 13.1 Important features of the Euclidean Hamiltonian

The following are some important features in the derivation and form of theEuclidean Hamiltonian H obtained in Eq. 13.64:

• The Euclidean Hamiltonian is not Hermitian due to the v∂/∂x term.• The Euclidean analysis of the Poisson brackets is consistent and one has to

explicitly keep track of the signatures of time that differ from the Minkowskiderivation.

• The acceleration Lagrangian (incorrectly) seems to have only one degree offreedom, namely x(t). On the other hand, the Hamiltonian H and its state spaceare each quantum systems with two degrees of freedom, namely x and v; this resultcan also be seen from the path integral since one needs two initial conditions and

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two final conditions to define the finite time path integral, reflecting that the initialand final state vectors have two degrees of freedom. The Dirac constraint analysisbrings out this feature of the acceleration (higher derivative) Lagrangian.

• The only difference between the initial canonical Hamiltonian Hc given in Eq.13.41 and the final result given by H in Eq. 13.64 is that the term ivλ in Hc hasbeen replaced by −v∂/∂x. The analysis of the constraints results in thereplacement of the nonphysical degree of freedom iλ by the physical degree offreedom operator −∂/∂x.

• It is shown in the path integral analysis of H , analyzed in Section 13.6 below, thatthe term −v∂/∂x has the remarkable property of constraining, in the path integral,v to be equal to −dx/dt , as required by Eq. 13.1.

• The fact that −v∂/∂x is a constraint operator explains why the term is independentof the coupling constants, since the constraint is on the phase space itself and noton the forms of interaction allowed in this phase space.

13.6 Acceleration path integral

The path integral derivation of K(xf , xf ; xi, xi) given in Eq. 13.14 was done en-tirely in terms of the coordinate degree of freedom x(t); in particular, the velocitydegree of freedom v(t) did not appear in the expression for the path integral.

One would like to interpret the evolution kernel K(xf , xf ; xi, xi) as the proba-bility amplitude from an initial state of state space to a final state. Such an interpre-tation of course needs both a state space and a Hamiltonian.

Based on the boundary conditions given in Eq. 13.6, it can be seen that the statespace has to have two independent degrees of freedom, corresponding to the twoinitial conditions given by the initial position x and velocity x. Hence, to have astate space interpretation of the path integral, state space V is taken to have two de-grees of freedom, namely (position) coordinate degrees of freedom x and a degreeof function reflecting x in the state vector. The Hamiltonian has to be chosen in sucha manner that an independent velocity degree of freedom v is is introduced and it isconstrained to be equal to the velocity −x of the coordinate degree of freedom x.

Consider two independent degrees of freedom x and v. The completeness equa-tion for the basis states is given by

I =∫ +∞

−∞dxdv|x, v〉〈v, x|, (13.65)

〈x, v|x ′, v′〉 = δ(x − x ′)δ(v− v′).

A state space representation of the evolution kernel K(xf , xf ; xi, xi) is derivedbelow. It will be shown in this section that the evolution kernel is closely related

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13.6 Acceleration path integral 287

to the probability amplitude of going, in time τ , from the initial state |xi, vi〉 to thefinal state 〈xf , vf | and is given by

KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉. (13.66)

A similar definition is adopted for the path integral of a pseudo-Hermitian Hamil-tonian in Kandirmaz and Sever (2011).

It remains to be seen what is the relation of the probability amplitude KS(xf , vf ;xi, vi ) defined using the state space and Hamiltonian to the probability amplitudeK(xf , xf ; xi, xi) defined using the path integral. In particular, as it stands in Eq.13.66, the initial vi and final velocity vf have no relation to the coordinate degreeof freedom x. The Hamiltonian has to implement a constraint to set the initial andfinal state in Eq. 13.66 to have the same boundary conditions given in Eqs. 13.6and 13.7 for K(xf , xf ; xi, xi).

The Hamiltonian, for infinitesimal time τ = ε, is given by the Dirac–Feynmanformula as

〈x, v|e−εH |x ′, v′〉 = C(ε)eεL(x,x′;v,v′), (13.67)

where C(ε) is a normalization constant that depends only on ε. In terms of v = −x,the discrete time Lagrangian, from Eq. 13.2, is given by

L(x, x ′; v, v′) = −γ2

(x − x ′

ε

)2

− α

2v2 − 1

2[�(x)+�(x ′)]. (13.68)

A straightforward generalization of the Hamiltonian given in Eq. 13.64 yields aEuclidean Hamiltonian with an arbitrary potential �(x),

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ 1

2αv2 +�(x) = H

(x,

∂

∂x, v,

∂

∂v

). (13.69)

The Hamiltonian and its conjugate both act on a state space V that has two degreesof freedom, namely position coordinate x and velocity degree of freedom v. Thestate function |�〉 is given by

|�〉 ∈ V, 〈x, v|�〉 = �(x, v). (13.70)

The transition probability amplitude is given by defining ε = τ/N and insertingN − 1 complete sets of states given in Eq. 13.65. Hence, for boundary conditionsgiven by x0 = xi, v0 = vi; xN = xf , vN = vf , the transition amplitude is given by

K(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉

=N−1∏n=1

∫dxndvn

N∏n=1

〈xn, vn|e−εH |xn−1, vn−1〉

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=N−1∏n=1

∫dxn〈xN, vN |e−εH |xN−1, vN−1〉

×{ N−1∏

n=1

∫dvn〈xn, vn|e−εH |xn−1, vn−1〉

}. (13.71)

The differential operator H given in Eq. 13.69, for xn = x, v = vn; x ′ =xn−1, v′ = vn−1, yields

〈x, v|e−εH |x ′, v′〉 = e−εH(x, ∂∂x;v, ∂

∂v )〈x|x ′〉〈v|v′〉= e−εH

∫dp

2π

dq

2πeip(x−x

′)+iq(v−v′)

=∫

dp

2π

dq

2πe− ε

2γ q2+iq(v−v′)− εα

2 v2eip(x−x

′+εv)−ε�(x)

= Cδ(x − x ′ + εv) exp{− γ

2ε(v− v′)2 − εα

2v2 − ε�(x)

}.

(13.72)

The appearance of the δ-function in Eq. 13.72 yields the following constraint inthe path integral:

δ(x − x ′ + εv) ⇒ v = −x − x ′

ε, (13.73)

⇒ limε→0

v = −dxdτ

. (13.74)

Equation 13.72 yields the remarkable result that the term−v∂/∂x in the Hamiltonianyields a constraint on the degree of freedom v so that it is constrained to be −xdegree of freedom, namely v = −dx/dt . It is the delta function constraint onthe velocity degree of freedom that leads to its complete elimination in the pathintegral.


〈xn, vn|e−εH |xn−1, vn−1〉 = Cδ(xn − xn−1 + εvn) exp{εLn}, (13.75)

where Ln, from Eqs. 13.72 and 13.73, is given by

Ln = − γ

2ε2(v− v′)2 − α

2v2 − 1

2[�(x)+�(x ′)]. (13.76)

The path integral and Lagrangian that appear in Eq. 13.4 make no reference tothe integration over the velocity variables. Hence, all the velocity integrations

∫dvn

need to be carried out in order to obtain the expression in Eq. 13.4. Remarkablyenough, all the

∫dvn integrations can be done exactly using the δ-functions that

appear in Eq. 13.72.

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13.7 Change of path integral boundary conditions 289

Equations 13.71 and 13.75 yield

N−1∏n=1

∫dvn〈xn, vn|e−εH |xn−1, vn−1〉 = CN−1

N−1∏n=1

∫dvnδ(xn − xn−1 + εvn)

× exp{εLn}. (13.77)

The full path integral, from Eqs. 13.71 and 13.77, can be heuristically written, inthe continuum notation, as

K(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉

=∫

DxDvτ∏t=0

δ[x(t)+ v(t)] exp

{∫ τ

0dtL(x, v)

}, (13.78)

L(x, v) = −γ2

v2 − α

2v2 −�(x),

and we recover the expression for the Lagrangian given in Eq. 13.68. To obtain theprecise content of the continuum expression given in Eq. 13.78, one needs to go tothe lattice and carefully address the issue of the boundary conditions.

13.7 Change of path integral boundary conditions

The initial and final time steps need to be examined carefully to see how the initialand final velocity, v0 and vN respectively, can be expressed solely in terms of thecoordinate degree of freedom. The four boundary conditions given in Eq. 13.11are solely in terms of the coordinate degree of freedom x(t), whereas the bound-ary conditions given in Eq. 13.66, the defining equation for KS(xf , vf ; xi, vi), aregiven in terms of final and initial positions and velocities xf , vf ; xi, vi respectively.

Note that the integrand of Eq. 13.77, for n = 1, yields, from Eq. 13.76,∫dv1δ(x1 − x0 + εv1) exp{εL1}

L1 = −γ2

(v1 − v0

ε

)2

− α

2v2

1 −1

2[�(x1)+�(x0)]. (13.79)

On performing the∫dv1 integration, the delta function constrains v1 = −(x1 −

x0)/ε; hence, L1 has the value

exp{εL1} = exp{− γ

2ε

(x1 − x0

ε+ v0

)2

− α

2ε(x1 − x0)

2 +O(ε)}

= exp{− γ

2ε

(x1 − x0

ε+ v0

)2

− αε

2v2

0 +O(ε)}

= C(ε)δ(x1 − x0 + εv0)+O(ε) = C(ε)δ(x1 − xi + εvi), (13.80)

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where x0 = xi and v0 = vi are the initial position and velocity.The final time boundary term for the action, from Eq. 13.75, yields

〈xN, vN |e−εH |xN−1, vN−1〉 = C(ε)δ(xN − xN−1 + εvN) exp{εLN }= C(ε)δ(xf − xN−1 + εvf ) exp{εLN },

since xN = xf and vN = vf .The path integral over the velocity degrees of freedom yields, in addition to the

expected acceleration action, two extra delta functions. These delta functions arecrucial in changing the boundary conditions for the path integral

∫Dx over the

coordinate degree of freedom.Collecting all the results yields the discrete time path integral for the transition

probability amplitude expressed solely in terms of the coordinated degrees of free-dom, namely

KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉

= CN−1∏n=1

∫dxnδ(xf − xN−1 + εvf )δ(x1 − xi + εvi)

× exp

{ε

N∑n=1

Ln

}, (13.81)

where C is a normalization.Note that the path integral given in Eq. 13.81, due to the two delta functions in

the integrand, has four boundary conditions for the coordinate degree of freedom,namely xi, x1, xN−1, xf ; the delta functions, in effect, remove two integrations,namely

∫dx1dxN−1 in the path integral given in Eq. 13.81 by fixing the value of

x1, xN−1. In the derivation of the probability amplitude carried out by Kleinert(1986), the x boundary conditions are finally implemented in the discretized pathintegral by constraining the variables near the end points, namely x1 and xN−1.

To take the continuum limit we define

x(t) = dx(t)

dt= xn − xn−1

ε, t = nε. (13.82)


vi = x1 − x0

ε→−xi , (13.83)

vf = xN − xN−1

ε→−xf , (13.84)

since xi = dx(0)/dt and xf = dx(τ)/dt .

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13.8 Evolution kernel 291

From Eq. 13.77,

εLn = − γ

2ε(xn − xn−1)

2 − εα

2x2n − ε�(xn). (13.85)

Taking the limit of ε → 0 and using x = [xn − xn−1]/ε yields

L = −γ2x2 − α

2x2 −�(x). (13.86)

The delta functions for the boundary values of the functional integral∫Dx are

constraints that change the boundary conditions on the path integral, converting thetwo boundary conditions each for x and v in

∫DxDv to four boundary conditions

for x in the path integral∫Dx. Hence, one obtains the continuum result

KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉 =∫

Dxδ(vi + xi)δ(vf + xf )eS∣∣∣(xi ,xf )

=∫

DxeS∣∣∣(xi ,xi=−vi ;xf ,xf=−vf )

= K(xi, xi = −vi; xf , xf = −vf )

⇒ KS(xf , vf ; xi, vi ) = K(xf ,−xf ; xi,−xi).Hence

KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉=∫

DxeS = K(xf ,−xf ; xi,−xi). (13.87)

The representation of the path integral in terms of the transition amplitudeKS(xf , vf ; xi, vi) is necessary to have the amplitude obey the composition lawderived in Section 4.8. This is discussed by Hawking and Hertog (2002) andFontanini and Trodden (2011).

13.8 Evolution kernel

The Hamiltonian in Eq. 13.69 [for the Lagrangian in Eq. 13.86] is given by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ α

2v2 +�(x).

In terms of the path integrals, from Eqs. 13.4 and 13.78

L(x) = −γ2x2 − α

2x2 −�(x), L(x, v) = −γ

2v2 − α

2v2 −�(x),∫

Dx exp{∫ τ

0dtL(x)} =

∫DxDv

τ∏t=0

δ[x(t)+ v(t)] exp{∫ τ

0dtL(x, v)}.

Implicit in the path integrals above are the appropriate boundary conditions.

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The symmetry of the evolution kernel given in Eq. 13.19 using the path integral,namely

K(xf , xf ; xi, xi) = K(xi,−xi; xf ,−xf )has a Hamiltonian derivation. Note that H † can be obtained from Eq. 13.69 byconjugation and is given by

H † = − 1

2γ

∂2

∂v2+ v

∂

∂x+ α

2v2 +�(x) �= H. (13.88)

From Eq. 13.87

K∗S(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉∗ = 〈xi, vi |e−τH † |xf , vf 〉. (13.89)

For the Hamiltonian given in Eq. 13.69, the only difference in the path integralbetween H and H † is to change the sign of v∂/∂x to −v∂/∂x; this in turn yieldsthe constraint δ(v− x) for the path integral representation of Eq. 13.89. Hence, onperforming the velocity path integral

∫Dv for Eq. 13.89, the only change from the

earlier case is that the boundary on velocity now has a minus sign and yields

〈xi, vi |e−τH † |xf , vf 〉 = KS(xi,−vi; xf ,−vf ). (13.90)

For the case of KS being real, which is the case considered in Eq. 13.19, Eqs. 13.89and 13.90 yield

KS(xf , vf ; xi, vi ) = KS(xi,−vi; xf ,−vf ), (13.91)

from which Eq. 13.19 follows due to Eq. 13.87. The symmetry of the evolutionkernel has been shown in Hawking and Hertog (2002) to be correct by a directevaluation of the classical action S[xc].

The evolution kernel has an implicit time ordering; explicitly putting in the initialti and final tf time coordinates, the kernel can be written out explicitly in the form

KS(xf , vf , tf ; xi, vi , ti) = 〈xf , vf |e−|tf−ti |H |xi, vi〉. (13.92)

Using the identity

∂

∂texp{−α|t |} = −α exp{−α|t |} + δ(t)

yields, from Eq. 13.92, the following Schrödinger equation with a source:[∂

∂tf+H

]KS(xf , vf , tf ; xi, vi , ti) = δ(xf − xi)δ(vf − vi)δ(tf − ti). (13.93)

This result has been derived for the Euclidean case by Kleinert (1986) and for theMinkowski case by Mannheim (2011b).

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13.9 Summary 293

13.9 Summary

The Euclidean acceleration Lagrangian leads to a number of new results. The the-ory has a state space that has two degrees of freedom, namely x and its velocityv. The theory has three branches shown in Figure 13.1. The first branch, given byα > αc, has a pseudo-Hermitian Hamiltonian and a state space with a positivedefinite norm. The second branch, given by α < αc, has complex parameters inits Hamiltonian and the third critical branch α = αc – separating the other twobranches – consists of an essentially non-Hermitian Hamiltonian consisting of Jor-dan blocks.

To obtain the Hamiltonian, the acceleration Lagrangian is written as a con-strained system and the Dirac constraint method was utilized to obtain a non-Hermitian Hamiltonian. The Hamiltonian in turn was then used for obtaining apath integral representation of the evolution kernel and was shown to reproducethe earlier result, confirming the result obtained for the Hamiltonian. In particular,a derivation was given of the change of boundary conditions in going from the x, vdegrees of freedom for the Hamiltonian to the solely x degree of freedom in termsof which the acceleration action is directly defined.

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14

Pseudo-Hermitian Euclidean Hamiltonian

There is no conservation of probability in the Euclidean formulation of theacceleration Lagrangian. A narrow definition of a Hamiltonian, and the one thatholds in quantum mechanics but is not suitable for our case, is that the Hamiltoniandefines a probability conserving unitary evolution. A broader definition of theHamiltonian is adopted that is consistent with both statistical mechanics and quan-tum mechanics. Namely, the Hamiltonian – which is equivalent to the transfermatrix in statistical mechanics – is the generated of infinitesimal translations intime for both the Minkowski and Euclidean cases.

The Hamiltonian derived for the acceleration Lagrangian in Chapter 13 is stud-ied in this chapter using the concepts of the state space and operators. The Hamilto-nian for unequal and real frequencies ω1, ω2 is pseudo-Hermitian and is explicitlymapped to a Hermitian Hamiltonian using a nontrivial differential operator. Anexplicit expression is obtained for the matrix element of the differential operator.The mapping fails for a critical value of the coupling constants.

The Hamiltonian and state space are shown to have real and complex branchesthat are separated by the critical point; both branches have a well-behaved statespace and algebra of operators. At the critical point, the Hamiltonian is inequivalentto any Hermitian operator, and is shown in Chapter 15 as being equal to an infinitedimensional block diagonal matrix, with each block being a Jordan matrix.

For the real branch of the theory, the state space of a single quantum degreeof freedom described by the acceleration action is shown to be a function of twodistinct degrees of freedom, namely velocity v and position x, with the two beingrelated by a constraint equation. The state vector for the acceleration Hamiltonianis given by ψ(x, v), and is quite unlike the state vector ψ(x) of a quantum degreeof freedom described by a Lagrangian having only the velocity term – namely,without an acceleration term.

The acceleration Lagrangian and its path integral are well defined for all valuesof ω1, ω2, including complex values. The Euclidean Hamiltonian obtained for the

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14.1 Pseudo-Hermitian Hamiltonian; similarity transformation 295

acceleration Lagrangian shows that the state space of the acceleration Lagrangianis only well defined for real ω1, ω2. For equal frequency ω1 = ω2, the Hamiltonianbecomes essentially non-Hermitian, and this is analyzed at length in Chapter 15.For the case of complex ω1, ω2 the Hamiltonian is neither Hermitian nor pseudo-Hermitian and the state space is no longer a positive normed space.

In Section 14.1 a similarity transformation eQ/2 is discussed that shows thatthe Hamiltonian H is pseudo-Hermitian. In Section 14.2 the Hamiltonian H ismapped, by a similarity transformation, to a Hermitian Hamiltonian that is the sumof two decoupled oscillators. In Section 14.3 the matrix elements of the similaritytransformation are evaluated and are shown in Section 14.4 to yield the expectedresults. The eigenfunctions of the equivalent Hermitian Hamiltonian are evaluatedin Section 14.5 and a few of the excited states of the acceleration Hamiltonianare derived in Section 14.9. The state space for unequal frequencies constitutes aHilbert space, with all state vectors having a positive norm.

The general features of the state space of the pseudo-Hermitian Hamiltonian arediscussed in Section 14.11, and in particular the necessity of introducing a metricon state space that is required for defining the conjugate state vector and scalarproduct for the state space. The eigenfunctions of the acceleration Hamiltonian arestudied in Section 14.6: the propagator for the degree of freedom is evaluated inSection 14.12 using operator methods and in Section 14.13 using the state spaceanalysis. It is shown that the propagator cannot be obtained from any HermitianHamiltonian.

14.1 Pseudo-Hermitian Hamiltonian; similarity transformation

The acceleration Hamiltonian, from Eq. 13.64, is given by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ γ

2(ω2

1 + ω22)v

2 + γ

2ω2

1ω22x

2. (14.1)

For H to be pseudo-Hermitian, similarity transformation eQ/2 is required suchthat

e−Q/2HeQ/2 = HO, (14.2)

where HO

is a Hermitian Hamiltonian; it is shown in Section 14.2 that HO

consistsof a system of two decoupled harmonic oscillators, one each for degree of freedomx and v.

The Hermitian conjugate Hamiltonian H †, from Eq. 14.2, is given by

H † = e−Q/2HOeQ/2

⇒ H † = e−QHeQ : pseudo-Hermitian. (14.3)

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296 Pseudo-Hermitian Euclidean Hamiltonian

Note that (e−τH

)† = e−τH† = e−Qe−τH eQ. (14.4)

Equation 14.3 is a definition of a pseudo-Hermitian operator, that differs fromits Hermitian conjugate by a similarity transformation. It can be shown that theenergy eigenvalues of a pseudo-Hermitian Hamiltonian are either real or appear incomplex conjugate pairs, and vice versa. All Hamiltonians that are equivalent to aHermitian Hamiltonian up to a similarity transformation, as is the case for Eq. 14.2,are automatically pseudo-Hermitian. However, all pseudo-Hermitian Hamiltoniansare not equivalent to a Hermitian Hamiltonian, and Chapter 15 analyzes such acase.

There is another class of Hamiltonians that have real eigenenergies and theenergy eigenstates are complete; these Hamiltonians can be brought to a Hermitianform by a similarity transformation. Such Hamiltonians are referred to by Scholtzet al. (1992) as being quasi-Hermitian. All quasi-Hermitian Hamiltonians arethus also pseudo-Hermitian, but not all pseudo-Hermitian Hamiltonians are quasi-Hermitian.

A Q is obtained for Euclidean time by analytically continuing the remarkableresult obtained by Bender and Mannheim (2008b), and this yields

Q = axv− b∂2

∂x∂v. (14.5)

For the real domain where ω1, ω2 are real, both coefficients a, b are real and hence

Q = Q† : Hermitian for ω1, ω2 real. (14.6)

The equation for the commutator

e−QOeQ =∞∑n=0

1

n! [[[O,Q],Q] . . .Q] (14.7)

needs to be applied to O = x, v, ∂/∂x, ∂/∂v.To obtain the commutator, note that the n-fold commutator of Q with x, v, ∂/∂x

and ∂/∂v follows a simple pattern that repeats after two commutations. In particu-lar, note that

[x,Q] = b∂

∂v, [[x,Q],Q] = abx, . . .[

∂

∂x,Q

]= av,

[[∂

∂x,Q

],Q

]= ab

∂

∂x, . . .

[v,Q] = b∂

∂x, [[v,Q],Q] = abv, . . .

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14.2 Equivalent Hermitian Hamiltonian HO 297[∂

∂v,Q

]= ax,

[[∂

∂v,Q

],Q

]= ab

∂

∂v, . . .

Carrying out the nested commutators to all orders and summing the result yields,for a, b > 0,

e−τQxeτQ = cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v,

e−τQ∂

∂xeτQ = cosh(τ

√ab)

∂

∂x+√a

bsinh(τ

√ab)v,

e−τQveτQ = cosh(τ√ab)v+

√b

asinh(τ

√ab)

∂

∂x,

e−τQ∂

∂veτQ = cosh(τ

√ab)

∂

∂v+√a

bsinh(τ

√ab)x. (14.8)

Note that the definitions of the values of a and b are chosen based on ω1 > ω2,which makes the operator Q Hermitian the domain where ω1, ω2 are real. The defi-nition ofQ continues to hold for the complex domain butQ is no longer Hermitian.

14.2 Equivalent Hermitian Hamiltonian HO

The fundamental commutation relations given in Eq. 14.8 can be applied to theacceleration Hamiltonian, and the coefficients a, b can be chosen to decouple thex and v degrees of freedoms. Consider the equation

e−Q/2HeQ/2 = C1∂2

∂v2+ C2x

∂

∂v+ C3v

∂

∂x+ C4

∂2

∂x2+ C5x

2 + C6v2. (14.9)

To obtain the factorization of the Hamiltonian into two de-coupled oscillators,following Bender and Mannheim (2008b), we choose the following values for aand b:√

a

b= γω1ω2; sinh(

√ab) = 2ω1ω2

ω21 − ω2

2

⇒ √ab = ln

(ω1 + ω2

ω1 − ω2

). (14.10)

We define

A = cosh

(√ab

2

)= ω1√

ω21 − ω2

2

, B = sinh

(√ab

2

)= ω2√

ω21 − ω2

2

,

C =√a

b= γω1ω2. (14.11)

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Using the result of Eq. 14.8 and the definitions in Eq. 14.9 yields

C1 = − 1

2γA2 + γ

2

(B

C

)2

ω21ω

22

= − 1

2γ

[cosh2

(√ab

2

)− sinh2

(√ab

2

)]= − 1

2γ.

Similarly, after some simplifications

C2 = −CABγ

+ γω21ω

22

(AB

C

)= 0,

C3 = −(A2 + B2)+ γ (ω21 + ω2

2)

(AB

C

)= 0.

The constants a and b were chosen so that C2 = C3 = 0; hence one has

C2 = C3 = 0 ⇒ determines a and b.

The remaining coefficients are given by

C4 = −ABC+ γ

2(ω2

1 + ω22)

(B

C

)2

= − 1

2γω21

,

C5 = − 1

2γB2C2 + γ

2ω2

1ω22A

2 = γ

2ω2

1ω22,

C6 = −ABC + γ

2(ω2

1 + ω22)A

2 = γ

2ω2

1.

Collecting all the results yields

e−Q/2HeQ/2 = HO

⇒ HO= − 1

2γ

∂2

∂v2− 1

2γω21

∂2

∂x2+ γ

2ω2

1v2 + γ

2ω2

1ω22x

2. (14.12)

14.3 The matrix elements of e−τQ

The Q-operator is given from Eq. 14.5 by

Q = axv− b∂2

∂x∂v,

where, from Eq. 14.10,√a

b= γω1ω2, sinh(

√ab) = 2ω1ω2

ω21 − ω2

2

.

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14.3 The matrix elements of e−τQ 299

The finite matrix elements of the Q-operator are required for many calculationsinvolving the state vectors. The exact matrix elements can be obtained by not-ing that the Hermitian Q-operator exactly factorizes into two decoupled harmonicoscillators by an appropriate change of variables.

Consider the change of variables given by

α = 1√2(x + v), β = 1√

2(x − v), (14.13)

∂

∂x= 1√

2

(∂

∂α+ ∂

∂β

),

∂

∂v= 1√

2

(∂

∂α− ∂

∂β

). (14.14)

In these coordinates the α, β sectors completely factorize and yield

Q = a

2(α2 − β2)− b

2

(∂2

∂α2− ∂2

∂β2

), (14.15)

= −1

2b∂2

∂α2+ 1

2aα2 + 1

2b∂2

∂β2− 1

2aβ2. (14.16)

Consider the Hamiltonian of a quantum oscillator given by

Hsho = − 1

2m

∂2

∂z2+ 1

2mω2z2, (14.17)

with the transition amplitude given by

K(z, z′; τ) = 〈z|e−τH |z′〉=√

mω

2π sinhωτexp

{− mω

2 sinh(ωτ)

[(z2 + z′2) cosh(ωτ)− 2zz′

]}.

(14.18)

Comparing the α- and β-sectors of Q with Hsho shows that, for a, b real, the α

sector is the usual quantum oscillator but the β sector yields a divergent transitionamplitude. The result for the β sector is assumed to be given by the analytic contin-uation of the oscillator transition amplitude; this assumption will later be verifiedby an independent derivation.

To exploit the quadratic form of the α and β sectors, consider extending therange of m to the real line. This yields

α − sector : b = 1

m; a = mω2 ⇒ ω = √ab, (14.19)

β − sector : b = − 1

m; a = −mω2 ⇒ ω = √ab. (14.20)

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Hence Eqs. 14.18, 14.19, and 14.20 yield

〈α, β|e−τQ|α′, β ′〉

= NαNβ exp

{− 1

2b

√ab

sinh(τ√ab)

[(α2 + α′2) cosh(τ

√ab)− 2αα′

]}

× exp

{1

2b

√ab

sinh(τ√ab)

[(β2 + β ′2) cosh(τ

√ab)− 2ββ ′

]}

= N (τ ) exp{− 1

2G(τ )(α2 + α′2 − β2 − β ′2)+H(τ )(αα′ − ββ ′)

},

where

G(τ ) =√a

bcoth(τ

√ab), H(τ ) =

√a

b

1

sinh(τ√ab)

(14.21)

and

N (τ ) =√ √

ab

2πb sinh(τ√ab)

·√ √

ab

−2πb sinh(τ√ab)

⇒ N (τ ) = i

2π

√a

b

1

| sinh(τ√ab)| =

i

2π

∣∣H(τ )∣∣. (14.22)

Hence a heuristic derivation for Q yields

〈x, v|e−τQ|x ′, v′〉 = N (τ ) exp{−G(τ )(xv+ x ′v′)+H(τ )(xv′ + vx ′)

}. (14.23)

The normalization constant N (τ ) will be seen to play a crucial role in the normal-ization of all the state vectors of the Hamiltonian H .

The result obtained by the heuristic method will be verified to be indeed exactlycorrect, including the normalization constant.

For the real branch, both coefficients G(τ ) and H(τ ) are real and hence Q = Q†

is Hermitian.Note that the form obtained for e−τQ in Eq. 14.23 is a major simplification since

in general, for Hermitian Q, one would need to evaluate 4 + 12 = 16 real coeffi-cients. Instead, the form that has been heuristically derived in Eq. 14.23 has reducedthe determination of e−τQ to that of computing two real coefficient functions G(τ )and H(τ ) and a normalization constant N (τ ).

The operator Q is unbounded and many of the manipulations are only formallyvalid. For example, the identity e−τQeτQ = I holds for the matrix elements only ina formal sense. Explicitly evaluating the matrix elements of the product e−τQeτQ

using Eq. 14.23 yields

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14.4 e−τQ and similarity transformations 301

〈x, v|e−τQeτQ|x ′, v′〉 =∫

dξdζ 〈x, v|e−τQ|ξ, ζ 〉〈ξ, ζ |eτQ|x ′, v′〉

= N 2(τ )

∫dξdζe−G(τ )(xv+ξζ )+H(τ )(xζ+vξ)eG(τ )(ξζ+x

′v′)−H(τ )(ξv′+ζx′)

= N 2(τ )e−G(τ )(xv−x′v′)∫

dξdζ exp{iH(τ )iξ(v′ − v)+ iH(τ )iζ(x ′ − x)}

= N 2(τ )

(2π

iH(τ )

)2

δ(x − x ′)δ(v− v′) = δ(x − x ′)δ(v− v′)

⇒ e−τQeτQ = I.

To make the above derivation more rigorous one can analytically continue τ backto Minkowski time t = −iτ , do the computation and then analytically continueback to Euclidean time τ . This would give the result above.

14.4 e−τQ and similarity transformations

The heuristic derivation for G(τ ) and H(τ ) was obtained by an analogy with theoscillator Hamiltonian and cannot be assumed to be correct since the β-sectoryields an unstable Hamiltonian. The result needs to be independently verified.

That the matrix elements of e−τQ given by Eq. 14.23 are in fact correct is nowverified in this section. The fundamental similarity transformations of e−τQOeτQfor operators O = x, ∂/∂x, v and ∂/∂v are directly obtained using the result givenin Eq. 14.23 and shown to be identical to the defining equations for Q given in Eq.14.8.

Recall from Eq. 14.8 that eτQ yields the following similarity transformations:

I. e−τQxeτQ = cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v,

I I. e−τQ∂

∂xeτQ = cosh(τ

√ab)

∂

∂x+√a

bsinh(τ

√ab)v,

I II. e−τQveτQ = cosh(τ√ab)v+

√b

asinh(τ

√ab)

∂

∂x,

IV . e−τQ∂

∂veτQ = cosh(τ

√ab)

∂

∂v+√a

bsinh(τ

√ab)x.

Consider the operator equation

e−τQxeτQ = cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v. (14.24)

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The matrix element of e−τQxeτQ is given by

〈x, v|e−τQxeτQ|x ′, v′〉

=(

cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v

)δ(x − x ′)δ(v− v′). (14.25)


〈x, v|e−τQ|x ′, v′〉 = N exp{−g(xv+ x ′v′)+ h(xv′ + vx ′)}, (14.26)

where

g = G(τ ) =√a

b

cosh(τ√ab)

sinh(τ√ab)

,

h = H(τ ) =√a

b

1

sinh(τ√ab)

,

N = N (τ ) = i

2πH(τ ).

The left hand side of Eq. 14.25 yields

〈x, v|e−τQxeτQ|x, v〉 =∫

dξdζ 〈x, v|e−τQ|ξ, ζ 〉〈ξ, ζ |ξeτQ|x ′, v′〉

= N

∫dξdζe−g(xv+ξζ )+h(xζ+vξ) ξ〈ξ, ζ |eτQ|x ′, v′〉.

But

ξ exp{−g(xv+ ξζ )+ h(xζ + vξ)}

=[√

b

asin(τ

√ab)

∂

∂v+ cosh(τ

√ab)x

]e−g(xv+ξζ )+h(xζ+vξ). (14.27)

Hence, we obtain the expected result, given in Eq. 14.25, that

〈x, v|e−τQxeτQ|x ′, v′〉 =[

cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v

]

× 〈x, v|e−τQeτQ|x ′, v′〉

⇒ I. e−τQxeτQ = cosh(τ√ab)x +

√b

asinh(τ

√ab)

∂

∂v.

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Hence we have verified that the expression for 〈x, v|e−τQ|x ′, v′〉 given inEq.14.23 is in fact correct and reproduces all the basic similarity transformations,from I to IV .

14.5 Eigenfunctions of oscillator Hamiltonian HO

The Euclidean Hamiltonian has two distinct and independent state spaces, namelythe state space V of the non-Hermitian Hamiltonian H and the state space of theHermitian Hamiltonian HO , namely VO .

The oscillator state space VO is a Hilbert space, with the norm of two statevectors |�〉 and |χ〉 given by

〈χ |�〉 =∫

dxdvχ∗(x, v)�(x, v). (14.29)

The oscillator Hamiltonian HO

, from Eq. 14.12, is given by

HO= − 1

2γ

∂2

∂v2− 1

2γω21

∂2

∂x2+ γ

2ω2

1v2 + γ

2ω2

1ω22x

2.

The oscillator structure of the Hamiltonian yields two sets of creation anddestruction operators, given by

av =√γω1

2

[v+ 1

γω1

∂

∂v

], a†

v =√γω1

2

[v− 1

γω1

∂

∂v

],

ax =√γω2

1ω2

2

[x + 1

γω21ω2

∂

∂x

], a†

x =√γω2

1ω2

2

[x − 1

γω21ω2

∂

∂x

],

⇒ [ai, a†j ] = δi−j , i, j = 1, 2. (14.30)

Note that, similarly to the case for Minkowski time, the creation operator a†i , for

i = 1, 2, is the Hermitian conjugate of the destruction operator ai .From the above

v =√

1

2γω1(av + a†

v ),∂

∂v=√γω1

2(av − a†

v ),

x =√

1

2γω21ω2

(ax + a†x),

∂

∂x=√γω2

1ω2

2(ax − a†

x).

The oscillator Hamiltonian is given by

HO= ω1a

†vav + ω2a

†xax +

1

2(ω1 + ω2).

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14.6 Eigenfunctions of H and H† 305

The vacuum is defined, as usual, by requiring that there be no excitations, namely

av|0, 0〉 = ax |0, 0〉 = 0,

HO |0, 0〉 = E0|0, 0〉, E0 = 1

2(ω1 + ω2).

The coordinate representation of the oscillator vacuum state is given by

〈x, v|0, 0〉 =(γ 2ω3

1ω2

π2

)1/4

exp

{−1

2[γω1v2 + γω2

1ω2x2]}.

The energy eigenfunctions |n,m〉 and eigenenergies Enm of HO

are given by

HO|n,m〉 = Enm|n,m〉, 〈n,m|HO

= Enm〈n,m|, (14.31)

|n,m〉 = a†nv√n!

a†mx√m! |0, 0〉, 〈x, v|m, n〉 = 〈m, n|x, v〉 : real,

Enm = nω1 +mω2 + E0 = nω1 +mω2 + 1

2(ω1 + ω2). (14.32)

The dual eigenfunctions 〈n,m| satisfy the orthonormality equation

〈n′,m′|n,m〉 = δn−n′δm−m′ . (14.33)

Hence, the spectral representation of HO

is given by

HO=∑mn

Emn|m, n〉〈m, n|. (14.34)

All the state vectors and operators have been defined entirely on state space VOwith no reference to state space V . The operator eQ/2 maps state space VO into V .

14.6 Eigenfunctions of H and H†

The Euclidean Hamiltonian is given, from Eq. 14.1, by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ γ

2(ω2

1 + ω22)v

2 + γ

2ω2

1ω22x

2,

and from Eq. 14.12, the equivalent Hermitian Hamiltonian HO

is given by

H = eQ/2HOe−Q/2, H † = e−Q/2H

OeQ/2, (14.35)

HO= − 1

2γ

∂2

∂v2− 1

2γω21

∂2

∂x2+ γ

2ω2

1v2 + γ

2ω2

1ω22x

2. (14.36)

The left and right eigenfunctions of H are different since H is pseudo-Hermitian; let us denote the right eigenfunctions by |�mn〉 and left dual eigen-functions by 〈�D

mn|. The notation 〈�Dmn| is used to denote the dual of the state

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14.6 Eigenfunctions of H and H† 307

The evolution kernel has the spectral decomposition given by

〈x, v|e−τH |x ′, v′〉 =∑mn

e−τEmn�mn(x, v)�Dmn(x

′, v′). (14.44)

14.6.1 Dual energy eigenstates

Note that unlike the case for a Hermitian Hamiltonian, the dual eigenfunction 〈�|is not the complex conjugate of |�〉. The dual eigenstates �D

mn(x, v) are given by

〈�Dmn| = 〈m, n|e−Q/2 ⇒ �D

mn(x, v) = 〈�Dmn|x, v〉 = 〈m, n|e−Q/2|x, v〉.

To obtain the matrix element of a non-Hermitian operator, it is important to notethat all operators act only on the dual space. This fact is unimportant for Hermitianoperators, since acting on the state space or its dual space is equivalent. But this isnot true for non-Hermitian operators, since the result depends on which space theoperators act on. In particular, for operator O(x, v, ∂x, ∂v) one has

〈x, v|O|�〉 = O(x, v, ∂x, ∂v)�(x, v).

Since all operators are defined by their action only on the dual space, the matrixelement of the operator O(x, v, ∂x, ∂v) acting on the dual state vector 〈�| is givenin terms of the conjugate operator O†(x, v, ∂x, ∂v) as

〈�|O|x, v〉 ≡(〈x, v|O†|�〉

)∗= [

O†(x, v, ∂x, ∂v)�(x, v)]∗.

Given the general rule that all operators for a non-Hermitian system must act onthe dual coordinate 〈x, v|, to obtain the matrix element 〈�D

mn|H |x, v〉 consider theeigenfunction equation

Emn�Dmn(x, v)∗ = 〈�D

mn|H |x, v〉∗ = 〈x, v|H †|�Dmn〉 = H †�D

mn(x, v).

Note that the general form of the Hamiltonian H given in Eq. 14.1 implies that

H †(x, v, ∂/∂v, ∂/∂) = H(x,−v,−∂/∂v, ∂/∂x)

= H(−x, v, ∂/∂v,−∂/∂x).Since all the eigenfunctions are real, there are two possibilities,

H †�Dmn(x, v) = H †�mn(x,−v) = Emn�mn(x,−v)

or

H †�Dmn(x, v) = H †�mn(−x, v) = Emn�mn(−x, v).

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Hence, the only difference between �Dmn(x, v) and �mn(x, v) is that v → −v or

x →−x and one obtains the general result that

�Dmn(x, v) =

⎧⎨⎩�mn(x,−v)or�mn(−x, v)

. (14.45)

As will be seen for the first two excited states given in Eq. 14.82, the dual eigenvec-tors are determined precisely in a manner that guarantees a positive norm (Hilbert)state space for the pseudo-Hermitian Hamiltonian H .

This remarkable feature of how the dual state vector is defined can be expectedto hold for all the eigenstates. Of the two possibilities stated in Eq. 14.45, it is theQ operator that determines which choice is made for the dual state vector.

The probability amplitude, from Eq. 13.91, is given by

KS(x, v; x ′, v′) = KS(x′,−v′; x,−v)

⇒ 〈x, v|e−τH |x ′, v′〉 = 〈x ′,−v′|e−τH |x,−v〉. (14.46)

From Eqs. 14.46 and14.44∑mn

e−τEmn�mn(x, v)�Dmn(x

′, v′) =∑mn

e−τEmn�mn(x′,−v′)�D

mn(x,−v)

⇒ �mn(x, v)�Dmn(x

′, v′) = �mn(x′,−v′)�D

mn(x,−v). (14.47)

To prove that the duality relation given in Eq. 14.45 yields Eq. 14.47 we need todiscuss the parity operator P , defined by

P : x →−x, v →−v, P2 = I

⇒ P�mn(x, v) = �mn(−x,−v). (14.48)

It can be readily shown that P commutes with H ,

[P, H ] = 0.

Hence all the eigenfunctions �mn(x, v) are also eigenfunctions of parity witheigenvalue s; Eq. 14.48 yields

P�mn(x, v) = �mn(−x,−v) = s�mn(x, v), s = ±1. (14.49)

Let us consider the two distinct cases for the dual vector given in Eq. 14.45, andEq. 14.47 is verified for each case:

• �Dmn(x, v) = �mn(x,−v). Eq. 14.47 yields

Left hand side : �mn(x, v)�Dmn(x

′, v′) = �mn(x, v)�mn(x′,−v′),

Right hand side : �mn(x′,−v′)�D

mn(x,−v) = �mn(x′,−v′)�mn(x, v),

hence, Eq. 14.47 is verified.

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14.7 Vacuum state; eQ/2 309

• �Dmn(x, v) = �mn(−x, v). Eq. 14.47 yields, using Eq. 14.48,

Left hand side : �mn(x, v)�Dmn(x

′, v′) = �mn(x, v)�mn(−x ′, v′),Right hand side : �mn(x

′,−v′)�Dmn(x,−v) = �mn(x

′,−v′)�mn(−x,−v),

= s2�mn(−x ′, v′)�mn(x, v).

Since s2 = 1, Eq. 14.47 is verified.

It is the nontrivial behavior of the dual state vector, as given in Eq. 14.45, that leadsto results that are not allowed for Hermitian Hamiltonians, such as the expressionobtained for the propagator and discussed in Section 14.13.

14.7 Vacuum state; eQ/2

The vacuum state and the dual vacuum state of the Euclidean Hamiltonian areobtained using the expressions for eQ/2 and e−Q/2 respectively. The vacuum can beverified to be correct by the direct application of the Hamiltonians H and H † andhence provides an independent verification of the matrix elements obtained for eτQ

in Section 14.3.The vacuum state is given by

|�00〉 = eQ/2|0, 0〉, H |�00〉 = E0|�00〉, E0 = 1

2[ω1 + ω2]. (14.50)

The coordinate representation of the vacuum state can be directly obtained fromthe Hamiltonian H by inspection and is given by

�00(x, v) = 〈x, v|�00〉 = 〈x, v|eQ/2|0, 0〉= N00 exp

{−γ

2(ω1 + ω2)ω1ω2x

2 − γ

2(ω1 + ω2)v

2 − γω1ω2xv}.

(14.51)

The vacuum state |�00〉 is real valued and normalizable, withN00 the normalizationconstant. The dual vacuum state obeys

〈�D00| = 〈0, 0|e−Q/2 ⇒ �D

00(x, v) = 〈�D00|x, v〉 = 〈0, 0|e−Q/2|x, v〉.

It can be directly verified that the dual ground state, from Eq. 14.51, is

�D00(x, v) = N00 exp

{−γ

2(ω1 + ω2)ω1ω2x

2 − γ

2(ω1 + ω2)v

2 + γω1ω2xv}.

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For τ = −1/2, Eq. 14.23 yields the matrix elements

〈x, v|eQ/2|x ′, v′〉

= N (1

2) exp

⎧⎨⎩√a

bcoth

(√ab

2

)(xv+ x ′v′)−

√a

b

(xv′ + vx ′)

sinh(√

ab2

)⎫⎬⎭ . (14.52)

Note that√a

bcoth

(√ab

2

)= γω2

1,

√a

b

1

sinh(√

ab2

) = γω1

√ω2

1 − ω22, (14.53)

N (1

2) = i

2π

√a

b

1

sinh(√

ab2

) = i

2πγω1

√ω2

1 − ω22. (14.54)


〈x, v|eQ/2|x ′, v′〉 = N (1

2) exp

{γω2

1(xv+ x ′v′)− γω1

√ω2

1 − ω22(xv′ + vx ′)

}.

From Eq. 14.27, the oscillator vacuum state is given by

〈x, v|0, 0〉 =(γ 2ω3

1ω2

π2

)1/4

exp{−γ

2(ω1v2 + ω2

1ω2x2)}.

Hence

�00(x, v) = 〈x, v|eQ/2|0, 0〉 =∫

dξdζ 〈x, v|eQ/2|ξ, ζ 〉〈ξ, ζ |0, 0〉

= N (1

2)

(γ 2ω3

1ω2

π

)1/4 ∫dξdζeS,

where

S = −γ2(ω1ζ

2 + ω21ω2ξ

2 − 2c1ξζ )− γ c2(ζx + vξ)+ γ c1xv,

c1 = ω21, c2 = ω1

√ω2

1 − ω22.

Performing the Gaussian integrations over ξ and ζ yields

�00(x, v) = N00eF ,

where

F = γ 2c22

2γ (ω31ω2 − c2

1)[ω1v2 + ω2

1ω2x2 + 2c1xv] + γ c1xv

= −γ2[(ω1 + ω2)ω1ω2x

2 + (ω1 + ω2)v2] − γω1ω2xv,

which is the expected result.

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14.7 Vacuum state; eQ/2 311

To determine N00 note that the Gaussian integration has the matrix

M = γ

[ω2

1ω2 −c1

−c1 ω1

],√

det(M) = iγ ω3/21

√ω1 − ω2.

The determinant of matrix M is negative and hence leads to the correct sign for theexponent F of the vacuum state vector.

Using Eq. 14.54, Gaussian integration yields

N00 = N (1

2)

(γ 2ω3

1ω2

π

)1/42π√

det(M)

= i

2πγω1

√ω2

1 − ω22

(γ 2ω3

1ω2

π2

)1/42π

iγω3/21

√ω1 − ω2

⇒ N00 = (ω1ω2)1/4

√γ

π(ω1 + ω2). (14.55)

To derive the dual vacuum state, note from Eq. 14.52

〈x, v|e−Q/2|x ′, v′〉 = N (1

2) exp

{−γω2

1(xv+ x ′v′)+ γω1

√ω2

1 − ω22(xv′ + vx ′)

}.

A derivation similar to the one for the vacuum state �00 yields the dual vacuumstate

�D00(x, v) = 〈0, 0|e−Q/2|x, v〉

= N00 exp{−γ2(ω1 + ω2)ω1ω2x

2 − γ

2(ω1 + ω2)v

2 + γω1ω2xv}.(14.56)

The norm of a state is defined by the scalar product of the state with its dual andyields the following norm for the vacuum state:

〈�D00|�00〉 =

∫dxdv�D

00(x, v)�00(x, v)

= N200

∫dxdv exp{−γ (ω1 + ω2)ω1ω2x

2 − γ (ω1 + ω2)v2} = 1.

Recall that Euclidean velocity is related to Minkowski velocity by v = dx/dτ =−idx/dtM = −ivM . Hence, the vacuum state |�00〉 when analytically continuedto Minkowski time has a divergent norm. Furthermore, the eigenstates generatedby applying the creation operators on the vacuum state all have a divergent norm.The problem of rendering the Minkowski time state space convergent has beenaddressed by Bender and Mannheim (2008b).

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14.8 Vacuum state and classical action

The definition of the probability amplitude given in Eq. 13.66,

KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉,yields, from Eq. 14.44, the infinite τ limit given by

limτ→+∞KS(xf , vf ; xi, vi) � e−τE0〈xf , vf |�00〉〈�00|e−Q|xi, vi〉 +O(e−τ(E1−E0))

= e−τE0�00(xf , vf )�D00(xi, vi). (14.57)

Hence, from Eq. 14.57, the infinite limit of transition amplitude is the product ofthe dual vacuum state �D

00(xi, vi) and the vacuum state �00(xf , vf ). On the otherhand, the classical solution yields

limτ→+∞K(xf , vf , xi, vi) = lim

τ→+∞N ′eSc(xf ,vf ,xi ,vi ). (14.58)

The infinite τ limits of matrix elements M , given in Section 13.2, are given by

limτ→+∞M11 = 2ra(r2 + ω2), lim

τ→+∞M22 = 2ra, limτ→+∞M12 = 2r2a − b,

limτ→+∞M13 = lim

τ→+∞M14 = limτ→+∞M23 = 0. (14.59)

Therefore, the vacuum state is obtained from Eqs. 13.28 and 14.59 and yields

�(x, v) = limτ→+∞N exp

(−1

2M11x

2 − 1

2M22v2 −M12xv

)= N exp

(−ra2(r2 + ω2)x2 − rav2 − (2r2a − b)xv

), (14.60)

where xf = x, vf = v, and N is fixed by normalizing �(x, v).To make connection with the earlier parameterization, Let us rewrite the

Lagrangian in ω1 and ω2 parameterization (the only difference from the parame-terization given in Eq. 13.29 is in the overall factor of a instead of γ )

L = −1

2a

(x2 + 2

b

ax2 + c

ax2

)= −γ

2

(x2 + (ω2

1 + ω22)x

2 + (ω21ω

22)x

2), (14.61)

where ω1 and ω2 are1

ω1 = 1√2γ

(√b +√ac +

√b −√ac

),

1 Note that ω1 and ω2 in Eq. 13.30 have a different parameterization, given by

ω1 = 1

2√γ

(√α + 2

√γβ +

√α − 2

√γβ

), ω2 = 1

2√γ

(√α + 2

√γβ −

√α − 2

√γβ

).

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14.9 Excited states of H 313

ω2 = 1√2γ

(√b +√ac −

√b −√ac

).

The definition variables r and ω in terms of ω1 and ω2 are given by

ω1 = r + iω, ω2 = r − iω.

The two parameterizations have the relationship

r2 + ω2 = ω21ω

22, r = 1

2(ω1 + ω2), b = γ

2(ω2

1ω22). (14.62)

Equations 14.60 and 14.62 hence yield – replacing a by γ to conform to the nota-tion given in Eq. 13.30 – the vacuum state given in Eq. 14.51, namely

�00(x, v) = N exp(−γ

2(ω1 + ω2)ω1ω2x

2 − γ

2(ω1 + ω2)v

2 − γω1ω2xv).

The definition of the evolution kernel KS(xf , vf ; xi, vi) = 〈xf , vf |e−τH |xi, vi〉given in Eq. 13.66 is seen to be correct since the evolution kernel obtained fromthe classical solution also gives the same result for the vacuum state as given bythe Hamiltonian in Eq. 14.51.

In particular, there is no need to include the state space metric e−Q in the defi-nition of KS(xf , vf ; xi, vi), since the expression 〈xf, vf |e−τH |xi, vi〉 for the proba-bility amplitude is the matrix elements in a complete basis, and not the probabilityamplitude between physical states. However, in obtaining the vacuum state theoperator e−Q appears, since it defines the dual of the vacuum state |�00〉 via theexpression 〈�00|e−Q.

14.9 Excited states of H

To illustrate the general features of the eigenfunctions, the first few eigenfunctionsare evaluated. Note

a†v =

√γω1

2

(v− 1

γω1

∂

∂v

), (14.63)

a†x =

√γω2

1ω2

2

(x − 1

γω21ω2

∂

∂x

). (14.64)

In general, one can find the explicit coordinate representation of any eigenfunc-tion by the following procedure:

�nm(x, v) = 〈x, v|eQ/2

{a†n

v√n!

a†mx√m!}|0, 0〉

= 〈x, v|eQ/2 a†nv√n!

a†mx√m!e

−Q/2eQ/2|0, 0〉

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= eQ/2 a†nv√n!

a†mx√m!e

−Q/2�00(x, v), (14.65)

where, using Eqs. 14.69 and 14.76 given below, one can explicitly evaluate

eQ/2a†nv a†m

x e−Q/2 = (eQ/2a†

ve−Q/2

)n (eQ/2a†

xe−Q/2

)m. (14.66)

14.9.1 Energy ω1 eigenstate �10(x, v)

The ω1 single excitation energy eigenstate is the following:

�10(x, v) = 〈x, v|eQ/2{a†

v |0, 0〉}= 〈x, v|eQ/2a†

ve−Q/2eQ/2|0, 0〉

⇒ �10(x, v) = eQ/2a†ve−Q/2�00(x, v). (14.67)

The fundamental similarity transformation given in Eq. 14.8 yields

eQ/2ve−Q/2 = Av− B

C

∂

∂x,

eQ/2 ∂

∂ve−Q/2 = A

∂

∂v− BCx, (14.68)

where, from Eq. 14.11 the coefficient functions are given by

A = ω1√ω2

1 − ω22

, B = ω2√ω2

1 − ω22

, C = γω1ω2.


eQ/2a†ve−Q/2 =

√γω1

2eQ/2

[v− 1

γω1

∂

∂v

]e−Q/2

=√γω1

2

[Av+ BC

γω1x − B

C

∂

∂x− A

γω1

∂

∂v

]

⇒ eQ/2a†ve−Q/2 =

√γω1

2(ω21 − ω2

2)

[ω1v+ ω2

2x −1

γω1

∂

∂x− 1

γ

∂

∂v

]. (14.69)


�10(x, v) =√

γω1

2(ω21 − ω2

2)

[ω1v+ ω2

2x −1

γω1

∂

∂x− 1

γ

∂

∂v

]�00(x, v).

Using the explicit representation of the vacuum state �00(x, v) given in Eq. 14.50yields the final result

�10(x, v) =√

2γω1

ω21 − ω2

2

(ω1 + ω2) [v+ ω2x]�00(x, v). (14.70)

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14.9 Excited states of H 315

The dual energy eigenstate is defined by

〈�D10| = 〈1, 0|e−Q/2 = 〈0, 0|ave

−Q/2

⇒ |�D10〉 = e−Q/2a†

v |0, 0〉 = e−Q/2a†ve

Q/2e−Q/2|0, 0〉⇒ �D

10(x, v) = 〈x, v|�D10〉 = e−Q/2a†

veQ/2e−Q/2�D

00(x, v). (14.71)

Similarly to the derivation of Eq. 14.69, Eq. 14.8 yields

e−Q/2a†ve

Q/2 =√

γω1

2(ω21 − ω2

2)

[ω1v− ω2

2x +1

γω1

∂

∂x− 1

γ

∂

∂v

], (14.72)

and, from Eq. 14.71, we obtain the dual eigenfunction

�D10(x, v) =

√γω1

2(ω21 − ω2

2)

[ω1v− ω2

2x +1

γω1

∂

∂x− 1

γ

∂

∂v

]�D

00(x, v).

Using the representation of the vacuum state �D00(x, v) given in Eq. 14.56 yields

the final result

�D10(x, v) =

√2γω1

ω21 − ω2

2

(ω1 + ω2) [v− ω2x]�D00(x, v). (14.73)

14.9.2 Energy ω2 eigenstate �01(x, v)

The ω2 one excitation energy eigenstate is given by

�01(x, v) = 〈x, v|eQ/2{a†x |0, 0〉

}= 〈x, v|eQ/2a†

xe−Q/2eQ/2|0, 0〉

⇒ �01(x, v) = eQ/2a†xe−Q/2�00(x, v). (14.74)

The similarity transformation given in Eq. 14.8 yields

eQ/2xe−Q/2 = Ax − B

C

∂

∂v,

eQ/2 ∂

∂xe−Q/2 = A

∂

∂x− BCv. (14.75)


eQ/2a†xe−Q/2 =

√γω2

1ω2

2eQ/2

[x − 1

γω21ω2

∂

∂x

]e−Q/2

=√γω2

1ω2

2

[Ax + BC

γω21ω2

v− B

C

∂

∂v− A

γω21ω2

∂

∂x

]

⇒ eQ/2a†xe−Q/2 =

√γω2

2(ω21 − ω2

2)

[ω2v+ ω2

1x −1

γω2

∂

∂x− 1

γ

∂

∂v

]. (14.76)

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�01(x, v) =√

γω2

2(ω21 − ω2

2)

[ω2v+ ω2

1x −1

γω2

∂

∂x− 1

γ

∂

∂v

]�00(x, v).

Using the representation of the vacuum state �00(x, v) given in Eq. 14.50 yieldsthe final result

�01(x, v) =√

2γω2

ω21 − ω2

2

(ω1 + ω2) [v+ ω1x]�00(x, v). (14.77)

The dual energy eigenstate, similarly to Eq. 14.78, is defined by

�D01(x, v) = e−Q/2a†

veQ/2e−Q/2�D

00(x, v). (14.78)


e−Q/2a†xe

Q/2 =√

γω2

2(ω21 − ω2

2)

[−ω2v+ ω2

1x −1

γω2

∂

∂x+ 1

γ

∂

∂v

]. (14.79)

Using the explicit representation of the vacuum state �D00(x, v) given in Eq. 14.56

with Eq. 14.78 yields the dual eigenfunction

�D01(x, v) =

√2γω2

ω21 − ω2

2

(ω1 + ω2) [−v+ ω1x]�D00(x, v). (14.80)

We collect the results for the first two one excitation states; the normalizationconstants are separated out for later convenience; using the value of N00 given inEq. 14.55 yields

�10(x, v) = N10 [v+ ω2x]ψ00(x, v), �D10(x, v) = N10 [v− ω2x]ψD

00(x, v),

�01(x, v) = N01 [v+ ω1x]ψ00(x, v), �D01(x, v) = N01 [−v+ ω1x]ψD

00(x, v),

N10 = γ√

2(ω1 + ω2)√π(ω1 − ω2)

ω3/41 ω

1/42 , N01 = γ

√2

(ω1 + ω2)√π(ω1 − ω2)

ω1/41 ω

3/42 ,

E10 = ω1 + E00, E01 = ω2 + E00. (14.81)

The eigenstates are orthogonal and normalized, namely

〈�D10|�10〉 = 1 = 〈�D

01|�01〉, 〈�D10|�01〉 = 0.

Note the remarkable result that under a duality transformation, the dual eigen-states have a transformation that depends on the eigenstate, as discussed in Eq.14.45; in particular

�D10(x, v) = �10(−x, v), �D

01(x, v) = �01(x,−v). (14.82)

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14.10 Complex ω1, ω2 317

This feature generalizes to all the energy eigenstates and guarantees that the statespace, for ω1 > ω2, always has a positive norm.

The first two energy eigenstates of the pseudo-Hermitian Hamiltonian H are thefirst excitation of the position degree of freedom x, given by �01, and the veloc-ity degree of freedom v, given by �10. Note that in the limit of ω1 → ω2, theeigenstates �01 and �10 become degenerate. This important property of the energyeigenspectrum will be studied in some detail when the limit of ω1 → ω2 is takenin Section 15.1.

14.10 Complex ω1, ω2

The application of the acceleration Lagrangian to the study of equities, as shownby Baaquie et al. (2012), requires that the parameters ω1, ω2 are complex. FromEq. 13.31, consider the parameterization given by

ω1 = Reiφ, ω2 = Re−iφ : R > 0, φ ∈ [−π/2, π/2], (14.83)

⇒ ω1 + ω2 = 2R cos(φ), ω1ω2 = R2. (14.84)

The eigenenergies are given by

E00 = 1

2(ω1 + ω2) = R cosφ,

Emn = mω1 + nω2 + E00 = mReiφ + nRe−iφ + R cosφ.

As discussed in Section 14.2, the eigenenergies of a pseudo-Hermitian Hamiltoniancome in conjugate pairs and are given by

Emn = mReiφ + nRe−iφ + R cosφ,

Enm = nReiφ +mRe−iφ + R cosφ = E∗mn.

In other words, energy eigenvalues Emn and Enm are complex conjugates of eachother.

From Eq. 14.10, the parameters are given by√a

b= γω1ω2 = γR2, sinh(

√ab) = 2ω1ω2

ω21 − ω2

2

= −icos(2φ)

.

Since a, b are now complex, the Q-operator is given from Eq. 14.5,

Q = axv− b∂2

∂x∂v

is no longer Hermitian.Since the eigenenergies are complex, there is no sense of the ordering of energies

and the concept of a ground state is no longer valid. One can still consider the

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real part of the eigenenergies and the lowest real energy E00 has the followingeigenstate given by the continuation of the vacuum state

�00(x, v) = N00 exp{−γR3 cos(φ)x2 − γR cos(φ)v2 − γR2xv

}. (14.85)

Equation 14.85 yields a positive norm state that is well defined for

cos(φ) > 0 ⇒ − π/2 < φ < π/2.

Since ω1, ω2 are complex, the algebra of the creation and destruction operatorsgiven in Eq. 14.30 is no longer valid. Hence, for the complex branch of the param-eters, these operators can no longer generate the spectrum of states for H .

14.11 State space V of Euclidean Hamiltonian

The state space of the non-Hermitian Hamiltonian H , namely V , has a non-trivialmetrical structure. This is most clearly seen in the manner in which the dualstate vectors are defined as well as the rules of conjugation for operators, calledQ-conjugation, that is different from the usual Hermitian conjugation.

To define orthonormality of the eigenfunctions |�nm〉, one needs to define thescalar product of a state vector |�〉 with a dual vector 〈χ |, that are elements ofV . In particular, one needs to define a dual that is consistent with the similaritytransformation given in Eq. 14.12 that maps the non-Hermitian Hamiltonian H tothe Hermitian HamiltonianH

O. Figure 14.1 schematically shows the mapping from

the state space of HO

denoted by V0 to the state space of H denoted by VH.

e+Q

VH

V0

Figure 14.1 A mapping between two state spaces.

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14.11 State space V of Euclidean Hamiltonian 319

A consistent quantum mechanics can be defined by generalizing the scalar prod-uct by using a metric W on state space in the following manner [Ballentine (1998)]:

〈χ |�〉W= 〈χ |W |�〉, W † = W. (14.86)

In analogy with the result for Minkowski time [Mannheim and Davidson(2005)], the metric for the state space V is given by

W † = W = e−Q ⇒ 〈χ |�〉Q = 〈χ |e−Q|�〉 : scalar product. (14.87)

Hence two state vectors in V are orthogonal if

〈χ |e−Q|�〉 = 0 : orthogonal. (14.88)

The norm of a state vector, as exemplified by Eq. 14.38, is given by

|�|2Q= 〈�|e−Q|�〉. (14.89)

The definition of the scalar product given in Eq. 14.87 is equivalent to defining thedual vector of |�〉 by the rule

|�〉 → Dual → 〈�|e−Q : dual vector. (14.90)

In particular, using the definition of the scalar product given in Eqs. 14.87 and14.90, it follows that all the eigenfunctions |�nm〉 are orthonormal as shown in Eq.14.39.

The completeness equation for state space V is

I =∫ +∞

−∞dxdveQ/2|x, v〉〈x, v|e−Q/2

=∫ +∞

−∞dxdveQ/2|x, v〉

(〈x, v|eQ/2

)e−Q,

with completeness and orthogonality obtained using Eq. 13.65

14.11.1 Operators acting on VThe conjugation of operators in state space V depends on the state space metrice−Q. Q-conjugation for operator O is defined by

〈χ |e−QO|�〉∗ = 〈�|O†e−Q|χ〉= 〈�|e−Q[eQO†e−Q]|χ〉 = 〈�|e−QO†

Q|χ〉,

where the Q-conjugate of operator O is defined by

O†Q= eQO†e−Q

and O† is the usual Hermitian conjugation.

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All operators in state space V must be Q-conjugated under conjugation. In par-ticular, a Q-self-conjugate operator is given by

O†Q= eQO†e−Q = O

⇒ 〈χ |e−QO|�〉∗ = 〈�|e−QO|χ〉 : Q-self-conjugate operator. (14.91)

A Q-self-conjugate operator has real eigenvalues since, from Eq. 14.91, for a nor-malized eigenfunction |�〉

E|�〉 = O|�〉, O†Q= O

⇒ E〈�|e−Q|�〉 = E = 〈�|e−QO|�〉,E∗ = 〈�|e−QO|�〉∗ = 〈�|e−QO†

Q|�〉

= 〈�|e−QO|�〉 = E ⇒ E : real.

An obvious but important result is that the state space metric e−Q isQ-conjugate,namely that

e−QQ= eQ/2e−Qe−Q/2 = e−Q

⇒(e−QQ

)† = e−Q/2e−QeQ/2 = e−Q : Q-self-conjugate operator.

The pseudo-Hermitian Hamiltonian H is Q-self-conjugate; from Eq. 14.12

H = eQ/2HOe−Q/2

⇒ H †Q= eQH †e−Q = eQe−Q/2H †

OeQ/2e−Q (14.92)

= eQ/2HOe−Q/2 = H : Q-self-conjugate operator. (14.93)

All operators acting on state space V need to be Q-conjugated to obtain theconjugated form of any operator equation. The presence of the operatorQ indicatesthat the operators and state vectors belong to state space V . Examples of operatorequations that are only valid on state space V are given in Eq. 14.8. It can be seenthat taking the Hermitian conjugate of any the equations in Eq. 14.8 leads to aninconsistency, whereas taking the Q-conjugate of these same equations leads to aconsistent result.

Note that all the properties of the state space V depend on the operator Q beingwell behaved. From Eq. 14.10 it can be seen that for ω1 = ω2, the operator Qdiverges and hence the Hamiltonian and state space need to be studied from firstprinciples.

To illustrate the role of the Hilbert space metric e−Q consider the time-orderedvacuum expectation value of the Heisenberg operators at two different (Euclidean)times τ > 0. The vacuum state is given by |�00〉 with H |�00〉 = E00|�00〉;

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14.11 State space V of Euclidean Hamiltonian 321

E00 ≡ E0 = (ω1+ω2)/2; using the rule for forming the dual vector of the pseudo-Hermitian Hamiltonian yields

G(τ) = 〈�D00|xH (τ)xH (0)|�00〉 = 〈�00|e−QxH(τ)xH (0)|�00〉

= 〈�00|e−QeτHxe−τHx|�00〉 = 〈�00|eτH †e−Qxe−τHx|�00〉

= 〈�00|e−Qxe−τ(H−E0)x|�00〉. (14.94)

The term G(τ) in Eq. 14.94 is the propagator and is analyzed in detail in Section14.12.

Recall that the probability amplitude is given in Eq. 13.66,

KS(x, v; x ′, v′) = 〈x, v|e−τH |x ′, v′〉,and the matrix elements of the the Hilbert space metric e−Q are given in Eq. 14.23,

〈x, v|e−τQ|x ′, v′〉 = N exp{−G(τ )(xv+ x ′v′)+H(τ )(xv′ + vx ′)

}. (14.95)

For both the operators e−τH and e−τQ, there is no need for an extra metric e−τQ

since Eqs. 13.66 and 14.23 are the matrix elements of the operators in a completebasis 〈x, v| and its dual |x ′, v′〉. If the matrix elements of operators e−τH and so onare determined for eigenstates and dual eigenstates of H , then the metric e−Q isrequired.

One can evaluate the matrix elements of e−τH in Hilbert space, using the basisstates of V , namely eQ/2|x, v〉 and the dual basis

(〈x, v|eQ/2)e−Q = 〈x, v|e−Q/2,

which yields

〈x, v|e−Q/2e−τH eQ/2|x ′, v′〉 = 〈x, v|e−τH0 |x ′, v′〉. (14.96)

The symmetry of the matrix elements of 〈x, v|e−Q/2e−τH eQ/2|x ′, v′〉 – given abovein Eq. 14.96 – is not the symmetry given in Eq. 13.91 for 〈x, v|e−τH |x ′, v′〉; hencethe correct expression for the kernel is given by KS(x, v; x ′, v′) = 〈x, v|e−τH |x ′, v′〉,which in turn yields the correct matrix elements given by the path integral as seenin the results obtained in Section 14.8.

14.11.2 Heisenberg operator equations

In Schrödinger’s formulation of quantum mechanics, the time dependence of aquantum system arises solely due to the time evolution of the state vector |�(t)〉 –with the operators O being taken to be time-independent. For a pseudo-HermitianHamiltonian, the Schrödinger equation yields, for Minkowski time t

M,

|�(t)〉 = exp{−itMH }|�〉, 〈χ(t)| = 〈χ | exp{it

MH †}. (14.97)

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For Euclidean time t = itM

, one has the expressions

|�(t)〉 = e−tH |�〉, 〈χ(t)| = 〈χ | etH †.

To obtain the Heisenberg operator equations of motion for the pseudo-HermitianHamiltonian, it is necessary to define the dual state vectors using Q-conjugation.In particular, for Euclidean time, the time dependent expectation value, using Eq.14.4, yields the result

E[O; t] = 〈χ(t)|e−QO|�(t)〉 = 〈χ |etH †e−QOe−tH |�〉

= 〈χ |e−QetHOe−tH |�〉 = 〈χ |e−QOH(t)|�〉. (14.98)

Hence, from Eq. 14.98, due to the choice of the Hilbert space metric e−Q, theHeisenberg time-dependent operator OH (t) for the pseudo-Hermitian Hamiltonianis given by the same expression as for the Hermitian Hamiltonian, namely

OH(t) = etHOe−tH ⇒ ∂OH(t)

∂t= [H,OH(t)].

In particular, for the Hamiltonian of the acceleration action given, from Eq. 13.69,by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ 1

2αv2 +�(x),

the Heisenberg equation for the coordinate operator xH (t), with x being theSchrödinger coordinate operator, yields

xH (t) = etHxe−tH ,

xH (t) = ∂xH (t)

∂t= [H,OH(t)] = −vH(t). (14.99)

Hence, the identification made in the path integral derivation, namely x(t) =−v(t), is seen to hold also as an operator equation for the Heisenberg operatorsxH (t), vH (t), as shown by Eq. 14.99.

14.12 Propagator: operators

Constructing the propagator by inserting the complete set of states yields a realiza-tion of the propagator in terms of the state space and Hamiltonian. The state spacedefinition of the propagator is given by Eq. 11.6 and yields

G(τ) = limT→∞

1

Ztr(e−(T−τ)Hxe−τHx

), τ = |t − t ′|.

Note that

limT→∞ e−TH � e−T E0 |�00〉〈�00|e−Q = e−T E0eQ/2|0, 0〉〈0, 0|e−Q/2,

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14.12 Propagator: operators 323

Z = tr(e−TH ) = e−T E0 .

Since

H = eQ/2HOe−Q/2, |�00〉 = eQ/2|0, 0〉, 〈�00| = 〈0, 0|eQ/2,

the propagator is given by

G(τ) = limT→∞

1

Ztr(e−(T−τ)H xe−τHx

), τ = |t − t ′|,

= 〈�00|e−Qxe−τ(H−E0)x|�00〉, (14.100)

= 〈0, 0|e−Q/2xeQ/2e−τ(H0−E0)e−Q/2xeQ/2|0, 0〉. (14.101)

Note that Eq. 14.100 has been obtained earlier in Eq. 14.94 based on a Hilbertspace derivation of the propagator.

From Eq. 14.8,

e−Q/2xeQ/2 = Ax + BC∂

∂v

= A

√1

2γω21ω2

(ax + a†x)+ BC

√γω1

2(av − a†

v ). (14.102)


e−Q/2xeQ/2|0, 0〉 = A

√1

2γω21ω2

|0, 1〉 − BC

√γω1

2|1, 0〉,

〈0, 0|e−Q/2xeQ/2 = A

√1

2γω21ω2

〈0, 1| + BC

√γω1

2〈1, 0|. (14.103)


G(τ) = A2 1

2γω21ω2

〈0, 1|e−τ(H0−E0)|0, 1〉 − (BC)2 γω1

2〈1, 0|e−τ(H0−E0)|1, 0〉

= 1

2γω21ω2

A2e−ω2τ − γω1

2(BC)2e−ω1τ .

Note that all the operators and state functions in the equation above are definedsolely in state space V0. However, the coefficients of the various matrix elements,in particular the negative sign on the second matrix element, are a result of theproperties of the conjugation operator eQ and reflect the presence of the underlyingstate space V; the result obtained could not have been generated by working solelyin Hilbert space V0.

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A2 = cosh2

(√ab

2

)= ω2

1

ω21 − ω2

2

,

(BC)2 = b

asinh2

(√ab

2

)= 1

γ 2ω21ω

22

· ω22

ω21 − ω2

2

= 1

γ 2ω21

· 1

ω21 − ω2

2

.

Hence, collecting all the results yields the expected result that

G(τ) = 1

2γ

1

ω21 − ω2

2

[e−ω2τ

ω2− e−ω1τ

ω1

]. (14.104)

14.13 Propagator: state space

Recall from Eq. 14.100, that the propagator is given by

G(τ) = 〈�00|e−Qxe−τ(H−E0)x|�00〉= 〈�D

00|xe−τ(H−E0)x|�00〉. (14.105)

The completeness equation for H , from Eq. 14.40, is given by

I =∞∑

m,n=1

|�mn〉〈�Dmn|, (14.106)

and yields, from Eq. 14.105,

G(τ) =∞∑

m,n=1

〈�D00|xe−τ(H−E0)|�mn〉〈�D

mn|x|�00〉

= e−τω1〈�D00|x|�10〉〈�D

10|x|�00〉 + e−τω2〈�D00|x|�01〉〈�D

01|x|�00〉,(14.107)

= e−τω1G1 + e−τω2G2. (14.108)

The vacuum state and its normalization, from Eqs. 14.50, 14.55, and 14.56 is

�00(x, v) = N00ψ00(x, v), �D00(x, v) = �00(x,−v) = �00(−x, v),

ψ00(x, v) = exp{−γ2(ω1 + ω2)ω1ω2x

2 − γ

2(ω1 + ω2)v

2 − γω1ω2xv},

N00 = (ω1ω2)1/4

√γ

π(ω1 + ω2).

Recall from Eq. 14.81,

�10(x, v) = N10 [v+ ω2x]ψ00(x, v), �D10(x, v) = N10 [v− ω2x]ψD

00(x, v),

�01(x, v) = N01 [v+ ω1x]ψ00(x, v), �D01(x, v) = N01 [−v+ ω1x]ψD

00(x, v),

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14.13 Propagator: state space 325

N10 = γ√

2(ω1 + ω2)√π(ω1 − ω2)

ω3/41 ω

1/42 , N01 = γ

√2

(ω1 + ω2)√π(ω1 − ω2)

ω1/41 ω

3/42 .

Using the coordinate representation for the state vectors yields

G1 = 〈�D00|x|�10〉〈�D

10|x|�00〉= N2

10N200

∫dxdv x(v+ ω2x)ψ

D00(x, v)ψ00(x, v)

×∫

dxdv x(v− ω2x)ψD00(x, v)ψ00(x, v)

= −N210N

200ω

22

[π

2γ (ω1 + ω2)2(ω1ω2)3/2

]2

= − 1

2γ (ω21 − ω2

2)2ω1

. (14.109)

Similarly

G2 = 〈�D00|x|�01〉〈�D

01|x|�00〉= N2

10N200

∫dxdv x(v+ ω1x)ψ

D00(x, v)ψ00(x, v)

×∫

dxdv x(−v+ ω1x)ψD00(x, v)ψ00(x, v)

= N210N

200ω

21

[π

2γ (ω1 + ω2)2(ω1ω2)3/2

]2

= 1

2γ (ω21 − ω2

2)2ω2

. (14.110)

Hence, Eqs. 14.108, 14.109, and 14.110 yield the expected result given in Eq.14.104, namely that

G(τ) =∞∑

m,n=1

〈�D00|xe−τ(H−E0)|�mn〉〈�D

mn|x|�00〉

= e−τω1〈�D00|x|�10〉〈�D

10|x|�00〉 + e−τω2〈�D00|x|�01〉〈�D

01|x|�00〉= e−τω1G1 + e−τω2G2

= 1

2γ

1

ω21 − ω2

2

[−e

−ω1τ

ω1+ e−ω2τ

ω2

]. (14.111)

There are a number of remarkable features of the state space derivation. The nega-tive sign that appears in the propagator for the termG1 is usually taken to be a proofthat no unitary theory can yield this result. The reason for this is the following:

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consider any arbitrary Hermitian Hamiltonian such that HA = H†A; the spectral

resolution of this Hamiltonian in terms of its eigenstates |χmn〉 is given by

I =∞∑

mn=1

|χmn〉〈χmn|. (14.112)

Note that 〈χmn| = 〈χDmn|, since for a Hermitian Hamiltonian HA the left and right

eigenstates are complex conjugates of each other. Hence, the propagator for theHermitian Hamiltonian HA is given by

GA(τ) =∞∑

m,n=1

〈χD00|xe−τ(HA−E0)|χmn〉〈χD

mn|x|χ00〉

= e−τω1〈χD00|x|χ10〉〈χD

10|x|χ00〉 + e−τω2〈χD00|x|χ01〉〈χD

01|x|χ00〉= e−τω1

∣∣〈χ00|x|χ10〉∣∣2 + e−τω2

∣∣〈χ00|x|χ01〉∣∣2.

The result above shows that a Hermitian Hamiltonian defined on a Hilbert spacecannot have a propagator such as the one given in Eq. 14.104, except by allowing∣∣〈χ00|x|χ10〉

∣∣2 < 0, which implies that |χ10〉 is a ghost state that has a negativenorm.

In contrast, the pseudo-Hermitian Euclidean Hamiltonian H has a positive normfor all the states in its state space; the duality transformation in going from |�mn〉 to〈�D

mn| provides the negative signs that allow for the propagator given in Eqs. 14.104and 14.111.

14.14 Many degrees of freedom

Consider the generalization of a Hamiltonian with a quadratic potential given inEq. 13.69 to many degrees of freedom. For degrees of freedom xn, n = 1, 2, . . . Nthe acceleration action, in matrix notation, is chosen to be

H = −1

2

∂

∂vST

1

γS∂

∂v− v

∂

∂x+ 1

2vST αSv+ 1

2xST βSx, (14.113)

that yields the Lagrangian

L = −1

2

[xST γ Sx + xST αSx + xST βSx

]. (14.114)

A very special choice has been made for H and L in that all the couplings havethe same similarity transformation S connecting the different degrees of freedom.In particular, choosing S = I would decouple all the different degrees of freedom.

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14.14 Many degrees of freedom 327

The (real) orthogonal matrix S and the diagonal matrices are given in matrixnotation as

SST = I, γ = diag(γ1, γ2, . . . , γN),

α = diag(α1, α2, . . . , αN), β = diag(β1, β2, . . . , βN). (14.115)

Let us define new variables, in matrix notation

x = ST z, v = ST u. (14.116)

The Hamiltonian in Eq. 14.113 is given by

H = −1

2

N∑n=1

1

γn

∂2

∂u2n

−N∑n=1

un∂

∂zn+ 1

2

N∑n=1

αnu2n +

1

2

N∑n=1

βnz2m. (14.117)

The many degrees of freedom Lagrangian, from Eq. 14.114 and similar to Eq.13.29, is given by

L = −1

2

N∑n=1

γn[z2n + (ω2

1n + ω22n)z

2n + ω2

1nω22nz

2n

]. (14.118)

The parameterization is a generalization of Eq. 13.30 and is given by

ω1n = 1

2√γn

(√αn + 2

√γnβn +

√αn − 2

√γnβn

), (14.119)

ω2n = 1

2√γn

(√αn + 2

√γnβn −

√αn − 2

√γnβn

), (14.120)

ω1n > ω2n for ω1n, ω2n real.

The Hamiltonian H given in Eq. 14.113 is diagonalized by the following gener-alization of the Q-operator given in Eq. 14.5:

Q =N∑n=1

Qn, (14.121)

Qn = anznun − bn∂2

∂zn∂un, (14.122)

with the following values for an and bn:√an

bn= γnω1nω2n,

√anbn = ln

(ω1n + ω2n

ω1n − ω2n

). (14.123)

The diagonal Hamiltonian HO

is given similarly to the earlier case. The groundstate is given by generalizing Eq. 14.51 and yields

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�00(x, v) =

NN∏n=1

exp{−γn

2(ω1n + ω2n)ω1nω2nz

2n −

γn

2(ω1n + ω2n)u

2n − γnω1nω2nznun

}.

In terms of the original coordinates,

�00(x, v) = N exp

{−1

2

N∑m,n=1

(Pmnxmxn +Qmnvmvn + 2Rmnxnvn

)},

(14.124)

where, again in matrix notation

P = SpST , pn = γn(ω1n + ω2n)ω1nω2n,

Q = SqST , qn = γn(ω1n + ω2n),

R = SrST , rn = γnω1nω2n.

14.15 Summary

The Euclidean acceleration Hamiltonian, for the real branches, was shown to bepseudo-Hermitian and was mapped to a Hermitian Hamiltonian, which consists oftwo decoupled harmonic oscillators. The similarity transformation Q was shownto be an unbounded differential operator and the matrix elements of e±τQ wereexactly evaluated. All the defining commutation equations were evaluated usingthe matrix elements of e±τQ to confirm the result obtained.

The state space of the pseudo-Hermitian Euclidean Hamiltonian has a state spacemetric that is a natural generalization of the state space of quantum mechanics.The Heisenberg operator equations were analyzed to conclude that a state spacemetric is required for making the theory consistent and leads to a generalized scalarproduct and to the concept of Q-conjugation discussed by Mostafazadeh (2002).

The state vector and its dual vector were analyzed. It was shown that the asym-metry of the evolution kernel obtained using the Lagrangian and classical actioncan be derived using the parity symmetry of the Hamiltonian and duality propertyof the state functions.

The propagator was evaluated using the algebra of the creation and destruc-tion operators. The first two excited states were computed and the propagator wasevaluated a second time using the properties of state space. It was seen that thepropagator has a form that is forbidden for a Hermitian Hamiltonian and exists forthe acceleration Hamiltonian due to the properties of Q-conjugation required for apositive norm state space.

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14.15 Summary 329

As was seen in many of the derivations in this chapter, the results fromMinkowski time serve as a useful guide for the derivations; but – given a plethoraof i and various + and − signs that differ between the Euclidean and Minkowskiresults – all the derivations for the Euclidean have to be done from first principlesand independently from the Minkowski case.

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15

Non-Hermitian Hamiltonian: Jordan blocks

The Euclidean action with acceleration has been analyzed in Chapters 13 and 14 forits path integral and Hamiltonian. In this chapter, the acceleration Hamiltonian isanalyzed for the case when it is essentially inequivalent to a Hermitian Hamiltonian:for critical values of the parameters, given by ω1 = ω2 and shown in Figure 13.1,the mapping of the Hamiltonian to an equivalent Hermitian Hamiltonian becomessingular. The Hamiltonian continues to be pseudo-Hermitian but is no longerequivalent to a Hermitian Hamiltonian.

The Hamiltonian for real ω1 > ω2 is pseudo-Hermitian as well as being equiva-lent to a Hermitian operator H0 = H †

0, due to the existence of a similarity transfor-

mation Q such that

H = e−Q/2H0eQ/2 ⇒ H † = e−QHeQ, ω1 > ω2.

For the case ω1 = ω2 the coefficients in Q diverge and the Hamiltonian H can nolonger be mapped to a Hermitian Hamiltonian. The Hamiltonian is essentially non-Hermitian and is shown in this chapter to be equal to a direct sum of block diagonalmatrices, with each block being a Jordan block matrix. The Jordan block itself canbe shown to be pseudo-Hermitian, but the transformation cannot be obtained fromthe diverging Q. The case of the Hamiltonian being a Jordan block is analyzed indetail in this chapter as it forms a quantum mechanical system with its own uniquefeatures that is not encountered in Hermitian quantum mechanics.

In Section 15.1 the equal frequency Hamiltonian is obtained; in Sections 15.2and 15.3 propagator is analyzed and the two lowest lying state vectors for the sin-gular case are evaluated. In Section 15.5 the equal frequency propagator is eval-uated using the state vectors of the Jordan block. In Sections 15.6 and 15.7 theHamiltonian for the Jordan block is derived and the Schrödinger equation for thissystem is studied.

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15.2 Propagator and states for equal frequency 331

15.1 Hamiltonian: equal frequency limit

The critical Hamiltonian is given for the equal frequency limit. In the equal fre-quency limit of ω1 = ω2 (α = 2

√βγ ) the parameters of the Q-operator, namely

a and b, given in Eq. 14.10 become divergent and a well defined Q-operatorno longer exists. Although the Hamiltonian has special properties for the equal-frequency point, the path integral is well behaved for all (complex and real) valuesof ω1, ω2, including the equal-frequency critical point at ω1 = ω2. Moreover, thenon-Hermitian Hamiltonian H remains well defined at the critical point.

The singularity for the Q-operator is due to the fact that the accelerationHamiltonian H cannot be mapped to an equivalent Hermitian Hamiltonian H0.For the case of ω1 = ω2, non-Hermitian Hamiltonian H is no longer pseudo-Hermitian but instead, H is essentially non-Hermitian and has been shown to beexpressible as a Jordan-block matrix by Bender and Mannheim (2008a).

The general analysis of the equal frequency Hamiltonian has been carried outfor Minkowski time in the pioneering work of Bender and Mannheim (2008a) andthe analysis for Euclidean time is similar to their analysis, but with many detailsthat are quite different.

15.2 Propagator and states for equal frequency

To illustrate the general features of the equal frequency limit, the propagator isanalyzed from the point of view of the underlying state space. As mentioned atthe end of Section 14.9, in the limit of ω1 = ω2 the single excitation eigenstates�10, �01 become degenerate, with both eigenstates having energy 2ω.

The purpose of analyzing the propagator is to extract the state vectors thatemerge in the limit of ω1 → ω2.

Since ω1 > ω2, consider the limit of ε → 0+ with

ω1 = ω + ε, ω2 = ω2 − ε, (15.1)

which yields, from Eq. 14.32

E00 = 1

2(ω1 + ω2)→ ω, E10 → 2ω + ε, E01 → 2ω − ε.

Consider the limit of ω1 → ω2 for the state vector expansion of the propagatorgiven by Eq. 14.107,

G(τ) = e−τω1〈�D00|x|�10〉〈�D

10|x|�00〉 + e−τω2〈�D00|x|�01〉〈�D

01|x|�00〉= e−τω[G10 +G01], (15.2)

where, defining∫dxdvdx ′dv′ = ∫

x,v,x′,v′ yields

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332 Non-Hermitian Hamiltonian: Jordan blocks

G10 = e−ετ 〈�D00|x|�10〉〈�D

10|x|�00〉= N2

00N210

∫x,v,x′,v′

x(v+ ω2x)ψD00(x, v)ψ00(x, v)

× x ′(v′ − ω2x′)ψD

00(x′, v′)ψ00(x

′, v′)

= N200N

210

∫x,v,x′,v′

xx ′(v+ ω2x)(v′ − ω2x

′)P (x, v)P (x ′, v′) (15.3)

and

G01 = eετ 〈�D00|x|�01〉〈�D

01|x|�00〉= N2

00N201

∫x,v,x′,v′

x(v+ ω1x)ψD00(x, v)ψ00(x, v)

× x ′(−v′ + ω1x′)ψD

00(x′, v′)ψ00(x

′, v′)

= N200N

201

∫x,v,x′,v′

xx ′(v+ ω1x) (−v′ + ω1x′)P (x, v)P (x ′, v′). (15.4)

In the limit of ω1 → ω2 the vacuum state has the well-defined limit given by

limε→0

ψ00(x, v) = ψ00(x, v) = exp{−γω3x2 − γωv2 − γω2xv} (15.5)

and yields

P(x, v) = ψD00(x, v)ψ00(x, v) = exp{−2γω3x2 − 2γωv2}.

To leading order in ε the normalization constants yield the following:1

N210 → C

ω1

ε, N2

10 → Cω2

ε, C = 4γ 2ω3

π, (15.6)

and from Eq. 14.55

N200 = lim

ε→0N2

00 =2γω2

π. (15.7)

Hence, collecting the above equations,

G10 +G01 = CN200

∫x,v,x′,v′

xx ′F(x, v; x ′, v′)P (x, v)P (x ′, v′), (15.8)

where

F(x, v; x ′, v′) = e−ετω1

ε(v+ ω2x)(v

′ − ω2x′)+ eετ

ω2

ε(v+ ω1x)(−v′ + ω1x

′).

(15.9)

1 The definition of the constant C in this chapter is different from the constant with the same notation used inChapter 14.

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15.2 Propagator and states for equal frequency 333

Expanding F(x, v; x ′, v′) to leading order in ε yields

F(x, v; x ′, v′) = 1

ε

[(1− ετ)(ω + ε){v+ (ω − ε)x}{v′ − (ω − ε)x ′}+ (1+ ετ)(ω − ε){v+ (ω + ε)x}{−v′ + (ω + ε)x ′}

]+O(ε)

= 2[(ωτ − 1)(v+ ωx)(−v′ + ωx ′)+ ωx(−v′ + ωx ′)

+ ω(v+ ωx)x ′]+O(ε)

= 2[− (v+ ωx)(−v′ + ωx ′)+ ω{x + τ(v+ ωx)}(−v′ + ωx ′)

+ ω(v+ ωx)x ′]. (15.10)

The expression for F(x, v; x ′, v′) given in Eq. 15.10 carries information on thestate vectors that determine the propagator; to extract this information, the statevectors need to read off from the equation. We recall that the state vectors and theirduals are polynomials of x, v multiplied into the vacuum state.

Hence, Eqs. 15.2, 15.8, and 15.10 yield the following [the factor of e−2ωτ hasbeen included in the definitions of the state functions for convenience later]:

ψ1(x, v; τ) = 〈x, v|ψ1(τ )〉 = (v+ ωx)ψ00(x, v)e−2ωτ , (15.11)

ψD1 (x, v; τ) = 〈ψD

1 (τ )|x, v〉 = (−v+ ωx)ψD00(x, v)e−2ωτ , (15.12)

ψ2(x, v; τ) = 〈x, v|ψ2(τ )〉 = ω{x + τ(v+ ωx)}ψ00(x, v)e−2ωτ , (15.13)

ψD2 (x, v; τ) = 〈ψD

2 (τ )|x, v〉 = ω{x + τ(−v+ ωx)}ψD00(x, v)e−2ωτ . (15.14)

The dual state vector is defined by v →−v, namely2

ψD1 (x, v; τ) = ψ1(x,−v; τ), ψD

2 (x, v; τ) = ψ2(x,−v; τ).Note that the subtlety of conjugation for the unequal frequency case – with a dif-ferent rule for each excited state as given in Eq. 14.82 – has been lost for the equalfrequency case since the two excited states have become degenerate.

Collecting the results from Eqs. 15.2, 15.8, 15.9–15.14 yields

G(τ) = 2CN200e

ωτ 〈ψD00|x

[− |ψ1(τ )〉〈ψD

1 (0)|+ |ψ2(τ )〉〈ψD

1 (0)| + |ψ1(τ )〉〈ψD2 (0)|

]x|ψ00〉.

(15.15)

2 Bender and Mannheim (2008a), for the case for Minkowski time, define the dual state vector by x →−x,which gives an incorrect result for Euclidean time.

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The result in Eq. 15.15 shows that the eigenstates |�10〉, |�01〉 that gave the resultfor the propagator in Eq. 15.2 have been replaced, in the limit of ω1 → ω2, by newstate vectors that are well defined and finite for ε = 0.

In state vector notation, Eqs. 15.2 and 15.15 yield

limω1→ω2

e−τE10 |�10〉〈�D10| + e−τE01 |�D

00〉〈�D01|

= 2CN200e

ωτ[− |ψ1(τ )〉〈ψD

1 (0)| + |ψ2(τ )〉〈ψD1 (0)| + |ψ1(τ )〉〈ψD

2 (0)|].

(15.16)

15.3 State vectors for equal frequency

The Hamiltonian for the equal frequency case, from Eq. 14.1, is given by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ ω2v2 + γ

2ω4x2. (15.17)

The vacuum state is an energy eigenstate with

Hψ00(x, v) = ωψ00(x, v), 〈ψD00|ψ00〉 = 1

N200

= π

2γω2. (15.18)

The state vectors |ψ1(τ )〉, |ψ2(τ )〉 were obtained by analyzing the equal fre-quency propagator. The state vectors have the interpretation set out below.

15.3.1 State vector |ψ1(τ)〉The state vector |ψ1(τ )〉 is an energy eigenstate given by the average of the twounequal frequency eigenstates that become degenerate, namely

ψ1(x, v; τ) = limε→0

1

2[e−τE10ψ10(x, v)+ e−τE01ψ01(x, v)]

⇒ ψ1(x, v; τ) ≡ 〈x, v|ψ1(τ )〉 = e−2τω(v+ ωx)ψ00(x, v)

Hψ1(x, v; τ) = 2ωψ1(x, v; τ). (15.19)

The first sign of the irreducible non-Hermitian nature of the equal frequencyHamiltonian appears with |ψ1(τ )〉; unlike the norm of all the energy eigenstates,the norm of |ψ1(τ )〉 is zero; namely

〈ψD1 (τ )|ψ1(τ )〉 = e−4ωτ N2

00

∫dxdv(−v+ ωx)(v+ ωx)ψD

00(x, v)ψ00(x, v)

⇒ 〈ψD1 (τ )|ψ1(τ )〉 = 0. (15.20)

The norm of the eigenstate being zero is a general feature of a Hamiltonian that isof the form of a Jordan-block and, in particular, is not pseudo-Hermitian (Bender

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15.3 State vectors for equal frequency 335

and Mannheim, 2008a). The fact that the eigenstate has zero norm does not preventthe eigenstate being included in the collection of state vectors that, taken together,yields a resolution of the identity operator.

15.3.2 State vector |ψ2(τ)〉The second state vector |ψ2(τ )〉 that appears for the equal frequency case canbe written as the difference of the two unequal frequency eigenstates that becomedegenerate; for dimensional consistency, the pre-factor of ω is introduced in theε → 0; hence

ψ2(x, v; τ) = limε→0

ω

2ε

[e−τE01ψ01(x, v)− e−τE10ψ10(x, v)

]= ω

2εe−2ωτ

[(1− ετ)(v+ (ω − ε)x)− (1− ετ)(v+ (ω − ε)x)

]× ψ00(x, v)

⇒ ψ2(x, v; τ) = 〈x, v|ψ2(τ )〉 = e−2τωω[x + τ(v+ ωx)

]ψ00(x, v). (15.21)

Time-dependent state vector |ψ2(τ )〉 is not an (energy) eigenstate of H ; however,since it results from the superposition of two energy eigenstates, it can be explicitlyverified that |ψ2(τ )〉 satisfies the time dependent Schrödinger equation, namely

− ∂ψ2(x, v; τ)∂τ

= Hψ2(x, v; τ) ⇒ ψ2(x, v; τ) = exp{−τH }ψ2(x, v; 0).

Initial value ψ2(x, v; 0) = ωxψ00(x, v). (15.22)

Note that |ψ2(τ )〉 has a finite norm and a nonzero overlap with |ψ1(τ )〉; namely,using Eq. 15.6,

〈ψD2 (τ )|ψ2(τ )〉 = e−4τω

2C= 〈ψD

2 (τ )|ψ1(τ )〉, C = 4γ 2ω3

π. (15.23)

The equal frequency state space continues to have a nonnegative norm; in partic-ular, the norm of the time dependent state |ψ2(τ )〉 is positive definite. Of course,since one is working in Euclidean time, probability is not conserved and one cansee from Eq. 15.23 that the norm of the states decays exponentially to zero.

In summary, on taking the equal frequency limit, the two energy eigenstates|�10〉, |�01〉 coalesce to yield a single energy eigenstate |ψ1(τ )〉; a second timedependent state |ψ2(τ )〉 appears in this limit, also from the two eigenstates, andtakes the place of the loss of one of the eigenstates.

An analysis similar to that carried out for the single excitation level holds forall levels, and has been discussed by Bender and Mannheim (2008a). At eachlevel, all the energy eigenstates collapse into a single energy eigenstate of the

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Energy eigenstate y1

Time dependent y2

y10

y01

w1 → w2

Figure 15.1 The equal frequency limit yields two new states from two energyeigenstates.

equal frequency Hamiltonian; the eigenstates that are “lost” are replaced by time-dependent state vectors that are a superposition of the eigenstates of the unequalfrequency Hamiltonian. The time-dependent states together with the single eigen-state provide a resolution of the identity. This structure of the equal frequency statespace is illustrated in Figure 15.1.

15.4 Completeness equation for 2 × 2 block

We now discuss how the time-dependent state replaces the lost energy eigenstateto provide the complete set of states for the equal frequency case.

The example of the single excitation states, created by applying a creation oper-ator a†

v or a†x to the harmonic oscillator vacuum state |0, 0〉, showed that in the limit

of ω1 = ω2 the two energy eigenstates |�10〉, |�01〉 were superposed to create newstates |ψ1(τ )〉, |ψ2(τ )〉.

Since the orthogonality of the eigenstates is maintained in the superposition, themixing of states is only amongst states of a fixed excitation; in other words, stateshaving two excitations consisting of applying the creation operator twice, namely(a†

v )2, (a†

x)2, or a†

va†x yield three eigenstates that only mix with each other in the

limit of ω1 = ω2, and so on for all the higher excitation states.Hence, the resolution of the identity – which is an expression of the completeness

of a set of basis states – as shown in Figure 15.2, breaks up into a block-diagonalform, with states of a given excitation mixing with each other and not with thestates of the other blocks.

To illustrate the general result, consider the 2× 2 block for the single excitationstates. In light of the result obtained in Eq. 15.15, consider the following Hermitian

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15.5 Equal frequency propagator 337

ansatz for the 2×2 block identity operator, with all the state vectors taken at initialtime τ = 0. For notational simplicity, let

|ψ1(0)〉 = |ψ1〉 = (v+ ωx)|ψ00〉, 〈ψD1 (0)| = 〈ψD

1 |, (15.24)

|ψ2(0)〉 = |ψ2〉 = x|ψ00〉, 〈ψD2 (0)| = 〈ψD

2 |. (15.25)

Then the identity operator, which is Hermitian, has the following representation forthe 2×2 block of Hilbert space:

I2×2 = −P |ψ1〉〈ψD1 | +Q

[|ψ2〉〈ψD

1 | + |ψ1〉〈ψD2 |]= I

†2×2. (15.26)

Recall from Eqs. 15.20 and 15.23,

〈ψD1 |ψ1〉 = 0, 〈ψD

2 |ψ2〉 = 1

2C= 〈ψD

2 |ψ1〉.Hence, from Eq. 15.26 and the above equations

I2×2 = I22×2

= 1

2C

{(−2PQ+Q2)|ψ1〉〈ψD

1 | +Q2[|ψ2〉〈ψD

1 | + |ψ1〉〈ψD2 |]}

,

and this yields from Eqs. 15.26 and 15.23

P = 1

2C

{2PQ−Q2

}, Q = 1

2CQ2

⇒ P = Q = 2C = 8γ 2ω3

π.

Hence, the completeness equation for the 2 × 2 block single excitation states isgiven by

I2×2 = 2C[− |ψ1(0)〉〈ψD

1 (0)| + |ψ2(0)〉〈ψD1 (0)| + |ψ1(0)〉〈ψD

2 (0)|]. (15.27)

The completeness equation above is equal, up to a normalization, to Eq. 15.16.

15.5 Equal frequency propagator

The defining equation for the propagator is, from Eq. 14.105,

G(τ) = 〈�D00|xe−τ(H−ω)x|�00〉

⇒ G(τ ) = limε→0

G(τ) = N200〈ψD

00|xe−τ(H−E0)x|ψ00〉. (15.28)

The completeness equation can be used to give a derivation of the equal fre-quency propagator from first principles. Inserting the completeness equation given

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in Eq. 15.27 into the expression for the equal frequency propagator given in Eq.15.28 yields

G(τ ) = 2CN200〈ψD

00|xe−τ(H−ω)×[− |ψ1(0)〉〈ψD

1 (0)| + |ψ2(0)〉〈ψD1 (0)| + |ψ1(0)〉〈ψD

2 (0)|]x|ψ00〉.

(15.29)

Note that Eq. 15.29 is equivalent to the earlier expression given in Eq. 15.15.It follows from Eqs. 15.19 and 15.21 that

〈x, v|e−τH |ψ1(0)〉 = 〈x, v|ψ1(τ )〉 = e−2τω(v+ ωx)ψ00(x, v), (15.30)

〈x, v|e−τH |ψ2(0)〉 = 〈x, v|ψ2(τ )〉 = e−2τωω{x + τ(v+ ωx)}ψ00(x, v). (15.31)

It can be shown that the first and last terms inside the square bracket in Eq. 15.29cancel. Hence, from Eqs. 15.29, 15.30, and 15.31

G(τ ) = e−τω2CN200〈ψD

00|x|ψ2(τ )〉〈ψD1 (0)|x|ψ00〉

= e−τω2CN200

∫dxdv ωx{x + τ(v+ ωx)}P(x, v)

×∫

dxdv x(−v+ ωx)P (x, v)

= e−τω2Cω2N200 [1+ ωτ ]×

[∫dxdv x2P(x, v)

]2

. (15.32)

Performing the Gaussian integrations yields∫dxdv x2P(x, v) = 1

2ω2C. (15.33)

Hence

G(τ ) = e−τωN2

00

2ω2C[1+ ωτ ] = 1

4γ

1

ω3e−ωτ [1+ ωτ ] , (15.34)

where C = (4γ 2ω3)/π is given in Eq. 15.6 and the normalization constant N200 =

2γω2/π is given in Eq. 15.7.To verify the equal frequency result obtained for the propagator, consider taking

the limit of ω1 → ω2 in Eq. 14.104, the result being shown in Figure 15.2b. Thepropagator has the well-defined and finite limit

G(τ ) = 1

4γ ε

1

ω1 + ω2e−ωτ

[eετ

ω − ε− e−ετ

ω + ε

]

= 1

4γ

1

ω3e−ωτ [1+ ωτ ] , (15.35)

and agrees with the result obtained in Eq. 15.34.

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15.6 Hamiltonian: Jordan block structure 339

11 1

1 1 11 1 1 1

11

1

10

0

w1 w2

(a) (b)

12x2

3x3

4x4

NxN

0

0

Figure 15.2 (a) Completely diagonal Hamiltonian H for the unequal frequencycase. (b) Block-diagonal structure of the Hamiltonian in the equal frequency limit,with each N ×N block being given by an N ×N Jordan block.

The path integral yields a propagator for all values of ω1, ω2. The state space ap-proach requires a lot more effort to find the propagator and, in particular, the equalfrequency result needs a calculation quite distinct from the unequal case. Further-more, it is not clear how the state space approach can be used to evaluate the prop-agator for the case when ω1, ω2 are complex. These results show the utility of thepath integral which, among other things, allows us to have a deeper understandingof the underlying state space and operator structure of quantum mechanics.

15.6 Hamiltonian: Jordan block structure

In the limit of equal frequencies ω1 = ω2, there is a re-organization of state spaceinto a direct sum of finite dimensional subspaces, one subspace for each blockdiagonal component of H , as shown in Figure 15.2. The breakdown of the pseudo-Hermitian property of the Hamiltonian H is due to the fact that, for equal frequen-cies, H becomes a direct of sum of Jordan blocks.

The total Hilbert space V , for the equal frequency case, breaks up into a directsum of finite dimensional vector spaces Vn, and is given by

V = ⊕∞n=1Vn, (15.36)

where V1 is one dimensional, V2 is two dimensional and so on.The coordinate x and velocity v operators are not block diagonal in this repre-

sentation; the matrix elements of these operators connect the vectors of differentsubspaces Vn. This feature of the coordinate operator comes to the fore in the cal-culation of the propagator in the block diagonal basis.

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Denoting the finite dimensional matrix representation of the Hamiltonian by Hn,as shown in Figure 15.2, yields the block diagonal decomposition

H = ⊕∞n=1Hn = ⊕∞n=1anJλn,n. (15.37)

The coefficients an are real constants; Jλn,n is an n× n Jordan block, specified byits size n and eigenvalue λn, and given by3

Jλn,n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

λn ±1 0 0. . .

. . .

0 λn ±1 0. . .

. . .

. . . 0 λn ±1 0. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . . 0 λn ±1. . .

. . .. . .

. . . 0 λn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (15.38)

The Hamiltonian is analyzed for the first two blocks; H1 is one dimensional andH2 is a 2×2 matrix.

The ground state forms an invariant subspace V1 with a single element |e0〉 pro-portional to |ψ00〉; for dimensional consistency and to preserve the correct normal-ization, the mapping is

|e0〉 = N00|ψ00〉, 〈e0|e0〉 = 1. (15.39)

The eigenvalue equation H |ψ00〉 = ω|ψ00〉 yields the Hamiltonian on V1 given by

H1 = ω, H1|e0〉 = ω|e0〉. (15.40)

15.7 2×2 Jordan block

A derivation is given of the 2×2 Jordan block structure of the Hamiltonian andstate space.

The result given in Eq. 15.40 together with Eq. 15.37 yields

H = H1 ⊕H2 ⊕ . . . (15.41)

= ω ⊕ 2ωJ2 ⊕ . . . (15.42)

It will be shown in this section that

J2 =[

1 −10 1

]. (15.43)

3 The ±1 terms in the super-diagonal in Eq. 15.38 are allowed, since multiplying Jλn,n by −1 can switch thesign of the super-diagonal from 1 to −1, and in so doing re-define the eigenvalue to be −λn.

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15.7 2×2 Jordan block 341

The Jordan block Hamiltonian has been shown by Bender and Mannheim (2008a)to be pseudo-Hermitian; for the case of J2, it can be seen that

σ1J2σ1 =[

1 0−1 1

]= J †

2 , σ1 =[

0 11 0

].

The result above demonstrates that although the Jordan block matrix is pseudo-Hermitian, it is essentially inequivalent to a Hermitian matrix.

Bender and Mannheim (2008a) derive the 2×2 Jordan block for the MinkowskiHamiltonian by defining creation and destruction operators that have a finite limitwhen ε → 0. In this section, the 2×2 Jordan block for the Euclidean Hamiltonianis directly derived from the state vectors, and the completeness equation is obtainedby taking ε → 0 following the procedure in Sections 15.3 and 15.4.

Recall from Eqs. 15.17, 15.24, and 15.25, that the Hamiltonian and state vectorsfor the equal frequency limit are given by

H = − 1

2γ

∂2

∂v2− v

∂

∂x+ ω2v2 + γ

2ω4x2,

|ψ1〉 = |ψ1(0)〉 = (v+ ωx)|ψ00〉, 〈ψD1 | = 〈ψD

1 (0)|,|ψ2〉 = |ψ2(0)〉 = ωx|ψ00〉, 〈ψD

2 | = 〈ψD2 (0)|.

The fact that the state vectors |ψ1〉, |ψ2〉 form a closed subspace under the actionof H points to an invariant 2×2 subspace of the total Hilbert space.

In the 2×2 block space, the Hamiltonian can be represented by a 2×2 Jordanblock in a basis fixed by the representation of |ψ1〉, |ψ2〉 by two-dimensional col-umn vectors. To obtain this finite dimensional representation, note that H |ψ00〉 =ω|ψ00〉 and from Eq. 15.19 H |ψ1〉 = 2ω|ψ1〉; hence, the action of H on the statevectors |ψ1〉, |ψ2〉 is given by

H |ψ1〉 = 2ω|ψ1〉, (15.44)

H |ψ2〉 = −ωv|ψ00〉 + ω2x|ψ00〉 = −ω(v+ ωx)|ψ00〉 + 2ω|ψ2〉⇒ H |ψ2〉 = −ω|ψ1〉 + 2ω|ψ2〉. (15.45)

Since |ψ1〉 is an eigenvector of the Jordan block it is natural to make the identi-fication

|ψ1〉 ∝[

10

]. (15.46)

Recall from Eqs. 15.20 and 15.23,

〈ψD1 |ψ1〉 = 0, 〈ψD

2 |ψ2〉 = 1

2C= 〈ψD

2 |ψ1〉. (15.47)

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Since |ψ1〉 has zero norm, its normalization is fixed by its overlap with |ψ2〉. Choos-ing the normalization consistent with Eq. 15.47 yields

√C|ψ1〉 = |e1〉 =

[10

],√C|ψ2〉 = |e2〉 =

[1/21/2

], (15.48)

with the dual vectors given by√C〈ψD

1 | = 〈eD1 | =[0, 1

],√C〈ψD

2 | = 〈eD2 | =[1/2, 1/2

]. (15.49)

Note that 〈eD1 | is not the transpose of |e1〉.The completeness equation for the state space of the 2×2 block has a discrete

realization; recall from Eq. 15.27

I2×2 = 2C[− |ψ1〉〈ψD

1 | + |ψ2〉〈ψD1 | + |ψ1〉〈ψD

2 |]

⇒ I2×2 = 2[− |e1〉〈eD1 | + |e2〉〈eD1 | + |e1〉〈eD2 |

]. (15.50)

The completeness equation for the Jordan block shows that there is a nontrivialmetric on the discrete state space V2.

Using Eqs. 15.48 and 15.49, Eq. 15.50 yields

I2×2 = 2

{−[

0 10 0

]+ 1

2

[1 10 0

]+ 1

2

[0 10 1

]}=[

1 00 1

],

and we have obtained the expected result.

15.7.1 Hamiltonian

Let H2 denote the realization of the Hamiltonian as a discrete and dimensionlessmatrix acting on the two-dimensional state space of the 2×2 Jordan block. Apply-ing Eq. 15.48 to Eqs. 15.44 and 15.45 yields the 2×2 representation

H2|e1〉 = 2ω|e1〉 ⇒ 〈eD1 |H2 = 2ω〈eD1 |H2|e2〉 = −ω|e1〉 + 2ω|e2〉 ⇒ 〈eD2 |H2 = −ω〈eD1 | + 2ω〈eD2 |.

The Hamiltonian H2 – in the |e1〉 and |e2〉 basis – is proportional to the 2×2 Jordanblock matrix and is given by4

H2 = 2ω

[1 −10 1

]. (15.51)

The definition of the discrete vectors |e1〉 and |e1〉 given in Eq. 15.48 requires arescaling by

√C because of dimensional consistency; in contrast, there is no need

to rescale H2 since it has the correct dimension set by ω.4 The Euclidean Hamiltonian given in Eq. 15.43 has a−1 for the superdiagonal, unlike the case for the

Minkowski Hamiltonian (Bender and Mannheim, 2008a), where it is +1.

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15.7 2×2 Jordan block 343

The Jordan block Hamiltonian given in Eq. 15.43 has only one eigenvalue andthis is the reason that the two different eigenstates for the unequal frequenciescollapsed into a single eigenstate. The Jordan block limit of H2 (for equal fre-quency) shows that H2 continues to be pseudo-Hermitian but is inequivalent to anyHermitian Hamiltonian since the Jordan block is inequivalent to any Hermitianmatrix.

The right eigenvector of H2 is |e1〉 and the left eigenvector of H is the dual 〈eD1 |;namely

H2|e1〉 = 2ω|e1〉, 〈eD1 |H2 = 2ω〈eD1 |,⇒ 〈eD1 |e1〉 = 0 = 〈ψD

1 |ψ1〉.Hence, the Jordan block structure shows why the equal frequency eigenstate has azero norm.

15.7.2 Schrödinger equation for Jordan block

The Schrödinger equation for an arbitrary vector |e〉 is given by

− ∂

∂τ|e(τ )〉 = H2|e(τ )〉.

For eigenvector |e1〉 the time-dependent solution is

− ∂

∂τ|e1(τ )〉 = H2|e1(τ )〉 = 2ω|e1(τ )〉,

⇒ |e1(τ )〉 = e−2ωτ |e1〉, |e1〉 =[

10

]. (15.52)

The time dependence of the state vector |e2(τ )〉 is given by

− ∂

∂τ|e2(τ )〉 = H2|e2(τ )〉, |e2(0)〉 = |e2〉 =

[1/21/2

]. (15.53)

In the 2×2 block representation |e2(τ )〉 is given from the solution obtained in Eq.15.21, which yields

ψ2(x, v; τ) = e−2τωω[x + τ(v+ ωx)

]ψ00(x, v)

⇒ |e2(τ )〉 = e−2τω[|e2〉 + ωτ |e1〉

]= e−2τω

[1/2+ ωτ

1/2

]. (15.54)

It can be directly verified using the explicit form for the Hamiltonian given inEq. 15.43 that the solution for |e2(τ )〉 given in Eq. 15.54 satisfies the Schrödingerequation given in Eq. 15.53.

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15.7.3 Time evolution

The Jordan block Hamiltonian is given by Eq. 15.43; a simple calculation yieldsthe evolution operator

e−τH2 = e−2ωτ

[1 2ωτ0 1

]. (15.55)

The time dependence of the state vectors follows directly from the evolutionoperator. For eigenvector |e1〉 the time-dependent solution is

|e1(τ )〉 = e−τH2 |e1〉 = e−2ωτ |e1〉,which is the expected result as in Eq. 15.52.

The time dependence of the state vector |e2(τ )〉 is given by

|e2(τ )〉 = e−τH2 |e2〉 = e−2ωτ

[1/2+ ωτ

1/2

], (15.56)

which is the expected result as in Eq. 15.54.

15.8 Jordan block propagator

The equal frequency propagator is given in Eq. 15.28,

G(τ ) = N200〈ψD

00|Xe−τ(H−ω)X|ψ00〉.The position operator X, unlike the Hamiltonian, is not block diagonal for the

equal frequency case and connects different subspaces Vn. To determine the prop-agator, the representation of the position operator X needs to be determined in the3×3 subspace given by V1 ⊕ V2, which includes the ground state and the 2×2Jordan block. The operator X has the following matrix elements:

〈ψD00|X|ψ00〉 = 0 = 〈ψD

1 |X|ψ1〉 = 〈ψD2 |X|ψ2〉 = 〈ψD

2 |X|ψ1〉,〈ψD

00|X|ψ1〉 = 1

2ωC= 〈ψD

00|X|ψ2〉. (15.57)

Note that the matrix elements of the operator X are zero within a block and arenonzero only for elements that connect vectors from two different blocks.

Since X acts on the V1 ⊕ V2 we need to extend the vectors defined on the sub-spaces |e0〉 ∈ V1 and |e1〉, |e2〉 ∈ V2 to the larger space; let us define the followingthree-dimensional vectors:

|e0〉 = 1, |E0〉 = |e0〉 ⊕ |0〉 =⎡⎣1

00

⎤⎦ ,

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15.8 Jordan block propagator 345

|E1〉 = |0〉 ⊕ |e1〉 =⎡⎣0

10

⎤⎦ , |E2〉 = |0〉 ⊕ |e2〉 =

⎡⎣ 0

1/21/2

⎤⎦ . (15.58)

The dual vectors are given by the transpose, except for 〈ED1 | given by

〈ED1 | =

[0, 0, 1

]. (15.59)

Let the position operator in the block diagonal space be denoted by X ; from Eq.15.57, since all the elements are dimensionless in the Jordan block representation,

〈ED0 |X |E0〉 = 0 = 〈ED

1 |X |E1〉 = 〈ED2 |X |E2〉 = 〈ED

1 |X |E2〉,〈ED

0 |X |E1〉 = 1 = 〈ED0 |X |E2〉, (15.60)

which yields the following representation for the Hermitian matrix X :

X =⎡⎣0 1 1

1 0 01 0 0

⎤⎦ . (15.61)

Since X is dimensionless, its mapping to the coordinate position operator Xneeds a dimensional scale; let X = ζX. From Eqs. 15.39, 15.49, 15.57, and 15.60,

1 = 〈ED0 |X |E1〉 = ζ N00

√C〈ψD

00|X|ψ1〉 = ζN00

2ω√C, (15.62)

⇒ X = N00

2ω√C

X , ζ = 2ω√C

N00

. (15.63)

Extending the Hamiltonian to the V1 ⊕ V2 space yields, from Eq. 15.43,

H = H1 ⊕H2 = ω ⊕ 2ω

[1 −10 1

]= ω

⎡⎣1 0 0

0 2 −20 0 2

⎤⎦ .

The evolution kernel is given by

exp{−τH} = e−2ωτ

⎡⎣eωτ 0 0

0 1 2ωτ0 0 1

⎤⎦ . (15.64)

The completeness equation from Eq. 15.50 has the following extension toV1 ⊕ V2:

I3×3 = |E0〉〈ED0 | + 2

[− |E1〉〈ED

1 | + |E2〉〈ED1 | + |E1〉〈ED

2 |]. (15.65)

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In the block-diagonal basis, the propagator is given by

G(τ ) = N200〈ψD

00|Xe−τ(H−ω)X|ψ00〉

=(

N00

2ω√C

)2

〈ED0 |X e−τ(H−ω)X |E0〉. (15.66)

Using the completeness equation given in Eq. 15.65 yields

G(τ )

= N200

4ω2C〈ED

0 |X e−τ(H−ω)

×(|E0〉〈ED

0 | + 2[− |E1〉〈ED

1 | + |E2〉〈ED1 | + |E1〉〈ED

2 |])

X |E0〉

= N200

2ω2C〈ED

0 |X[− e−ωτ |E1〉〈ED

1 | + eωτ |E2(τ )〉〈ED1 | + e−ωτ |E1〉〈ED

2 |]X |E0〉

= N200

2ω2Ceωτ 〈ED

0 |X |E2(τ )〉〈ED1 |X |E0〉 = N2

00

2ω2Ceωτ 〈ED

0 |X |E2(τ )〉, (15.67)

since

〈ED1 |X |E0〉 = 1 = 〈ED

2 |X |E0〉.From Eq. 15.53, the time dependence of |E2(τ )〉 is given by

|E2(τ )〉 = e−τH|E2〉 = e−2ωτ

⎡⎣ 0

1/2+ ωτ

1/2

⎤⎦ ⇒ 〈ED

0 |X |E2(τ )〉 = 1+ ωτ,

which yields, from Eq. 15.67, the expected result for the propagator, namely

G(τ ) = N200

2ω2Ce−ωτ (1+ ωτ) = 1

4γω3e−ωτ (1+ ωτ).

A direct derivation can be given using the matrix representation of the evolutionkernel; from Eq. 15.66

G(τ ) =(

N00

2ω√C

)2

〈ED0 |X e−τ(H−ω)X |E0〉.

Using

X |E0〉 =⎡⎣0

11

⎤⎦ , 〈ED

0 |X = [0, 1, 1

]

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15.9 Summary 347

yields, from Eq. 15.64, the expected answer

G(τ ) =(

N00

2ω√C

)2

e−ωτ (2+ 2ωτ) = 1

4γω3e−ωτ (1+ ωτ).

All the N × N blocks for the Hamiltonian can be analyzed one by one and it canbe shown that they are all equal to a corresponding Jordan block matrix. However,the higher order blocks may not be as simple as J2 as they can include the directsum of lower order Jordan blocks.

15.9 Summary

The equal frequency case of the acceleration Lagrangian leads to a Hamiltonianthat is essentially inequivalent to any Hermitian Hamiltonian. A carefully chosenlimit for the equal frequency leads to a Hamiltonian that is block diagonal, witheach block consisting of Jordan block matrices. The state space has zero normstate vectors, even for the Euclidean theory, showing that the fundamental inequiv-alence of the acceleration Hamiltonian to a Hermitian Hamiltonian holds for bothMinkowski and Euclidean time.

The equal frequency propagator was evaluated using various techniques to high-light the different aspects of the Jordan block system that provide a representationof the essentially non-Hermitian sector of the theory. The specific form of the prop-agator for equal frequency reflects the presence of time-dependent states that areessential in evaluating the propagator. In particular, these time dependent-statesappear in the completeness equation and hence are required for spanning out acomplete basis for the state space of the equal frequency case.

The quantum mechanics of the 2×2 Jordan block was worked out, with thesolution of the Schrödinger equation having only one eigenstate and another timedependent state. A calculation for the propagator was done, block by block, basedon the discrete subspaces and finite Jordan block matrices.

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Part six

Nonlinear path integrals

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16

The quartic potential: instantons

The simple harmonic oscillator is exactly soluble because it has a quadraticpotential and yields a linear theory in the sense that the classical equation ofmotion is linear. Nonlinear path integrals have potentials that typically have aquartic or higher polynomial dependence on the degree of freedom, or the po-tential can be a transcendental function, such as an exponential. The techniquesdiscussed for quadratic Gaussian path integrals needed to be further developed fornonlinear path integrals. In general, to solve these nonlinear systems, one usuallyuses either a perturbation expansion or numerical methods.

A perturbation expansion is useful if the theory has a behavior that is smoothabout the dominant piece of the action or Hamiltonian; in practice a smooth expan-sion of the physical quantities yields an analytic series in some expansion parame-ter, say a coupling constant g, around g = 0.

There are, however, cases of physical interest for which nonperturbative effectschange the qualitative behavior of the theory Two examples where nonperturbativeeffects dominate are the following:

• Tunneling through a finite barrier. If one perturbs about the lowest lying eigen-states inside a well, one cannot produce the tunneling amplitude.

• The spontaneous breaking or restoration of a symmetry cannot be produced byperturbing about an incorrect ground state.

Tunneling and symmetry breaking are nonperturbative because these effects de-pend on g as exp{−1/g}, which is nonanalytic about g = 0, and hence cannot beobtained by perturbation theory.

The semi-classical expansion is an approximation that is a useful tool for study-ing nonperturbative effects. The general features of this approximation scheme arediscussed for a nonlinear Lagrangian in Section 16.1. Section 16.3 covers non-trivial classical solutions of the double-well potential, called instantons. In Section16.4 the so called zero mode features of the instantons are discussed, and in Section

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352 The quartic potential: instantons

16.5 the Faddeev–Popov analysis is applied to the zero mode problem. In Section16.6 the multi-instanton solutions are obtained, and the transition amplitude, cor-relators and the dilute gas approximation are then discussed in Sections 16.7, 16.8and Section 16.9. The double-well potential, in the strong coupling limit, is shownin Section 16.10 to be equivalent to an Ising model, and in Section 16.11 a nonlocalIsing model is shown to produce the double-well potential. Sections 16.12–16.14discuss spontaneous symmetry breaking and symmetry restoration.

16.1 Semi-classical approximation

Consider the evolution kernel for Euclidean time,

K(x, x ′; τ) =∫

DxeS[x]/�,

where

S[x] =∫ τ

0Ldt (16.1)

and∫Dx is the path integration measure. For � → 0, the classical trajectory, for

which S is a maximum, that is

δS[xc]δx(t)

= 0, (16.2)

dominates the path integral with the next leading term yielding an expansion in apower series in �.

One can expand the paths about the classical path, that is

x(t) = xc(t)+ η(t), η(0) = 0 = η(τ),

and expand the action about the classical path. Thus, we have

S[x] = S[xc + η]= S[xc] + 1

2

∫dt1dt2ηt1ηt2

δ2S[xc]δx(t1)δx(t2)

+ 0(η3)

= Sc + S2, Sc = S[xc].The classical action is a maximum of the Euclidean action, namely

δ2S[xc]δx(t1)δx(t2)

≤ 0. (16.3)

Hence the quadratic approximation of the action yields a convergent expansion forthe path integral,

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16.2 A one-dimensional integral 353

K(x.x ′; τ) � N eSc/�∫

DηeS2/�

= N eSc/�

/√det

[1

�

δ2S[xc]δx(t)δx(t ′)

]. (16.4)

Note the approximation breaks down if det[δ2S[xc]/δx(t)δx(t ′)

] = 0.Consider a typical Euclidean action

S = −∫ τ

0dt

(1

2mx2 + V (x)

). (16.5)

Then

δ2Sc

δx(t)δx(t ′)= −

(−m d2

dt2+ V ′′(x)

)δ(t − t ′), (V ′ = ∂V/∂x), (16.6)

and hence

det

(1

�

δ2Sc

δxδx

)∝ det

(1

�

(−m d2

dt2+ V ′′(x)

)). (16.7)

To evaluate the determinant, the earlier discussion on the simple harmonic oscil-lator needs to be generalized.

16.2 A one-dimensional integral

The semi-classical expansion is identical to the saddle point method for finite di-mension integrals. To see this, consider the integral

I (g2) =∫ +∞

−∞dx√2π

exp

{−1

2x2 − g2

8x4

}.

The integral is well defined for g2 > 0, but can it be analytically continued tog2 < 0? A perturbation expansion around g2 > 0 cannot answer this question anda saddle point expansion will be used to go beyond the perturbation expansion.

The minimum is at x0 = 0, and expanding the integrand about this yields

I �∫ +∞

−∞dx√2π

e−12 x

2(

1− g2x4

8+ g4x6

128

)

= 1− 3

8g2 + 105

128g4 + 0(g5) (16.8)

: analytic about g = 0.

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Let us rewrite I as

I =∫

dxdy

2πe−

12 x

2e−

12 y

2e−

i2 gyx

2

=∫

dy√2π

e−12 y

2− 12 ln(1+igy). (16.9)

If one expands about g2 = 0 one obtains the same result as Eq. 16.8. However,consider expanding about the minimum of

S(y) = −1

2y2 − 1

2ln(1+ igy). (16.10)

The minimum y0 is given by

S ′(y0) = 0 = y0 + ig

2(1+ igy0)

⇒ iy0(±) = 1

2g

(−1±

√1+ 2g2

).

For g→ 0, the limiting values are

iy0(−)→−1

g− g

2→∞,

iy0(+)→ g

2→ 0.

The action about the saddle point y0(−) yields a negative divergent action, namely

limg→0

S(y0(−))→− 1

2g2+ 1

2ln

(−g

2

2

)→−∞ divergent.

Hence, the y0(+) branch is chosen for doing the expansion since it yields thefinite action

S(y0(+)) = 1

8g2(√

1+ 2g2 − 1)2 − 1

2ln

(1+√1+ 2g2

2

), (16.11)

≡ S(0)+ → 0 as g→ 0. (16.12)

Furthermore, the action has a Taylors expansion about y0(+) given by

S(y) = −1

2y2 − 1

2ln(1+ igy)

=∞∑n=0

S(n)+n! (y − y0(+))n, (16.13)

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16.3 Instantons in quantum mechanics 355

where the notation S(n)+ = ∂n

∂ynS(y0(+)) is being used:

S(0)+ = 1

8g2(√

1+ 2g2 − 1)2 − 1

2ln

(1+√1+ 2g2

2

),

S(2)+ = − 2

√1+ g2

1+√1+ 2g2> 0, S

(3)+ = − 8ig3

(1+√1+ 2g2)3,

S(4)+ = 48g4

(1+√1+ 2g2)4, S

(n)+ ∼ 0(gn).

Hence, shifting y by y0(+), Eqs. 16.9 and 16.13 yield

I (g2) � eS(0)+∫ +∞

−∞dy√2π

e12S

(2)+ y2

[1+ S

(4)+4! y

4 + 0(g6)

]

∼= e1

8g2 (√

1+2g2−1)2

(1+ 2g2)1/4

[1+ 3g4

2(1+ 2g2)(1+√1+ 2g2)2+ · · ·

]. (16.14)

Note that the second term above is O(g4), whereas in the expansion given in Eq.16.8 the second term is only O(g2).

The nonanalytic structure of the integral about g2 = −1/2, namely that it has abranch cut, is captured in the semi-classical expansion, but is missed in the expan-sion about g2 = 0 as given in Eq. 16.8. In particular, from Eq. 16.14 it can be seenthat the semi-classical expansion provides an analytic continuation of the integralI (g2) to the range −1/2 ≤ g2 ≤ +∞, whereas the perturbation expansion cannotbe extended to g2 ≤ 0.

The semi-classical expansion is a technique that can capture nonperturbativeproperties of the path integral and is very useful in quantum mechanics and inquantum field theory.

16.3 Instantons in quantum mechanics

Consider the nonlinear action and Lagrangian for Euclidean time given by

S =∫

dtL, L = −1

2mx2 − V, (16.15)

V = g2

8(x2 − a2)2. (16.16)

The potential term V is shown in Figure 16.1. The Lagrangian L has the paritysymmetry of being invariant under the transformation x →−x.

Note that if one expands the potential into a polynomial, the quadratic term hasa coefficient that is positive; hence any expansion about the quadratic Lagrangian

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-a +a

V

x

Figure 16.1 The double-well potential for Euclidean time.

will be unstable and lead to a divergent expansion. The reason is that the minimaof the Lagrangian are at ±a, as shown in Figure 16.1, and hence the Lagrangianneeds to be expanded about one of these minima.

The transition amplitude for Euclidean time τ is given by

K(x, x ′; τ) =∫

DxeS = 〈x|e−τH |x ′〉, (16.17)

where H is the Euclidean Hamiltonian. Writing the time integration in a symmetricmanner yields the action

S = −∫ + τ

2

− τ2

dt

(1

2mx2 + g2

8(x2 − a2)2

). (16.18)

The action is expanded about one of the classical solutions, say x = a, similarlyto the exercise carried out for a single variable in Section 16.2. Consider the shiftof variable x = y − a; the action is given by

S = −∫ + τ

2

− τ2

dt

(1

2my2 + g2

8y2(y + 2a)2

)

� −∫ + τ

2

− τ2

dt

(1

2my2 + g2

2a2y2

)+ 0(y3). (16.19)

The theory has small oscillations about the minimum at x = a, as shown inFigure 16.2; these oscillations spontaneously break the symmetry of x → −x, asis evident from Figure 16.2 and Eq. 16.19.

It is known that a one-dimensional quantum mechanical system cannot sponta-neously break any continuous symmetry. However, the system can, in principle,break discrete symmetries. The question that needs to be addressed is whether theperturbative breaking of discrete symmetry of x → −x by the ground states cen-tered at x = a or x = −a, referred to as x = ±a, is only an approximation – with

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x=-a x=+a

Figure 16.2 The double-well potential for Euclidean time.

nonperturbative effects restoring the parity symmetry? And if symmetry restora-tion takes place, what is the mechanism by which, in particular for the x4 potential,the discrete parity symmetry is restored?

The semi-classical expansion for the path integral reveals that there are largequantum fluctuations that are encoded in the neighborhood of an instanton config-uration. [A fluctuation is defined as one possible path for the degree of freedom.] Itis these large quantum fluctuations – far from the perturbative vacuum defined byx = ±a – that are responsible for restoring the parity symmetry that is apparentlybroken by the perturbative vacuum.

Consider the equation of motion [x = dx/dt, x = d2x/dt2]

0 = δS[xc]δx(t)

= mxc − V ′(xc) (16.20)

or

mxc − g2

2xc(x

2c − a2) = 0. (16.21)

The minimum at x = ±a are two solutions of the classical equations of motion,and expanding about either of them gives an analytical expansion around g = 0.

The equation of motion given in Eq. 16.21 can be re-written as the conservationof energy in the following manner:

d

dt

[1

2mx2

c −g2

8(x2

c − a2)2

]= 0 = dE

dt

⇒ E = 1

2mx2

c −g2

8(x2

c − a2)2. (16.22)

Hence, from Eq. 16.22, the Euclidean energy is given by

E = 1

2mx2 − V (x).

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–V(x)

0 0

a x

(a) (b)

V(x)

–a

a

classical

x

–a

Figure 16.3 The potential for Euclidean time (a), and for Minkowski time (b).

In Minkowski time, the potential is a double well, as shown in Figure 16.3(b).The equations of motion have no classical solutions that go from one well to an-other when the classical particle does not have an energy E sufficient to cross thepotential barrier given by g2a4/8. In particular, for E = 0, which corresponds tothe vacuum solutions, there is no trajectory from one well to another.

The Euclidean equations of motion have a nontrivial solution, since it is −Vthat appears in eS ; hence, the sign of the potential for the Euclidean case has beenreversed from the Minkowski case. The potential wells correspond to maxima oftwo “hills,” as shown in Figure 16.3(a), and the classical Euclidean solution corre-sponds to the particle starting at x = −a and rolling down the hill to reach x = a;there is no energy barrier separating the two maxima.

Two zero energy classical solutions are given by the particle being stationaryat the two wells x = ±a. The question is, are there other zero energy classicalsolutions to the equations of motion that contribute to the path integral as wellas the x = ±a solution? The answer is in the affirmative, and we look for thesesolutions.

For zero energy the classical trajectory, from Eq. 16.22, is given by

E = 0 = 1

2mx2 − V (x) ⇒ 1

2mx2 − g2

8(x2 − a2)2 = 0. (16.23)

The trivial classical solution is given by xc = ±a with xc = 0 for all time. Thenontrivial solution has nonzero velocity and is given by

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x = ± g

2√m|x2

c − a2| = ± g

2√m(a2 − x2

c ), x2 ≤ a2

⇒∫ x±

0

dy

a2 − y2= ± g

2√m

∫ t

tc

dt.

Integrating the above equation yields

1

atanh−1

(x±a

)= ± g

2√m(t − tc).

Hence the instanton and anti-instanton classical solutions xc are given by

xc = x± = ±a tanh

(ω(t − tc)

2

), ω2 = g2a2

m. (16.24)

The particle starts off with zero velocity at ±a at t = −∞ and travels very slowlyuntil time tc when it rapidly crosses over towards ∓a and then slowly rolls to ∓aat t = +∞.

Figure 16.4 shows the shape of the instanton for two different values of ω. Asω becomes large, the particle almost “jumps” in an instant from −a to +a at timetc. Figure 16.5(a) shows a kink (instanton) solution that tunnels from −a to +a,and Figure 16.5(b) shows an antikink (anti-instanton) solution that tunnels from a

to −a.For this reason, the classical solution is called an instanton or an anti-instanton,

depending on whether it tunnels from left to right or from right to left; it also calleda kink or an antikink due to the kink-like shape of the classical solution.

tc

x

t

-a +a

ω: largeω: small

Figure 16.4 The classical instanton solution for different values of ω2 = g2a2/m.

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(a)

t

Kink Anti-Kink

t

x xa a–a–a

(b)

Figure 16.5 Classical solution for large coupling constant g that is (a) a kink orinstanton, and (b) an anti-kink or anti-instanton.

The instanton classical action – from the zero energy classical equation given inEq. 16.23 – is the following:

S[xc] =∫ ±τ/2

∓τ/2dt[−m

2x2c − V (xc)] = −m

∫ ±τ/2

∓τ/2dtx2

c

== −m∫ ±τ/2

∓τ/2dtxcxc = −m

∫ ±τ/2

∓τ/2dtdxc

dtxc

= −m∫ ±a

∓adxcxc.

Hence, from Eq. 16.23, the classical action for one kink or one anti-kink is given by

S[xc] = −m∫ ±a

∓adxc

(∓ ω

2a(x2

c − a2))

= ±mω2a

(1

3x3c − xca

2)|±a∓a = −2

3mωa2.

Hence, for one kink or anti-kink

Sc = −2m2ω3

3g2.

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The evolution kernel for the particle to travel from one well to the other isgiven by

K(±a,∓a; τ) = 〈±a|e−τH | ∓ a〉 � N eSc = N e− 2m2ω3

3g2

≡ N e− c

g2 , c = 2

3m2ω3

: essential singularity around g = 0.

A perturbation expansion of the path integral about g = 0 to any order cannotproduce an essential singularity at g = 0.

The classical Lagrangian for a kink or an antikink is given by

Lc = −1

2mx2

c − V (xc) = −mx2c

= −ma2ω2

4sech4ω

(t − tc

2

).

The classical kink Lagrangian, as shown in Figure 16.6, is sharply localizedaround t = tc with a width of

�t ∼ 1

ω=√m

ga.

–L

t0 t

∼ 1/ω

Figure 16.6 Classical Lagrangian for a kink.

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The time at which the tunneling occurs, namely tc, is arbitrary and reflects thefact that the theory, for τ → ∞, is invariant under translation in time, namelyunder a shift given by t → t+const.

At first glance, it seems that, for g � 0, 〈+a|e−τH | − a〉 ∼ eSc = e−c/g2is neg-

ligibly small and hence cannot contribute significantly to the transition amplitude.However, this is not true since there is a whole collection of classical solutions, onefor each tc, that are all close to each other, and they all contribute to the transitionamplitude. We will show that in fact a more accurate analysis yields

〈+a|e−τH | − a〉 ∼ N e−c/g2∫ + τ

2

− τ2

dtc = N τe−c/g2.

Hence, as long as τ < e−c/g2the contribution from the classical trajectory is

indeed negligible; but for large τ the contribution becomes very large.There are also multi-kink and multi-antikink solutions with multiple tunneling

across the barrier that all contribute in the limit of large time.There is no exact solution known due to the kink–antikink interaction, but for

g→ 0, an approximate solution is to compose the multiple kink–antikink solutionsfrom the product of single kink and antikink solutions.

16.4 Instanton zero mode

The double-well action is given by

S = −∫

dt

{1

2mx2 + g2

8(x2 − a2)2

}, (16.25)

with the kink, anti-kink classical solution given by (ω2 = a2g2/m)

xc = ±a tanh

(ω(t − tc)

2

).

Consider the semi-classical expansion for the transition amplitude. Let the(false) vacua at ±a be denoted by �±. Then

〈�+|e−τH |�−〉 =∫

dbdb′〈�+|b〉〈b|e−τH |b′〉〈b′|�〉∼= �+(a)〈+a|e−τH | − a〉�−(a). (16.26)

The last equation has been obtained using the fact that the false vacua �± areapproximately delta-functions (well localized) at points ±a.

Let x(t) = xc(t) + η(t); then, since η(τ/2) = 0 = η(−τ/2), one has from Eq.16.26

〈+a|e−τH | − a〉 =∫

DηeS[xc+η]. (16.27)

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16.4 Instanton zero mode 363

The action has the following semi-classical expansion:1

S[x] = S[xc + η]= S[xc] + 1

2

∫dtdt ′ηtηt ′

δ2S

δxtδxt ′+ 0(η3)

= Sc − 1

2m

∫dtη2 − g2

8

∫dt (6x2

c η2 − 2a2η2)+ 0(η3)

= Sc − 1

2

∫ +τ/2

−τ/2dtη

(−m d2

dt2+ 3

2g2x2

c −g2a2

2

)η + 0(η3)

= Sc − 1

2

∫ +τ/2

−τ/2dtdt ′η(t)M(t, t ′)η(t ′)+ 0(η3), (16.28)

where the matrix elements of the operator M are defined by

M(t, t ′) = 〈t ′|M|t〉 ≡(−m d2

dt2+ 3

2g2x2

c −g2a2

2

)δ(t − t ′).

To do the∫Dη path integration one diagonalizes the action using the normal

mode expansion of η(t) = ∑∞n=0 ψn(t)ηn. The normal modes ψn are defined by

the eigenfunction equation

M|ψn〉 = λn|ψn〉. (16.29)

The explicit eigenfunction equation, with ψn ≡ 〈t |ψn〉, is given by

λnψn =(−mψn + 3g2

2x2cψn − g2a2

2ψn

)

=(−m d2

dt2+ 3g2a2

2tanh2

(ω(t − tc)

2

)− g2a2

2

)ψn. (16.30)

We recall that the classical equation of motion given in Eq. 16.21 yields

mxc − g2

2x3c +

g2a2

2xc = 0.

Differentiating the above equation with respect to t , and defining eigenfunctionψ0 as

ψ0 = 1

||xc|| xc, ||xc|| =√∫ +τ/2

−τ/2dtx2

c , (16.31)

1 Note that (x2 − a2)2 = (x2c − a2)2 + 6x2

c η2 − 2a2η2 + 0(η3).

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yields

mψ0 − 3g2

2x2cψ0 + g2a2

2ψ0 = 0. (16.32)

Hence, from Eqs. 16.30 and 16.32, ψ0 is an eigenfunction such that

λ0 = 0 for ψ0.

The zero eigenvalue eigenfunction, also called the zero mode ψ0, arises in theaction due to the time translation invariance of the action: the instanton can tunnelfrom−a to+a at any time tc. This is the reason that the eigenvalue equation for ψ0

is a direct result of the equations of motion, which is also time translation invariant.The operator M , since λ0 = 0, has the spectral representation

M =∞∑n=0

λn|ψn〉〈ψn| =∞∑n=1

λn|ψn〉〈ψn|, (16.33)

detM =∞∏n=0

λn = 0.

Note that M is singular, having no inverse, since detM = 0 due to the zero eigen-value λ0 = 0. Hence, we obtain the path integral given in Eq. 16.27, since

〈+a|e−τH | − a〉 =∫

DηeS[xc+η] � N eSc√detM

= ∞. (16.34)

16.5 Instanton zero mode: Faddeev–Popov analysis

The Faddeev–Popov method is of great generality and is indispensable in the quan-tization of Yang–Mills gauge fields. The elimination of a divergence in the pathintegral due to asymmetry is analyzed using the Faddeev–Popov approach; the ad-vantage of carrying out this analysis is that it illustrates the main features of theFaddeev–Popov approach in a relatively simple context.

The analysis yields what are called the collective coordinates. The fundamentalingredient in the Faddeev–Popov approach is to explicitly introduce a constraintinto the path integral that breaks the symmetry, thus removing the singularity inthe action; a term has to be introduced to compensate for constraint so as to leavethe path integral invariant.

We recall that the zero mode arises due to the invariance of the action undershifting the time coordinate, namely t → t+constant. This symmetry is strictly aninvariance of the action only for τ → ∞, with corrections that are exponentiallysmall and can be ignored in our analysis. In particular, from Eq. 16.18 and forτ >> t∗, the double-well action has the symmetry

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16.5 Instanton zero mode: Faddeev–Popov analysis 365

S = −∫ +τ/2

−τ/2dt

(1

2mx2(t)+ g2

8

{x2(t)− a2

}2)

� −∫ +τ/2

−τ/2dt

(1

2m ˙x2(t)+ g2

8

{x2(t)− a2

}2), x(t) = x(t + t∗)

⇒ S[x] = S[x]. (16.35)

For the double well potential, in the Faddeev–Popov approach, one breaks theinvariance under t → t + t∗ by directly introducing a fixed value for time, denotedby t0 directly into the action in the following manner. Following Zinn-Justin (2005),consider the identity

1 =∫ +τ/2−τ/2 dt0δ

[ ∫dtxc(t)x(t + t0)− λ

]∫ +τ/2−τ/2 dt0δ

[ ∫dtxc(t)x(t + t0)− λ

] . (16.36)

To perform the integration in Eq. 16.36, consider the following change of variable:

ξ =∫

dt xc(t)x(t + t0)− λ, (16.37)

dξ = dt0

∫dt xc(t)x(t + t0), (16.38)

which yields∫ +τ/2

−τ/2dt0δ

[∫dtxc(t)x(t + t0)− λ

]=

∫dξδ[ξ ]∫ +τ/2

−τ/2 dtxc(t)x(t + t0)

= 1∫ +τ/2−τ/2 dtxc(t)x(t + t0)

. (16.39)

Hence, from Eqs. 16.36 and 16.39,

1 =[∫ +τ/2

−τ/2dtxc(t)x(t + t0)

]· δ[∫ +τ/2

−τ/2dt0δ[

∫dtxc(t)x(t + t0)− λ]

].

(16.40)

The path integral given in Eq. 16.27 is re-written, using Eq. 16.40, in the followingmanner:

〈+a|e−τH | − a〉 = K(a,−a; τ) =∫

DxeS

=∫

DxeS[∫ +τ/2

−τ/2dtxc(t)x(t + t0)

]

×[∫ +τ/2

−τ/2dt0δ

[∫dtxc(t)x(t + t0)− λ

]].

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The crucial step is to define a change of integration variables as follows:2

x(t)→ x(t) = x(t − t0). (16.41)

Due to Eq. 16.35, the action is invariant under this change of variables and

S[x] = S[x]. (16.42)

In particular, note that S[x] is independent of t0. Hence, dropping the tilde on x

yields

K(a,−a; τ) =∫

DxeS∫ +τ/2

−τ/2dtxc(t)x(t)×

∫ +τ/2

−τ/2dt0δ

[∫dtxc(t)x(t)− λ

]

=[∫ +τ/2

−τ/2dt0

]·∫

DxeS∫ +τ/2

−τ/2dtxc(t)x(t)δ

[∫dtxc(t)x(t)− λ

].

The zero mode, namely the variable t0, has been completely factorized out of thepath integral, which no longer has any divergence due to the delta-function con-straint.

The zero mode is a physical quantity that reflects that tunneling can take placeat any instant in the interval [−τ/2,+τ/2]. In contrast, for the case of path inte-grals for gauge fields, the Faddeev–Popov procedure leads to a factorization of thenonphysical gauge degrees of freedom.

It is more convenient to re-write the delta-function constraint as another termin the action. To do this note that the above equation is valid for all λ; hence,multiplying both sides with exp{−αλ2/2) and integrating over λ yields

K(a,−a; τ) = τ

√α

2π

∫Dx

∫ +τ/2

−τ/2dtxc(t)x(t)e

S− α2

( ∫ +τ/2−τ/2 dtxc(t)x(t)

)2

. (16.43)

Note that since α is an arbitrary parameter, the evolution kernel K(a,−a; τ) mustbe independent of it.

The classical field equation for the path integral given in Eq. 16.43 is given by[δS

δx(t)− αxc(t)

∫ +τ/2

−τ/2dt ′xc(t ′)x(t ′)

] ∣∣∣∣x=xc

= 0. (16.44)

From Eq. 16.20,

δS[xc]δx(t)

= 0 (16.45)

2 Note xc(t) is unchanged since, unlike x(t), it is not an integration variable.

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and ∫ +τ/2

−τ/2dtxc(t)xc(t) = x2

c (τ/2)− x2c (−τ/2) = 0.

Hence, the classical trajectory xc(t) defined by the double-well action given by Eq.16.45 is also a classical solution (maximum) of the modified action given in Eq.16.44.

Note that, from Eq. 16.31, the modified action can be written as

S ′ = S − λ0

2〈x|ψ0〉〈ψ0|x〉, λ0 = α||xc||2, (16.46)

ψ0 = 1

||xc|| xc, 〈ψ0|x〉 ≡∫ +τ/2

−τ/2dtxc(t)x(t).

Expanding the path integral about the classical solution,

x(t) = xc(t)+ η(t), (16.47)

yields, from Eq. 16.28, the action

S[x] = Sc − 1

2

∫ +τ/2

−τ/2dtdt ′η(t)M(t, t ′)η(t ′)− λ0

2〈η|ψ0〉〈ψ0|η〉, (16.48)

= Sc − 1

2

∫ +τ/2

−τ/2dtdt ′η(t)M(t, t ′)η(t ′), (16.49)

where

M = λ0|ψ0〉〈ψ0| +∞∑n=1

λn|ψn〉〈ψn|.

Note from Eq. 16.29 that ψ0 is a vector that is orthogonal to all the eigenfunctionsof operator M and yields the completeness equation

I = |ψ0〉〈ψ0| +∞∑n=1

|ψn〉〈ψn|.

Hence M is a nonsingular operator with determinant given by

det M = λ0

∞∏n=1

λn �= 0.

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K(a,−a; τ) = τ ||xc||2√

α

2πeSc∫

Dη exp{−1

2

∫ +τ/2

−τ/2dtdt ′η(t)M(t, t ′)η(t ′)}

= τ ||xc||2√

α

2πeSc × 1√

det M= τ ||xc||2

√α

2πeSc × 1√

λ0∏∞

n=1 λn

= τ ||xc||eSc × 1√2π

∏∞n=1 λn

, finite, (16.50)

where Eq. 16.46 has been used for the value of λ0.The evolution kernel obtained above is indeed convergent, unlike the earlier re-

sult derived in Eq. 16.34 for which it was divergent. Furthermore, K(a,−a; τ)given in Eq. 16.50 is indeed independent of α, as expected, since all the eigenval-ues λn and the constant are independent of α.

16.5.1 Instanton coefficient NThe one instanton transition amplitude is given by

〈−a|e−τH |a〉 = N eSc∫ +τ/2

−τ/2dtc = N τeSc ,

which is obtained from

〈−a|e−τH |a〉 = τeSc√det(−m d2

dt2+ V ′′(xc))

.

The determinant is apparently zero due to the zero mode giving rise to a zero eigen-value.

Instead of the Faddeev–Popov approach, another way of factoring out the sin-gularity due to the zero mode is to do an expansion of η that directly takes intoaccount the zero mode of the action. This is accomplished by the expansion

x(t) = xc(t − tc)+∞∑n �=0

ψn(t − tc)cn,

where tc is now considered an arbitrary variable. Note that there is no coefficientc0 multiplying the classical solution xc(t − tc); this is because c0 is replaced byan arbitrary variable tc, namely the instant at which the particle tunnels from one

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vacuum to the other. The change of variables from x(t) to cn, tc yields [Das (2006)]

Dx = ||xc||dtc∞∏n=1

dcn ⇒∫

Dx = ||xc||∫

dtc

∞∏n=1

∫dcn. (16.51)

By eliminating the zero eigenvalue in the eigenfunction expansion of η(t) thezero mode in the determinant has been eliminated and the nonzero and finite deter-minant is given by

det(−m d2

dt2+ V ′′(xc)) =

∞∏n=1

λn. (16.52)

Hence, the semi-classical expansion in the one instanton approximation yields

〈−a|e−τH |a〉 = const eSc

[ ∞∏n=1

∫dcne

−λnc2n

2

]∫ +τ/2

−τ/2dtc

= N τeSc . (16.53)

The coefficient N that appears in the transition amplitude is evaluated. For thetrivial classical solution given by xc = ±a in which the particle sits at one of thewells, the classical action is zero. The effective action about one of the wells isgiven by the simple harmonic oscillator action

S0 = −∫ +τ/2

−τ/2dt

(1

2mη2 + g2

2a2η2

),

boundary conditions: η(−τ/2) = 0 = η(+τ/2).

The zeroth order transition amplitude is given by Eq. 11.21 for the simple harmonicoscillator,3

〈0|e−τH |0〉(0) = eSc(±a)∫

DηeS0 =√

mω

2π� sinhωτ, (16.54)

� e−ωτ/2(mωπ�

)1/2 +O(e−ωτ )

≡ A, (16.55)

where Sc(±a) = 0 and we recall that ω2 = g2a2/2. A careful analysis by Das(2006) shows that

〈−a|e−τH |a〉 = Ar

∫ +τ/2

−τ/2dtc = Arτ,

3 Recall that for the simple harmonic oscillator – restoring the dependence on � – is given by

K(x′, x; τ) =√

mω

2π� sinhωτexp

{− mω

2� sinhωτ

[(x2 + x′2) coshωτ − 2xx′

]}.

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where A is the coefficient given in Eq. 16.55. The factor r comes from the instantondeterminant given in Eq. 16.52 and has been shown by Das (2006) to be

r = 2a

√2m

�ω3/2eSc .

16.6 Multi-instantons

Exact multi-instanton classical solutions, characterized by multiple tunneling, havenot yet been found. However, for tunneling at times that are widely separated, andfor coupling g→∞ one can build approximate multiple-instanton solutions fromthe single kink/antikink solutions. Consider the two kink–antikink ansatz

x(2)c (t) = ±a2 tanh

(ω(t − t1)

2

)tanh

(ω(t − t2)

2

), t2 � t1 (16.56)

as shown in Figure 16.7.It has been shown by Das (2006) that x(2)c satisfies the equations of motion up to

terms of O(e−ω(t2−t1)).In general a string of widely separated instantons and anti-instantons satisfies

the classical equations of motion. In the dilute instanton gas approximation, whichis valid for strong coupling g � 11, an N -instanton classical solution is approxi-mately composed of the product of N instantons and anti-instantons,

xNc (t) =N∏i=1

xc(t − ti)

with− τ

2≤ tn ≤ tn−1 · · · ≤ t1 ≤ τ

2.

For g2 → ∞, one can make further approximations of the N -instanton solu-tion: the cross over from one well to another is instantaneous. Furthermore, theN -instanton configuration, similarly to Eq. 16.56, is constructed by multiplying

-a

t2

t1

t2

t1

-a+a +a +a-a

≈ =

Figure 16.7 A two-kink configuration for the particle’s trajectory.

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16.7 Instanton transition amplitude 371

XN (t)

–t/2 ≤ t N ≤ t N–1 ≤ ...... ≤ t 1 ≤ t/2

t/2 t/2

t

t 2 t 1t 3t 4t 5t 6t 7t 8

Figure 16.8 A multiple kink–antikink configuration for the particle’s classicaltrajectory.

the configuration of N single instantons, with each instanton tunneling taken to beindependent of the other tunnelings. In this approximation, since

limω→∞ tanh(ωθ) = sgn(θ), sgn(θ) =

{−1 θ < 01 θ > 0

,

the N -instanton classical solution is given by

x(N)c (t) = ±aN

N∏j=1

sgn(t − tj ). (16.57)

A typical N -instanton configuration is shown in Figure 16.8.

16.7 Instanton transition amplitude

In the dilute instanton gas approximation, the transition amplitude is given by

〈−a|e−τH |a〉N = ArN∫ τ/2

−τ/2dt1

∫ t1

−τ/2dt2 · · ·

∫ tn−1

−τ/2dtn

= ArN1

N !(∫ τ/2

−τ/2dt

)N

= A1

N !rNτN,N − instanton contribution. (16.58)

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Hence, the full transition matrix element is given by summing over odd N multipleinstanton configurations, and yields

〈−a|e−τH |a〉 =∞∑N=0

〈−a|e−τH |a〉2N+1 = A

∞∑N=0

r2N+1

(2N + 1)!τ2N+1

= A sinh(rτ ) = 1

2

√mω

π�

(e−(

ω2−r)τ − e−(

ω2+r)τ

), (16.59)

where A is given by Eq. 16.55,

A = (mω

π�)1/2e−

ωτ2 .

16.7.1 Lowest energy states

For the double well potential, 〈a|e−τH |a〉 is given by the sum over all even N

multiple instanton configurations, starting with N = 0. Hence, from Eq. 16.58

〈a|e−τH |a〉 = A

∞∑N=0

(rτ )2N

(2N)! = A cosh(rτ )

= 1

2(mω

π�)1/2[e−( ω2−r)τ + e−(

ω2+r)τ ].

Let the two lowest excited states, as given in Figure 16.9, be denoted by |−〉 and|+〉; the completeness equation in this two-state sector is given by

I � |−〉〈−| + |+〉〈+| +O(e−4/g2)2. (16.60)

Hence, the completeness equation given in Eq. 16.60 yields

〈a|e−τH |a〉 ≈ 〈a|e−τH |+〉〈+|a〉 + 〈a|e−τH |−〉〈−|a〉= e−τE+|〈−|a〉|2 + e−τE−|〈+|a〉|2

= 1

2(mω

π�)1/2[e−( ω2−r)τ + e−(

ω2+r)τ ].

Hence, the two lowest energy eigenvalues are given by

E− � �(ω

2− r), E+ � �(

ω

2+ r), (16.61)

|+Ú

|–Ú

Figure 16.9 Ising configuration and an instanton.

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16.8 Instanton correlation function 373

yielding the level splitting given by

E = E+ − E− = 2�r = 4ω3/2√

2m� eSc .

16.8 Instanton correlation function

The correlation of the double-well potential can be obtained far from the linearizedtheory by expanding the path integral about the instanton solutions [Polyakov(1987)]. Consider the two point correlator

G(T1, T2) = 1

Z

∫Dx x(T1)x(T2)e

S.

A normal mode expansion of the correlation function about x(�)c , the � instantonconfiguration, yields

x(t) = η�(t − tc)+ x(�)c (t − tc) =∞∑n=1

cnψn(t − tc)+ x(�)c (t − tc).

The fluctuations about the x(�)c are given by η�. In the semi-classical expansion, thecorrelator is given by

G(T1, T2)

�∑

�

∫Dx[η(T1 − tc)+ x(�)(T1 − tc)][η(T2 − tc)+ x(�)(T2 − tc)]eS[η+x(�)]∑

�

∫DxeS[η+x(�)]

.

Note that∑

� is a sum over the �-instanton and anti-instanton classical solutions,denoted by x(�). The expansion of the action yields, to leading order

S[x] = S[η� + x�c ] = S[x�c ] + S[η�] + . . .

The first two classical solutions are S[x0c ] = 0, S[x1

c ] = Sc.Summing over the trivial classical (� = 0) solution given by x = ±a and the

� = 1 one instanton and anti-instanton yields, to leading order in �,

G(T1, T2) � a2Z0 + eScZ1∫dtcxc(T1 − tc)xc(T2 − tc)

Z0 + eScZ1∫dtc

= a2 + BeSc∫dtcxc(T1 − tc)xc(T2 − tc)

1+ BeSc∫dtc

, (16.62)

⇒ G(T1, T2) � a2 + BeSc∫ +∞

−∞dtc

[xc(T1 − tc)xc(T2 − tc)− a2

]. (16.63)

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Coefficient Z0 is given by

Z0 =∫

Dη0eS[η0] =

∞∏n=1

∫dcne

− 12 (λn,0)

2c2n,

where the classical solution x(0)c is given by x(0)c = ±a. The effective action S[η0]is given in Eq. 16.19 and λn,0 are the eigenvalues for the effective action about thetrivial classical solution ±a.

From Eq. 16.51, for the one instanton solution the path integral measure isDη1 = ||xc||dtc∏∞

n=1 dcn = dtcD′η1. The tc integration couples to the classi-

cal solutions and is included in Eq. 16.62 Coefficient Z1 is given by the remainingintegrations,

Z1 =∫

D′η1eS[η1].

The classical solution x(1)c is the one instanton solution and yields

Z1 = ||xc||∞∏n=1

∫dcne

S[η1] = ||xc||∞∏n=1

∫dcne

− 12 (λn)

2c2n,

where the eigenvalues λn for the single instanton are given in Eq. 16.30.Hence, from Eq. 16.62

B = Z1

Z0= ||xc|| ·

∏∞n=1

∫dcne

− 12λ

2nc

2n∏∞

n=1

∫dcne

− 12 (λn,0)

2c2n

. (16.64)

16.9 The dilute gas approximation

The dilute gas approximation, from Eq. 16.57, is based on an approximate descrip-tion of the double-well potential for large coupling, namely g � 1 and leads tothe simplification that xc(ti − tc) = ±a sgn(ti − tc). The dilute gas approximationconsiders the N -instanton solution to consist of a collection of N noninteractinginstanton and anti-instanton solutions. With these assumptions, the double well po-tential can be solved exactly.

From Eq. 16.63, for T2 > T1, the integral over the one instanton contribution isgiven by4

a2B

∫ +∞

−∞dtc [sgn(T1 − tc)sgn(T2 − tc)− 1]

= a2B

[∫ T1

−∞dtc −

∫ T2

T1

dtc +∫ +∞

T2

dtc −∫ +∞

−∞dtc

]4 The case of T1 > T2 is obtained in a straightforward manner by exchanging T1 with T2.

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16.9 The dilute gas approximation 375

= −2a2B|T1 − T2|, |T1 − T2| � ω−1. (16.65)

The integration range is shown in Figure 16.7.Hence, in the one instanton approximation, the correlator is given by

G(T1, T2) = a2(1− 2B|T1 − T2|eSc).In the dilute instanton gas approximation, the contributions of all the multi-instantons are simply products of the contribution of a single instanton and anti-instanton. The multi-instanton tunneling times tc,n yield

xNc (t) =N∏i=1

xc(t − tc,i)

with T1 ≤ tN ≤ tN−1 · · · ≤ t1 ≤ T2.

Note that the instantons are well localized; for large time t , the single instantonhas the behavior

xc(t) = ±a(1+O(e−ωt ).

The action for the superposition of an instanton (I) and an anti-instanton (AI), withseparation T12, is given by

S2 � Sc(I )+ Sc(AI)+O(e−ωT12)

= 2Sc, (16.66)

and hence the action for N -instantons is

SN � NSc. (16.67)

Similarly to the derivation of Eq. 16.59, the correlator for the dilute instantongas is given by

G(T1, T2) � a2∞∑N=0

(−2B)NeNSc

∫dtc,NdtN−1 · · · dt1

= a2∞∑N=0

(−2B)NeNSc(T2 − T1)

N

N != a2e−|T1−T2|/ξ ,

where correlation time ξ is given by

ξ � 1

2Be−Sc = 1

2Bexp

{2m2ω3

3g2

}→∞ as g→ 0.

The nonlinear double well potential gives rise to a nonperturbative correlationlength, with an essential singularity at g = 0. Any perturbation about the trivial

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vacua x = ±a would not yield the correlation length; in contrast, the semi-classicalexpansion yields a well-defined order by order procedure for evaluating the pathintegral.

16.10 Ising model and the double well potential

The N -instanton solution, in the dilute gas approximation, can be constructed fromN single instanton configurations. Since the overlap of the component one instan-ton configurations is taken to be negligible, the action is given by

S = NSc = −N c

g2,

with tunneling times given by

t1 < t2 < t3 · · · < tN.

From Eq. 16.57 we find that the N -instanton solution, for strong coupling givenby g2 � 1, reduces to

x(N)c (t) = ±a

N∏j=1

sgn(t − tj ).

A single instanton configuration is equivalent to a configuration of the Isingspins, as shown in Figure 16.10 The multi-instanton configuration is now equiva-lent to the one-dimensional Ising model, as the the trajectory of the particle has avalue of either +a or −a, and the tunneling between these two configurations isequal to a spin flip for the Ising spin.

To make the connection with the Ising model more explicit, consider the limitof ω = ag/

√m → ∞, or equivalently g2 → ∞; up to irrelevant constants, the

potential given in Eq. 16.15 yields

limg→∞ e−

g2

8 (x2−a2)2 → δ(x2 − a2) = 1

2a[δ(x − a)+ δ(x + a)].

-a

t1

+a -a

t1

+a

Figure 16.10 Ising configuration and an instanton.

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16.11 Nonlocal Ising model 377

Discretizing time t = nε, the path integral given in Eq. 16.18 yields

Z =∫

Dxe−m2ε

∑i (xi+1−xi )2

∏i

δ(x2i − a2)

=∫

Dxemε

∑i (xixi+1−x2

i )δ(x2 − a2)

=∑

{μi=±1}eK

∑i μiμi+1, Ising model,

where

μi = xi/a, K = ma2

ε.

Recall from Eq. 8.22, the Ising model’s transfer matrix is

L =(eK e−K

e−K eK

).

The eigenstates and eigenvalues, from Eqs. 8.27 and 8.25, are given by

|λ1〉 = 1√2

(11

), λ1 = eK + e−K,

|λ2〉 = 1√2

(1−1

), λ2 = eK − e−K.

Since λ1 > λ2, the two states have the following identification:

|λ1〉 = 1√2

(11

)≡ |−〉, vacuum state,

|λ2〉 = 1√2

(1−1

)≡ |+〉. (16.68)

The vacuum state is symmetric under the exchange of a→−a and hence preservesparity symmetry.

16.11 Nonlocal Ising model

Let us consider the periodic and non-local Ising model

S = K

2

N∑i,j=1

μie−α|i−j |μj

= K

2

∑ij

μiμjAij . (16.69)

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Using Gaussian integration, the action S has the representation

eS =N∏i=1

∫dxi exp[−1

2

∑ij

xiA−1ij xj +

∑i

μxi],

Aij = Ke−α|i−j |.

It can be verified that the inverse of Aij is given by

A−1ij =

1

K[Cδij − B(δi−j−1 + δi−j+1)].

Then

S = − 1

2K

N∑i,j=1

xi

[Cδij − B(δi−j−1 + δi−j+1)

]xj +

N∑i=1

μixi

= − 1

2K

N∑i=1

{B(xi − xi+1)2 + (C − 2B)x2

i } +N∑i=1

μixi.

Therefore

Z =∑{μ}

eS[μ] =N∏i=1

∫dxie

S[x],

where

S[x] = − B

2K

∑i

(xi − xi+1)2 − (

C − 2B

2K)∑i

x2i +

∑i

ln(2 cosh xi)

= − B

2K

∑i

(xi − xi+1)2 − V (x),

and the potential, shown in Figure 16.11, is given by

− V = −(C − 2B

2K)x2 + ln(2 cosh x) �

⎧⎨⎩−C−2B

2K x2 + 12x

2, x � 0

−C−2B2K x2 + |x|. x � 1

For (C − 2B)/2K > 0, V is a double well, as shown in Figure 16.12. Hence, theIsing model, for a certain choice of parameters, is equivalent to the double wellpotential.

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16.11 Nonlocal Ising model 379

Full range view Small scale of V

Figure 16.11 The potential for different m = −(C−2B)/2K: coupling strengths.

–V

Figure 16.12 Double well potential from an Ising model of a lattice site for (C −2B)/2K > 0.

To determine coefficients B and C, we note that

N∑l=1

AilA−1lj = Ce−α|i−j | − B(e−α|i−j+1| + e−α|i−j−1|).

For the case of i = j ,

RHS = C − B(e−α + eα) = 1 ⇒ C = 1+ 2Be−α.

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For the case of i > j ,

RHS = Ce−α(i−j) − B(e−α(i−j+1) + e−α(i−j−1))

= e−α(i−j)[C − B(e−α + eα)] = 0,

and for i < j ,

RHS = e−α(i−j)[C − B(eα + e−α)] = 0.

Therefore, the case for both i > j and i < j yields

B = C

2 coshα⇒ C = 1

tanhα, B = 1

2 sinhα.

(C − 2B)/2K = (coshα − 1)/2K sinhα > 0 for coshα > 1 and hence thenonlocal Ising model exhibits symmetry breaking for all values of α > 0.

16.12 Spontaneous symmetry breaking

Symmetry breaking is quite common in classical physics. For any potential withmultiple minima, the particle can be stationary at any of the minima of the potentialand consequently satisfy the equations of motion. In the case of the double-wellpotential given by V = g2

8 (x2 − a2), the particle can be classically at rest at either

of the two points given by x = ±a, which breaks the x →−x parity symmetry ofthe potential.

In quantum mechanics, the Mermin–Wagner theorem states that quantum me-chanics cannot break continuous symmetries, but leaves open the possibility ofbreaking discrete symmetries.

Symmetry in quantum mechanics means the following. Suppose H commuteswith some operator R, that is, [H,R] = 0. The theory is said to have an unbrokensymmetry, or a symmetric state space, if the ground state |�〉 is invariant under R,namely that it satisfies

R|�〉 = |�〉, symmetric theory.

The theory has spontaneously broken symmetry if

R|�〉 �= |�〉, spontaneously broken symmetry. (16.70)

In particular, if R is the parity operator defined by RxR = −x, one has

〈�|x|�〉 = −〈�|RxR|�〉.It follows from the above equations that

〈�|x|�〉{= 0, symmetric,�= 0, nonsymmetric.

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16.13 Restoration of symmetry 381

16.12.1 Infinite well

Let us consider two infinitely deep potential wells. If the particle is in well I, it hasan entire Hilbert space VI which is disjoint from the particle in well II; that is

〈VI |VII 〉 = 0,

and VI ∩ VII = φ, with |�I 〉, |�II 〉 being the vacuum states.The discrete symmetry of the potential V (−x) = V (x) is broken by the vacuum

since

R|�I 〉 = |�II 〉 �= |�I 〉, nonsymmetric vacuum.

16.12.2 Double well

Let us now consider the double-well potential Hamiltonian

H = − 1

2m

∂2

∂x2+ g2

8(x2 − a2)2.

Let R be the parity operator such that 〈x|R|ψt〉 = Rψt(x) = ψt(−x). Let the“false” vacuum, centered around x = ±a, be denoted by |�±〉. Then

R|�±〉 = |�∓〉, nonsymmetric vacuum.

Hence |�±〉 spontaneously breaks the R-symmetry. It will be shown that the truevacuum is invariant under R, namely that

R|�〉 = |�〉, symmetric vacuum. (16.71)

16.13 Restoration of symmetry

For the finite double-well, it is shown how the symmetry of x →−x is restored bytunneling. Consider only the ground state sector of the Hilbert space of the doublewell. To any order in perturbation theory, one has for the lowest energy states ofthe double well the lowest order Hamiltonian given by

H0 =(E0 00 E0

), H0| ± a〉 = E0| ± a〉,

with the degenerate eigenstates, as discussed in Section 16.7, given by

|a〉 �(

10

), |−a〉 �

(01

).

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In the limit of τ →∞, all the states except the two lowest lying states decouplefrom the transition amplitude 〈a|e−τH | − a〉. On this two dimensional subspace ofV , the Hamiltonian H is a 2× 2 matrix and hence yields the expansion

〈a|e−τH | − a〉 � 〈a|(1− τH)| − a〉� −τH(a,−a), 〈a|H | − a〉 ≡ H(a,−a).


−τH(a,−a) = τNeSc .

and hence (note the negative sign)

H(a,−a) � −Ne−c/g2.

The full Hamiltonian is consequently given by

H� =(

E0 −Ne−c/g2

−Ne−c/g2E0

),

with eigenstates and eigenvalues

|�〉 = 1√2(|a〉 + | − a〉), E = E0 −Ne−c/g

2, symmetric,

|1〉 = 1√2(|a〉 − | − a〉), E = E0 +Ne−c/g

2, anti-symmetric.

The vacuum state |�〉 was obtained earlier using the Ising approximation in Eq.16.68. The true vacuum state for the double well is symmetric under the parityoperator. The false vacua that to lowest order in perturbation theory start from eitherof the degenerate perturbative vacua, namely |±a〉, are corrected by the instantons,and symmetry is “restored” in the sense that a calculation for the vacuum statethat includes the instanton contributions yields the actual symmetric and uniquevacuum state.

Instantons represent large quantum fluctuations that are represented by tunnelingfrom one false vacuum to the other. The transition amplitudes could not have beenobtained by considering only small fluctuations about the false vacua. It is thelarge fluctuations obtained by perturbing about the N -instanton configurations thatrestore the symmetry of the false vacuum.

A superposed state is completely nonclassical: the point particle is in an in-determinate state, existing simultaneously at two distinct points ±a. This is howquantum mechanics restores parity symmetry.

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16.15 Summary 383

16.14 Multiple wells

By labeling the two wells as 1 and 2 one can write the Hamiltonian H as

HN = E0(|1〉〈1| + |2〉〈2|)− t (|1〉〈2| + |2〉〈1|), (16.72)

with t = Ne(−c/g2). The symmetry operator is then

R|1〉 = |2〉, R|2〉 = |1〉.Consider the potential that has minima at sites na, with n = 0, 1, 2, . . . N .A calculation similar to the double-well case gives to leading order that the

effective low energy Hamiltonian is

HN = E0

N∑n=1

[|n〉〈n| − t

(|n〉〈n+ 1| + |n+ 1〉〈n|

)], (16.73)

where for simplicity we assume space is periodic with |N+1〉 = |1〉. The symmetryof the Hamiltonian is given by the shift operator

R|n〉 = |n+ 1〉, [HN,R] = 0.

One can verify that the eigenstates of HN are given by

|θ〉 = 1√N

N∑n=1

einθ |n〉, HN |θ〉 = (E0 − 2t cos θ)|θ〉.

with the true ground state given by the symmetric combination

|�〉 = 1√N

N∑n=1

|n〉, E� = E0 − 2t.

The θ -parameter is a measurable quantity and occurs in many theories, includingthe Yang–Mills Lagrangian.

16.15 Summary

Nonlinear path integrals can have qualitative properties that cannot be obtained byperturbing about the linear part of the theory, which is defined by the quadraticpiece of the Lagrangian. The double-well potential was chosen to illustrate thisaspect of nonlinear path integrals.

Parity is a symmetry of the double-well potential that is broken by the falsevacuum of the linearized theory. The path integral was studied to ascertain whetherthe nonlinearities of the Lagrangian can restore parity symmetry.

The domain of integration over which the path integral is defined is an infinitedimensional function space. The double-well action has classical multi-instanton

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solutions that are configurations in function space far from the linear domain. Toprobe the nonlinearities of the theory, the path integral is expanded about the classi-cal solutions and is evaluated using perturbation theory about the classical solution.

The instantons have a zero mode, which is not constrained by the action, thatarises from the instanton being free to tunnel from one false vacuum to the other atany time. The zero modes lead to a summation over all instants that the instantoncan tunnel and produces a large effect. Examining the lowest energy states, it wasseen that tunneling creates an off-diagonal term in the Hamiltonian which cannotbe obtained by perturbing about the linearized theory. The off-diagonal term in turnleads to the Hamiltonian having a true vacuum (ground) state that preserves parity.

The multi-instanton expansion can be approximated by a dilute gas of noninter-acting instantons. This approximation scheme was used to evaluate the correlationfunction of the double well; the correlation length obtained shows that it dependson the coupling constants in a nonperturbative manner, further demonstrating theessential distinction between a linear and a nonlinear theory.

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17

Compact degrees of freedom

Quantum mechanical degrees of freedom come in many kinds and varieties andare variables that can take values in many different kinds of spaces (manifolds).Compact and noncompact degrees of freedom are two important types. In essence,a compact degree of freedom takes values in a finite range whereas the noncompacttake values in an infinite range.

The simplest compact degree of freedom is the Ising spin variable μ that takestwo values, namely μ = ±1, and which was studied in Chapters 8 and 9. InChapters 11 to 16, the noncompact degree of freedom x – taking values on thereal line, that is x ∈ [−∞,+∞] – was studied for various models.

Compact degrees of freedom have many specific properties not present for thenoncompact case, and the focus of this chapter is on these properties. The degreesof freedom that take values in compact spaces for two different cases are discussed,with each case having its own specific features, as follows:

• A degree of freedom taking values on a circle S1. This case has a nontrivialtopological structure that occurs in a wide range of problems, with the simplestcase being of a quantum mechanical particle moving on a circle S1.

• A degree of freedom taking values on a two dimensional sphere S2, a com-pact space. This degree of freedom can represent a quantum particle movingon a sphere and is the simplest case of a quantum particle moving on a curvedmanifold.

In Section 17.1 the degree of freedom taking values on a circle is introduced, andin 17.2 its multiple classical solutions are derived. The degree of freedom takingvalues on a sphere S2 is discussed in Section 17.3 and its Lagrangian is derived inSection 17.4; a divergence arising from the curvature of the sphere is discussed inSection 17.5. Sections 17.6–17.8 apply the S2 degree of freedom to study of thestatistical mechanics of the DNA molecule.

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386 Compact degrees of freedom

17.1 Degree of freedom: a circle

The S1 degree of freedom takes its values on a circle. The space S1 is geometricallyflat, similarly to the real line, but is a topologically nontrivial manifold: S1 is notsimply connected; what this means is that a loop on the circle cannot smoothlybe contracted to a single point. One can wind around the circle n times, and allthese windings are inequivalent since they cannot be mapped to, say, a single loop.The existence of the winding number is the single most important reflection of thenontrivial topology of S1.

Consider a particle moving on a circle of radius R; a degree of freedom x thenhas the property that all points x+ 2πnR, with n = 0,±1,±2, . . .±∞ are equiv-alent. In particular, the state function ψ(x) has the symmetry1

ψ(x + 2πnR) = ψ(x), n = 0,±1,±2, . . .±∞.

The Hamiltonian is given by

H = − 1

2m

∂2

∂x2, x ∈ [−πR,+πR].

The state function has the normalization

1 =∫ +πR

−πRdx|ψ(x)|2.

The normalized eigenfunctions of the Hamiltonian are given by

Hψn = Enψn,

ψn = einx/R√2πR

, En = n2

2mR2, n ∈ Z. (17.1)

It is convenient to define a new angular variable θ such that

θ = x

R∈ [−π,+π].

Then

H = − 1

2mR2

∂2

∂θ2,

with the orthonormal state functions given by∫ +πR

−πRdxψ∗n (x)ψm(x) = R

∫ +π

−πdθψ∗n (θ)ψm(θ) = δn−m.

1 This condition can be relaxed to

ψ(x + 2πnR) = eiφψ(x), n = 0,±1,±2, . . .±∞.

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17.1 Degree of freedom: a circle 387

The evolution kernel, for Euclidean time τ , is given by

K(θ ′, θ; τ) = 〈θ ′|e−τH |θ〉 =∑n

e−τEnψn(θ′)ψ∗n (θ)

= 1

2πR

∞∑n=−∞

e− 1

2

(τ

mR2

)n2

ein(θ′−θ). (17.2)

17.1.1 Poisson summation formula

The Poisson summation formula is a useful result in the study of a quantum particlemoving on a circle, and is given by the identity

∞∑n=−∞

δ(ξ − n) =∞∑

�=−∞e2πi�ξ . (17.3)

To prove the Poisson summation formula, let

f (ξ) =∞∑

n=−∞δ(ξ − n) ⇒ f (ξ +m) = f (ξ), m = integer.

Let ξ = 0 and m = 1; then f (1) = f (0). Hence f (ξ) is a periodic functionon the interval 0 ≤ ξ < 1; note that the point ξ = 1 has been excluded, sincedue to periodicity, it is identical to the point ξ = 0. Consider the following Fourierexpansion of f (ξ):

f (ξ) =∞∑

�=−∞e2πi�ξf�,

⇒ f� =∫ 1

0dξe−2πi�ξf (ξ) =

∞∑n=−∞

∫ 1

0dξe−2πi�ξ δ(ξ − n) = 1,

⇒ f (ξ) =∞∑

�=−∞e2πi�ξ ,

which proves Eq. 17.3. The identity yields

∞∑n=−∞

fn =∫ ∞

−∞dξ

∞∑n=−∞

δ(ξ − n)f (ξ) =∞∑

�=−∞

∫ ∞

−∞dξe2πi�ξf (ξ), (17.4)

where f (ξ) is the extension to a continuous argument ξ of the function fn definedon a lattice.

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17.1.2 The S1 Lagrangian

The Lagrangian is given by the Dirac–Feynman formula

K(θ ′, θ; τ) = 〈θ ′|e−εH |θ〉 = N (ε)eεL(θ,θ′). (17.5)

Taking the limit ε → 0 in Eq. 17.2 is hopeless since higher and higher order termsare necessary to evaluate the sum. To take the limit one needs to invert τ → 1

τin

Eq. 17.2. The Poisson summation formula given in Eq. 17.4 yields

〈θ ′|e−τH |θ〉 = 1

2πR

∞∑n=−∞

e− τ

2mR2 n2

ein(θ′−θ)

= 1

2πR

∫dξ

∞∑�=−∞

e2πi�ξ e− τ

2mR2 ξ2

eiξ(θ′−θ)

=√

m

2πτ

∞∑�=−∞

e−mR2

2τ (θ ′−θ−2π�)2 . (17.6)

Hence, taking the limit of τ = ε yields

〈θ ′|e−εH |θ〉 �√

m

2πεe−

12ε mR

2(θ ′−θ)2 +O(e−

1ε

)= N (ε)eεL, (17.7)

and the Lagrangian is given by

L = −1

2mR2

(θ ′ − θ

ε

)2

= −1

2mR2

(dθ

dt

)2

, θ ∈ [0, 2π ],

with N (ε) =√

m

2πε.

17.2 Multiple classical solutions

The evolution kernel is defined by

K(θ ′, θ; τ) =∫

Dθe∫ τ

0 L. (17.8)

The path integral measure, for τn = nε ,Nε = τ , is given by

∫Dθ = lim

N→∞

+N/2∏n=−N/2

∫ +π

π

√m

2πεdθn.

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17.2 Multiple classical solutions 389

The action for the particle on a circle is given by

S = −1

2mR2

∫ τ

0dt

(dθ(t)

dt

)2

, θ ∈ [0, 2π ].

The action is similar to that of a free particle, except that now θ is a compactvariable, taking values on a circle.

To evaluate the evolution kernel, a semi-classical expansion is carried out aboutall the classical solutions. The equations of motion are given by

θ = 0 ⇒ θcl = (a + bt)mod 2π,

B.C. θc(0) = θ, θc(τ ) = θ ′ + 2π�, � : winding number. (17.9)

The multiple classical solutions are shown in Figure 17.1 by displaying θ as anoncompact variable. The winding number, shown in Figure 17.2, reflects the factthat θ(t) is a compact variable that can satisfy the boundary conditions by winding

Figure 17.1 Classical solutions satisfy the boundary conditions, shown for zero,one and two windings around the compact direction.

Figure 17.2 Winding number; the unbroken line is the classical solution with nowinding and the dashed line is the classical solution with one winding around thecompact direction.

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around the compact direction. Eq. 17.9 shows that the final position is equal to θ ′

modulo 2π , since the compact quantum variables θ(t) are periodic. Hence

θc(t) =θ + (θ ′ − θ)

τt + 2π�

t

τ, θcl = 1

τ

(θ ′ − θ + 2πl

).

The classical action for the �th winding number is given by

Sc(�) =− mR2

2

∫ τ

0θ2cldt = −

mR2

2·(θ ′ − θ + 2πl

)2

τ.

As expected, the more the classical solution winds, the larger is the velocity itrequires; hence, classical solutions with higher and higher winding numbers havemore and more negative values for the classical action, as given by Sc(�).

A semi-classical expansion of the path integral about all the classical solutions,similar to the one carried in Section 16.6 for the double well potential, yields

Z =∫

DθeS = N(τ)

∞∑�=−∞

eSc = N(τ)

∞∑�=−∞

e−mR2

2τ (θ ′−θ+2π�)2 .

To obtain the prefactor N (τ ) one needs to analyze the path integral over thecompact variable, namely

∫DθeS ; consider the change of variables

θ(t) = θc(t)+ z(t), B.C. z(0) = z(τ ) = 0. (17.10)

This yields, from Eq. 17.9, the action

S = −1

2mR2

∫(θcl + z)2dt = −1

2mR2

∫ τ

0θ2cldt −

1

2mR2

∫z2dt

= −1

2mR2

(θ ′ − θ + 2πl

)2

τ− 1

2mR2

∫z2dt. (17.11)

The quantum variable z(t) is noncompact, taking values of−∞ ≤ z(t) ≤ +∞; thefluctuations about the classical solutions are noncompact, making the evaluation ofthe prefactor an exact Gaussian path integral. The prefactor N (τ ), in fact, is thesame as that for a free particle on the real line and it has been evaluated earlier inSection 5.8.

Since z(t) does not depend on the classical solutions, it completely factorizesfrom the sum over the classical solutions and yields the final result

K(θ ′, θ; τ) =[∫

Dze−12mR

2∫z2]∑

�

eScl(�) = N (τ )∑�

eScl (�)

=√

m

2πτ

∞∑�=−∞

e−mR2

2τ (θ ′−θ+2π�)2 . (17.12)

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17.3 Degree of freedom: a sphere 391

The result obtained in Eq. 17.12 was obtained earlier, in Eq. 17.6, using the Poissonsummation formula.

The summation n in Eq. 17.2 is a summation over all the discrete momentumstates of the particle whereas � in Eq. 17.12 is a summation over all the windingnumbers. For a compact variable, the momentum and winding number are “dual”to each other. For noncompact variables, the concept of winding numbers does notexist and hence there is no duality.

17.2.1 Large radius limit

The limit of the radius of S1 R → ∞ takes the circle into the real line. We recallfrom Eq. 17.2

K(θ ′, θ; τ) = 1

2πR

∞∑n=−∞

e− 1

2τm

n2

R2 einR·R(θ ′−θ). (17.13)

Let p = n/R; for R→∞ we have

−∞ ≤ p ≤ +∞,

∞∑n=−∞

= R ·∫

dp.

Let us define Rθ = x, Rθ ′ = x ′. Hence

K(x ′, x; τ) = 1

2πRR

∫ ∞

−∞dze−

τ2mz

2eiz(x−x

′) =√

m

2πτe−

m2τ (x

′−x)2 .

Furthermore, from Eq. 17.12

limR→∞K =

√m

2πτ

∞∑�=−∞

e−12mR2τ

(θ ′−θ−2π�)2 =√

m

2πτ

∞∑�=−∞

e−m2τ (x

′−x−2π�R)2

⇒ K =√

m

2πτe−

m2τ (x

′−x)2 . (17.14)

We have recovered the result for the evolution kernel of a free particle moving onthe real line, given in Section 5.8.

17.3 Degree of freedom: a sphere

The main new feature of a degree of freedom that is a circle, for example a quan-tum particle moving on a circle S, is the winding number arising from the factthat the space is not simply connected. The degree of freedom taking values on asphere is exemplified by a quantum particle moving on a two dimensional sphereS2. The two dimensional sphere is simply connected in that any closed loop on

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Time

Figure 17.3 Particle moving on a sphere.

S2 can be continuously contracted to a point, with every intermediate loop in themapping being points in S2. Hence S2 has a trivial topological structure and has nowinding number; in particular, unlike the S1 case, the S2 degree of freedom doesnot have multiple classical solutions.

Although S2 is topologically simple, it has another important geometrical fea-ture, namely that of curvature: the two dimensional sphere is a Riemannian mani-fold with constant curvature; for a sphere of radius R the Ricci scalar curvature isgiven by 2/R2.

A particle moving on a sphere is shown in Figure 17.3; the position of the particleon the sphere is specified by two angles, which in spherical coordinates shown inFigure 17.4 are given by the polar and azimuthal angles θ, φ respectively.

The three-dimensional Laplacian, in spherical polar coordinates r, θ, φ, is givenby

�∇2 = ∂

∂x2+ ∂

∂y2+ ∂

∂z2= 1

r

∂

∂r2r − L2

r2,

where

L2 = 1

sin2 θ

∂

∂ϕ2+ ∂

∂θ2+ cot θ

∂

∂θ, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.

The position of the particle on the sphere is given by

R (cos θ sinϕ, cos θ cosϕ, sin θ) .

A particle moving on a sphere of fixed radius r = R has ∂∂r= 0. Two particles at

the ends of a rigid rod undergoing rotations form a (linear) rigid rotor. A particleconfined to move on the surface of a sphere is equivalent to the motion of a rigid

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17.4 Lagrangian for the rigid rotor 393

Figure 17.4 Spherical coordinates.

Figure 17.5 A quantum mechanical rigid rotor.

rotor (for example, a diatomic molecule), where I is the moment of inertia of therigid rotor, which is shown in Figure 17.5.

The Hamiltonian of the rigid rotor is given by

H = − �2

2mR2L2 = −�

2

2IL2, I = mR2. (17.15)

The eigenfunctions are the spherical harmonics Ylm (θ, ϕ), given by Gottfriedand Yan (2003), and satisfy

HYlm =�2

2Il(l + 1)Ylm (θ, φ) , l = 0, 1, 2, . . .∞, − l ≤ m ≤ l.

17.4 Lagrangian for the rigid rotor

The rigid rotor is described by two angular degrees of freedom, namely θ, φ; theHilbert space has coordinate basis states |θ, φ〉 and the dual (“momentum”) coor-dinates are |l, m〉 with

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〈θ, ϕ|l, m〉 = Ylm (θ, ϕ).

Note that ∫ π

0dθ

∫ π

−πdϕ sin θ |θ, ϕ 〉〈 θ, ϕ| = I.

The Lagrangian is given by the Dirac–Feynman formula

N (ε) eε�L = 〈θ ′, ϕ′|e− ε

�H |θ, ϕ〉.

It is more convenient to use the momentum basis p, q since, for infinitesimaltime ε, it is equivalent to the spherical harmonic basis |l, m〉. The completenessequation is given by∫ +∞

−∞dpdq

4π2|p, q〉〈p, q| = I, 〈θ, ϕ|p, q〉 = eipθ+iqφ.

The completeness equation yields, from Eq. 17.15

N (ε) eε�L =

∫dpdq

4π2eip(θ

′−θ)+iq(ϕ′−ϕ)e− ε

2�I �

(q2

sin2 θ+p2−ip cot θ

)

=∫

dpdq

4π2eip(θ ′−θ+ ε�

2I cot θ)+iq(ϕ′−ϕ)

e− ε�

2I

(q2

sin2 θ+p2

)

=(

2I

ε�

)sin θe−

sin2 θ2ε� (ϕ

′−ϕ)2

e− I

2ε�

(θ ′−θ+ ε�

2I cot θ)2

.

Hence for ε → 0, the Lagrangian of the rigid rotor is given by

L = −I2

sin2 θϕ2 − I

2

(θ + �

2Icot θ

)2

. (17.16)

The normalization

N (ε) =(

2I

ε�

)sin θ (17.17)

depends on θ and is not a constant. This is a typical case for all path integrals oncurved manifolds. The pre-factor N (ε) cannot be obtained from the Lagrangianand is a result of the state space of the degree of freedom taking values on a curvedmanifold.

The classical Lagrangian is given by the velocity of the particle on a sphereand is

L = −I2

(sin2 θϕ2 + θ2

), I = mR2.

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17.5 Cancellation of divergence 395

Compared to the classical Lagrangian above, the extra term �

2I cot θ in Eq. 17.16 isa quantum correction to L.

The path integral is2

K(θ ′, ϕ′; θ, ϕ; t) = 〈θ ′, ϕ′|e−τH |θ, ϕ〉

= NN−1∏n=0

∫ ⟨θn+1, ϕn+1

∣∣e−εH ∣∣ θn, ϕn⟩ N−1∏n=1

dθndϕn sin θn

= NN−1∏n=1

∫dθndϕn sin θne

S, (17.18)

S = ε

N−1∑n=0

L (θn+1, ϕn+1, θn, ϕn, ε),

where N is a normalization constant.

17.5 Cancellation of divergence

The path integral measure, from Eq. 17.18, is given by

N−1∏n=1

sin θn = e∑N−1

n=1 ln(sin θn).

Note that there is a problem with the measure term as it is apparently divergentsince, in the limit ε → 0,

N−1∑n=1

ln (sin θn)→ 1

ε· ε∑n

ln (sin θn) = 1

ε

∫ τ

0dt ln (sin θ (t))→∞.

The only way to obtain a finite result from the path integral is if all the divergentterms that appear in the path integral exactly cancel. To show the cancellation ofthe 1/ε term, a perturbation expansion is carried out for the path integral in powersof θ . The functional integral

∫Dφ is performed while keeping the functional inte-

gral∫Dθ fixed. Performing the functional integral

∫Dφ generates a term for the

θ variable that precisely cancels the divergent term − 12ε

∫θ2 that comes from the

path integral measure.Note that 0 ≤ θ ≤ π ; the minimum value of the action S is around θ = π/2.

Hence, the angle θ is shifted to θ as shown in Figure 17.6, so that the shifted angle,in the path integral, is constrained to be near zero due to the action.

2 Henceforth, unless required, we set � = 1.

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Figure 17.6 Spherical coordinates with angle θ measured from the “forward”y-axis.

Hence, the angle θ is shifted to θ = θ − π/2, with −π/2 ≤ θ ≤ π/2; droppingthe tilde yields, from Eq. 17.16

L = −I2

cos2 θφ2 − I

2

(θ + �

2Itan θ

)2

.

Let τ →∞; the path integral is given by

Z =∫

DθDφ

(∏t

cos θt

)exp

{∫ ∞

−∞dtL(t)

}.

Consider the case of the moment of inertia I � 1 and the action is expanded inpowers of 1/I . Hence

L � −I2

(1− θ2

2+ · · ·

)2

φ2 − I

2θ2 +O(1).

Note that

cos θ � 1− θ2

2⇒

∏t

cos θt � e−12ε

∫ τ0 dtθ2

.

Using the notation ∫ ∞

−∞dt ≡

∫,

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17.6 Conformation of DNA 397

the partition function is given by

Z �∫ ∞

−∞DθDφe−

12ε

∫θ2

exp

{−I

2

∫φ2 − I

2

∫θ2 + I

2

∫θ2φ2 + · · ·

}.

(17.19)

The generating function for a free particle, namely

1

Z

∫Dφ exp

{−I

2

∫φ2 +

∫jφ

}= exp

{1

2I

∫jtDt−t ′jt ′

},

has been evaluated in Eq. 12.1.The correlation function is given by

G(t − t ′) = E[φtφt ′ ] =∫

dω

2π

eiω(t−t ′)

ω2, (17.20)

and yields

E[φt φt ′ ] = 1

I∂t∂t ′ 〈φtφt ′ 〉 = 1

I

∫dω

2πeiω(t−t

′) = 1

Iδ(t − t ′

).

Hence, for discrete time

E[φ2t ] =

1

Iε, (17.21)

and from Eqs. 17.19 and 17.21

Z �∫

Dθe−12ε

∫θ2[

1+ I

2

∫θ2 1

Iε− I

2

∫θ2

]

�∫

Dθ exp

{−I

2

∫θ2

},

1

εterms exactly cancel!

The divergent piece of the measure term is canceled by a term generated from thekinetic term of the ϕ variable.

It can be shown that the cancellation holds to all orders in powers of 1/ε[Zinn-Justin (1993)]. In summary, all the singular terms in the path integral cancel,yielding perfectly finite results for all computations.

17.6 Conformation of DNA

A polymer is a linear chain of molecules and can be considered to be a string laidout in space. The conformation (shape) of a polymer is statistically determined bythe likelihood of it taking the various allowed shapes. The statistical mechanicsof the system – in equilibrium at temperature T – is described by the competitionbetween the entropy of the chain of length L and the energy required for bending it.

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x

y

z

t1

t2

Figure 17.7 The trajectory of a particle specified by its tangent vectors.

A polymer in space is shown in Figure 17.7. If one does not need to know thelocation of the polymer in three-dimensional space, then as shown in Figure 17.7,the polymer’s configuration is completely determined by specifying its three di-mensional tangent vector along its length.

Let us consider a DNA molecule, with total length L, that is free to move in asolute, discussed by Phillips et al. (2008). The curve of the DNA in three dimen-sional space is parameterized by a parameter s; the vector t(s) that is tangential tothe shape of the DNA specifies the shape of the DNA, as shown in Figure 17.8.The parameterization is chosen so that, for every s, the tangential vector has unitlength, namely t(s) · t(s) = 1, and takes values on a two-dimensional sphere S2.

In terms of the spherical polar angles given in Figure 17.4, the tangent vectorhas the coordinates

t = (sin θ cosφ, sin θ sinφ, cos θ) .

The degree of freedom for the DNA is taken to be the tangent vector t, whichtakes all possible allowed values along the curve occupied by the polymer (DNA);the random configurations of the DNA can be modeled by assigning a probabilitydistribution for the different configurations of t(s).

The statistical mechanics of the DNA molecule is modeled by the following“action” and Lagrangian which, to leading order in ξ , are given by

S =∫ L

0dsL, L = −ξ

2

(dtds

)2

= −ξ2

sin2 θφ2 − ξ

2θ2.

The role of time is played by the parameter s that runs along the curve.

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17.7 DNA extension 399

Figure 17.8 DNA polymer in three-dimensional space.

The energy of bending is given by

H = −kBT S = kBT × ξ

2

∫ L

0ds

(dtds

)2

,

and the partition function

Z =∫

Dt exp

{− H

kBT

}=∫

DteS.

Since the action is a nonlinear functional of the degrees of freedom, the partitionfunction can be evaluated perturbatively by expanding the action about θ = φ = 0.

The correlation function of the z-component of the tangent vectors is given by

G(s, s ′

) = E[tz(s)tz(s ′)].In order to calculate the force required to stretch the DNA we need to include a

force term applied along z in the action S that is given by f∫ L

0 tzds. Hence

L = −ξ2

(d�tds

)2

+ f tz = −ξ2

(sin2 θφ2 + θ2

)+ f cos θ.

17.7 DNA extension

The chain’s extension is given by P = ∫ L0 cos θ(s)ds. Note that if all the tangent

vectors are parallel, then θ = 0 and z = L leading to the full extension of theDNA. The average extension is given by

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P = E[P] = 1

Z

∫Dt∫ T

0cos θdseS = ∂ lnZ

∂f.

For the small force limit f � 1, which yields the expansion

Z(f ) =∫

DteS[

1+ f

∫ L

0cos θ(s)ds + f 2

2

∫ L

0dsds ′ cos θ(s) cos(s ′)+ · · ·

]= 1+G1 +G2 +O(f 3).

For ξ � 1 the Lagrangian is approximately given by

L � −ξ2φ2

(1− 1

2θ2

)2

− ξ

2θ2 � −ξ

2φ2 − ξ

2θ2 +O(θ4).

For making the computation well defined we introduce a regulator ω andconsider

L = −ξ2φ2 − ξ

2

(θ2 + ω2θ2

).

The limit ω → 0 will be taken at the end of the calculation. The propagator, fromEq. 11.9, is approximately given by

E[θ(s)θ(s ′)] � e−ω|s−s′|2ωξ

.

For the first term note that

G1 = f

∫ L

0E[cos θ(s)]ds,

E[cos θ(s)] = 1

2Z

∫Dθ

(eiθ(s) + e−iθ(s)

)eS0 = 1

Z

∫Dθeiθ(s)eS0 .

Hence, for jl = δ (s − l), we have from Eqs. 12.1 and 17.20

〈cos θ〉 = e−12

∫jlDll′ jl′ = e

− 14ωξ ⇒ lim

ω→0〈cos θ〉 = e

− 14ωξ → 0

⇒ G1 = 0.

The second term is given by

G2 = f 2

2

∫ L

0dsds ′G

(s, s ′

), G

(s, s ′

) = E[cos θ(s) cos(s ′)].

The path integral is approximated by

G(s, s ′

) � 1

Z

∫DθDφeS cos θ(s) cos θ(s ′)

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17.8 DNA persistence length 401

� 1

Z

∫Dθe−

∫ L0

ξ2 (θ

2+ω2θ2)ds cos θ(s) cos θ(s ′)

= 1

2E[ei{θ(s)+θ(s

′)} + ei{θ(s)−θ(s′)}].

Let us define the external current for the two terms, respectively, by∫dlji(l)θ(l), i = 1, 2

⇒ j1(l) = δ (l − s)+ δ(l − s ′

), j2(l) = δ (l − s)− δ

(l − s ′

).

The first term for G(s, s ′

)is zero since

limω→0

exp

(− 1

4ωξ

∫dldl′ (δl−s + δl−s′) e−ω|l−l′| (δl′−s + δl′−s′)

)

= limω→0

exp

(− 1

2ωξ

(1+ e−ω|s−s′|

))→ 0.

Hence, the correlator is given by

G(s, s ′

) = limω→0

1

2exp

(− 1

4ωξ

∫dldl′ (δl−s − δl−s′) e−ω|l−l′| (δl′−s − δl′−s′)

)

= limω→0

1

2exp

(− 1

2ωξ

(1− e−ω|s−s′|

))= 1

2exp

(−∣∣s − s ′

∣∣2ξ

).

The correlation length is 2ξ , where ξ is called the persistence.

17.8 DNA persistence length

DNA can be thought of as a jointed chain having rigid links of persistence lengthξ , as shown in Figure 17.9. Typically, ξ �50 nm. The distance between base pairsis approximately 0.33 nm, hence ξ is about 150 base pairs. Total length of the DNAis L � 16μm and f � 0.1 pN [Phillips et al. (2008)].

ξ

ξ

ξ

DNA Freely Jointed Chain

Figure 17.9 A DNA polymer with finite correlation length is equivalent to a freelyjointed rigid chain.

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For f � 0, the correlator yields the partition function

Z(f ) = 1+ f 2

2

∫ L

0dsds ′G

(s, s ′

) = 1+ f 2

4

∫ L

0dsds ′e−

|s−s′|2ξ

∼= 1+ 1

2f 2Lξ for L� ξ.

Hence, from the equation above

f � 0 ⇒ E[P]L

= 1

L

∂ lnZ(f )

∂f= f ξ, DNA extension per unit length.

The other limit is that of a large force f � 1, in which case the action isapproximated by

L � −ξ2φ2 − ξ

2θ2 + f

(1− 1

2θ2 + · · ·

)= −ξ

2φ2 − ξ

2

{θ2 + f

ξθ2

},

P =∫ L

0ds cos θ � L− 1

2

∫ L

0ds θ2.

In this approximation, using the result of the simple harmonic oscillator given inEq. 11.10 (with m = ξ and ω2 = f/ξ ),

E[θ(s)θ(s ′)] = 1

2ξ√fe−(

√f /√ξ)|s−s′|

yields the result

E[P] = 1

Z

∫DθDφ

∫ L

0ds cos(θ(s))eS � L− 1

2

∫ L

0ds θ2(s)eS

= L− 1

2

∫ L

0ds

1

2ξ√f

⇒ E[P]L

= 1− 1

4ξ√f.

For intermediate f , the following equation interpolates between small f andlarge f :

f ξ � E[P]L

+ 1

4 (1− E[P]/L)2 −1

4.

The graph of force versus extension for the DNA is shown in Figure 17.10.The path integral for the DNA’s conformation can answer more complicated

questions such as what is the likelihood of the DNA looping and interacting.Intersections are sometimes crucial for obtaining the full information encoded inthe base pairs as the interaction. When the DNA loops, there are special proteinssitting on the DNA that lock the intersections and hence bringing otherwise distantbase pairs into close proximity [Phillips et al. (2008)].

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17.9 Summary 403

E[P] / L

Figure 17.10 Extension of the DNA.

17.9 Summary

The two cases of compact degrees of freedom that were studied, namely S1 and S2,have qualitative features that occur widely in nonlinear theories.

The S1 degree of freedom is periodic and hence is a topologically nontrivialtheory that has multiple classical solutions, classified by the number of times theclassical path winds around S1. Furthermore, the momentum is discretized due tothe periodicity of the degree of freedom.

The path integral was defined using the Hamiltonian. The semi-classical expan-sion of the path integral yielded the exact result; the S1 theory has the simplifyingfeature that the semi-classical expansion yields the exact path integral. The Poissonsummation formula allowed the semi-classical expansion to be exactly re-summedand leads to the Lagrangian description of the degree of freedom. In effect, thePoisson summation formula interchanges the sum over the winding number of theclassical solutions with the discrete momentum of the periodic degree of freedom.

The conformational properties of the DNA molecule – in equilibrium at finitetemperature – were modeled by an S2 degree of freedom. It was seen that themathematics for describing the statistical mechanics of a system by a path integralis identical to that used for describing a quantum system, with the difference lyingonly in the interpretation of the results.

The key new feature of the S2 degree of freedom is that S2 is a manifold withconstant nonzero curvature. This leads to a nontrivial measure for the path integral,

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which, in turn, gives rise to apparent divergences in the path integral. The diver-gence arising from the nontrivial measure is a generic feature of all path integralsdefined for degrees of freedom that take values in a curved manifold.

An expansion in powers of the inverse of the moment of inertia I showed that,to lowest order, all the divergences exactly cancel; it can be shown that this cancel-lation takes place to all orders [Zinn-Justin (1993)]. The path integral yields finiteand well-defined results for all physical quantities. Various experimentally observ-able properties of the DNA were derived to illustrate the flexibility and utility ofmodeling the system using a path integral.

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Index

Q-operatormatrix elements, 298, 300similarity transformation, 297, 299, 301, 303

S1

classical solutionmultiple, 387, 388, 390winding number, 389, 390, 392

Lagrangian, 387, 388, 390S2

cancellationdivergence, 394–397, 399

correlator, 398, 399, 402DNA

polymer, 396, 397, 399

accelerationevolution kernel

symmetry, 292, 294, 309, 311Hamiltonian

classical, 283, 285Lagrangian

coordinate, 274, 276domains, 278, 280

path integralcoordinate, 274, 276

propagatorequal frequency, 331, 333

action, 105acceleration

completeness equation, 286, 288equations of motion, 276, 278

antifermion, 204conjugation, 205Hilbert space, 206state space, 204

Bohr, 7boundary conditions

acceleration, 289, 291, 293

canonical equations, 80circle

Lagrangian, 387, 388, 390classical action

acceleration, 277, 279magnetic field, 268oscillator, 233

classical equation of motiondouble-well, 356, 357, 359

classical solutionacceleration, 275, 277

vacuum, 312, 314instanton, 358, 359, 361oscillator, 231

coherent statespath integral, 99

commutation equationEuclidean, 89Minkowski, 89

completeness equation, 32, 74Jordan block 3×3, 345, 347coherent states, 32, 35, 39, 98coordinate basis, 31dual eigenstate, 324, 326eigenstates, 37fermion, 203fermion–antifermion, 207Ising, 163

magnetic field, 181Jordan block, 336–339Jordan block 2×2 , 342, 344matrix elements, 32momentum basis, 42pseudo-Hermitian, 319, 321

conditionality probabilityoscillator, 239

conservation lawsenergy, 82symmetry, 82

Copenhagen interpretation, 7correlatorS2, 398, 399, 402

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410 Index

correlator (con’t)linear regression, 185

creation operator, 226

degree of freedom, 10S1, 385, 386, 388S2, 390, 391, 393circle, 385, 386, 388compact, 384, 385, 387continuous, 30periodic, 39sphere, 390, 391, 393

density matrixmixed states, 50pure states, 50

destruction operator, 226Dirac

bracket, 90bracket notation, 12constraint

commutation equation, 91, 92two constraints, 91

Dirac bracketsaction

acceleration, 280, 282Dirac delta function, 33Dirac–Feynman formula, 67, 74

continuous paths, 69discrete paths, 69

DNAcorrelation length, 400, 401, 404extension, 398, 399, 402force

extension, 401, 402, 405jointed

chain, 401persistence length, 400, 401, 404statistical mechanics, 397, 398, 401

double-wellkink, 358, 359, 361multi-kink, 361, 362, 364

double-well potentialIsing model, 375, 376, 378

eigenfunctionsevolution kernel, 249

eigenfunctionspseudo-Hermitian

left, 306, 308right, 306, 308

pseudo-Hermitian H , 305, 307pseudo-HermitianH †, 305, 307

evolution kernel, 63S1, 390, 391, 393

large radius, 390, 391, 393indeterminate final position, 256circle, 386, 387, 389constant source, 260double-well

singularity, 360, 361, 363eigenfunctions, 249free particle, 93magnetic field, 267oscillator, 230

Faddeev–Popov analysisinstanton zero mode, 363, 364, 366

Fermi pseudo-potential, 59fermion, 198

antifermioneigenstates, 214

calculus, 198complex, 207

Gaussian, 209Gaussian integration, 207generation function, 209Hamiltonian, 214Hilbert space, 201integration, 200Lagrangian, 217normal ordering, 212path integral, 211real, 207variables, 199

fermion–antifermionconjugation, 206

Feynman path integral, 61, 70evolution kernel, 72see path integral, 61

Fokker–Planckpath integral, 156

free energyoscillator, 243

functional differentiation, 115chain rule, 115

Gaussianpath integrals, 251

Gaussian integration, 129N -variables, 131fermions, 207

Gaussian random variable, 130generating function

oscillator, 234

Hamiltonian, 20, 22, 64acceleration

operator, 285, 287eigenfunctions, 51Euclidean, 84, 85fermionic, 214Fokker–Planck

ground state, 156pseudo-Hermitian, 155

Jordan block, 330, 3322×2 , 342, 344

mechanics, 80oscillator, 226

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Index 411

path integral, 75phase space

quantization, 94pseudo-Hermitian, 155, 295–298

critical, 331, 333eigenfunctions, 304–307equivalent, 297, 299excited states, 313, 315similarity transformation, 296, 298

quadratic momentum, 87quasi-Hermitian, 296, 298rigid rotor, 392, 393, 395self-adjoint

extension, 55Hamiltonian: Fokker–Planck, 151

pseudo-Hermitian, 153harmonic oscillator

forcedcoherent states, 102

coherent states, 101Heisenberg, 7Heisenberg algebra

unitary representation, 49Heisenberg commutation equation, 47Heisenberg equations

pseudo-Hermitian Hamiltonian, 321, 323Hilbert space, 14

fermionic, 201

indeterminatepaths, 23

instanton, 354, 355, 357classical solution, 358, 359, 361coefficient, 367, 368, 370correlation function, 372, 373, 375dilute gas, 373, 374, 376expansion

spectral representation, 363, 364, 366multi-, 369, 370, 372transition amplitude, 370, 371, 373zero mode

double-well, 361, 362, 364Faddeev–Popov analysis, 363, 364, 366

Ising2×N lattice, 176block spin, 191correlator

open chain, 167periodic chain, 169

degree of freedom, 161magnetic field, 180

correlator, 184evolution kernel, 182magnetization, 190partition function, 189transfer matrix, 181

magnetization, 183model, 161

nonlocal, 376, 377, 379partition function, 172path integral, 171periodic lattice, 168renormalization, 191spin, 161state space

binary, 163Ising model, 161

double-well potential, 375, 376, 378magnetic field, 180

Ising spinBloch sphere, 164Heisenberg operator, 167Schrödinger operator, 167

Itochain rule, 137discretization, 135

Ito calculus, 136

Jordan block2×2

Schrödinger equation, 343, 3452×2 , 340, 342completeness, 336–339Hamiltonian, 339, 341propagator, 337, 339, 344, 346–349

kinkdouble-well, 358, 359, 361

Kolomogorov, 127

Lagrangian, 69, 105S1, 387, 388, 390acceleration, 273, 275Euclidean, 84fermionic, 217path integral, 75rigid rotor, 393, 394, 396

Langevin equationlinear, 140nonlinear, 145potential, 143

Laplacian, 391, 392, 394Legendre transformation, 81

magnetic fieldpath integral, 267

measurement, 18momentum

Euclidean, 84momentum basis

path integral, 243multi-kink

double-well, 361, 362, 364multi-instantons, 369, 370, 372

normal ordering, 98normal random variable, 129, 130

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412 Index

normalizationrigid rotor, 393, 394, 396

objective reality, 15, 23, 127operator

Hamiltonian, 43acceleration, 285, 287

momentumdomain, 52

self-adjointdomain, 52, 54

Weyl, 43multiplication, 44shift, 44

operators, 14, 30exponential, 252fermionic, 211position

momentum, 50self-adjoint, 51

oscillatorclassical action, 233classical solution, 231

source, 234conditionality probability, 239eigenstates, 226evolution kernel, 230finite lattice, 241generating function, 234Hamiltonian, 226infinite time, 230normalization, 233, 254simple harmonic, 225transfer matrix

lattice, 246over-complete basis

coherent states, 98fermion, 203

path integralacceleration, 286, 288coherent states, 99continuum limit, 76Euclidean, 86evolution kernel, 73fermionic, 211free particle, 240Gaussian, 251Hamiltonian, 106indeterminate positions, 261Lagrangian, 106Minkowski, 85momentum basis, 243periodic, 253phase space, 85quantization, 105time lattice, 75

pfaffian, 208phase space

path integralacceleration, 283, 285

Poisson bracket, 87Euclidean, 88Jacobi identity, 88

Poisson summation formula, 386, 387, 389polymer

tangent vector, 397, 398, 400potential

acceleration, 274, 276quadratic, 274, 276

delta function, 57double-well

Ising model, 375, 376, 378quartic, 350, 351, 353

probabilityconditional, 63, 128joint, 128marginal, 128

probability amplitude, 10, 24composition rule, 76, 291, 293time evolution, 61

probability theoryclassical, 127

propagatoracceleration

path integral, 279, 281Jordan block, 337, 339, 344, 346–349lattice

oscillator, 245oscillator

finite time, 227pseudo-Hermitian

operators, 322, 324state space, 324, 326

pseudo-Hermitiandual eigenstates, 307, 309

quantum entity, 26definition, 27

quantum mechanicsoperator formulation, 22three formulations, 25

quantum numbers, 46quantum paths

infinite divisibility, 70quantum state, 11quantum superstructure, 8quartic potential, 350, 351, 353

random paths, 142random variable, 127renormalization

recursionIsing, 193

block spinmagnetic field, 195

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Index 413

flowIsing, 194

Ising spin, 191

sample space, 127Schrödinger equation, 19, 64

Jordan block2×2, 343, 345

measurement, 8properties, 21

semi-classical approximation, 351, 352, 354semi-classical expansion

double-well, 362, 363, 365integral, 352, 353, 355

simple harmonic oscillatorsee oscillator, 225

spectral representationinstanton expansion, 363, 364, 366

spherical harmonics, 392, 393, 395spin decimation

Ising, 175spontaneous symmetry breaking, 379, 380, 382state space

basis states, 35continuous degree of freedom, 30degree of freedom, 11pseudo-Hermitian Hamiltonian, 318, 320

state vectorstatistical, 8

state vector collapse, 19stochastic, 125

stochastic quantization, 148stock price, 137

geometric mean, 138superposition

indeterminate paths, 65quantum, 21

symbol, 11symmetry breaking

multiple wells, 382, 383, 385symmetry restoration, 380, 381, 383

ground state, 382, 383, 385

tangent vectorpolymer, 397, 398, 400

tensor productstate space, 17

timeEuclidean, 84

transfer matrixIsing spin, 165oscillator, 246

transition amplitude, 63fermionic, 219

vacuum stateHamiltonian

pseudo-Hermitian, 309, 311

white noise, 132Wilson expansion, 139

Ito calculus, 138

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