IC/92/81

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

QUANTUM GRAVITY, QUANTUM COSMOLOGY

AND LORENTZIAN GEOMETRIES

Giampiero Esposito

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MIRAMARE-TRIESTE

IC/92/81

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUANTUM GRAVITY, QUANTUM COSMOLOGYAND LORENTZIAN GEOMETRIES

Giampiero Esposito

International Centre for Theoretical Physics, Trieste, Italy,

Scuola Intemazionale Superiote di Studi Avanzati, Trieste, Italia,

Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Gruppo IV, Napoli, Italia

and

Dipartimento di Scienze Fisiche, Universita di Napoli, Napoli, Italia.

MIRAMARE - TRIESTE

May 1992

PREFACE

This book is aimed at theoretical and mathematical physicists and mathematicians

interested in modern gravitational physics. I have thus tried to use languages familiar

to readers working on classical and quantum gravity, paying attention both to difficult

calculations and to existence theorems, and discussing in detail the current literature.

The first aim of the book is to describe recent work on the problem of boundary

conditions in one-loop quantum cosmology. The motivation of this research was to under-

stand whether supersymmetric theories are one-loop finite in the presence of boundaries,

with application to the boundary-value problems occurring in quantum cosmology. Indeed,

higher-loop calculations in the absence of boundaries are already available in the litera-

ture, showing that supergravity is not finite. I believe, however, that one-loop calculations

in the presence of boundaries are more fundamental, in that they provide a more direct

check of the inconsistency of supersymmetric quantum cosmology from the perturbative

point of view. It therefore appears that higher-order calculations are not strictly needed,

if the one-loop test already yields negative results. Even though the question is not yet

settled, this research has led to many interesting, new applications of areas of theoretical

and mathematical physics such as twistor theory in flat space, self-adjointness theory, the

generalized Riemann zeta-function, and the theory of boundary counterterms in super-

gravity. I have also compared in detail my work with results by other authors, explaining,

whenever possible, the origin of different results, the limits of my work and the unsolved

problems.

The second aim of the book is to present a recent study of the singularity problem

for space-times with torsion. Indeed, the singularity problem in cosmology and theories

of gravity with torsion play a fundamental role in motivating respectively quantum cos-

mology and supergravity. The reader can thus find the nonperturbative continuation of

the first two parts of my work within the framework of classical theories of gravitation.

It was my intention to write a treatise not too specialized in a single topic, but dealing

with some fundamental problems of both classical and quantum gravity. I hope this will

Preface

stimulate further interaction between these two branches of theoretical and mathematical

physics. In fact, research workers interested in classical gravity are not always aware of the

conceptual and technical problems of quantum gravity, whereas those working on quantum

cosmology do not frequently use the elegant, powerful and rigorous global techniques of

general relativity.

The book is based on the author's J.T. Knight Prize Essay and Ph.D. Thesis at

Cambridge University, plus a paper published in Fortschritte der Physik, a series of papers

written with Dr. Peter D'Eath, other papers of mine on classical gravity and my post-

doctoral work in Napoli and in Trieste. The chapters contain more details than the papers,

and the presentation of the arguments is different. Much work appearing in my manuscript,

however, has not been previously published. Where appropriate, sections on background

material appear because I tried to write this book in self-contained form. I have chosen

to present my results in the order they were derived. In many cases, problems have

been initially formulated in the simplest possible way, and finally presented and solved at

increasing levels of complexity. Quantum cosmology is not an isolated field of research,

but it lies at the very heart of fundamental theoretical physics. I hope the reader will

appreciate this after reading the book.

I am especially indebted to Dr. Peter D'Eath for encouraging, correcting and super-

vising my work on quantum cosmology over many years, to Professors W. Beiglbock, J.

Ehlers and J. Wess of the Springer-Verlag Editorial Board for several suggestions which

led to a substantial improvement of the original manuscript, and to Professor Stephen

Hawking fox inspiring all my work on classical and quantum cosmology.

Special thanks are also due to Professors John Beem, James Hartle, Friedrich Hehl,

Chris Isham, Giuseppe Marmo, Renato Musto, Cesare Reina, Abdus Salam and Dennis

Sciama; Drs. Stewart Dowker, Gary Gibbons, Domenico Giulini, Jonathan Halliwell,

David Hughes, Bernard Kay, Gerard Kennedy, Jorma Louko, Hugh Luckock, Ian Moss,

Richard Pinch, Stephen Poletti and Kristin Schleich; graduate students Giuseppe Bimonte,

Eduardo Ciardiello, Paola Diener and Hugo Morales Tecotl, and undergraduate student

Gabriele Gionti, for enlightening conversations. At last, but not at least, I gratefully

acknowledge lots of help from Professors Ruggiero de Ritis and Giovanni Platania, and

11

Preface

Drs. Paolo Scudellaro, John Stewart and Cosimo Stornaiolo in understanding theories

with torsion and related literature, and I thank Drs. Marcos Bordin, Mark Manning and

Paolo Lo Re for solving computer problems while I typed the manuscript.

Over the past five years my research has been financially supported from St. John's

College, the Istituto Nazionale di Fisica Nucleare and the International Atomic Energy

Agency.

in

HOW TO READ THIS BOOK

Readers interested in a general overview of classical and quantum gravity should study

the first three chapters, all sections from 6.1 to 6.4, all sections from 10.1 to 10.6. They

can thus find the detailed description of the motivations for studying quantum gravity

and quantum cosmology, with achievements and unsolved problems; Dirac's theory of con-

strained Hamiltonian systems and its application to the quantization of general relativity;

the background-field method and the one-loop approximation in perturbative quantum

gravity; the Batalin-Pradkin-Vilkovisky and Faddeev-Popov methods of quantizing gauge

theories; the mathematical foundations of classical general relativity, i.e. Lorentzian geom-

etry, spinor structure, causal structure, asymptotic structure and Hamiltonian structure

of space-time.

Note that the reader may well study Part III before Part I, or at least before chapter

2. However, I found it more satisfactory to write a continuous sequence of chapters on

quantum gravity before Part III. The first three chapters enable one to become familiar

with the basic took of canonical and perturbative quantum gravity (nonexperts may limit

themselves to these chapters and to Part III). It is then possible to understand the one-

loop calculations presented from chapter four to chapter nine. The asymptotic heat kernel

for manifolds with a boundary is first studied in the case of the Dirac operator subject

to global boundary conditions. This is the completion of previous work by theoretical

physicists on the role of fermionic fields in quantum cosmology, and is motivated by the

mathematical study of spectral asymmetry and Riemannian geometry.

However, only local boundary conditions respect supersymmetry. One possible set of

supersymmetric local boundary conditions involves field strengths for spins 1, | and 2,

the undifferentiated spin~| field, and a mixture of Dirichlet and Neumann conditions for

scalar fields. The corresponding one-loop properties, and the relation of these boundary

conditions to twistor theory in flat space, are derived in chapters five, seven and eight.

A detailed proof of self-adjointness of the boundary-value problem for the Dirac operator

with these local boundary conditions is also given in chapter five.

IV

How to Read this Book

An alternative set of boundary conditions can be motivated by studying transforma-

tion properties under local supersymmetry, as in chapter nine; these are in general mixed,

and involve in particular Dirichlet conditions for the perturbed three-metric of pure gravity,

Dirichlet conditions for the transverse modes of the vector potential of elect romagnetism,

a mixture of Dirichlet and Neumann conditions for scalar fields, and local boundary con-

ditions for the spin-1 field and the spin-| potential. Remarkably, the one-loop results for

fermionic fields are equal to the ones obtained using nonlocal boundary conditions (chap-

ters eight and nine). Moreover, no exact cancellation of one-loop divergences is found,

in the presence of boundaries, for simple supergravity and extended supergravity theories

(chapter nine). All one-loop calculations are performed in great detail, so as to enable

graduate students and research workers to learn these techniques.

However, evidence exists that restriction of gauge theories to a set of physical degrees

of freedom leads to different one-loop results with respect to the quantization of the full

theory in Becchi-Rouet-Stora-Tyutin-invariant fashion. This problem is also investigated

in chapter six (section 6.5), in the case of electromagnetism.

New results on the singularity problem for space-times with torsion are finally derived

and discussed in sections 10.7-8, whereas the research results obtained in the whole book

are summarized in chapter eleven.

At the end of Part IV, I have proposed a series of problems for the reader. I encourage

all readers to work very hard on these problems, since this is the best way to make sure

they have learned the techniques and the ideas described in the book.

The only prerequisites are the knowledge of the basic differential geometry described

in chapter two of Hawking and Ellis 1973, and of the path-integral formalism for quantum

field theory at the level of many introductory textbooks in the current literature. I have

omitted the treatment of these topics since I believe they are very well described by other

authors. I have instead focused on quantum gravity, quantum cosmology and Lorentzian

geometries, since there are not many textbooks which study all of them. I hope the

resulting monograph will be useful to a very large audience.

CONTENTS

PART I : QUANTUM GRAVITY

Page1. QUANTUM GRAVITY, QUANTUM COSMOLOGY 1

AND CLASSICAL GRAVITY1.1 Quantum Gravity : Approaches, Achievements 1

and Unsolved Problems1.2 Quantum Cosmology : Motivations and Some 7

Recent Developments1.3 Introduction to Supergravity 111.4 An Outline of This Work 14

2. CANONICAL QUANTUM GRAVITY 252.1 Hamiltonian Methods in Physics .262.2 Dirac's Quantization of First-Class 33

Constrained Hamiltonian Systems

2.3 Dirac's Quantization of Second-Class 35Constrained Hamiltonian Systems

2.4 ADM Formalism and Constraint Algebra 42in Canonical Quantum Gravity

2.5 Mathematical Theory of Wheeler's Superspace 48

3. PERTURBATIVE QUANTUM GRAVITY 523.1 The One-Loop Approximation 54

3.2 Zeta-Function Regularization of Path Integrals 583.3 Gravitational Instantons 603.4 Perturbative Renormalization of Field Theories 70

VI

Contents

PART II : ONE-LOOP QUANTUM COSMOLOGY

4. GLOBAL BOUNDARY CONDITIONS AND C(0) VALUE 76FOR THE MASSLESS SPIN-A FIELD

4.1 PDF One-Loop Results for Pure Gravity 774.2 Mathematical Foundations of Global Boundary Conditions 814.3 How to Deal with First-Order Differential Operators 864.4 Detailed Calculation of the Infinite Sums 894.5 The Heat Kernel and the Prefactor 95

4.6 From Global to Local Boundary Conditions 97

5. CHOICE OF BOUNDARY CONDITIONS IN ONE-LOOP 100QUANTUM COSMOLOGY

5.1 General Form of the Action of the Spin-| Field 1025.2 Hamiltonian Form of the Action and Supersymmetry Constraints 104

5.3 Gauge Condition 106

5.4 Final Form of the Action 1095.5 PDF Prefactor of the Semiclassical Wave Function 109

with Global Boundary Conditions5.6 Basic Results of Twistor Theory in Flat Space 1145.7 Local Boundary Conditions and Spin-Lowering Operators 1165.8 Application of Local Boundary Conditions to the Spin-| Field 1215.9 Spin-1 and Spin-0 Fields 1355.10 Preservation in Time of the Gauge Constraint 147

6. GHOST FIELDS AND GAUGE MODES IN ONE-LOOP 149QUANTUM COSMOLOGY

6.1 Main Ideas about Gauge Theories and their Quantization 1516.2 Extended Phase Space for the Spin-1 Field 1536.3 Formal Equivalence to the Faddeev-Popov Result 1566.4 Physical Degrees of Freedom and Redundant Variables 1576.5 A More Careful Study of the Spin-1 Problem 161

7. LOCAL BOUNDARY CONDITIONS FOR THE WEYL SPINOR 1717.1 Local Boundary Conditions for the Spin-2 Field Strength 1737.2 One Cannot Fix the Linearized Electric Curvature on S3 174

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Contents

7.3 Calculation of the PDF ((0) when the Linearized 183Magnetic Curvature is Vanishing on S3

7.4 Comparison of Different Techniques 193

8. ONE-LOOP RESULTS FOR THE SPIN-f FIELD 197WITH LOCAL BOUNDARY CONDITIONS

8.1 General Structure of the C(0) Calculation for the Spin- 1985 Field Subject to Local Boundary Conditions on S3

8.2 Contribution of W& and w £ } 204

8.3 Effect of w£\ W£} and W& 2058.4 Vanishing Effect of Higher-Order Terms 2068.5

Contents

PART IV : SUMMARY

11. CONCLUSIONS 28111.1 Foundational Issues 281

11.2 Our Results 283

11.3 Our Unsolved Problems 285

PROBLEMS FOR THE READER 288

APPENDICES

Appendix A : Two-Component Spinor Calculus and its Applications 292Appendix B : The Generalized Zeta-Function 299Appendix C : Euler-Maclaurin Formula and Free Part of the 301

Heat Kernel for the Spin-| FieldAppendix D : Complex Manifolds 304Appendix E : Lorentzian ADM Formulae for the Curvature 306Appendix F : £(0) Calculations 308

REFERENCES 313

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