introduction to a realistic quantum physics

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Giuliano Preparata



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Physics

Giuliano Preparata

V f e World Scientific

« • New Jersey London • Singapore •ew Jersey • London • Singapore • Hong Kong



In memoriam



P r e f a c e

It is almost thirty years that I have been teaching university courses in

the f ield that has (almost) total ly absorbed my research act ivi ty: Quantum

Field Theory (QFT) and part icle interact ions, today known as the Stan

dard Model (SM). I have thus had innumerable occasions to observe the

uneasiness , indeed the embarassment of s tudents when making the jump

from Quantum Mechanics (QM) to QFT, the only sensible quantum de

script ion of the relat ivis t ic world, where the number of part icles — quantacannot be kept fixed. Sensible, that is a good word, but can it be really

applied to QM itself? When pressed on this point the s tudents , emerging

from their ins t i tut ional courses on non-relat ivis t ic QM, without exception

showed how uncertain and uneasy their feeling was about a physical theory

which is more than seventy years old, and permeates large sections of mod

ern technology. As to their intellectual attitude toward QM, that is also

without exception "conventionalistic" totally centered on rules and proce

dures , largely based on the C openhag en interp retat ion and i ts subjectivis t icprobabil i ty approach.

I have always regarded (and obvioulsy I have not been alone in this)

this state of affairs as very unsatisfactory, including the fact that a crit

ical debate on such fundamental issues has remained confined to a small

community of "fundamentalists", at the frontiers of physics, metaphysics

and philosophy. I have thus tried to devote a (necessarily small) part of my

lectures to present my point of view, which tr ies to reap pro priate Q ua nt um

Physics (QP) to a s trongly real is t ic view of the world, in the great t radi

t ion from Gali lei to Einstein that has shaped the deep s tructure of modern

science. But without a systematic exposition I felt that my efforts were

doomed to have little or no effect at all . Thus I decided to subtract a part

of my vacation time (summer 1998) to more or less futile beach talks, and

V l l



viii Preface

devote the following Essay to my stude nts (and to whoever cares ab ou t th e

problems addressed in it) who, I hope, will benefit from being presented

with ideas and analyses that are not usually found in the l i terature that isavailable to them.

This Essay is written in English for two main reasons, first this language

is the common language of science, and physics students should know all

too well how useful it is to read (and speak) it fluently. The second reason

is that I nurture the hope that, some day it will be felt desirable to have

access to its ideas in the wide world.

Milano, October 1998.

Giuliano Preparata was born in Padova, Italy on March 10, 1942. The-

oretical physics was the focus o f his insatiable intellectual curiosity. He

succumbed to cancer in Frascati, Italy on April 24, 2000.



C o n t e n t s

Preface vii

1. T he Fads and Fal lacies of Q ua ntu m Mechanics 1

2. Kine m atics: Th e Descriptive Framew ork of Physical Real i ty 11

2.1 Sta tes an d observables in Classical Phy sics (C P) 13

2.2 States an d observables in Q ua ntu m Physics (Q P) 14

2.3 T he impo ssibili ty of a traje cto ry is th e im possibili ty

of a realistic QM 19

2.4 Q uan tum fields are the only reali st ic physical objects . . . . 21

3. Dyn am ics: T he Laws of Evolut ion of Physical Rea l i ty 27

3 .1 The Ham i lton-Lagrange theory of c lass ical dynamics . . . . 27

3.2 T he Ha mil tonian operator: the gene rator of qu an tum

dynamics 31

3.3 T he P a th Integra l (P I): classical trajecto ries a nd

Q uan tum Physics 33

4. Quantum Field Theory: The Only Real ist ic Theory of the

Q uan tum World 39

4.1 A prelim inary discussion of coherent sta tes 40

4.2 T he Vacuum , the tem plate of physical real i ty 45

4.3 T he "classical" l imit of Q F T : the em ergence

of cohe rence 50

4.4 T he "quantum -mec hanical" l imit of Q F T : the

Schro dinger wave -function 57

ix



X Contents

5. Final Considerat ions 63

Appendix 65

Bibl iography 73



Ch ap te r 1

T h e F a d s a n d F a l l a c i e s o f Q u a n t u m

M e c h a n i c s

Q ua ntu m Physics is abo ut one hund red years old, i t represents a new image

of the physical world, that sprang from the irresolvable difficulties in which

classical physics (CP) sank at the end of the XIX century. And yet, and

this is the peculiarity of this (unachieved) scientific revolution, after one

hundred years the basic ideas and achievements of this new and profound

approach to physical reality do not belong at all to the background of our

culture or shape our collective und erstan ding of and exp ectat ions a bo ut theworld.

The reason for this new and s trange s i tuat ion, that did not occur in

other scientific revolutions, like the Copernican Heliocentrism or the Ein-

steinian Relativity, is to my mind due to the peculiar interpretation of its

basic laws and mathematical results , that s ince the 1930's imposed itself,

and is mainly associated with the name of Niels Bohr: the Copenhagen

interpretation. Such point of view suffers from a number of strange and,

in the end, untenable "dogmas" about the s tructure of the physical world.Let 's t ry and identify them.

Firs t of al l , according to the Copenhagen interpretat ion, quantum

physics , or bet ter , Quantum Mechanics (QM) addresses the microscopic

world only, i .e. the world of atoms, molecules and the plethora of particles

that have been the subject of intensive studies in the second part of this

Century. The macroscopic world, the world of our immediate experience,

is instead fully and consistently described by the venerable ideas and laws

of CP. To such world there belong all physical observers, whose measuring

devices have thus the sharpness and the determinism of classical physics. In

this way, according to the Copenhagen interpretat ion, the physical world

becom es intrinsically dualistic: on one side the m icroscopic physical sy stem s

with their chancy motions, no more describable in terms of well defined

1



2 An Introduction to a Realistic Quantum Physics

trajectories, on the other the classical observers with their classical devices,

obtaining information about the microscopic world that can only be statis

tical in character and, more oddly, can in no way represent the reality ofthe microscopic system, for the random, unpredictable interaction with the

observing apparatus is an essential aspect of the physical situation, in which

the observed and the observer are inextricably entangled. And the statis

tical distributions — the square of the wave-functions — which obey the

quantum laws of evolution, do not tell the tale of the observed, but rather

of the knowledge of that tale that the observer may acquire through his

devices. J. A. Wheeler once put this peculiar status of affairs quite vividly

when he stated that the Schrodinger wave-function bears to (the unknow

able) physical reality the same relationship that a weather forecast bears

to the weather. In such way the Copenhagen physicist has given up his

ambition to describe the physical world as is, the fundamental aim of any

realist (like Galileo Galilei and all the g reat classical physicists, A lbe rt Ein

stein included), contenting himself to account for what he may say ab o u t

the world, whose reality remains fundamentally inaccessible. Thus he com

pletely embraces the posit ion of Cardinal Bellarmino, that cost Galilei the

ordeal of the process by the Inquisition an d of its abject con dem natio n. It is

interesting, but far f rom amusing, that with the Copenhagen QM modern

science, born from the intellectual courage and achievements of a group of

"realists" like Galilei, comes full circle to subscribe to the epistemological

theses of "conventionalists" like Bellarmino himself.

The reason for such drastic epistemological choice, that has caused so

much turmoil in the physics of this Century and is responsible for the

"marginalization" of quantum physics from the main cultural trends of our

t ime , is the realisation that particles like the electron may show, in the ex

perim ents of the Davisson-G ermer type, character is t ic wave-like b ehaviours

such as interference, which intermingle two aspects of reality that appear

completely irreducible, that of a particle and its discontinuous behaviour

with that of a wave with its fundamentally continuous character. And it

is in the typical notion of "wave-particle complementarity" of the Copen

hagen "vulgata" that any hope of a realistic, objective interpretation of

QM fades away, leaving in its place a well defined set of rules to "com

p u t e " the statistical distributions of the outcomes of given observations (or

observables) on a statistical ensemble of identical physical systems.

As the aim of this Essay is to show the existence and describe the

structure of a realis t ic approach to quantum physics (which does without

the deeply engrained fallacies of the Copenhagen School, by denying an



The Fads and Fallacies of Quantum Mechanics 3

independent, self consistent status of "bona-fide" physical theory to Quan

tum Mechanics, which only belongs to Quantum Field Theory (QFT)), i t

seems appropriate to give a brief historical account of the early developments of quantum physics, that in the first two decades of our Century

appeared to point in a direction completely different from what came to be

general ly accepted through the rest of the Century.

The bir th of quantum physics is usual ly at t r ibuted to Planck's theory

of the Black Body's radiat ion. The appl icat ion by J. Jeans of the theorem

of energy equiparti t ion to the wave-modes of a classical electromagnetic

field inside a "Hohlraum", i.e. an oven, led to the well known and depre

cated "ul t raviolet catastrophe", according to which the energy densi ty of

the electromagnetic field inside the oven comes out to be infinite, due to

the contributions of the high frequency (v ) modes, whose energy content is

instead experimentally found to be highly suppressed. Planck's solution, in

spectac ular ag reement w ith experim ents, was really revolut ionary in th at i t

assumed that , contrary to Maxwell theory, the energy exchanges between a

given mode of frequency v and the "matter oscillators" of the oven's walls

could not be cont inuous, whence the equipart i t ion theorem, but could only

occur in quantit ies, or packets or quanta of energy e = hu, where h is the

famous Planck's constant {h = 6.626 10~2 7

erg . sec) .

Such "q uant izat ion" of the energy negotiat ions between the electromag

netic (e.m.) field and matter was completely at variance with the continuous

character of the Maxwell field and of i ts energy density, and dramatically

called for a completely new theory of both i ts kinematics and dynamics.

In the meantime, a few years later (1905) Albert Einstein proposed that

in such energy negotiations the main actor is a new strange particle-like

physical object, the "quantum" of the e.m. field, which he baptized "pho

ton" . According to Einstein 's view, when energy (and momentum) get ex

changed between the e.m. field and an atomic system, a given e.m. mode

of frequency u and wave vector k can only exchange energy in quantit ies

which are integer multiples of e = hu , while the mo m entu m m ust be the

same integer multiple of p = hk (H — ^ ) . Other exchanges are s t r ic t ly for

bidden. In this way he could explain the oddity of the photoelectric effect,

in which the abili ty of an electromagnetic radiation to extract electronsfrom a given metal surface depended not on the energy deposited on the

surface but on the frequency of the radiation. The particle-like attributes

of the photon were later (1921) confirmed by experiments on the diffusion

of electromagnet ic radiat ion by a charged part icle, known as the Compton

effect.




To a "realist", such as Einstein has been throughout his l ife, the "pho

tons" , which invariably appear when the e.m. field interacts with matter,

mus t exist in the field, and the field must be made of them. Yes, but how?How can the particle properties of the photon(s) be reconciled with the

wave chara cter of the t ransve rse Maxwell f ield? This was the di lemm a th at

could, perhaps, be resolved in a fully realistic theory of the quantum world

had Einstein's realism been a bit less "naive", and Planck's ideas received

a bit more attention. It is not very well known, in fact, that while the

great advancements of the Black Body's theory were followed by Einstein's

theo ries of th e pho toele ctric effect (1905) and of th e specific h ea t of a solid

(1908), a sharp debate went on between Planck and Einstein upon the physical meaning and propert ies of the quanta and their relat ionship to their

quantum fields.*

According to Planck, Einstein 's at t i tude to at t r ibute the "photons",

with their particle-like properties, a well defined physical reality beyond

the acts of interaction in which they manifest themselves — discontinu

ous "avatars" of a wave-field — was too naive in the sense that in this

way one could only account correctly for one side of reality, the interac

tion field-matter (observer), but totally ignored the other side of reality,namely the dynamical evolution of the field when nobody observes i t , and

no energy-momentum negotiat ion takes place with other (matter) f ields.

An d th e wave-like asp ects of the la tter , such as diffraction an d interference,

bear witness to the fact that the photons cannot be the whole story, for

one side of reality cannot contradict another: the real world can't be but

one! Planck's real ism was thus more sophist icated than Einstein 's in that i t

suggested the idea that the quantum field was not a well defined collection

of "photons", a picture that suffices to account for the thermodynamics

of the Black Body, but a coherent physical system defined in all space for

all t imes (much as classical physics describes i t) , whose kinematics and

dynamics, however, must be redefined so as to incorporate, under appro

priate condit ions, the appearance of photons. Of course at the t ime ( the

first decade of this Century) Max Planck did not know how to realize his

ideas, but the real remarkable aspect of his realism is that i t suggested in

an unerring manner the way to avoid the type of nonsense that tragically

convinced most of the physicists of this Century to give up the glorious

tradition of realism, which led to modern science.

*For in the theory of the specif ic heat of a sol id Eins te in describes the a toms of a

sol id by a con t inuo us aco us t ic f ie ld , whose "qu ant iza t i on " d la Planck produces a new

t y p e of q u a n t a , t h e " p h o n o n s " .



The Fads and Fallacies of Quantum Mechanics 5

We may now, perhaps, begin to understand how the Copenhagen inter

pretation came to be perceived, at the end of the 1920's, as the only way

"to make sense of the nonsensical". Soon after World War I (1923) Louisde Broglie proposed to describe in a wave-like way any (microscopic) piece

of matter , thus turning upside-down Planck's (and Einstein 's) approach.

Now the classical physical object, the particle, gets somehow endowed with

a wave-like behaviour much as the quantum object , the photon, der ives

such properties from the classical physical system it belongs to, the e.m.

field. W hereas , however, in th e qu an tum f ield app roach of Plan ck a nd Ein

stein the field aspect is essential (and remains so after "quantization") in

de Broglie's idea th e field aspe ct is tota lly extrinsic, it is ju st an ab str ac t

me ntal construction grafted u pon a very concrete physical object , Ne wto n's

material point. Whereas for the quantum field it was the reality of the no

tion of photon that had to be assessed and understood in the light of the

reality of the field, in de Broglie's quantum mechanics it is the reality of

the matter point that ostensibly clashes with that of i ts "associated wave",

leading inevitably to the Copenhagen's brand of conventionalism.

But the spectacular successes in the realm of atomic physics of the

newborn QM of Schrodinger and Heisenberg, that soon followed (1925)

de Broglie 's proposal , were judged as the best demonstration of the use-

fulness and adequacy of QM to account for the innumerable observations,

that since a few decades challenged any understanding based on classical

physics. Even though, it was clear, de Broglie waves had nothing to do with

real space—time physical processes, but could only be used to calculate th e

stat is t ical dis tr ibution s of the ou tcomes of any observation on the par t icu lar

microscopic system.

One may now see the reasons which steered physics into a new, un

savoury course that severed all ties with the great intellectual tradition of

realism that marks the development of modern science. Famous is the de

bate between Albert Einstein and Niels Bohr* at the Solvay Conference of

1927, in which the irreducible realist (Einstein) tr ied to oppose with all his

intellectual might the new conventionalistic (Bohr 's) , and basically sceptic,

approach that the "stars" of QM (Bohr, Heisenberg, Schrodinger , Born,

Pauli and Jordan) were then boldly shaping. To Einstein 's objections, that

were all based o n his philosophical choices, Bo hr could oppo se a rem arka bly

effective logical system that, once its odd postulates were accepted, would

TFor an account by the pro tago nis ts of w hat ha pp ene d du r ing the Solvay Conference

see Schi lpp , [Schi lpp (1957)] .




sh o w n o g a p n o r w e a k n e s s . A n d t o a s c i e n t i f i c c o m m u n i t y , w h o se i n t e r e s t s

i n t h e a t o m i s t i c a s p e c t s o f t h e p h y s i c a l w o r l d w e r e d r a m a t i c a l l y r i s i n g , t h e

s m o o t h w o r k i n g of C o p e n h a g e n ' s Q M w a s m o r e t h a n e n o u g h t o a t t r i b u t et h e v i c t o r y t o B o h r , a n d p u s h E i n s t e i n o u t of t h e q u a n t u m p h y s i c s ' s t a g e ,

a n d t o s t o p t o li s t e n t o h i s o b j e c t i o n s , t h a t b e c a m e p a r t i c u l a r l y s h a r p i n

t h e p a p e r ( 1 9 3 5 ) w r i t t e n i n c o l l a b o r a t i o n w i t h R o s e n a n d P o d o l s k y ( w h i c h

e x p o s e s t h e s o c a l le d E P R p a r a d o x ) .

I n t h i s w a y t h e t r u l y su cc e s s fu l r e v o l u t i o n o f o u r C e n t u r y , t h e a t o m i s t i c

r e v o l u t i o n , a p p e a r s t o h a v e p r e v e n t e d t h e fu ll d e v e l o p m e n t o f q u a n t u m

p h y s i c s . E i n s t e i n , o f c o u r se , w a s t o t a l l y j u s t i f i e d i n h i s d e e p u n h a p p i n e s s

a b o u t B o h r ' s e x t r e m e e p i s t e m o l o g i c a l p o s i t i o n , b u t t h e r e e x i s t e d n o w a y

to cha l l enge i t i n pure ly logical t e r m s , n o m o r e t h a n B e l l a r m i n o ' s a r g u

m e n t s a g a i n s t Co p e r n i c u s c o u l d b e l o g i c a l l y d i sp r o v e d b y G a l i l e i . H o w e v e r

G a l i l e i ' s r e a l i sm p r o v e d t h e n a b e t t e r g u i d e fo r t h e p u r su i t of s c ie n t if ic t r u t h

t h a n t h e l o gi c al ly u n a t t a c k a b l e c o n v e n t i o n a l i s ti c s u b t l e t i e s of t h e C a r d i n a l .

T h e q u e s t i o n n o w i s w h e t h e r E i n s t e i n ' s r e a l i sm c o u l d n o t b e a s su c c e s sf u l

a g u i d e i n h i s s t r u g g l e a g a i n s t B o h r ' s t r i u m p h a n t c o n v e n t i o n a l i s m . T h e a n

swer is , un for tun a te ly , no .* Surp r i s ing ly , in E ins te i n ' s que s t fo r a r ea l i s t i c

a p p r o a c h t o q u a n t u m p h y s i c s , h e s e e m s t o h a v e f o r g o t t e n t h e t h e m e s o f

h i s y o u t h a n d , i n p a r t i c u l a r , h i s d e b a t e w i t h M a x P l a n c k , a l l f o c u s se d o n

t h e field a sp e c t s of t h e q u a n t u m w o r l d . A n d i n s t e a d of g o i n g b a c k t o t h e

m a i n f ie ld t h e o r e t i c a l a sp e c t s of e a r l y q u a n t u m p h y s i c s , a n d t o su b j e c t t o a

d e e p a n a l y s i s t h e r e l a t i o n sh i p s b e t w e e n f i e l d s a n d q u a n t a , i n s e a r c h o f n e w

( n o n - c l a s s i c a l ) p r i n c i p l e s a n d l a w s g o v e r n i n g p h y s i c a l r e a l i t y , h e b e l i e v e d

t h a t t h e o d d s t a t i s t i c a l s t r u c t u r e o f Q M w a s b u t a m a n i f e s t a t i o n o f t h e f a ct

t h a t Q M is a n incomplete t h e o r y o f t h e q u a n t u m p h e n o m e n a , a ri s i n g f ro m

a ye t to be d i scovered averag ing over hidden v a r i a b l e s , w h o s e n a t u r e h a d

t o b e r e s e a r c h e d a n d u n d e r s t o o d . W e n o w k n o w t h a t E i n s t e i n ' s r e s e a r c h

p r o g r a m m e ( w h i c h , h o w e v e r , d i d n o t a r o u s e m u c h in t e r e s t ) w a s d o o m e d t o

f a i lu r e f or , a s J . Be l l ( 1 9 6 4 ) sh o w e d , a n y c o m p l e t i o n of Q M t h r o u g h h i d d e n

v a r i a b l e s c a n b e e x p e r i m e n t a l l y t e s t e d , y i e l d i n g r e su l t s a t v a r i a n c e w i t h t h e

p r e d i c t i o n s o f Q M. A n d t h e e x p e r i m e n t a l e v i d e n c e ( A sp e c t , 1 9 8 0 ) i s d e f i

n i t e ly in f avour o f QM, in sp i t e o f the imposs ib le d i f f i cu l ty o f under s t and ing

t h e n o n - v a n i s h i n g particle c o r r e l a t i o n s a t sp a c e - l i k e d i s t a n c e s , i m p l i e d b y

t h e q u a n t u m m e c h a n i c a l d e s c r i p t i o n o f s u c h e x p e r i m e n t s .

I t t h u s a p p e a r s q u i t e p l a u s i b l e t h a t t h e k e e n i n t e r e s t t h a t h a s su r

r o u n d e d p a r t i c l e p h y s i c s d u r i n g t h i s C e n t u r y h a s g r e a t l y c o n t r i b u t e d t o

*And this, I contend, is the reason of the utter failure of Einstein's struggle.



The Fads and Fallacies of Quantum Mecha nics 7

place the particle at the centre of theoretical investigation, endowed with a

basical ly autonomous kinematics and dynamics, in contrast to the complete

depend ence of the dyn amics of the q ua ntu m of early qua ntu m physics fromthat of its field. And in fact the theory of the quantization of a classical

field, accomplished at the end of the 1920's, came to be known as "second

quantization", to distinguish it from the quantization of a system with a

finite number of degrees of freedom, i.e. QM. It is again the centrality of

the notion of isolated part icle in the physical thought of our t imes that , to

my mind, has prevented an in-depth crit ical analysis of the relationships

between QM and the "second quant izat ion", or Quantum Field Theory

(QFT). For to the mind that embraces a wider physical realm the Copen

hagen epistemology is not only hard to swallow but i t is also at odds with

physical observations.

Let 's begin with the idea that quantum behaviour has to do with the

microscopic world only. T he well-known collective ph eno m en a of Ferrom ag-

netism, Superconductivity and Superfluidity, whose classical impossibili ty

is easy to demonstrate,§ bear witness to the fact that there exist macro

scopic pieces of matter whose behaviour cannot be described by classical

physics. If not classical, what is then the relevant physics?

Most contem porary physicists have no dou bt th at Q F T m ust be involved

in the sti l l mysterious^ workings of these fascinating phenomena, but, be

yond a few phenomenological at tempts, such as the Landau-Ginsburg ap

proach, no real headway has ever been made into this kind of physics. And,

I contend, the fallacious philosophy of QM is largely responsible for this un

fortunate state of affairs. Conversely, couldn't the situation improve, indeed

change drastically if we were to realize the centrali ty of QFT and find thatQM is but some kind of approximation of QFT in a well defined, l imiting

physical situation? This is precisely what this Essay proposes to show. But

let 's go on.

We have just seen that i t is thus not t ru e tha t , as the Cop enha gen

physicists claim, the quantum world is the microscopic world: there is a

peculiar macroscopic system, whose behaviour can only be accounted for

by quantum physics. Let 's now address the other basic assumption of QM,

§The c lass ica l imposs ibi l i ty of Ferromagnet ism is nice ly expla ined in Feynman's Lec

t u r e s , [Feynman et. al (1965)] .

^N a tura l ly , th i s re fers to the genera l ly accep ted theory o f condensed m a t t e r phys ics ,

based on e lec t ros ta t i c in te rac t ions . For the succes s o f a new approach to condensed

m a t t e r , b a s e d o n Q u a n t u m E l e c t ro D y n a m i c s s ee G i u l i a n o P re p a r a t a , Q E D C o h e re n c e

in Matter (World Scient i f ic , S ingapore , 1995) .




namely the dichotomy of the world in microscopic quantum systems and

macroscopic classical observers. I wish to show the fallacy of this Copen

hagen's "tenet" by simply recalling that classical physics is in fact logicallyinconsistent, and can thus in no way provide the theoretical description of

even a part of reality, that of the observer.

The inconsistency I am referring to is "the entropy crisis", a notable

example of which is the "ultraviolet catastrophe", whose resolution led to

Planck's discovery of quantum physics. The fundamental works in the sec

ond half of the XIX century of J. C. Maxwell and L. Boltzmann on the

statistical theory of the perfect gas exposed the fundamental connection

between the statistics of the configurations in classical phase space of alarge ensemble of (almost) non-interacting matter points and their thermo-

dynamical behaviour. In particular Clausius' entropy was shown by Boltz

mann to be proportional (by the proportionali ty constant that r ightly bears

his name) to the logarithm of the number of different configurations that

correspond to a thermodynamical (equilibrium) state; defined by the ther-

modynamical var iables: temperature, volume, and other possible macro

scopic labe ls.

The wonderful achievement of directly relating the great laws of Thermodynamics to the structure and the combinatorics of the classical phase-

space points — the system's configurations — hid, however, a cadeau em-

poisonne that emerged when the l imit of zero (absolute) temperature came

to be studied in detail . T he system atic analysis th at W alter Nerns t [Nernst

()] conducted across the XIX and XX Centuries on the low temperature

behaviour of a large number of thermodynamic systems, that finally led

to his celebrated "Heat theorem" (known today as the third principle of

Thermodynamics) , could exclude that nature exhibits the entropy singularity that for T —> 0 classical s tatis t ical mechanics inevitably predicts .

The reason of this singularity is quite simple: at zero (absolute) tempera

ture there exists only one possible configuration, where all particles are at

rest in their equilibrium positions, hence the entropy vanishes; but if we

change the temperature by a small but f inite amount, the number of clas

sical configurations corresponding to such thermodynamical states increase

by an infinite amount, thus clearly exposing an unregularizable singularity.

The lack of "sufficient reason" for such zero-temperature entropy singularity excludes classical statistical mechanics, and with it classical physics

altogether, from the realm of possible descriptions of the physical world.

And this is evidently very bad news for the Copenhagen's view, which sees

its classical world of observers lose any meaning when the temperature



The Fads and Fallacies of Quantum Mechan ics 9

becomes sufficiently low; a paradox that unfortunately is little considered

and understood by the quantum mechanicians of our t imes.

It is amusing that at the same time (mid 1920's), when the formalismof QM was being laid out and i ts conventionalis t ic interpretation tuned

u p , Einstein together with the young Indian physicist S. N. Bose [Einstein

(1924)] succeeded in finally showing how the quantum "perfect" gas could

avoid the "entropy crisis" of the classical system, going through a phase

transit ion, the Bose-Einstein condensation, which for T —> 0 brought with

continuity all atoms to the unique, discrete ground state, and their entropy

to vanish. And in order to accomplish this they had to revisit the early quan

tum theory of Planck and Einstein, where the f ield concept is pr imary and

the qu an ta, the particles, jus t s tem from th e peculiar interactions between

different fields, having thus no independent, autonomous reality, which is

totally engendered in the new dynamics of the quantum field.

Had more attention been paid by the physicists ' community to such

developments and their deep meaning, one may legit imately dream that

the story would have been quite different, and this Essay could have been

writ ten a long t ime ago.



Ch ap te r 2

K i n e m a t i c s : T h e D e s c r ip t i v e

F r a m e w o r k of P h y s i c a l R e a l i t y

KCvrjiia, Kivr\iia.T<x, in the Greek language means motion. Kinematics in

m ode rn physics denotes the science of mo tion, i .e . the m ath em atica l descr ip

t ion of the correlat ions between the space domains spanned by the generic

physical system and time. Thus central for its development is a suitable

mathematical theory of space and t ime, which Bernhard Riemann (1854)

has taught us to be deeply connected to and influenced by the physical

reality one wishes to describe.

In order to keep our discussion as general as possible by space we mean

a three-dimensional manifold whose points can be put in a one-to-one cor

respondence with the points (coordinates) x € R3, the three-dimensional

continuum of real numbers. Any such correspondence, as we know well,

defines a particular observer and the principle(s) of relativity can be for

mulated in general by specifying the allowed class of observers and their

relationships, i .e. the coordinate transformations among different observers.

As for time, it is described by a one-dimensional continuum of real numbers,

which depends in general on the observer , so that in the coordinate trans

formations from one observer to another the time coordinates can in general

change: only in the Gali leo-Newtonian Physics t ime is total ly independent

of the observer, i .e. there exists an "absolute" time.

A physical system can then be defined by first identifying a space-region

(with respect to any allowed observer) which differs from what is perceived

as empty space — the Vacuum — and where some observable properties

can be identified and given numerical values. In classical physics the sim

plest of physical systems is the matter-point , whose space-domain reduces

to a single point x and its defining physical property is the mass, a positive

number depending on the chosen units. A complete set of physical observ-

ables for the matter-point is the quadruple (x,m) t h a t at the given time

11



12 An Introduction to a Realistic Quan tum Physics

exhaustively describes the configuration of this physical object. However

such quadruple is insufficient to describe the state of motion of the matter-

point, for, in order to do this, we need to specify the correlations betweenthe matter-point 's configurations at different times t, and in part icular be

tween two infinitesimaHy close tim es. It would appe ar th at to "label" th e

state of motion of the matter-point one needs to assign (at a given time)

x, p = m i (the m om entum ) and a num ber of higher t ime-derivatives of x,

but dynamics, i .e. the equations of motion, shows that all t ime-derivatives

of order higher than one are well defined functions of x a n d p: thus any

state of motion of the matter-point can be uniquely represented by a point

of the six-dimensional classical phase-space (x, p), which with the flowingof time describes a well defined trajectory (x(t), p(t) = mx (t)) .

By "glueing" together with appropriate "internal forces" a number n

of such points we may construct physical objects of increasing complexity

whose phase-space dimensionality is 6n, and the basic physical observables

are Xi,p% (i = 1,2,... ,n). But the discrete systems of Newton's Principia

do not exhaust the domain of physical phenomena, in classical physics a

great deal of attention has been paid to continuous systems, the fields,

whose space-domain is continuous and the number of degrees of freedom aswell as of the dimensions of phase-space (PS) is infinite. Up to the great

Maxwellian synthesis of electric and magnetic phenomena into electromag-

netism, the field concept could be looked at as a convenient approximation

of a discrete system (a collection of atoms and molecules) with a very large

number (of the order of the Avogadro number, M = 6.02 102 3

) of matter-

points, and this was precisely the point of view of Maxwell-Boltzmann

stat is t ical mechanics . But with the theory of electromagnetism and the

utter failure of a mechanical description based on a hypothetical Aetherthere emerges in physics a new object, irreducible to any "mechanistic"*

description, defined in all space for all t imes, a truly all-encompassing phys

ical entity, which affects electrically charged matter where its intensity is

perceptibly different from zero. With Einstein's General Relativity (1916)

classical physics gets enriched with another genuine field, the gravitational

field, which mutatis mutandis bears to the masses of the universe the

same relation that the e.m. field bears to the electric charges. Thus, just

when quantum physics (QP) was moving i ts f irs t uncertain s teps towarda new image of the world, classical physics (CP) had reached its zenith,

"H ere , as in com mo n par lance , "m echanist ic" den otes any desc r ip t ion th a t t r ies to

reduce a g iven physica l system to the juxtaposi t ion of a ( la rge) number of Newton ' s

m a t t e r p o i n t s .



Kinematics 13

delivering us a fully dychotomic world where matter is described by the

relativistic generalization of the discrete (and discontinuous) Newton's me

chanics and the fields represent that kind of continuous "alterations" of the

physical Vacuum that account for the dynamical evolution of electrically

charged and massive matter. This kinematical separation between matter

(the world of "little balls") and fields (the "stressed state" of the Vacuum)

and, therefore, the irreducibility of the motion of particles belonging to the

former, and of the propagation of waves characterizing the latter, is perhaps

the most significant feature of the picture of the world painted by classical

physics, one that is deeply rooted in our immediate sensory experience.

Quantum physics has totally subverted all this, and it is thus quite

remarkable that the classical distinction between matter (physical systems

with a finite number of degrees of freedom) and fields has survived in the

distinction between QM and the "second quantized" systems, i.e. QFT. As

already anticipated, in the following I shall argue that it is just this act of

hybris, which has been haunting quantum physics for almost a century, that

must be finally repaired, dropping once and for all the distinction between

matter and field, and recognizing that in quantum physics there exists only

one consistent type of physical object: the quantum field.

2.1 States and observables in Classical Physics (CP)

Any physical system, that we think of at a given time t, is, i.e. it exists in

a particular state. Our preliminary discussion has already indicated what

a state of a classical system consists of: it is just a point of the classicalphase-space (PS) whose dimension is 2 / , if / is the number of (Lagrangian)

degrees of freedom that are necessary to describe its configuration. Thus

the classical PS is the space of vectors of components q%,Pi (i = 1,2, . . . , / ) .

For instance, for the isolated point the PS is six-dimensional, comprising

the vectors (x,p), while for a rigid body it is 12-dimensional, comprising

(XCM , 4>,Q,i>iPc'M,L), when XQM and PCM are the center of mass coor

dinates and momentum respectively, <f>,6,ip the Euler angles and L the

components of angular momentum. For a field we can also define a classicalPS , which is now oo-dimensional,comprising the values of the field and of

its conjugate momentum (see later) at any space point.

As for the physical observables, they are represented by all sufficiently

well behaved functions 0(<ft,Pi), which at a given time t assume a well

defined real numerical value, once the totality of q%{t) and Pi(t) have been




determined. Thus in classical physics there is no basic distinction between

the s tate of a system at th e t ime t an d its class of physical observables at the

same time: for the state is uniquely determined by the values of the q^s andPi's, which thus constitute a complete set of observables, in the sense that

any other observable can be unambiguously ob tained from those values. T he

possibili ty of uniquely associating to a state of a classical system a point

of the classical PS clearly stems from the fundamental hypothesis that the

process of measuring the q^s and pi's has no perturbing effect whatsoever

on the system, and the experimental indeterminacy can be made, with due

care, arbi t rar i ly sm all. Th us we may conclude th at in C P between state s

and observables there is basically no distinction, for a state, a point ofclassical phase-space (PS), is uniquely determined by the values that the

observables qi's and p^s at ta in in the s ta te .

2 .2 S t a t e s a n d o b s e r v a b l e s i n Q u a n t u m P h y s i c s ( Q P )

The main point of departure between CP and QP is that in the quantum

world the re is no unlimited observability. As a result , states and observablesare no more in a one-to-one correspondence: the process of measurement

of the basic kinematical variables, the q^s and p^s of classical PS, loses its

classical neutrality to become a fundamental element in the determinat ion

of the phenomenic properties of a given physical object.

Without going through the chain of logical arguments and the set of

experim ental facts th at have led to the discovery of the m athe m atica l st ru c

ture of QP,t i t suffices to recall that this latter postulates that the states of

a quantum system belong to a complex vector space (Hilbert space) whilethe observables are just He rmit ian op erators in such space. Thu s in Q P a

measurement process is represented by the acting on the state vector of the

system of the Hermit ian operator corresponding to the physical observable

being measured. Only if the state vector happens to be an eigenvector of

the observable O, i.e. if*

O|o) = o\o), (2.2.1)

with o a real number, the measurement of O yields the well defined value

o, belonging to its spectrum, i.e the set of all possible values that O can

tFo r th i s a ve ry good book to consu l t s t i ll is P .A.M . Di rac , The P r in c ip les of Q ua n t um

Mechanics [Dirac (1958)] .

* Th rou gh ou t th i s Es say we sha l l ado p t Di rac ' s no ta t io n , which has becom e s t an da rd .



Kinematics 15

assume. Formally this is obtained if

• the sta te vectors are norm alized, i.e.

<V#> = 1

• the value of O, technically the "expectation value" (O),/ , , in the

s tate \ip) is given by the "matrix-element"

( O ) ^ = ( V W > • (2.2.2)

However if, in general, the state vector \tp) is not an eingenvector of O,

the measurement of O will not give a sharply predictable outcome, forthe appl icat ion of the operator O to \ip) discont inuously changes the state

vector into an eigenstate \o n) of O, with eingenvalue on. It is here that our

understanding (intuition) of the quantum world is put to a severe test: while

we can easily understand (2.2.1), that attributes O a well defined value o,

what does (2.2.2) real ly mean? We know from the general mathematical

theory of l inear, Hermitian operators in Hilbert spaces that l^) can be

uniquely decomposed as the l inear superposition:

\^) = Y,c>n), (2.2.3)

n

whe re{ | on )} is a complete orthonormal set of eigenvectors of O, and cn are

complex numbers. Then from (2.2.2) we obtain:

(O)^, = 2_, \cn\

2on I /_ , | c n |

2= l , from the state no rma lizat ion J , (2.2.4)

n \ n /

for which we m ay adv ance the following in terp reta tion : |c „|2

is the probabi l

i ty that the measurement of O reduces the state-vector l^) to \o n), yielding

the outcome on. As a result the "expec tat ion value" (O )^ is jus t the sta

tist ical average of the outcomes of the measurements of the observable O

upon an appropriate ensemble of ident ical copies of the quantum system,

all in the state \ij)).

We may now clearly see that allowing the measuring process to per

turb the state of the system has produced a profound metamorphosis inour picture of the world: the complete determinacy of the observables, the

sharp ness of the values they assum e in a given stat e of CP , gives way now to

quantum observat ions whose outcomes can only be predicted statistically,

th e st a te ve ctor IV'), th ro ug h its coefficients c „, being only a ble to tell us

the probabi l i ty ]c n |2

of the outcome on. Very odd, isn't i t? But the evidence




that such is the real architecture of the world is so overwhelming that we

are well advised to get r id at once of our (classical) intuitio n a nd prejudices,

and to second within ourselves the cogency and the majesty of the imageof the Universe that QP has finally disclosed us.

Another dramatic consequence of the "limited observability" of QP is

that classical PS loses all meaning in the quantum world. For convenience

let 's consider a system with only one degree of freedom, a particle moving

in one dimension; its basic classical observables are its position x and i ts

m om e n tum p. Its generic sta te is a point in the tw o-dimen sional classical P S

(x,p). x a n d p, which we shall now denote X and P to remind us that they

are not real numbers (historically referred to as c-numbers) but operators(or q-numbers , in the same terminology) , are also quantum observables ,

whose spectra may or may not coincide with the classical values, which

belong to the open real line. Such coincidence occurs if the particle motion

is unconstrained, i.e. is allowed to wander over the full real line (—00, +00).

According to quantum kinematics the generic s tate can be writ ten:

\rp)= Jdxip{x)\x) (2.2.5)

where \x) is a complete set of eigenvectors of X, orthonormalised in the

cont inuum as (<5 is the Dirac ^-function) ,

(x'\x)=S(x-x'), (2.2.6)

which thus provide a basis for the Hilbert space of the physical states. If

x is the outcome of the measurement of the observable X, what can we

say about observing P? From the operator nature of X and P it is clear

that P can have a well defined value on the vector \x ) only if this is alsoan eingenvector of P , and this can only ha pp en if X and P com m ute, i .e.

if [X, P ] = X P — P X = 0. Bu t this can never hap pen , for a fundam ental

quan tum pos tu la te 5 demands tha t (i is the imaginary unit , i2

= - 1 )

[ X , P ] = i f t , ( 2 . 2 . 7 )

(h = ^ , h is Planck's constant) which i l luminates the pivotal role the

Planck constant plays in determining the s tructure of the Hilber t space

of s tates an d of the canonical observables X and P . Indeed, due to the

§The ge ne ra l fo rmu la t ion of t h i s pos tu l a t e i s t h a t i n a ny La g ra ng ia n sys t e m wi th /

degrees of freedom Qf, call ing Pf t he c on juga te mome n ta , i . e . Pf = •§£-, t h e q u a n t u m

obse rva b les a ssoc ia t e d to the m obe y the fo llowing c om m uta t io n e qua t ion s [Q f . Q f ] =

[ P f . P f ] = 0 a nd [ Q f , P f / ] = ihS ffi.



Kinematics 17

non -com m utativity of X and P , the value of the latte r on the eigenvector

\x ) of the former turns out to be totally unpredictable, in the sense that

any outcome p is equiprobable. This is a special case of the celebratedHeisenberg principle (1926), which can be thus derived from the "canonical

co m m uta tion relation " (CC R) (2.2.7). Le t's define th e "dispersio ns" of th e

canonical variables on the generic state \i[>)

(Axf = «V|X 2 |V> - (V'lXIV)2) = <V|(X - z)2|V> = <V>|U2|V> (2.2.8a)

(Ap)2

= ( M P2|V> - (V'IPIV')

2) = M ( P - P ) V > = (V -|V

2|V) (2.2.8b)

where by x and p we have denoted (iplXlip) and (V'IPIV')) and by U and V

the operators (X — x) and (P — p) respectively. From the CCR we have

(V»|[U,V]|V> = tft (2.2.9 )

which, inserting a complete basis ^ n | n ) ( n | be tw een the ope r a to r s and

writ ing {n\\J\ip) = a n e ' * n and {n\V\tp) = /3„e1^", gives

J2 « r A sin(V»„

-<t>n)

= ^ (2.2.10)

leading to the inequality,

^ an(3 n > ^2a^n sin (ipn - 4>n) = - . (2.2.11)

Noting that inserting the same complete set of states in (2.2.8) yields

( A z )2

= J >2

, (2.2.12a)

n

and

( A p )2

= £ / 32

, (2.2.12b)n

only simple geometrical considerations (the Schwarz inequality) are needed

to derive from (2.2.1) the Heisenberg inequality:

(Ax) (Ap) > \ (2.2.13)

Thus if Arc = 0, as it happens when \ip) = \x), an eigenstate of X, (2.2.3)

requires th at Ap —• oo, i.e. th e outco m e of a m easu rem ent of P is com pletely

uncertain. The equiprobabili ty of any outcome can be fur ther assessed by

working out the unitary transformation from the (complete, or thonormal)



18 An Introduction to a Realistic Qua ntum Physics

basis of the eigenstates \x) of X to the bas is of the eige nstate s of P . Calling

\p) the eigenvector of P with eigenvalue p, we have

\p) = Jdx\x)(x\p), (2.2.14)

and sandwiching the CCR (2.2.7) between (x \ and \p) we easily get:

dx'(x\P\x')(x'\p)x' = (-ih + px){x\p), (2.2.15)

whose immediate consequences are:

(x\P\x') = -ih—6(x - x1) (2.2.16)

ox

/ <

and

whose solution is

-ih—(x\p) = p(x\p), (2.2.17)

(zip) = 4 = ^ = (2.2.18)

where the normalizat ion stems from the orthonormali ty of the \x) basis:

(x\x') = J dp(x\p) (p\x') = 6{x-x'). (2.2.19)

Thus the probabi l i ty (densi ty) of the outcome p in \x) is uniform, being

given by

MP)? = ^ (2-2.20)

in full agreement with Heisenberg's principle.

T h e kinematical discussion of Q P c arried in this section ma y seem excessively simplified and sketchy, but actually is quite complete, for its general

ization to systems of any number of degrees of freedom, including quantum

fields, encounters absolutely no new concept , only mathematical compli

cations that can in no way represent a major reason of concern for the

realist .



Kinematics 19

2 . 3 T he i m po ss i b i l i t y o f a t ra j e c t o r y i s t h e i m po ss i b i l i t y o f

a rea l i s t i c QM

In C P the notion of a m atter-p oint is inseparable from t ha t of i ts trajec tory

x = x(t), a continuous spatial curve that the point describes as time flows.

And the value of the momentum at each instant measures the "quantity"

of such motion, i.e. how fast or slow the trajectory is spanned. Also, the

continuity of the trajectory allows us to identify the par t icu la r m at te r -

point we are actually observing and to tell it apart from any other point

that is moving in the vicinity of its trajectory. In other words, its degrees

of freedom are unambiguously defined, and more importantly, observable.Thus the reality of the classical mechanics of a system with a finite number

of degrees of freedom (finite set of matter points) is strongly rooted both

in i ts theoretical descr iption — the trajectory — and in i ts experimental

observation.

In QM the si tuation is drastically and dramatically different. The im

possibil i ty of attr ibuting simultaneously well defined values to the coordi

nate X and the momentum P implies that in principle no t ra jectory can

be assigned to the quantum particle. To remain in the simplest possible

framework of a single degree of freedom, we can write for the generic state,

as we have seen,

|V>) = / dxip(x)\x)

where

ip{x) = (x\ip),

the projection of the vector \tp) on the eigenstate|a;) of X, is called "wave-

function". According to the preceding discussion, at a given time ip(x) is

a^

that QM allows us to know about the state in which the one-dimensional

particle happens to f ind itself, and IVK^OI2

is the probabili ty (density) th at

a measur ing apparatus located a t x has to find an object that, like the clas

sical particle, is pointlike. In particular, if \ip) i s the momentum eigensta te

\p) the wave-function is given by (2.2.18), and it is thus completely delo-calized, like a plane-wave of the electromagnetic field. It is here that our

perception of the reality of the physical system we are interested in gets into

an irresolvable crisis: on one han d t he system is in the s ta te \p), whose space

s t ruc tu re (x\p) (see Eq. (2.2.18)) is highly suggestive of a wave-like charac

ter, on the other when we explore the space region where such system is we

(2.3.1)




find it endowed with the characters of a pointlike particle. There should be

no doubt th at bo th aspe cts pertain to th e realm of physical reali ty, the real

trouble is that they happen to be irreconcilable, i .e. they cannot belong tothe same physical object, whose reality and identity should persist through

the measuring process. And this is precisely the logical difficulty that the

Copenhagen's view resolves by flatly rejecting the constraints of realism,

and inventing the surprising notion of "wave—particle complementarity"

(N. Bohr) , by which the quantum mechanical part icle behaves sometimes

as a particle and sometimes as a wave, depending on the kind of measure

ments we perform upon i t . In this way the answer to the quest ion what

really is the quantum particle is pushed forever outside the reach of QM,whose task is restricted to the computations of the statistical predictions

of all different types of observations.

But as Einstein, Rosen and Podolsky (EPR) [Einstein et. al (1935)] re

marked in the 1930's, the predictions of QM become even more (if possible)

puzzling to even a mild realist , when the QM of a two-particle system is

analyzed. The wave-function for such system is now ip{x\,X2), a general

function of th e eigenvalues of the observables X i ^ , the "position opera

tors" of the two part icles. EPR showed that the general "entanglement"

of ip(x 1,22), i-e-

t n efact

that in general ip(xi,X2) ^ <f>(xi)x(x2), leads to

the prediction of non vanishing particle—particle correlations for space-like

sep ara tions , thu s violating the principle of causality. As mention ed, careful

experiments carried out in the 1980's have fully confirmed such quantum

predictions, exposing in a definite way the impossibility to reconcile QM

with causality as well .

The discussion of this section has, I believe, convinced us that the most

severe difficulties of QM do not lie at all in the general ideas of quantum

kinematics, in the fundamental distinction between the Hilbert space of

states and the operator-nature of physical observables, which stems from

the new "creative" nature of the observations or, more generally, of the in

terac tions am ong different physical system s. W her e such difficulties becom e

insurmountable is in the quantum-mechanical assum pt ion that throug h the

general ideas and postulates of quantum kinematics one can bui ld a the

ory of the isolated matter system, i.e. of the system with a finite number

of (Lagrangian) degrees of freedom. Without the possibility of defining,

through observat ion, a t rajectory qi — qi(t), (i = 1, . . . , / ) , as we have

seen above, there vanishes our ability to speak of a well defined, localized,

isolated matter system, whose identity is the very "condicio sine qua non"

for defining its degrees of freedom, in spite of QM' s pretenses. Thus we



Kinematics 21

may conclude that the impossibili ty of identifying a trajectory, inherent

in quantum kinematics, engenders the impossibili ty of uniquely identifying

the system with the finite number of degrees of freedom, whose evolution,classically, that very trajectory describes, and therefore to define the very

start ing point of QM. We shal l see that the only way to understand the

quan tum -me chan ical com putat ion s and to justi fy their undeniable success

is to real ize that QM is nothing but an approximation of QFT in the l imit

when its "field densities" become extremely small .1' Naturally, the eclipse

of QM as a basic physical theory will also mean the eclipse of the Copen

hagen's view as a new, unsavory image of the physical world, upon which

so much bad philosophy has been recently built .

2 .4 Q ua nt um fie lds are th e on ly rea l i s t i c ph ys ica l ob je ct s

Let us now look in detail into the kinematical structure of QFT. In order

to be able to concentrate on the fundamental conceptual aspects I shal l

try to keep the descriptive level as simple as possible, thus foregoing any

preten se of r igour and completeness which, again, cann ot be the m ain actu alconcern of the realist. The quantum field we shall focus on is the self-

interacting complex scalar field \ t(a; ,£) in one-dimensional space," whose

non-relativistic Lagrangian density can be written (from now on we shall

use units where h = c — 1)

wh ere m is th e m ass of th e field an d V is an ap pr op riat e function of '&*'$, th epotential . The quantization of \ t can be accomplished in full analogy with

the "first quantization" of QM, i .e. one identifies the momentum density

I I ( M ) = | £ = i t t * ( M ) , (2-4.2)

thus ^(x, t) and H(x, t), for each x (the space variable), are the variables of

the classical, oo-dimensional PS, which get promoted to the field operators

*&(x, t) and Ii{x,t) acting on a suitable Hilbert space of physical states and

"How small i s smal l wi l l be made c lear in due t ime.

II For a good in t roduc t ion to QFT and i t s re levan t app l ica t ions to modern phys ics s ee

J . D. Bjorken and S . D. Drel l , Rela t ivis t ic Quantum Fie lds , [Bjorken and Drel l (1965)]

For some app l ica t ions to condensed ma t te r phys ics s ee G. P repa ra ta , QED Coherence

in Matter (World Scient i f ic , S ingapore , 1995).




obeying the equal- t ime OCR's

[*(a;, t), U(x', t)] = iS(x - x'), (2.4.3)

which by vir tue of (2.4.2) reduces to

[ * ( z , t), &(x', t)] = S(x - x1). (2.4.4)

In order to keep the mathematical complexi ty to a minimum level, we quan

tize the fields in the interval — -§ < x < j , so t h a t the Fourier theorem

allows us to decompose the field into plane-waves, yielding

*(:M) = - ! = X > p ( i Kp x

, (2.4.5)p

where p = n^f-, n integer. It is immedia te to derive from (2.4.4) the cor

responding equal- t ime commutat ion re la t ions for the quantum ampl i tudes

a p(f ) and a^p^t):

[ap(t),aV(*)]= V - (2-4-6)

It is qui te remarkable that the CCR's (2.4.6) are capable to give us a com

plete pic ture of the extremely complex kinematical s t ructu re of the Hilbert

space of the s ta tes of the quantum field *, the Fock space. Indeed (2.4.6)

teaches us t h a t we can associate to each independent wave-mode p of the

f ield a one-dimensional quantum system, an oscillator, whose p and q are

related to a and a* t h r ough

a = ^ = ( p - » q ) , at = — ( p + iq ), (2.4.7)

whose CCR

[q ,p] = », (2.4.8)

readily implies

[ a . a ^ l . (2.4.9)

I t is well known that a complete or thonormal bas is of the Hilbert space

of the quantum osci l la tor is provided by the s ta tes (n is a non-negat ive

integer)

• ' ^ ( a t n O ) , (2.4.10)n !



Kinematics 23

where the ground state |0) is the unique state that is annihi lated by the

operator a, i.e. a|0) = 0. It is easy to check that {\n)} is the complete set

of the eigenvectors of the number-operator N = a^a and {n} the set of i tseigenvalues.

The kinematical independence of oscillators pertaining to different p's

implies that the Hilbert space of the physical states is nothing but the oo-

dimensional tensor product of the oo-dimensional Hilbert spaces for each

wave-mode p. An enorm ously complex struc ture w hich, surprisingly, adm its

a particularly simple description. But in order to have a better physical

understanding of the Fock space let us consider the free-field limit, i.e. the

case V(\&*\&) = 0. The classical Hamiltonian is now given by

' • /

dx(m - c) I dx1 d

2'

2m da;2 (2.4.11)

and to convert i t into the quantum operator H, we must prescribe the

ordering of the non-commuting operators St and ^*. General consistency

considerations demand that when going from the classical functions to the

quantum operators one makes the subst i tu t ion:

# * # - > ^ ( * t * + * ¥ * ) . (2.4.12)

Thus (2.4.11) yields for the quantum Hamiltonian

H= /

< t{ - i H *

,; S *

+G £ * H } '

(24-i3)

which through the Fourier decomposition (2.4.5) can be expressed as:

H=H) v + \) (2-4-14)

where use has been mad e of the co m m utators (2.4.6). As for the m om entu m

of the field, it is given by the operator

httE"• ' s *

+

( * • ) • '(2.4.15)

Th e remarkable feature of the opera tors H and P is tha t they are simultane

ously diagonal in the {n p}-basis (also called number representation) of the




Fock space; and on the generic vector (the index p denotes the wave-mode)

iw)=nw f. (2

-4

-1 6

)the total energy is given by

£(K}) = £f^(nP+i); ( 2 - 4 . 1 7 )p \ /

while the to ta l momentum is

P({nP}) = Y,p(nP + \)- (2-4-18)

The first remarkable aspect of the above results is t h a t the Ground

Sta te (GS), t h a t we may also call the Vacuum, which corresponds to the

wave-modes being all in their ground state (for which np = 0), has indeed

zero momentum (due to the cancellat ions between opposite momenta) but

its energy is given by:

* » - 5 £ ( £ ) ' (2A19)

a divergent quantity. Here we make our first encounter with a very unpala t

able aspect of the present formulation of QFT t h a t is usually referred to as

"ultraviolet divergence", and has subs tant ia l ly contr ibuted to confine the

use and s tudy of QFT to a ra ther res t r ic ted number of physicists, mainly

interested in particle physics. We note tha t the divergence of EGS s tems

from the sum extending over an infinite numbers ofwave-modes p. The ne

cessity of summing over an infinite range is dic ta ted upon us by our ( tacit)

assumption that space is cont inuous , i.e. t h a t we can dist inguish two dif

ferent space-points even when their distance is arbitrar i ly small . Naturally

the re is no experimental evidence for such an assumpt ion , the "continuity"

of space being presently tested down to space-distances of a bou t 10- 1 6

cm

only, while there is a strong theoretical evidence, based on the theory of

Quantum Gravi ty (QG) ,** that at the Planck dis tance (ap ~ 10- 3 3

cm)

such continuity gets lost due to the quantum f luctuat ions of the gravita

tional field. As a result a natura l momentum cutof f Pp = — ~ 1019

GeVJ

ap

is seen to emerge, delivering us a finite and finally realistic QFT.

**I am referr ing to a recent work [Caccia tor i et. al (1998)] which shows that a m o s t

l ikely phys ical rea l iza t ion of QG requ i res tha t space - t ime be a kind of a "foam" with

fulls and voids of the size of ap, the P l a n c k d i s t a n c e .



Kinematics 25

However, even without a serious analysis of the structure of space at

very short distances, the pragmatic at t i tude of the quantum field theorist

so far has been to ignore the ultraviolet divergence of Eos for it is in nofashion observable,^ the only observable energies being the energy differ

ences between the GS and all other states of the Fock space, which can be

wri t ten :

AE({np}) = E({np}) -E GS = ^ n J ^ ) , (2.4.20)

P ^ '

and are thus finite. We leave here this rather loaded problematics, by notingthat in the last few years the worst clouds on its real physical meaning have

been clearing. For the purposes of this Essay it is only important to note

that with QFT empty space, the Vacuum, ceases to be the negat ion of

being, l ike in Classical Physics, to become the Ground State, the state of

minimum energy, of all the quantum fields that exist in nature. In other

(metaphysical) words, in the quantum world the Vacuum does not precede

creation but is, actually, a fundamental piece of i t .

Getting back to Eqs. (2.4.17) and (2.4.18), we are finally able to recognize the "physical content" of the quantum field in the free (isolated) l imit.

Indeed, the generic state (2.4.16) is seen to correspond to a quantum field

configuration where the wave-mode p is "populated" by np "quanta" , whose

kinematical behaviour is identical to that of a system of np non- in teract ing2

classical particles, of momentum p and (kinet ic) energy ^ , and thu s of

mass m. Natural ly to the individual "quanta", which are observed in the

interactions of the field with external agents (other fields), wecannot

a ttribute any autonomous physical reality, i .e a physical Hilbert space of their

own, as QM pretends. Their reality begins and ends with that of their field,

whose interact ions with the other f ields they happen, under certain condi

t ions, to mediate. In part icular , one should note, the space-coordinate does

not appear in the theory as a quantum observable, l ike the momentum

(2.4.15), but rather as a simple space-label of the amplitudes of the quan

tum field: in QFT the not ions of quantum local izat ion and separat ion are

thus totally extraneous to the reality of the fields for all t imes in all space.

t+T his s ta tem en t is ac tu al ly inco rrect , for i t only refers to the energy exch ange s be

tween the f ie ld and any observer or other f ie lds , but i t neglects the interact ion with the

grav i ta t iona l f ie ld , which is coup led to th e ene rgy -m om en tum tensor th a t in th e GS ,

acco rding to (2 .4 .20) , is d iverg ent . How ever , even in th e f ini te v ers ion of the Q F T , t hi s

fac t poses s eve re p rob lems to the l a rge s ca le s t ruc tu re o f space - t ime by p red ic t ing a

cosmological constant some 120 o rde rs of ma gni tude l a rge r th an i t s p resen t l im i t .



26 An Introduction to a Realistic Qua ntum Physics

To conclude, the notions of localization and sep aration , th at realism de-

mands of any physical theory of the "quantum", and that are so patent ly

viola ted by both QM and Nature, imply that in any realis t ic physical the-ory of the "quanta" their clearcut objective definit ion must be structurally

and logically impossible. This s i tuat ion happens to hold in QFT, where

localization and separation are (approximate) physical properties of the

measur ing ap par atu s , and are in no way intimately con nected to the real ity

of the field. Thus, as far as we know today, quantum fields are the only

theoretical constructs that conform to a realis t ic picture of the world.



C h a p t e r 3

D yna mic s : The La w s o f Evo lu t ion o f

P h y s i c a l R e a l i t y

In the Greek language dvvafiL^, Svvdjitux; means force, power. Thus in

modern physics Dynamics has come to denote the science of physical pro-

cesses, the qu an titat ive d escription of the tim e evolution of physical syste ms

under the action of external forces. It is interesting that in Aristotle's (as

well as in the Medieval) view of the physical world no distinction could be

made between Dynamics and Kinematics, for motion was nothing but the

process by which the bodies of the sublunar world reach, once perturbed,

their "na tura l place". We owe it to th e intellectual "heroism" of Galileo

Galilei, at the beginning of the XVII century, that instantaneous motion

should not be viewed as a process but rather as a state of a mater ial sys-

tem, and that Dynamics deals quantitat ively with the physical causes that

change one state into ano ther as t ime flows. Th us we may r ightly asser t th at

modern science came into existence only when a clear understanding was

achieved of the fundamental dist inction between Kinematics and Dynam-

ics, a distinction which has profoundly shaped both classical and quantum

physics.

3 . 1 T h e H a m i l t o n - L a g r a n g e t h e o r y o f c l a ss i ca l d y n a m i c s

This section does not intend at all to give even a sketchy account of the

way the Analytical Mechanics of Hamilton and Lagrange is built from a set

of fundamental principles.* It aims, rather, at a recapitulation of ideas and

mathematical formulae of classical dynamics, which will prove necessary

*For th is the reader i s invi ted to consul t c lassica l tex tbo ok s such as E. Wh it ta ke r , A

Trea t i se on the Ana ly t i ca l Dynamics o f Pa r t i c l e s and R ig id Bod ie s [Whi t t ake r (1970) ] .

27




when their extension (and metamorphosis) to the quantum world wil l be

discussed.

We have already seen that the classical kinematics of a Lagrangian system of / degrees of freedom (the limit / —>• oocorresponding to a field

system) is fully described by its / Lagrangian coordinates :

qi(t),Q2{t),...,q f(t).

The Dynamics of such system is solved once the differential equations are

found that determine uniquely, given appropriate init ial conditions, the tra

jectories qi(t) (i= 1 , . . . , / ) . Th e g en e r a l me t h o d toderive such equat ions

of motion has its focal points in two fundamental not ions: the Lagrange

function

L(qi,Qi,t),

and the Action

ftfA= / dtL( q i,qi,t), (3.1.2)

Jt i

which is a "functional" inthe function-space of the trajectories <&(£), that

s t a r t at t ime U and end at t ime tf.

Given the Lagrange-function (or Lagrangian) L(qi , q\,t), which thus em

bodies al l the dynamics of the system, the equat ions ofmotion are those

which solve the "isoperimetric problem" of finding the trajectory that ren

ders the act ion A stat ionary, while keeping the boundary values

qk(U) =qki, 1k(tf) =q kj (3.1.3)

fixed. The calculus of variations shows that the sought equations are

d fdL\ dL ,

jt{wj-air°-(3-1-4)

the Lagrange equat ions: a set of / differential equations for qk(t), whose

highest t ime-derivatives are of the second order and which, according to

a well-known theorem of analysis , admit a unique solut ion both for the

boundary conditions (3.1.3), and for the more usual ones:

qk(U) = q , q'k(U) = qk • (3.1.5)

Equation (3.1.4) with the boundary conditions (3.1.3) or (3.1.5), give

mathematical substance to the complete determinism of classical dynamics,

and are the tools that the celebrated Laplace demon ut i l izes, once having

3.1.1



Dynamics 29

determined the form of the Lagrangian L, to predict with perfect certainty

the future. The Lagrangian formulation, whose fundamental variables, out

of which the Lagrangian is constructed, are qk, and q\ can be transformedinto the Hamiltonian one, whose variables are qk and pk, the coordinates

of classical Phase Space (PS), by means ofwhat is known as a Legendre

transformations. Let us first define the momentum

Pk = j^{qk,qk,t) (3.1.6)

"conjugate toqk" • The Hamilton function, or Hamiltonian, is then given

by

/

H(q k,Pk;t) =^Pkik -L(qi,qi;t). (3.1.7)

fc=i

It is clear from (3.1.6) that one can solve all qk (at fixed time t) in term s of qk

and pk and, once this isdone, the Hamiltonian (3.1.7) becomes a function

of the variables qk and pk- A very important construct in Hamil tonian

mechanics is t h a t of Poisson's bracket of tw o observables, which are well

defined functions of the "canonical" variables qk and pk- Let A(qk,Pk',t)

and B(qk,Pk', t) be such observables at a given tim e t, the Poisson's bracket

of A and B is then defined as

r , „, v^ fdA dB dB dA \

"*>-£(**-**)•(3'L8)

which is cleary antisymmetric for the interchange A <-> B, i.e

{A, B} = -{B, A}. (3.1.9)

By subst i tu t ing in(3.1.8) qi for A and pj for B we find

{qi,Pj} =8ij. (3.1.10)

Assuming, for simplicity, that neither L nor H explicitly depend on t, th e

Lagrange equations (3.1.4), by virtue of (3.1.6) and (3.1.7), become

a i r

oqk

while differentiating (3.1.7) with respect to pk we obtain:

OPk



Dynamics 31

invariant, i .e. setting

H'(Q,P) = H(q,p), (3.1.18)

for a "canonical transformation" it so happens that

9 Q f c ' (3.1.19)

WkdP k

It is easy to show that if (3.1.19) is satisfied the Poisson's brackets for the

new variables

fr i Vdij

dPi

dij

dPj J

are also left invariant. Thus the canonical transformations form that group

of (non-linear) transformations of the classical PS that leaves the Hamilto

nian dynamics invariant in form.

3 . 2 T h e H a m i l t o n i a n o p e r a t o r : t h e g e n e r a t o r o f q u a n t u m

d y n a m i c s

It was P.A.M Dirac in 1927 who discovered that once the quantum kinemat

ics was correctly described, the quantum dynamics could be directly and

simply inferred from the Hamiltonian classical dynamics. The remarkable

observation of the young Dirac was that the Poisson's bracket could be set

in direct correspondence with (equal time) commutators of the correspond

ing observables, i .e. between the Poisson's bracket {A, B} of two classical

observable and th e com m utator [A, B] of their corresponding He rmitian

operators , there exists the relation:

ih{A, B} -»• [A, B ] , (3.2.1)

which is fully compatible with the antisymmetry of the commutator , and

its vanishing in the classical limit h—>

0.The first successful testing ground of (3.2.1) is evidently the CCR's,

which directly follow from the canonical Poisson's bracket (3.1.10). In this

way Dirac's identification produces in an astonishingly simple fashion the

basic layout of quantum kinematics; but what is even more remarkable

is that also from his discovery quantum dynamics emerges in a perfectly




natural way. Indeed, from Eq. (3.1.15) the t ime derivative of the quantum

observable O is simply given by

i f i ^ O ( t ) = [ 0 ( t ) , H ] , ( 3 . 2 . 2 )

which is also known as Heisenberg's equation of motion. Equation (3.2.2)

exhibi ts in a part icularly t ransparent way the fundamental role that the

Hamil tonian operator plays in determining the dynamical evolut ion of the

quantum system. For a Hamiltonian explicit ly independent of t ime the so

lution of (3.2.2) is totally straightforward, namely

O ( i ) = e x p * H t O ( 0 ) e x p - K H t , (3.2.3)

showing that the t ime evolution of the Hilbert space of the physical states

of the quantum system is uniquely determined by the t ransformation

U(t)=exp-iHt

, (3.2.4)

obeying the uni tar i ty relat ions:

U(t)W(t) = U\t)U{t) =1, (3.2.5)

by vir tue of the Hermit ici ty of the Hamil tonian operator . As a result any

physical state evolves in t ime in a purely deterministic way according to

the equat ion:

h M ) = e x p - *H t

| V , 0 ) (3 .2 .6 )

which , due to the uni tar i ty of exp~*H t

, remains normalized at all t imes t.

The fundamental fact that the Hamil tonian is the generator of the uni

tary t ransformations of the Hilbert space of the physical states associated

with the dynamical evolut ion of the quantum system, puts the complete

set of the eigenvectors of H, the energy eigenvectors, in a special and privi

leged position. Let us in fact take an energy eigenvector \E , 0), with energy

eigenvalue E. According to (3.2.6) one has

\E,t)=exp-iEt

\E,0), (3.2.7)

i.e. with t h e flowing of tim e th e ch ange of \E , 0) consists only in th e m ultiplication by the phase factor exp — \Et. Such states , the "s tat ionary s ta tes" ,

are the only ones in the Hilbert space that donot experience a "defor

mation" during their t ime evolut ion, and are thus the good candidates to

describe the states of the quantum system between perturbat ions, includ

ing measurements. Differentiating (3.2.6) with respect to t ime allows us to



Dynamics 33

express quantum dynamics in the very familiar form of the Schrodinger

equat ion:

ih-^\tl>,t) = H\il>,t), (3.2.8)

the starting point of all quantum mechanical calculations. Finally a word

about the quantum analogs of the canonical t ransformations, which turn

out to be simply the total i ty of uni tary t ransformations. For equal t ime

canonical commutation relations are clearly left unchanged by any such

t ransformat ion V, and

Vte x p - *

H* V = e x p - *

v t i m, (3.2.9)

as can easily be checked through the following elementary argument:

V^exv-tutV = V^

^ f-i \ " H "

. 71= 0V /

V

^ • f l ^ H Q W ^ H V . .(„

1 0 )7 1 = 0

3 . 3 T h e P a t h I n t eg ra l ( P I ) : c la s s ica l t ra j e c t o r i e s a n d Q u a n -

t u m P h y s i c s

Thanks to Dirac [Dirac (1933)] and later Richard Feynman [Feynman

(1948)] that we possess today a very powerful and il luminating tool —

the Path Integral — to fully appreciate the conceptual steps involved inthe transition from classical to quantum physics, with respect to both the

kinematical and the dynamical aspects. The Path Integral (PI) proves also

very effective in giving straightforward and intuitive solutions to a few

quantum mechanical problems, where the operator formalism in a Hilbert

space tends to obscure the physical meaning of the calculations. The main

aim of the PI approach is to establish a connection between the transition

am pli tudes of qu ant um dynam ics and the t rajectories of classical dynam ics.

In order to keep the problem at i ts simplest level let us work again witha single Lag rangian degree of freedom, and call H ( Q , P ) the H am iltonian

operator. The quantum dynamical problem is completely solved if we know

the t ransi t ion ampli tudes

(Qf,tf\qi,U) = ( g / l e x p - ^ ' -4

' ) | f t ) , (3-3.1)




between any eigenstate j^) at the init ial t ime <j and any eigenstate \qf) a t

the final time tf of the operator Q. For, calling

\tf) = Jdq f\q f)i>(q f,t f), (3.3.2)

the s tate evolved at the t ime tf from

\k)= I dqilqiWq^U ) ( 3 . 3 . 3 )

the s tate at the init ial t ime ti , one clearly has:

~4>{<l},tf)= dqi(qf,tf\qi,ti)ilj{qi,ti). (3 .3 .4)

The PI approach yields a peculiar integral representation of the transi

tion amplitude (3.3.1) in the following way. Let us divide the time interval

tf — U in N equal subintervals of length At, with the intention to take the

limit At —> 0, or N —>• oo at the end. In this way the transit ion ampli

tude can be writ ten as the quantum composit ion of the evolutions over N

subintervals ;

< g / , * / k i , * i > = ( g / l e x p - * " ^ - * ^ 1 ^ )

= / dg jv - i ^ iV - 2 • • • dqi(q f\ e x p ~ x H A t |gjv-i>

x (g jv -i l ex p -* H A * |gAT_2> • • • ( gi | e x p - * H A t

\qi

). (3.3.5)

Let ' s concentra te our a t tent ion upon the matr ix e lement

(q i+ 1\ e x p - * H A t | f t ) ~ (q i+ 1\ (l - ^ H A i ) \qi), (3 .3 .6)

which, by inserting a com plete set of eigen states of P , yields:

( f t+ i | ex p - * H A t \qi) = I'dPi{q i+l\Pi){Pi\ M - J rHAiJ \q t) . (3.3.7)

I f the order ing ambiguity of the operators Q and P in the Hamiltonian is

solved by requiring th at in the general algebraic expression of the qu an tum

Hamiltonian all P's are to be put to the left of all Q 's , the last matrix



Dynamics 35

element in (3.3.7) is diagonal, and one has

(pi\ (l - ^U(P,Q)At\ \qi) =(pi\qi) (l - %jH {Pi ,qJM

= exp-iH(puii) {p.\qi)> ( 3 . 3 _ 8 )

and using the explicit form (2.2.16) for (p\q) we can write

dgx-idpN-i dqidpi dp0

2nh '" 2irh 2nh

v - i

exp - Y2 (Pk(qk+i - qk) - H(p k, qk)At). (3.3.9)

( ^ l e x p - ^ - ^ l

hk=0

In view of the smallness of At we may set qk+i — qk = 4kAt. We are now

in a position to take the limit At -4 0, thus giving a concrete meaning to

the functional integral (provided it converges)

f T T dqkdpk dp0 CJ 1 1 2 T T H 2 i x h " J

dqkdpk dp0 f [dq(t)dp(t)\, 27T?i

fc=i

(3.3.10)

we finally get

(q f, tf\qi,U) = J M ^ l l expl-j

tfdt\pq-H(p,q)}. (3.3.11)

Eq ua tion (3.3.11) is the sought out general connection between the qu an tu m

trans i t ion ampli tude {qf,tf\qi,ti) and the classical description in terms ofthe PS variables (q,p). It s t ipula tes that in order to obta in the quantum

transit ion amplitude one must "sum" over all phase-space trajector ies that

pass at ti th ro ug h <&; an d at tf t h r ough qf, assigning to each trajectory the

phase

®=\j dt\pq-H(p,q)}. (3.3.12)

B u t a physical ly much more t ransparent representat ion can be obtained

when the Hamil tonian depends quadrat ical ly on the momentum p, i .e. whe n

we may wr i te :

B=£+V(q). (3.3.13)




When this happens the p-integrat ion can be done analyt ical ly in the fol

lowing way

[N

f\ ^PLek^{Pk4k-^) . f[Mt)] ct ;:/*&-£)

J 11 2nh J 2irhk= 0

which is nothing but the product of N — 1 Gaussian integrals

/ _ * e* A * W - £ > = (-J2-) * « p (Ut^-) (3.3.14)J 2nh \2iirhAt) ^ \H 2 J

v'

Thus we arr ive at the resul t :

(qf,tf\qi,ti) = (const) [{dq(t)}eittdt[^-V{q)]

= (const) f[dq(t)}e^S''

diL(qA), (3.3.15)

which expresses the quantum transi t ion ampli tude as a sum over classi

cal t rajectories star t ing at qi and ending at qf of the phase factors, which

are just the action of the classical trajectory (3.1.2) in unit of h. W h e n

considered from the point of view of classical dynamics the quantum evolu

t ion, embodied in the t ransi t ion ampli tude {q/,tf\qi,ti), now fully reveals

its profound departure from the intuitive world of the classical trajectories

q(t). A ccording to (3.3.15) the evolution from qi at t ime U to qf at t ime

tf does not involve a single trajectory but the totality of trajectories t h a t

connect the two points at the two t imes, each t rajectory summed with the

"weight" exp j^A , determ ined by its classical action A. The wave-like char

acter of the quantum evolution is thus particularly transparent, for (3.3.15)

makes direct contact with the classical connection between the motion of a

wave and that of a bunch of particles whose trajectories are at right angles

w ith the wave-front. A lso the idea tha t to each trajecto ry one should as

sociate a phase related to the A ct ion does belong to classical Ha m il tonian

mechanics. What is peculiarly new, however, is the appearance of Planck

co nsta nt as a funda me ntal unit for the A ction. Th is fact ma kes us finally

understand quant i tat ively when a classical approximation to the quantum

real ity becomes adeq uate . Indeed when the A ct ions associated with the

classical paths q(t) are much larger than h, as i t happens for the motion of

the center of ma ss of a macroscopic body, the principle of stat io na ry pha se

shows that the only relevant trajectory in the PI is that for which the action

http://2iirhat/

http://2iirhat/

http://2iirhat/

http://2iirhat/

http://2iirhat/



Dynamics 37

is stationary, i.e. for which

SA = 0, (3.3.16)

whose solution leads, as we know, to the Lagrange equations of classical

mechanics. When the Actions involved are not large with respect to H th e

quantum evolution process involves a bundle of trajector ies around the

classical one, in full agreement with Heisenberg's uncertainty principle, and

the basic s tucture of quantum kinematics.

To conclude, we may say that in QP the notion of a trajectory (or of

a well defined classical field configuration <f>(x,t)) disappears because in i ts

evolution a quantum system, developing from a given init ial configuration

to a final one, du e to the fundam ental qu an tu m fluctuations, explores a

large number of trajector ies beyond the one of s tationary Action, which is

just the classical trajectory.



C h a p t e r 4

Q u a n t u m F i e l d T h e o r y : T h e O n l y

Rea l i s t i c Th eo ry o f t h e

Q u a n t u m W o r l d

In Sec. 1 we have am ply discussed th e reasons why Q M cannot be a realistic

theory of the quantum world, which may be summarized in the impossi

bility to give an objective (i.e. independent of the subject, the observer)

meaning to the key notion of wave-particle complementar ity . We have also

seen that the most problematic aspect of the picture of the quantum world

th at QM pain ts us is the physical natu re of the qu antu m par t ic le , an object

that, we should be aware, is quite distinct from the "quantum" of Einstein

and Planck. Whereas, in fact , the "quantum" is a par ticular manifestation

of the associated field and does not enjoy any dynamical autonomy, the

"quantum particle" is, according to QM, a well defined object, much like

the Newtonian mass-point, but for the fundamental, and puzzling, differ

ence that the very physical means to define it, by following its trajectory,

is in principle unavailable. In this sense, we may well say that the quan

tum particle is a truly metaphysical object, for no unique objective physical

observations exist to give it a real substance. On the other hand no such dif-

ficulties affect th e not ion of field, th a t describ es in wh ich way a given reg ion

of space differs from em pty space , whe re any physical obse rvation yields by

definition a null result. Localization and separability, two concepts that, we

have seen, haunt QM, have no fundamental relevance in field theory, for the

definition of space and time belongs to the observers through their measur

ing apparatus (including rigid rods and clocks), and not to the object of

field theory, which represents and describes the "physical condition" of the

particular region of space-time the observer focusses his attention upon.

In order to understand this lat ter point a l i t t le better , let us direct

our attention to the way in which a classical field gets measured, for in

stance an electric field. One moves a test charge around, in a region whose

points have been previously labelled through appropriately chosen (by the

39




observer) car tesian coordinates , and measures the acceleration that the test

charge is subject to at any particular point, which can be converted into

the numerical components of a vector, the electric field. Naturally in thisprocedure one heavily relies on classical physics: f irst the test particle, that

must be localized, then the measurement of acceleration, proportional to

the t ime der ivative of the momentum of the test par t icle at the same point .

But is i t really true that in order to detect and quanti tat ively determine

the presence of the field in an appropriately small region of space—time we

need a "classical measurement"? One may legit imately doubt this , for al l

we need to ascertain is the presence of the field able to detect an energy

exchange between the f ield and the measuring apparatus, whose interact ion with the f ield may be totally quantum mechanical , l ike i t happens,

for istance, in a photomultiplier. Thus we are led to identify in the pro

cess of energy-exchange between the field(s) and a general measuring ap

paratus localized in space and t ime, the fundamental means by which we

reveal how space is "modified" by field. As a result the main outcome

of a field measurement in an appropriately localized region of space—time

turns out to be the transit ion of the f ield between two states , whose energy

difference is the one that is measured in a "small" region of space, determined by the m easuring a pp ara tus . And, as we shall see, such is the way in

which the "quantum" gets "distilled" from the greatly wider reality of the

field.

4 . 1 A p r e l im in ar y d i s c u s s io n o f c oh e r e n t s ta t e s

As we are interested only in the main s tructural aspects of QFT we shallpursue our study of the simple one-dimensional field theory of Sec. 2.4. We

have seen that , through the Fourier decomposit ion (2.4.5) , the quantum

field S&(x, t) can b e reduced to t he en semble of its one-d imen sional qu an tu m

oscillators of amplitudes ap. It is thus very useful to have a better grasp of

the quantum mechanics of the s imple one-dimensional oscil lator .

Let us then get back to the quantum oscillator, a complete basis of whose

Hilber t space is spanned by the s tates \n ) (see Eq. (2.4.10)). I t turns out

to be advantageous, at this point , to spend some t ime to analyze in some

detail the basis of eigenstates of the operator a, satisfying the eigenvalue

equat ion

a|oj) = a\a), (4.1.1)



Quantum Field Theory 41

where a is a complex number. It is not difficult to show that*

|a) = exp ( - ^ j f ) @=\n) = exp ( - ^ j e x P ( aa t ) | 0 ) (4 .1 .2 )

obeys (4.1.1) and is normalized, i .e (a|o:) = 1. Due to the non-Hermitian

nature of the "creat ion operator" the basis \a ) is not orthonormal, for we

have:

(a\a ) = exp I a a —

1 1= exp— -\a — a'\

2exp—-(a*

1a — a*a'). (4.1.3)

The states (4.1.2) are called "coherent states" for, according to (4.1.1), the

phase of the quantum ampli tude is seen to tend for large ampli tudes to a

well defined value, the pha se of the com plex num ber a. Let us indeed define

the quantum phase $ as

a = e i * ( a t « O i = e i * N * , (4 .1 .4 )

where N = a*a is the number operator, diagonal with integer eigenvalues on

the basis \n). It is an easy exercise to show that the canonical commutation

rela tion [a, a*] = 1 implies th a t

[ N , $ ] = i (4.1.5)

which shows that the phase operator $ is conjugate to the number operator

N . Thus according to the Heisenberg principle

(A*)(AAT) > | , (4 .1.6)

implying that in the states where there is a fixed number of "quanta" (see

Sec. 2.4), i.e. AN = 0, the phase is completely undetermined. On the other

hand we compute

oo

< a|e **|a ) = ( a | a N ^ | a ) = £ — Q | a | 2 ™ e - M 2 , (4.1.7)^=J[n! (n + l ) ! ]5

which for |a | 3> 1 yields (a = |oj|e^ )

\a \2

(a |e** |a) » e - M 2a i - _ = e

i(t> (4.1.8)

' T h i s f ollows fr om the e a si ly e s t a b li s he d i de n t i ty [ a, e x p a a t ] = a e x p a a t .




stipulating that for |orj large (note that AiV = |a|) the phase is well defined

and equal to the phase of a. One can also easily show that the completeness

relation for coherent states can be expressed as

l = f > ) < n | = | ^ H « ) ( a | , (4.1.9)

71=0

which allows us to decompose any state \ip) of the quantum oscil lator as:

Th us to the generic s tate \ip) one can associate a holomorphic function ip(a)

of the complex variable a:

|V>) -+i/j(a) = e^{i>\a), (4.1.11)

in such representation

a|V> -> ^(a), (4.1.12)

while

aJ\il>) -+ail>(a), (4.1.13)

and the scalar product is given by

{ m =

/ ^ ^

e _ a a

> * ( « ) < A ( « )•

(4-1-14)

I t is very easy now to com pute the spectru m of th e num ber ope rato r N =

a* a. In deed th e eigenvalue equa tion is

ata | r /) = v\v), (4.1.15)

which is equivalent to

a—<t>v{a) = v<t>v{a) (4.1.16)

whose solution is obviously <f>u(a) = cva?. Now the holomorphy of (f)v(a)

restr icts v to the set of non-negative integers n = 0 , 1 , . . . , and the normal

ization condition (4.1.3) fixes cn = —?=.




The PI representation in this basis takes on a particularly simple form

(a f,t f\a uU) = / " ( a / l e - ^ ^ l a j v - i X ^ - i l e - ^ ^ l o j v - a )

fc=i7rz

Prescribing the ordering of a and a* so that all a* are to the left of all a:

(a k+ 1\e-iHAt

\a k) = e-iH

M +»^At

(a k+ 1\a k)

= e - ' ' *+ 1

a °f c + 1

e - ^ ea* + '

a >e -

< A t g<

a* + i '

a*>, (4.1.18)

t hus

.N-l

{a f,t f\ai,U) = / TTak

akexpiY](a*kia k - H(a*k,a k))At

> / i w — w j . e x p z / dtla (t)ia(t)

- f f ( a * , a ) ) . (4.1.19)

In the "free" theory, where H(a*,a) = wa*a, the PI provides a part icu

larly simple solution of the dynamics of the oscillator: by going to the new

variables

0(t) = a ( t ) e

f a

( ' -

t , )

(afM*i,U) = J[dP*Wf

{t)]exptjT ' dt(P*(t)i(3(t)

= (0f\l3i) = (Pf\°<i) = (<* fei»l

t'-V\ai)

= (a f\e-i^

tf-

t^a i), (4.1.20)

w here the last scalar prod uc ts have the form given in (4.1.3). Th is last res ult

has a very simple inter pre tation : the dynam ical evolution of a coherent sta te

in the "free" theory goes through coherent states whose amplitudes evolve

as

a(t) = e - ^ ' - ' ^ a ; (4.1.21)




It is also interesting to note that the latter result coincides with the classical

one . Inde ed the p rinciple of sta tio na ry action of classical phy sics, in this case

applied to (4.1.19), takes the form

6 dt[a*{t)ia(t)-u>a*(t)a(t)}=0, , (4.1.22)

whose Lagrange equation is

ia(t) = u;a(t), (4.1.23)

who se solutio n is ju st (4.1.21).

We may thus r ightly conclude that the coherent s tates represent the

"best" quantum approximation of classical physics .

As a last "exercise" in quantum oscillators, we shall consider the prob

lem solved long ago by N.N. Bogoliubov: the diagonalization of the Hamil-

tonian for two coupled oscillators (A positive)

H = (w + A)( a ja i + a 2a 2 ) + A ( a i a 2 + a ^ a 2 ) , (4.1.24)

which can be solved by finding first two new oscillator amplitudes:

A i = a a i + /3a2 , (4.1.25)

A 2 = 7 a 2 + 6a{ , (4.1.26)

such that

[ A 1 , A l ] = [ A 2 ) A i ] = l , [ A 1 , A 2 ] = [ A | , A 2 ] = 0 (4.1 .27)

It is easy to show that the conditions (4.1.27) are satisfied if

a = 7 = cosh6l /? = 6 = sinh 6. (4.1.28)

The solution is then obtained by finding the value of 9 such that

H = i /(At1 A 1 + A 2 A 2 ) + / i . (4.1.29)

A rather s imple, though tedious, calculation yields:

t a n h 2 0 = r- (4 .1 .30)w + A v

v = E = \/u2

+ 2Aw (4.1.31)

H = -~ / < 0 . (4 .1 .32)2 w + A + y ( u > + A ) 2 - A 2




The spectrum of (4.1.29) is

= (n 1+n2)E + p, (4.1.33)

corresponding to "quanta" of energy E > LJ, the per turbat ive energy, and

a ground s ta te , n i = n-i = 0, whose energy \i is negative, i .e. lower than the

per turbat ive ground s ta te (A —> 0) . One may wonder how does the s tate

|0)e, th at is annih ilated by A i and A 2, look in terms of th e eigenstates of

the number opera tor s a^a i and a 2 a2, the answer is

k=0

showing that the new ground state \0)g is a coherent superposit ion of s tates

with an indefinite number of pairs of quanta of both oscillators.

4 . 2 T h e V a c u u m , t h e t e m p l a t e o f p h y s i c a l r e a l i t y

Given a QFT, through i ts Lagrangian density and the CCR's , such as in

Sec. 3.4, we are confronted with the task of understanding i ts s tructural

and dynamical content. How does one begin? The answer is simple: f ind

the Vacuum, the Ground Sta te (GS) , the s ta te of minimum energy. Why

must one begin with the Vacuum? The answer is again s imple: due to the

fundamentally fluctuating nature of the quantum fields, if the Universe is

open (as we shall always assume) any quantum state in a finite region ofspace will decay af ter an appropriate t ime, depending on the interactions

among different fields (including those associated with the observers), to

the s tate of minimum energy, the only s table s tate.

Thus finding the Vacuum is the preliminary step to figure out in which

way that particular region of space, we focus our attention upon, will react

to our measuring devices, thereby informing us on the quantum states of

the fields present in that region. As we shall show below, the structure

of the Vacuum, which would seem by definition unknowable, determinesin a fundamental way the kind of physical excitat ions, or "quanta" , our

measuring devices detect or excite out of the quantum f ields , through their

energy nego tiat ions with th e f ields them selves. In other wo rds, th e Vacu um

turns out to be a kind of " template" of the physical reali ty of the quantum

fields.




For definiteness' sake, let us write the classical Hamiltonian of our one-

dimensional field theory as:

+ i f dxdyV(x - y)**(x, t)*(x, t)**(y, t)V(y, t)

where g is a real parameter, which can be either positive or negative. Thequartic term in (4.2.1) describes the field self-interaction through the poten

tial V(x — y) , propor t ional to the field densities $*$ at the two (arbi trary)

space points x and y.

Going to the quantum field St, i .e. quantizing the classical theory as

outlined in Sec. 2.4, we can express the quantum Hamiltonian as:

+ ^ T 2-J ^ ga

p i + qa

P 2 - qap i

aP 2 > (4 .2 .2 )

P l ] P 2 i 9

where the Fourier transform of the potential V(x) is defined by:

q

We have seen that for Vq —> 0 the Vacuum, the Ground State (GS) of the

QFT, is given by the tensor product of the ground states |0) p of the mode

oscillators a p . Such perturbat ive GS (PGS) is the start ing point of most

f ield theoret ical calculat ions, that t reat the quart ic term as a perturbat ion,

which seems rea sonab le if Vq is "small". However it may so happen that one

can approximate the interaction with other fields by a term g^*^, with

g < 0. In this case the PGS loses any meaning and the GS, the Vacuum,

changes its structure completely. Let us see why and how.

Let 's assume that the vacuum expectation value of the field \&, which

vanishes in the PGS, is different from zero, i.e.

(GS\*(x,t)\GS) = -?=, (4.2.3)




with a a complex number. From (2.4.5) this corresponds to shift ing the ap's

in (4.2.2) as

ap — > ap + 6Poa.

As a resu lt we are led to a new (effective) H am ilton ian :

(4.2.4)

H e / / = Epos + ga*a + X I (5

+ f~ )& ]

P&P

p^O ^m'

+ ^V0(a*a)2

+ -j-a*a ^ *£p*vp^O

+ -r j X ^ . P ^ a a ^ a p + a*2apa_p + a V p a t . p ] + • • • (4.2.5)

p # 0

where the dots denote the terms cubic and quart ic in the osci l lator ampli

tudes a p and a t p , which can be t reated perturbat ively i f the Vp's are small

enough. The minimizat ion of the GS energy yields,

ga + — a a a = 0 ,Lt

(4.2.6)

admitting a nontrivial solution if Vb > 0 and g < 0, i.e. if the energy of

interaction with other fields is negative. In this case, we have a minimum

for

a a =gL

V0

(4.2.7)

By a simple phase t ransformation we can replace a by a real number a,

and wri te

Heff = EpGs - -= jr + X2V0

P7 to2m

+ X P 3. n<k-p<*p

+9 X ^pi^-p +

a tP

a t- p ) "! >

w h e r e

- v P

(4.2.8)

(4.2.9)

which, at least for small p's, is positive. In Sec. 4.1 we have seen how to

diagonalize the quadratic part of the effective Hamiltonian. The Bogoliubov




t ransformation thus yields

t anh20„ =2 m 9tf

(4.2.10)

E„ 2L) -2gr-,2m) 2m

V 'and

M P = - - m '2m•9%+Ep

(4.2.11)

(4.2.12)

The model we have just analyzed, though very simple, gives us a rather

vivid picture of the massive rearrangement of the degrees of freedom of

the quantum field that occurs in presence of a negative (g < 0) energy

term, i .e of an instabili ty, stemming from the interaction with other fields.*

T h e non -van ishing of the e xp ec tatio n value (4.2.3) of th e field \& in th e

non-perturbative GS has the following fundamental consequences:

i t lowers the energy of the P G S by the qua nt i ty:

AE = EGS EPGS = ^ H P - W Q

P

(4.2.13)

implying th at th e P G S is unstab le and ca nnot represent a decent

approximation of the physics of the quantum field;

the particle spectrum drastically changes from the free one„2P

6plpGS=2m-'

id by

p —'(l_\

2_ P2_YL

\2m) 9mV0

— • ( * ) * ? - •

12

>

p-»0

(4.2.14)

(4.2.15)

(4.2.16)

tT h i s i s jus t wh a t h app ens to the qua rk f ie lds in in te rac t ion wi th th e co lour ga uge

f ie lds of Q CD , l ead ing to the i r "conf inement". See G. P re pa ra ta , II Nuovo C ime nto ,

[ P r e p a r a t a (1 9 8 6) ].




W e may thus conclude that the instability ofthe PGS changes com

pletely the s t ruc tu re of the quan ta of the field \&, which do not have any

more the charac ter of (non-relativistic) point particles of m a s s m, but ap

pear ascollective excitations, similar to "phonons", with sound velocity

"» = ( : S ? ) i -

W e end this section with the analysis of the GS ofour QFT in the

condensed matter l imit , i.e when in our interval L the number of part icles

is fixed and equal to N. In this case the expectat ion value on the GS of th e

number opera tor

N = / d a ; * t ( x , t ) * ( x , t ) = ^ a t p a p (4.2.17)

is given by

(GS\N\GS) = N. (4.2.18)

The minimization equation (4.2.6) is now replaced by

a*a = N, g=- -N. (4.2.19)Li

A s a result the effective Hamiltonian is

V0N2

p#0

V0 N2

v-^ (N.. p2\ +

H.„ = EPGs - _ + £ + JL] atpap

+ rsYy v -v + ^P -P\ + •• • • (4-2.20)

2L p

which differs from (4.2.8) in the ground s tate , energy being now

VnN2

EGS =EPOS ~ y ^ - , (4.2.21)

and in the value of A p, now given by

A

P = J;V

P . (4.2.22)

leading to a "phonon" spectrum with sound velocity

m L(4.2.23)



50 An Introduction to aRealistic Quantum Physics

4 . 3 T he " c l a s si cal " l i m i t o f Q FT : t h e em er g en ce o f

c o h e r e n c e

When analyzed in terms of its mode oscillators, the classical limit of the

quantum field is seen to involve states |n) p with very large occupation num

ber n. Th is is a consequence of th e well-known "principle of corre spon den ce"

introduced in QM by N. Bohr, according to which in the high-n l imit the

quantum behavior should merge with the classical one.

Let 's consider again our s imple model in the condensed matter l imit ,

which forces the field to evolve in the subsector of its Hilbert space where

the number operator

N = /2

dx&(x,t)#(x,t) (4.3.1)

is diagonal and its eigenvalue is equal to JV. This immediately suggests to

"normalize" the field ip as

¥(x,t)=N*&(x,t), (4.3.2)

where the field <& is normalized in aSchrodinger-like way, i.e. to one.

Let us now consider the PI representation of the transit ion amplitude

( [ a P / ] , t / | [ a P i ] , ij ) betwe en the coherent sta tes of th e field oscillators ap at

t ime U and those at t ime tf. Astraightforward generalization of (4.1.17)

allows us to write

{[aPt],tf\[aPi],ti}

x ex p i I " dt Yl ap idp -H(ap>

al) (4.3.3)

Normalizing the field according to (4.3.2), i.e. going from the variables ap

to the new variables

PP = N2ap,

<[&/].*/H/U*i> ^ j\[[Nd(ip{t)d(3;{t)]

(4.3.4)

the PI can be rewr i t ten

x expiN f'Jti

dt Y,fPH3p-HN{PP,fi;)L P

, (4.3.5)




where from (4.2.2)

HN =T,(S+£) PPPP + \{T) £ W W%-,A* A* . (

4-3-6)

which in the fixed density ( ^ ) l imit is in fact iV-indepe ndent. T he inter

esting aspect of the representation (4.3.5) is the explicit appearance of the

large num ber TV in the sa m e position of ^ (which however in ou r un its

system has been set to one, and has thus dropped out of sight). Formally,

then, the condensed matter l imit (N —> oo) is completely equivalent to the

classical limit (h —> 0) of the diluted system.

This important observat ion immediately suggests to us the simple solu

tion of the PI: the field evolves along a classical trajectory, solution of the

s tat ionary phase equat ions:

' ^ ' 7 ) 1 . ( 1

n2

x'

N' Pi,q

around which i t performs quantum fluctuat ions of ampli tudes 0(77=).

We have seen in Sec. 4.1 that for a quantum system the classical trajec

tory gives (approximately) at each time the numerical values of the phases

of the coherent state in which it finds i tself at that t ime. Let us now analyze

what kind of informations an observer may acquire on the system by mea

suring for instance the charge (if the field's charge is e) in a small interval

Aa around the poin t X Q. The observed "observable" is thus

Q f s o - ^ . s o + W j H j dx&(x,t)*(x,t). (4.3.8)

Let 's assume for simplicity that only the mode with p = 0 is occupied and

that the charge "leaks" out of the interval (-•§,§) , so that the "classical

trajectory" is given by

ap(t) = 6P0Nie-% (4.3.9)

The expectat ion value of the charge on the coherent state with ampli tude

(4.3.9) is:

Q = <a(t) |Q|a(t )> = e (^f) \a(t)\2

, (4.3.10)



52 An Introduction to aRealistic Quantum Physics

while its dispersion is readily evaluated as

AQ = e(?f)\a(t)\ (4.3.11)

As long as |o:(t)|2

3> 1, i.e. t <C j£nN, the regime is essentially classical,

for the quantum fluctuations of the charge measurement (4.3.11) are much

smaller than the expectation value (4.3.10). Note, however, that according

to quantum principles the measuring device necessarily registers the charge

value en, with a probability distribution peaked at n = f and with a

dispersion An = -f-. This very important consequence of the quantumprinciples can be understood from the fundamental hypothesis that any

interaction between the field and the measuring device cannot involve a

non-integer number ofquanta, the elementary field excitations over the

GS, whose charge is e.

It is only when | a ( t ) |2

^ is 0(1), i.e. when the field is in avery "dilute"

state, that quantum reality begins toshine with its discontinuities and

probabilities, the latter being intimately and inextricably tied to the former.

In this limit the value ( ^) <1 represents only the probability that a chargemeasurement in the interval Aa yields the result e,and the expectation

value ofthe operator }&t(x,i)&(x, t) can be interpreted as the uniform

probability density to find a single charge in the interval Aa around xo, just

like the square of the single particle Schrodinger wave-function in ordinary

QM. More of this in the next section.

We end this section by analyzing in the condensed matter limit avery

important and simple model: the Dicke system of N two-level atoms, the

basic description of Laser physics. In our simple one-dimensional field the

ory its Hamiltonian is

L. k

H = Ei f* *l*xda; + E2 I'' ¥£$adz + (E2 - EI)^SL

+ — [2

dx{^!2^1ai + ^[^ 2a}), (4.3.12)

and the number operator:

N = (2

< & ( * ! * ! + * £ * a ) . (4.3.13)




Th e physical me aning of th e model is quite clear: S&i an d ^ 2 den ote th e

matter f ields , whose quanta are atoms in the energy s tate E\ and Ei re

spectively, a is the amplitude of the transverse e.m. field with frequencytv = (E2 — Ei) in resonance with the atomic transit io n 1 o 2, and the

fourth term represents the electromagnetic coupling of the two atomic lev

els. In (4.3.12) and (4.3.13) we have chosen the length of our interval L

to be smaller than the wave-length A = ^ j of th e resonant electromagn etic

m ode so as to be able to neglect the spatial var iat ion of bo th th e m at ter and

the e.m. field. This restriction, which appears as a choice of convenience,

has an interesting physical meaning and motivation, for it implies that the

spatial region, in which the matter and the e.m. fields are defined, gets nat

urally par t i t ion ed in "Coherence Dom ains" (CD) of th e s ize A, wh ere their

amplitudes are essentially constant, that rapid variations of the fields and

the quantum and thermal fluctuations being confined to their boundaries.*

Thus the analysis that follows focusses on the physics of a single CD.

An other impo rtan t "caveat" abou t the Ha milton ian (4.3.12) is th at th e

term wa^a of the Dicke model is an oversimplification of the full Hamil

tonian, based on the so-called "slowly varying envelope approximation",

which assumes that , writ ing the e.m. f ield amplitude, a = a(t)e~ lult, t h e

time-dependence of the envelope a(t) can be neglected. If one does not

make this approximation which, as we shall see, clearly breaks down in the

highly condensed matter l imit , the one-mode Lagrangian that appears in

the phase of the PI is§

i • 1 .Lem — r fa*a - a*a] + —a*a . (4.3.14)

2 2w

The PI representation of the transit ion amplitude ( / | i ) can then be writ ten

(w = (E 2 - E x))

m-J dai(t)da\(t) da2(t)da^(t) dada*(t)

lixi 2iri 2ni

e xp i / < a\idi + a^idi — E\d\ai — i?2 a2 a2 + ^ t a * a — aa*]2 L

+ ~ad* + - f = ( a * a ! a e -i w t

+ a ^ a V " " ) ) . (4.3.15)

*The physica l re levance of CD's in condensed mat te r i s d iscussed in the book of

Ref . [Prepara ta (1995)] .

§As show n in the above ment io ned boo k.




The condensed matter l imit s tems from the condit ion

( a | a i + a 2 a 2 ) = i V (4 .3 .1 6)

which advises us, as in (4.3.2), to "rescale" the amplitudes as

h,2 = ai ,2 iV~3 ,

0 = aN~i .

As a result the PI (4.3.15) gets transformed into

(4.3.17)

(f\i) ~ f \dbidb\db2dbldl3dl3*}exviN f ' dt

x J b\ib\ + b*ib2 - Ekhh* ~ E2b2b*2 + %-[(3*p~ - /3/i*] + ^ / 3 / 3 *

(4.3.18)g\/j(b*2b1f3e-i" t + b\b20*e+i^)

which can be further simplified through the "interaction representation"

t ransformat ion:

bi : - 'B l t

/ 3 l

b2 -> e~ iE*p 2

yielding the PI Action

dtPI=N fJtt

ftifa + (3*2i(32 + % -{(3*(3 - 0$*) + ±0*$

(4.3.19)

Apart from the factor N, which as we have alread y seen, controls th e classi

cal limit, Apj is remarkable for two more reasons: the disappearance of the

rapidly varying phase factor e±wt

, and the large amplification by y/N of

the coupling term, which shows its crucial importance in the high density

J^ l imit. The "classical" equations of motion are now obtained by the

stat ionari ty of the act ion with respect to the variat ions 6(3* 2 and 5(3* [the




variations of <5/3i,2 and dp yielding the complex conjugate equations]:

iPi+9\Jjp2p*=0, (4.3.19a)

i(32+9\TPiP = 0, (4 .3 .19b)

~ h&+

^+ 9

]/T^02 =

°' (4.3.19c)

with the condition [see (4.3.16)]

| /?i|2+| f t |

2= l (4 .3 .20)

I shall not analyze indetail the interesting consequences of the differen

t ial system (4 .3 .19) , which has been amply studied in the mentioned book

"QED Coherence in the Mat ter" . Here I will only demonstrate a funda

mental consequence of this simple, but extremely realistic model of the

quantum physics of matter coupled to the e.m. field: the so-called "Super-

radiant Phase Transi t ion" (SPT).^ Let us f i rst rewri te the system in termsof the adimensional t ime r =ujt;

iP - JjFlh, ( 4 . 3 . 2 1 a )

*/?2 =- ^ y j / 3 / J i , ( 4 . 3 . 2 1 b )

-lp + 0=-£.yfEft(32. ( 4 . 3 . 2 1 c )

Suppose that at t=0 the system is in the pertur bat iv e ground sta te (P G S) ,

as is universal ly assumed in today's condensed matter physics . We have

A(0)"'

+° U ) •

A(0) =° (is) • m

- ° O N ) • (4'3'2

the question we wish to give an answer to is: will the configuration (4.3.23)

rem ain at all t imes? W e are interested in the evolution of the sy stem (4 .3.21)

^This remarkab le fac t , fundamenta l forour u n d e r s t a n d i n g of c o n d e n s e d m a t t e r

phys ics , was discovered in th i s mode l in the ' 70 ' s byK . H e p p a n d E. Lieb [Hepp and

Lieb (1973) ] , but itss ignif icance was discarded due to the e r roneous op in ion tha t the

mode l v io la ted e lec t rodynamic gauge- inva r iance . For adiscuss ion of th is gigant ic b lund er

see the ment ioned book .




in the neighbourhood of r = 0, i .e. when to a good approximation /3i ~1.

The differential system simplifies as

< * - - »

-§ >+*=V I /TA -

w h i c h i s e q u i v a l e n t t o (G = ^ V x )

4 ^ +4 " + i G 2 "= ° .w i t h t h e i n i t i a l c o n d i t i o n s :

m-ofa), to-ofa).

( 4 .3 .2 2 a )

( 4 .3 .2 2 b )

(4 .3 .24)

The linear nature of the differential equation (4.3.24) assures us that both

foand /? remain of O ( -?= J if and only if the algebriac equation:

^ - - p2

+ G2

= 0 (4.3.25)

has no complex roots. This condition is violated when

2 _ g2N 16

G- ^ X > 2 7 ' ( 4 3 - 2 6 )

J V\_

16W

1

27 s;7 "mplying that , given gan d u, there is acri tical densi ty ( ^ )above which the system will "run away" from the perturbative configuration

(4.3.23). Where to? The answer is again quite easy, one can show that the

real GS, which we m ay call Coherent G round Sta te (CG S) a nd which obeys

the system (4.3.21) with the condition (4.3.20), is given by:

/ 3 i = c o s f l eW l

W , (32=sm6Je^T\ (3 = Ae*™ , (4.3.27)

with #1,2 and <f> linear function of r, satisfying the condition

0l-ei-<l>=~. (4.3.28)

What is most remarkable about the CGS is the phase-locking (4.3.28)

between the matter and the e.m. fields, presenting us avivid picture of

matter coherently and collectively oscillating between its two atomic levels




in tune with a coherent e.m. field which, unlike what happens in the Laser,

gets t rapped in the mat ter .

Finally a word about Lasers: it should be entirely evident that lasing

corresponds to a completely different configuration. Indeed, by means of

the population inversion achieved by the pumping mechanism, both /3i(0)

and /?2(0) are 0(1) and the short-time behaviour of the e.m. amplitude is

always of the "run-away" type corresponding to the Laser 's "switch-on", as

the reader may easily ascertain.

4 . 4 T h e " q u a n t u m - m e c h a n i c a l " l im i t o f Q F T : T h e

S c h r o d i n g e r w a v e - f u n c t i o n

In this section we shall s tudy QFT, again in the s imple one-dimensional

model, in a limit which lies at the opposite end of the classical one, namely

when the field is very diluted, i.e. it evolves in a subspace of the Hilbert

space of the eigenvectors of the number operator N in the interval L, t h a t

have small eigenvalues.

The cursory discussion of the dilute limit of the last section advises us

to concentra te our a t tent ion upon:

p s(x 1,x 2,...,xN,t)

= Nisl&ixut^ixut) • ••&(xN ,t)*(xN ,t)\s)N , (4.4.1)

which represents the equal-time correlation of N dens i ty measurements a t

th e different^ (disjoint) points xi,X2,..- ,XN, on t he st a te of th e field S&|S)JV, eigenvector of the number operator with eigenvalue N (in the interval

L). The po in t s XI,X2,...,XN being disjoint, we may write:

p s(xi,X2,.. .,Xff,t)

= J V M * ^ ! , t) • • • &(x N, t)¥(xN ,f) • • • *(x ut)\s)N , (4.4.2)

symm etr ic under any perm utat ion of the points xi,X2,---,xjf. By inser t inga com plete set of interm edia te state s |fc), eigen states of th e num ber op era tor

"Please no te th a t th e d is jo in tness condi t ion on the space poin ts i s ap pr op r ia te for

de sc r ib ing the qua n tu m m e a su re m e n t s of num be r de ns i ty upon a n " i so la te d" sys t e m o f

N q u a n t a .




with (integer) eigenvalue k, we have

pa(x 1,x 2,...,xN,t) = Y^N(s\^{xi,t)---^J

'{xN ,t)\k)k

(k\*(xN,t)---*(xut)\s)N. (4.4.3)

From the canonical commutation relat ions (2.4.4) it is straightforward to

show tha t

[N , ¥ ( a w , t) • • • * ( a ; i , *)] = -N¥(xN , t) • • • * ( x 1 ; t). (4.4.4)

Applying this identi ty to the generic matr ix element of (4.4.3)

(k\\N,*{x ff,t)---¥(x1,t)]\s)N

= -N(k\¥(xN ,t)---*(x1,t)\s)N

= (k- N)(k\9(x N ,t) • • • *(ari,*)l*>JV , (4-4.5)

we learn that the only non-zero matr ix element in the sum (4.4.3) corre

sponds to the Vacuum state |0), allowing us towri te ps in the factorizedform

Ps(xi,x2, ...,xN ,t)= N\\ip s(xi,x2,..., xN, t) \2 , (4.4.6)

where

A{xux2, ...,xN ,t ) = -j={0\ty(xN ,t) • • • *(x ut)\s)N . (4.4.7)

We shall now show that ip s(xi, x2,..., XN , t) has al l the properties of th e

Schrodinger wave-function of the N-particle system, and therefore is th e

Schrodinger wave-function.

Firs t of all the normalization. The dis jointness condit ion is equivalent

to expressing ps{x\,x2,..., xjy, t) as the Wick-product , (: ... :) where all

\&t operators are to the left of all \I> operators:

p s(xi,X2,...,Xn,t)

= N(s\ : &(x1,t)*{xi,t)---&(x N ,t)¥(xN ,t) : \s)N , (4.4.8)

and integrating over all spatial variables one has

dx!-- dxN ps(xi,x2, ...,xN ,t ) = N\, (4.4.9)




as can be readily shown in the following way:

/ dx l---dxNN(s\^{x l,t)---^{xN,t)^(xN,t)---'^(x1,t)\s)N

= / dxi • • • dxN_iN(s\^(x1,t) • • -N • • • ^(xi,t)\s)N

= / dxi • • • c t e j V - i i v ( s | *t ( x i , t ) • • * * ( x N _ i , t )

x * ( a ^ _ i , t ) - - - ^ ( x u t ) \ s ) N , (4.4.10)

which stems from the fact that, by an argument already used, the only

non-vanishing matrix element of the type

{k\^(xN_ut) • • -^(xut^N

is for k = 1, thus \k ) is an eigenstate of the number operator N with

eigenvalue 1. By a similar argument

/dxx • • • dxN-wisl&ixut) • • • &(x N-Ui)*(a;jv-i, t) • • • ^(x ut)\s)N

= / dx 1 - - - d a ; j V _ 2 J v ( s | *t0

ci >

i) - - - *

t(

a : :w - 2 , * ) N

x &(xN-2,t) • • • *(xi,t)\s)N

= 1.2 / dxi • • • dxN-2N {s\&(xi,t) • • • &(xN-2,t)

x * ( x ; v _ 2 , t ) • • • *(x ut)\s)N , (4.4.11)

and continuing the reduction until the last variable X\, one clearly gets

(4.4.9), which through the definition (4.4.5) coincides with the normaliza

tion condition of the iV-particle Schrodinger wave-function:

/\ip s(xi,x2,. .. ,xN,t)\2dxi • • • dxN = 1 . (4.4.12)

The iV-particle Schrodinger equation easily follows from the Heisenberg

equation of motion (3.2.2), when one sets 0(t) = <& (xN,t) • • • $?(xi,t). W e

have then (recall that in our units system h — 1)

i^[*(x tf,t)---*(x1,t)] = [¥(xN,t)---*(x1,t),U\ (4.4.13)




with the Hamil tonian:

— / •— r —

H= f2 dx-?-&(-V 2)* + l I dx\ dy

x &(x, t)*(x, t)V(x - y)&{y , t)*(y, t) . (4.4.14)

An easy calculation yields for the commutator in (4.4.13)

[ * ( * * , * ) • • • * ( * ! , * ) > H ]

fc=iK

+ \j2V(xh - xk)*(xN,t) • • • * ( a : i , t ) , ( 4 . 4 . 1 5 )

h,k

plus terms which vanish when sandwiched between (0| and |s)j \r- Sandwich

ing (4.4.13) and (4.4.15) between (0| and \s)N and dividing by -4^ we

obtain precisely the Schrodinger equation:

z— ip s(xi,x2,...,xN,t)

= Hip,(xi,X2,...,xN,t)

1 d2

2mdxl

N - Q2

k=\K

+ -^2,V{xh -x k)ips(xi,x2,...,xN,t). (4.4.16)h,k

To my m ind, t he results we have ju st derived lift finally th e veil up on the

mystery of the "unreasonable" effectiveness of QM in accounting for the

reality of the microscopic world: "unreasonable" in the light of its conven-

t ional ist ic nature, impermeable to any real ist ic interpretat ion, as we have

argued in the first Section of this Essay. What has come out of our anal

ysis is that QM, with i ts postulates and dynamical equat ions, is nothingbut an approximation of QFT in the l imit of extreme "di lut ion", i .e when

in the finite volume, where we observe the field, the number of particles,

the eigenvalue of the nu m ber o pe rato r N , is a small , finite nu m be r N.

When this is the case, and it is obviously so for a diluted gas of atoms or

molecules, QM with i ts formalism and dynamical equations is a rigorous



Quantum Field Theory 6 1

consequence of the fundamental pr inciples of QFT. What is , for tunately,

completely gone is instead the bizarre Copenhagen interpretation, hope

lessly entangled with the meaningless notion of "wave-particle" duality. Inparticular the alleged "definit ive" argument against the possibil i ty that the

Schrodinger wave-function describes a physical process in real space—time

(£,£), based on the multidimensional character of the support of the JV-

particle Schrodinger wave-function (4.4.7), is seen to evaporate due to the

fact that QFT now teaches us that the wave-function is nothing but the

m atr ix element of a s tr ing of N field operators it at N disjoint space points

and a t the same t ime between the Vacuum and a par t icular s ta te of the

Q F T where the operato r N , the space (which from now on resum es i ts

three-dimensional nature) integral of the density operator \&t(a?, t)^/(x, t),

is diagonal and its eigenvalue is equal to N. In this case the square of the

norm alized wave-function [See Eq . (4.4.9)] is an ap pro pria tely no rma lized

correlation function among the measurement of the f ield density at TV dis

joint points of real space at the real t ime t. And such correlation function

has exact ly the meaning that QM at t r ibutes to i t . But where QM is now

radically supplanted is in the fact that the space coordinates x[, X2,..., afjv

in no way can be a t t r ibuted to the N indis t inguishable quantum par t ic les ,

dynamical degrees of freedom of each particle, but are in a one-to-one cor

respondence with the posit ions in space where a possible** observer cares

to put a t the t ime t his density measuring devices. Whether the system

is observed or not, the wave-function, as well as physical reality, doesn't

care, i t continues to develop in t ime according to the Schrodinger equation

(4.4.16), t t unti l i t reaches the ground state, the state of minimum energy

EQ , solution of the stationary Schrodinger equation

E 0tp s{xi,x2,...,xN) =Hijjs(xi,X2,...,xN,t). (4.4.17)

Finally a few words about "entanglement". The EPR crit ique not only of

the Copenhagen interpretation, but of the asser ted completeness of QM, is

seen to boil down to the fact that the quantum mechanical wave-function

ip(x[,X2;t) of a particular two-particle system does not factorize in the

product of two functions 4>(x~i,t)x(x~2,t), when the region where the two

quanta are localized by the observation cannot be connected by a l ightray, and are therefore "causally disconnected". The lack of factorization,

" I t i s no t abso lu te ly necessa ry th a t t he observer be rea l , i t i s im po r tan t th a t he ex i s t s

as a "gedanken " observer ,

t tTo which we need to add an appropr ia te d i s s ipa t ive t e rm, tha t r e laxes exc i ta t ion

energy to the open wor ld .




or "entanglement" between the wave-functions of each quantum, can then

be seen to pose unsurmontable problems to any real ist ic interpretat ion of

QM, or, as Einstein liked to put it, to the "element of reality" requiringthat space-like separated particles should appear to any observer endowed

with their own properties only. On the other hand if factorization fails, and

in general it does fail in QM, one finds typical correlations between the

observations on the two particles.

The answer of QFT to EPR is very simple: the particles one is dealing

with here would possess their own "element of reality" only if QM were a

fundamental theory. The failure of QM to comply with EPR 's requirement

is thus only another way to see that QM cannot be a fundamental theory, aconclusion we have reached "ad nauseam" in this Essay. On the other hand,

in QFT the quantum particles have no independent physical reality, nor a

well denned physical localization, they just correspond to the peculiarity of

the localized quantum measurement, that can only reveal the presence of

the quantum field in a "quantized" way, i .e through their quanta. Thus in

QFT the "elements of reality" only belong to the quantum field, which is

neither localized nor separable, pace of EPR.



C h a p t e r 5

F i n a l C o n s i d e r a t i o n s

As explained in the Introduct ion the motivat ion of this Essay has been

mainly philosophical: a plea for and a contribution to the great tradition

of realism, to which we owe the incredible rise of modern Science. I have

shown that in order to secure a real ist ic approach to quantum physics one

must reject the idea that QM is a complete, self-consistent theory of the

quantum world, an idea that leaves us no choice but to subscribe to the

conventional ism of the Copen hagen interpre tat ion. A nd in the last sect ion I

have dem ons trated tha t QM is a r igorous consequence of the corresponding

QFT in the l imit of "infinite dilution", where the world is only populated

by a small , finite number of quanta.

I may easily imagine the "normal" scientist of our t imes sneering at this

lat ter state m ent: Big deal! W ha t have we gained? A predict ion of new phe

no m ena ? A new set of com pu tation al rules? Only a different inte rp reta tion

of the physical meaning of the mathematical symbols of the quantum me

chanical formulae, which does not take us an inch ahead in our long journey

through energy and matter . For such is the at t i tude of the posi t ivists , who

completely dominate modern science, generally uninterested in questions of

principle, philosophy in short , and keenly concentrating on the "practical"

aspects of their activity, whose range is, however, rigorously constrained by

a r igid "paradigm", that stands, without their knowing, on a pedestal of

philosophy, including Copenhagen's brand of conventionalism.

In the history of science it is not usually recognized that all nights away

from realism, beginning with Bellarmino and the Aristotelians and end

ing with Lorentz and Poincare, though difficult to attack on purely logical

grounds, did nevertheless do harm to the progress of science, and proved in

the end their inanity. Will this not be the case of QM conventionalism as

well? I surmise that the answer to this question is positive. According to

63



64 An Introduction to a Realistic Q uantum Physics

the discussion in Sec. 3.3, if one gives up a QM description of atomic and

molecular matter in favour of a full QFT, as any realist should do, then an

easy approach to the "classical" l imit of condensed matter becomes immediately available, thro ug h simple classical equ ations of m otion , tha t exhibit

the possibil i ty of a Superradiant Phase Transit ion (SPT). One may object

that the SPT was discovered in the Dicke two-level system, where mat

ter is treated in the fashion of QM; however, the reason why its remarkable

meaning was not appreciated, and i ts existence forgotten, to my mind must

be attr ibuted to the mathematical diff iculty of the approach followed by K.

Hepp and E. Lieb. In the full QFT treatment, on the other hand, the gen

erali ty of the SPT becomes at once evident, and with i t i ts compulsoryna tu r e .

Time will tell whether the discovery of the QED coherence of condensed

matter shall induce a momentous "paradigm shif t" in our picture of the

micro/m acroscopic wor ld, and propel human ity towards a new era of m at te r

and energy. What we can state with confidence now is that the realism of

QFT is opening our eyes to a wonderful world of phenomena, which has so

far remained hidden to our view, blocked by the prejudices and the narrow

expectations of a bizarre and conventionalis t ic Weltanschauung.

I wish to thank most warmly my collaborators E. Del Giudice and

G. Lo Iacono for their invaluable help.



A p p e n d i x

This Appendix is devoted to a br ief descr iption of the mathematics and the

formalism that lay at the basis of Quantum Physics (QP), and have been

util ized throughout this Essay.

We have seen in Sec. 2 that the bir th of QP has marked a dist inct de

parture from the kinematical descr iption of Classical Physics, where States

and Observables are in one-to-one correspondence in the classical configu

ration space, the Phase Space. In QP it turns out to be absolutely neces

sary to keep the two concepts , State and Observable, dist inct and to give

them different, appropriate mathematical realizations. We have seen that

the physical reason of this strange, revolutionary fact is to be found, as is

of ten emphasized, in the interaction between observer and physical system,

its magnitude being bounded from below by the Planck 's constant h, as

expressed by Heisenberg principle.

In order to understand what kind of mathematics must be at play to

realize the fundamental quantum distinction, one may recall the essential

wave character of QP and the great mathematics that th rough the bet ter

part of the XIX Century have been developed following the luminous ideas

of Joseph Fourier. As is well known, such developments culminated in the

theory of Hilbert spaces and of l inear operators operating upon them, to

which, as we shall see in a m om ent, t he re precisely correspo nd th e configura

tion space and the observables of QP respectively. Hilbert spaces and linear

operators, the basic tools of modern functional analysis, are but the gen

eralizations to infinite dimension of the more familiar algebraic structures

of finite dimensional complex vector spaces and of the matrices operating

on them. Thus for an easier understanding of the mathematical s tructure

of QP we shall proceed with finite N-dimensional complex spaces, with

the idea of taking the limit N —> oo at the end. Such limit poses several

65



66 An Introduction to a Realistic Qu antum Physics

subtle problems that have t roubled and occupied the mathematicians for

a large part of this Century, but as once reminded us Norbert Wiener, in

no way this should trouble and occupy the physicist , for he has the greatprivilege to have Nature as a kind of custodian angel to prevent him from

making mistakes and wasting his t ime. And, besides, i t is a fact that in the

m athe m atical descript ion of Na ture, tha t a physicist can make, zero and i ts

inverse, infinity, do not really exist, for they always imply an extrapolation

into uncharted terr i tory.

Suppose now that our quantum system ca n be found in N different

s ta tes , then the fundamental postulate of QP is that to each such state

one associates a complex N-dimensional vector v = (ci , C 2 , . . . , c?{) (cfc arecomplex numbers) that , for reasons that soon wil l become apparent , Dirac

baptized ke t and represented as

v - > » (A .l )

Before we go on, le t me emphasize that the above quantum postulate

is really revolutionary for i t implies, according to the algebraic structure of

a vector space, that if |vj) and \v2) are two such states (0:1,2 are complexnumbers)

( a i c p + a2cY>,..., aicff +a2e$) - » a i | u i ) + a2\v2) (A.2)

is also a possible sta te, thus realizing th e fund am ental

superposition principle.

The next assumption is that our N-dimensional complex vector space is

a unitary space, i .e. given any two vectors \vi) and \v2), one can define the

scalar product

E41)

*42) = ^ih ) = (^h)*, (A.3)

which implies that to each vector v one associates i ts dual v* =

[c 1,c 2, . • • ,c*N)

v* -> <v|, (A.4)

Dirac 's bra, so that the scalar product according to (4) has the structure

of a bra-ket, in English bracket. In order for the scalar product to have a

physical meaning, i t must have some kind of invariance, and in fact one

imposes that all allowed linear transformations U = \\Uij\\ t ha t map the



Appendix 67

vector space onto itself leaves the scalar product (3) invariant, i.e. is unitary.

In Dirac's symbolism this means that

(vilv*) = {v l/fa), (A.5)

where U^ = | |?7^|| is the Hermitian conjugate of U, which means that a

uni tary t ransformation obeys

U& = WU = 1. (A.6)

Physically the requirement of unitarity is necessary for the probabili ty in

terpretat ion of QP, as we shal l see in a moment . For the t ime being theinvariant scalar product allows us to normalize the physical states, i .e. to

impose that for any physical state

(v\v) = 1. (A.7)

This requirement implies that in the general l inear superposition:

|v} = ^ a* h> fc ) (A.8)

k

th e ak must satisfy

(v\v)=J2ah

ak(vh\vk) = l . (A.9)

kh

Furthermore the invariant scalar product gives us the possibili ty to define

the (N-dimensional) orthonormal basis {\vk)} for which

(v h\v k)= 8hk . (A.10)

so that any state |i») can be given the representation (8) with the condition

N

Y,<*tak = l. (A.ll)fc=i

It is easy to check that given an orthonormal basis, any other such basis

can be obtained from it by a unitary transformation. In fact let (\wk)) be

such a basis, with, \w k) = U\vk), t hen

{w h\w k) = (vh\UfU\v k) = (vh\v k) = 5hk . (A.12)

This much for the quantum States. Let us now turn our at tent ion to

the quantum Observables. The assumption is that to each Observable there




corresponds an Hermitian operator O, i .e . a l inear operator such that

0 + - 0 . (A.13)

This requirement insures that i ts eigenvalue spectrum is real and its eigen

vectors form an orthonormal basis. In fact the (normalized) eigenvector |o),

with eigenvalue o of the Observable O is defined by the equation

0 |o> = o|o) (A.14)

whence

( 0 | O | o ) = o , ( A . 1 5 )

o = (o\0\o) = <o|O t |o) = o* , (A.16)

showing that the eigenvalue is real. Thus, given two eigenvectors \oi) and

|o2) with different eigenvalues one has

O I , 2 = O I J 2 | O I , 2 > . (A.17)

Taking

( o 2 | 0 | o i ) = o i ( o 2 | o ! ) = o 2 ( o 2 | o i ) , (A.18)

one immediately derives the orthogonality of the two eigenvectors, i .e.

<o a |oi )= 0 . (A.19)

If the eigenvectors have the same (real) eigenvalue, one can by a standardprocedure (Schmidt 's or thogonalization) form appropriate l inear combina

tions of th e two so th at th e new eigenvectors are orthog on al. In fact supp ose

th a t (02I01) 7^ 0, th en |oi) an d

M _ w - y < « . w (A.20)V l - | ( o i | o 2 ) |

2

are easily shown to be or tho norm al. Thu s: the eigenvectors of an He rmitian

operator form an orthonormalbasis and the spectrum of its eigenvalues is

real. This result is central for the physical interpretation of the mathemat

ics of QP; indeed if the system is in a State described by the eigenvector

\o), the observation (measurement) of the Observable O will yield for sure

the value o, which can be expressed as the expectation value (o|0 |o) of O



Appendix 69

upon |o), as follow from (16). Let's now take a generic State \tp), given by

the linear superposition of the orthonormal basis of the Observable O

N

i t = i

what is the meaning of the expectation value (ip\0\ip)? We have

(ip\0\iP) = ^ ( o h | 0 | o k ) < c k = J ] o k | c k |2

, (A.22)

h k k

i .e. i ts value depends upon the square modulus of the (complex) superpo

sition coefficients Ck, whose sum is according to (11) normalized to 1. Thus

it is natural, and corroborated by experiments, to interpret |cfc|2

as the

probabil i ty that a measurement of O in the state \ip) yields the value Ok •

In this way the expec tat ion value of O upo n \ip) represents the average value

of O when a large number of measurements are performed on that State .

T he revolu tionary consequence of all this is, of course, th at wh en th e s ystem

is in the S tat e IV'), which is no t an eigenvec tor of O, th is O bse rvab le h as

no definite, sh arp value but only an average value (when a large n um ber

of measurements is collected), given by its expectation value (22). However

a single measurement always gives some well defined value o^, belonging

to the spectrum of i ts eigenvalues, the probability of this happening being

precisely |cfe|2. This bizarre, but real fact is usually referred to as the reduc

tion of the wave-function and has caused several headaches to those who

wish to give a realistic interp reta tion of QP . We now know, as I have arg uedat length in this Essay, that such reduction is the inescapable consequence

of the non-disposable presence of the observer, without whom no physical

meaning can be given to the Observable itself. As emphasized in the text ,

this looks really weird, but such is the architecture of the Universe!

One last aspect of the general description of a quantum system is how

to label uniquely the N kets of an orthonormal basis. We know in general

that given an Hermit ian operator O, i ts orthonormal basis defines a set of

subspaces of the original vector space, inside each of which its eigenvaluesdo not change. The problem of liftingthedegeneracy, i.e. of redu cing ea ch

subspace to a one-dimensional one, is solved by finding a complete set of

commuting operators, as Dirac calls i t . Indeed given two O bservables O i

and O2, they possess common eigenvectors if and only if they commute, i .e.

if [Oi, O2] = 0, as can easily be seen. Thus: let \a ) be one such eigenvector,




then

O i , a | a > = o i , a |a> (A.23)

[ O i , O a] | a> = (OiOa - 0 2 O x ) | a ) = ( 0 l o 2 - o 2 0 l ) | a ) = 0 , (A .24)

thus

[ O 1 , O 2 ] = 0 . (A .2 5)

As a result we may set \a) = \01O2), and if the orthonormal basis defined

by the commut ing O i i 2 does not lift the degeneracy, one continues until

the commuting Observables O I , 0 2 , . . . , O N do so. At this point we are

guaranteed that any other commuting Observable must be a l inear com

bination of the complete set, and the or thonormal kets \o\, o 2 , • • •, on) are

thus uniquely labeled.

Let us finally address the limit N -> 00. That this must be the general

case of a quantum system is easily understood by the fact that there ex

ist Observables with unbound and/or cont inuous spectra, and as in QFT

the number of independent commuting Observables is infinite. Thus when

passing from th e general finite trea tm en t I have jus t pres ente d to the l imit

of Hilbert spaces proper we do not encounter new concepts, only new math

ematical objects. Let us now see briefly what they are. The simplest case is

th at of a discrete unbound eigenvalue spectrum , which correspond to taking

the infini te l imit , that remains denum erable. In this case no new m ath em ati

cal object arises, only the discrete indices h,k ... span the infinite countable

set of non-negative integers. Things are different when among the Observ

ables of a complete commuting set there exist some that have a cont inuous

eigenvalue spectrum. This happens, for instance, to the momentum oper

ator of a quantum particle or field, defined in an unbound spatial region.

T h e trick th a t the physicist does in this case, as a consequence of his horror

infiniti, is to first discretize the cont inuum by introducing a fundamental

length a, and keeping in the interval (o 0 , on) only those points given by

Ofc=fca + oo I k = 0 , 1 , . . . , n ; —= a

) (A.26)

In this way one recovers the discrete case and all the above developments.

On the other hand the length a can be thought arbi t rar i ly small , and the

continuum limit is retrieved for a —> 0. Thus we must figure out how to

represent in this limit operations such as ^X^fc/fc>o r

symbols like 8hk]

modern functional analysis (and Dirac himself) gives the answer. We have

http://01o2/

http://01o2/

http://01o2/



Appendix 71

clearly

- £ / * = — V > / f c -> - ^ — fn dof(o). (A.27)

On the other hand for the Kronecker delta, from the identi ty:

Y,Shkfk = h (A.28)

k

we have

or

-+ J(o - o ') , (A.30)

the celebrated Dirac delta function, a strange function (indeed a distribu

tion) whose support is a single point , and being at that point infinite in

such a way that the integral

f do5{o - o') = 1. (A.31)

But for our purposes i t is t ime to stop here.



B i b l i o g r a p h y

P.A. Schi lpp Albert Einstein, Philosoph er Scientist, New York , (ed . ) Tudor Publ .

C o. (1957).

R.P. Feynman, R.B. Leigh ton , M. Sands , Feynman Lectures on Physics, Addison-

Wesley, Reading, Mass. (1965).

W. Nerns t , The New Heat Theorem, Methuen , London, 1926 .

A. Eins te in , Sitzungher. Klg. Preuss. Akad. Wiss. 1924 , 261 (1924); S.N. Bose,

Z. Phys. 26, 178 (1924).

P.A.M. Dirac , The Principles of Quantum Mechanics., Clarendon Press , Oxford ,

1958.

A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 4 7 (1935) 777.

S.D. Bjorken, S. Drell , Relativistic Quantum Fields, McGraw-Hil l , New York,

1965.

S. Cacciatori , G. Preparata, S. Rovel l i , I . Spagnolat t i , S.S. Xue, On the Ground

State of Qu antum Gra vity, Phys. Lett. B 4 2 7 (1998) 254.

E . T . W h i t t a k e r , A T reatise on the Anaytical Dyna mics of Particles and R igid

Bodies , Cambridge Universi ty Press, New York, 1970.

P.A.M. Dirac , Phys. Zeit. Sovietunion 3, 312 (1933).R .P . Feynman , Rev. Mod. Phys. 20, 267 (1948).

G. Prepa ra t a , / / Nuovo Cimento 96A (1986), 366.

G . P r e p a r a t a , QED Coherence in Matter, World Scientific, Singapore, 1995.

K. Hepp, E. Lieb , Ann. Phys. 76, 360 (1973); Phys. Rev. A 8 , 2517 (1973).

N . N . Bogoliubov, On the theory of Superfluidity, Izv. Akad. Nauk. SSSR. Ser.

Fiz. 11 (1957), 77; Soviet Phys. Jept 7 (1958), 41.

introduction to a realistic quantum physics

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