field quantization and wave particle duality

Field quantization and wave particle duality

M. Cini

Dipartimento di Fisica, Roma Istituto Nazionale di Struttura della Materia,

Universit�aa La Sapienza, Rome, Italy

Received 6 November 2002

Abstract

We owe to Pascual Jordan the first formulation of a theory of quantized fields, in the

framework of Heisenberg�s matrix mechanics. For Jordan it is quantization which creates par-

ticles, both photons and electrons. The purpose of the present paper is to show that a coherent

development of Jordan�s program leads to a formulation of quantum field theory in terms of

ensemble averages of the field�s dynamical variables, in which no reference at all is made to the

Schr€oodinger wave functions of ‘‘first quantization.’’ In this formulation the wave particle du-

ality is no longer a puzzling phenomenon. The wave particle duality is instead, in this new per-

spective, only the manifestation of two complementary aspects (continuity vs. discontinuity) of

an intrinsically nonlocal physical entity (the field) which objectively exists in ordinary three-

dimensional space. This theory, in which the field�s statistical properties are represented by

Wigner pseudoprobabilities deduced without any reference to Schr€oodinger wave functions,

is based on two postulates. The first one is the requirement of invariance under canonical

transformations of the probability distributions of a classical statistical description of the

field�s state. This invariance leads to an uncertainty relation for the conjugated variables of

the field�s oscillators. The second postulate is quantization. This means to assume that the in-

tensity of each monochromatic wave should only have discrete values instead of the continu-

ous range allowed by the classical theory. These two postulates can be satisfied only if the

system�s physical variables are represented by noncommuting numbers. In this way what is

generally assumed as a basic mathematical postulate in the standard formulation of quantum

mechanics, follows from the physical postulates of the theory.

� 2003 Elsevier Science (USA). All rights reserved.

Annals of Physics 305 (2003) 83–95

www.elsevier.com/locate/aop

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0003-4916/03/$ - see front matter � 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0003-4916(03)00042-3

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1. Introduction

We owe to Pascual Jordan [1], the first formulation of a theory of quantized fields,

in the framework of Heisenberg�s matrix mechanics. No wave function existed at

that time, because Schr€oodinger�s paper had not been published yet. For this reasonJordan�s contribution is particularly important. Actually, the concept of wave intro-

duced by de Broglie and developed by the Austrian physicist, in spite of its undispu-

tably central role played in giving to quantum mechanics its universally accepted

form, opened also the way to endless discussions which are still going on. At the or-

igin of wave mechanics there was in fact a misleading analogy: namely the extension

of Einstein�s relations E ¼ hm and p ¼ h=k for photons, where m and k are the fre-

quency and wavelength of an electromagnetic wave, to a wave of unknown nature

accompanying the motion of a particle. The completely different nature of the twowaves was later clarified by Born�s interpretation of the latter in terms of ‘‘probabil-

ity waves’’; at the heavy price, however, of introducing a concept whose physical na-

ture and meaning are still totally obscure.

Dirac [2] founded quantum electrodynamics a few months later by introducing a

‘‘second quantization’’ of the Schr€oodinger amplitudes, which had been in the mean-

time accepted as the basic tool for the ‘‘first quantization’’ of mechanics. One had to

interpret single particle Schr€oodinger waves as classical dynamical variables, which

were then quantized according to the rules of noncommuting quantum canonicalvariables, and to introduce successively Schr€oodinger probability waves defined in a

many particle system�s abstract space in order to compute the transition probabilities

between its different states. The reason for this cumbersome procedure was that Di-

rac wanted to describe the particlelike properties of photons as bosons, ignoring to

start with their nature of quanta of the electromagnetic field. Only at the end of his

paper the connection with the classical waves of this field is introduced. He was well

aware that a confusion might arise between the physical meaning of the two kinds of

waves, but in his case this confusion could be dissipated by keeping in mind their dif-ferent nature.

For Jordan, on the contrary, it is quantization which creates particles, both pho-

tons and electrons. According to him, therefore, rather than trying to explain phe-

nomena like diffraction and interference of single particles as properties of

‘‘probability waves’’ one should simply view them as primary properties of the field

of which they represent the quanta. The advantage becomes even more evident when

dealing with systems of two or more particles. In this case in fact one is forced, ac-

cording to the accepted wisdom, to interpret the physically observable wavelikeproperties of the system as a manifestation of waves defined in an abstract multi-di-

mensional space, existing perhaps only in the ‘‘observer�s’’ mind. From Jordan�spoint of view, instead, the wavelike behaviour of any field�s state with any number

of discrete quanta simply reflects the property of a physical nonlocal entity which ex-

ists objectively in ordinary three-dimensional space.

The purpose of the present paper is to show that a coherent development of Jor-

dan�s program leads to a formulation of quantum field theory which eliminates all

the problems raised during many decades of debates on the paradoxical aspects of

84 M. Cini / Annals of Physics 305 (2003) 83–95

quantum mechanics. In this formulation the wave particle duality is no longer a puz-

zling phenomenon. In order to carry on this program one should start from the obvi-

ous statement that this dual nature of the field implies a probabilistic description of its

properties from the beginning. Therefore the states of the field should be represented

by means of statistical ensembles in the phase spaces of its normal modes. In this de-scription only quantities endowed of physical meaning such as average values of the

field�s variables and of their correlations in any given state of the field should appear.

To this purpose an extension to field theory of the formulation of quantum me-

chanics in phase space introduced by Wigner [3] in 1932 is the appropriate tool.

In this formulation all the statistical predictions of the theory can be obtained with-

out recurring to the procedure, invented by Born, of taking the squared modulus of

‘‘probability wave’’ amplitudes. These predictions can instead be obtained directly

by means of the Wigner pseudoprobabilities which are the natural quantum exten-sions of the corresponding classical probability distributions when due account is ta-

ken—as argued convincingly by Feynman [4]—of the constraints imposed by the

uncertainty principle. In practice, however, the procedure proposed by Wigner in or-

der to obtain the pseudoprobability of a given quantum mechanical state is marred

with an essential drawback: it needs the preliminary derivation, by means of a

Schr€oodinger equation, of its wave function. This means that Jordan�s program can-

not be implemented by simply extending to the states of a field in this procedure.

In order to eliminate this drawback it is necessary to derive directly from firstprinciples, without having to recur to the previous determination of Schr€oodingerwave functions, the Wigner function of the field�s state, as a function in the phase

spaces of its normal modes. The main result of the present paper is to show that this

is indeed possible. This goal is attained by extending to classical fields the same two a

priori requirements—uncertainty principle and energy quantization—introduced in

a preceding paper [6] where it was shown that the Wigner functions of the states

of the one-dimensional motion of a single particle can be directly derived without

ever introducing Schr€oodinger wave functions.The paper is organized as follows. In Section 2 the statistical description of the

state of a classical field satisfying the uncertainty principle is introduced. In Section

3 the quantization of the free field�s oscillators is formalized. In Section 4 the prop-

erties of some standard ensembles of field�s quanta are discussed. In Section 5 the

emission and absorption of quanta from a source is treated. Some conclusions are

drawn in Section 6.

2. The statistical description of the state of a classical field satisfying the uncertainty

principle

Following Jordan, we take the simplest possible classical field represented by a

one-dimensional vibrating string of length l, whose motion is represented by the dis-

placement uðx; tÞ given by

uðx; tÞ ¼Xr

qrðtÞ sinxrx; xr ¼ pr=l; r ¼ 1; 2; . . . ð1Þ

M. Cini / Annals of Physics 305 (2003) 83–95 85

The Hamiltonian is

H ¼ ð1=2ÞZ l

0

dx½ðou=otÞ2 þ ðou=oxÞ2� ¼ ð1=2ÞXr

½ð2=lÞp2r þ ðl=2Þx2rq

2r �; ð2Þ

where

pr ¼ ðl=2Þðoqr=otÞ: ð3ÞEq. (2) expresses the conventional superposition of normal modes as an assembly

of harmonic oscillators of different frequencies and �mass� l=2. It is useful to intro-

duce from the beginning, in place of the conventional position and momentum vari-

ables, the complex variables ar; a�r expressed in terms of each wave�s amplitude N1=2

r

and phase hr by means of

qr ¼ ð�h=lxrÞ1=2ðar þ a�r Þ; pr ¼ ð1=2iÞð�hlxrÞ1=2ðar � a�r Þ; ð4Þ

ar ¼ N1=2r expð�ihr=�hÞ; a�r ¼ N1=2

r expðihr=�hÞ: ð5ÞThe constant �h has the dimension of an action and is introduced here to make

ar; a�r a dimensional. It will turn out, after quantization, to be equal to Planck�s con-

stant over 2p.In terms of these variables the Hamiltonian (2) takes the form

H ¼Xr

�hxra�rar ¼

Xr

�hxrNr: ð6Þ

We introduce now a statistical ensemble for each radiation oscillator r defined by

the constraint that the intensity Nrðq; pÞ has with certainty a given value vmr. The dis-

tribution function in phase space Pmðqr; prÞ of each ensemble will be given by (the suf-

fix r is omitted in the following in order to simplify notation):

Pmðq; pÞ ¼ hdðq� qÞdðp � pÞim

¼ ð2p�hÞ�2

Z Zdxdk eð�i=�hÞðkqþxpÞheði=�hÞðkqþxpÞim; ð7Þ

where h im represents the average over the variables p; q with the constraint that

Nðq; pÞ has the value m. The function heði=�hÞðkqþxpÞim of k; x is the characteristic function

of the ensemble m introduced by Moyal in [5] and we denote it by Cmðk; xÞ.Since the average of any field variable Mðq; pÞ in the ensemble m can be expressed

in terms of the characteristic function Cmðk; xÞ, it will be sufficient to write and solve

the equations satisfied by this function for any given ensemble in order to obtain all

the statistical properties of the field. If all the systems have the same value m of theintensity N it must be that (r always omitted)

hN2im ¼ m2; ð8Þwhere

m ¼ hNim ¼Z Z

dkdxnðk; xÞCmðk; xÞ; ð9Þ

where nðk; xÞ is the double Fourier transform of the function Nðq; pÞ.


It is easy to see, by making use of the property

eði=�hÞðkqþxpÞeði=�hÞðk0qþx0pÞ ¼ eði=�hÞ½ðkþk0Þqþðxþx0Þp� ð10Þ

that in order to satisfy Eq. (8) the function Cmðk; xÞ must satisfy the relationZ Zdy dhnðh� k; y � xÞCmðh; yÞ ¼ mCmðk; xÞ: ð11Þ

Eq. (11) is a homogeneous integral equation for the determination of the eigen-

values m of N and the corresponding eigenfunctions Cmðk; xÞ. Its solutions can be im-

mediately obtained from its inverse double Fourier transform. In terms of Nðq; pÞand of Pmðq; pÞ Eq. (11) shows to be no longer an integral equation but a simple al-

gebraic functional equation:

Nðq; pÞPmðq; pÞ ¼ mPmðq; pÞ; ð12Þwhich has the solutions

m ¼ Nðq; pÞ; ð13Þ

Pmðq; pÞ ¼ fmðhðq; pÞÞdðNðq; pÞ � mÞ ð14Þwith f ðhÞ an arbitrary function of the phase variable h conjugated to N . This arbi-

trariness reflects the fact that theremay be an infinity of ensembles inwhich the variable

Nm has the value m and h any probability distribution. Eq. (13) implies that, given a

couple of values q; p of the variables q; p, the variableN has necessarily the value m givenby (13). This seems a trivial statement but it will turn out to be essential later.

We impose now the requirement that the distribution (14) should be invariant un-

der canonical transformations of the form

N0 ¼ Nþ �fN;MgPB; ð15Þwhere fN;MgPB is the Poisson Bracket of N with any arbitrary variable M, namely

that

hfN;MgPBim ¼ 0: ð16ÞIt is easy to see that only if Pm does not depend on h, namely if ðoPm=ohÞ ¼ 0, Eq.

(16) holds. This means that only if f ¼ constant the required invariance holds. In this

particular classical ensemble the variable h can assume any value with equal proba-

bility and is therefore completely undetermined.

From Eq. (16) it follows immediately that, for the dispersion free ensemble in

which N has the value m and h is completely undetermined, the characteristic functionsatisfies, in addition to (11), also the equationZ Z

dy dhnðh� k; y � xÞðky � hxÞCmðh; yÞ ¼ 0 for all k; x: ð17Þ

Invariance under canonical transformations ensures that Eq. (17) is valid for any

variableM. Eq. (17) yields therefore the formal expression of a ‘‘classical uncertainty

principle,’’ representing the condition to be fulfilled by classical ensembles having the

property that when a given variable M has the value l its conjugate variable is un-


determined. Only the distribution functions of these ensembles are invariant under

canonical transformations. Conversely, if we impose that the characteristic function

of an ensemble satisfies Eqs. (11) and (17) we select only the ensembles in which the

‘‘uncertainty principle’’ is satisfied.

3. The quantization of the field oscillators

Our procedure of field quantization will be based on the assumption of the existence

of discrete field quanta. More precisely we assume (Quantum Postulate) that the spec-

trum of the quantum variable NN of each field oscillator should be discrete. This feature

can only be ensured if Eq. (11), which, according to (13) yields a continuous spectrum nfor the eigenvalues of the classical variable N, is modified to become a true Fredholmhomogeneous integral equation with a nonseparable kernel, allowing for the existence

of a discrete set of eigenvalues mn (the index n, labelling the discrete set of the eigenvalues

of NN, should not be confused with the suffix r, introduced previously to distinguish the

different radiation oscillators). Eq. (11) should be replaced therefore byZ Zdy dhnðh� k; y � xÞgðky � hxÞCmðh; yÞ ¼ mnCmðk; xÞ; ð18Þ

whereCmðk; xÞ is now the quantum characteristic function of the ensemble with NN ¼ mn.The functional dependence on k; x and h; y of the nonseparable part of the kernelgðky � hxÞ is justified [6] by dimensional requirements and by the limits for k ¼ x ¼ 0

and k ¼ h, y ¼ x. It should be stressed that gðky � hxÞmust be universal, namely that it

should be independent of the variable considered and of the ensemble chosen. In order

to transport in the quantized theory the functional dependence of the classical variable

N of the variables q; p, the function nðk; xÞ remains the same double Fourier transform

of the classical variable Nðq; pÞ. We shall see in a moment that for more general vari-

ablesMðq; pÞ also the expressionmðk; xÞ should be modified in going from the classical

to the quantum expression. The modification, however, is unambiguously defined,without extra assumptions, once that the function gðky � hxÞ (together with the

function f ðky � hxÞ of Eq. (20)) will have been determined.

Eq. (18), however, does not follow from Eq. (8), which still should hold in quan-

tum theory in order to preserve the physical meaning of the ensembles, if we main-

tain the validity of the rule (10) for the multiplication of exponentials. The only way

to obtain (18) from (8) is therefore to replace the exponentials in Eq. (10) of the clas-

sical variables q; p, with functions CCðk; xÞ (depending on the parameters k; x) of newquantum variables qq; pp, with the property

ð1=2Þ½CCðk; xÞCCðh; yÞ þ CCðh; yÞCCðk; xÞ� ¼ gðky � hxÞCCðk þ h; xþ yÞ: ð19ÞThis equation, however, cannot be satisfied by ordinary numbers. This means

that, if we want to allow for the existence of discrete values of the intensity NN we

are forced to represent the quantum variables by means of noncommuting quantities(Dirac�s q-numbers). Therefore the mathematical nature of the quantities needed to

represent the quantum variables is a consequence of the physical property of the ex-


istence of field quanta, and not vice versa, as the conventional view of reality under-

lying the axiomatic formulation of Quantum Mechanics assumes.

We only need, however, to assume for the moment that these new variables CCðk; xÞexist and that (19) is satisfied.Weneednot give nowany explicit representationof them,

because on the one hand the physical meaning of their averages Cmðk; xÞ is by definitionthe same as their classical counterparts heði=�hÞðkqþxpÞim and on the other hand the explicit

expressions of the functions Cmðk; xÞwill be derived by solvingEq. (18) togetherwith the

quantum generalization of Eq. (17) which we will now proceed to write down.

The next step is therefore to find the quantum correspondent of Eq. (17). It is easy

to see that, in order to ensure the invariance under canonical transformations of (8)

and (15), not only the multiplication rule (10) should be changed to the form (19),

but also that the classical Poisson Bracket appearing in Eq. (15) should be replaced

by a Quantum Poisson Bracket modified in the form

fCCðk; xÞ; CCðh; yÞgQPB ¼ f ðky � hxÞCC½ðk þ hÞ; ðy þ xÞ� ¼ 0; ð20Þ

where f ðkÞ is an odd function of its argument, satisfying, for consistency with the

classical Poisson Bracket, the condition limk!0 ¼ �k=�h2.

From (15) and (20) we now obtain immediatelyZ Zdy dhnðh� k; y � xÞf ðky � hxÞCmðh; yÞ ¼ 0: ð21Þ

This is the required generalization of Eq. (17).The final step to complete our formalism is the determination of the functions

f ðkÞ and gðkÞ. This goal has been attained in [6] by imposing the condition that both

relations (18) and (21) should be invariant under the canonical transformations gen-

erated by the QPBs. This condition leads to the following solutions:

gðky � hxÞ ¼ cos½ðky � hxÞ=2�h�; f ðky � hxÞ ¼ ð2=�hÞ sin½ðky � hxÞ=2�h�: ð22ÞIt is remarkable that the variables CCðk; xÞ with the property (19) turn out to be the

same exponentials appearing in (10):

CCðk; xÞ ¼ eði=�hÞðkqqþxppÞ; ð23Þwhere the classical q and p are replaced by quantum variables qq and pp satisfying the

conventional commutation relations

½qq; pp� ¼ i�h ð24Þof the variables of Quantum Mechanics.

In fact one easily checks that

eði=�hÞðkqqþxppÞeði=�hÞðhqqþyppÞ þ eði=�hÞðhqqþyppÞeði=�hÞðkqqþxppÞ

¼ 2 cos½ðky � hxÞ=2�h�eði=�hÞ½ðkþhÞqqþðxþyÞpp�

� eði=�hÞðkqqþxppÞeði=�hÞðhqqþyppÞ � eði=�hÞðhqqþyppÞeði=�hÞðkqqþxppÞ

¼ ð2iÞ sin½ðky � hxÞ=2�h�eði=�hÞ½ðkþhÞqqþðxþyÞpp�: ð25Þ

Eq. (25) yields Eq. (19) and gives the standard expression (20) for the QPBs.


Once that this correspondence has been established, the double Fourier transform

mðk; xÞ of any quantum variable MMðqq; ppÞ can be worked out using (25), by inverting

the definition

MMðqq; ppÞ ¼Z Z

dxdkmðk; xÞeði=�hÞðkqqþðxppÞ: ð26Þ

Eqs. (23)–(25) show that our theory reproduces all the results of the standard for-

mulation of Quantum Mechanics. All the statistically meaningful predictions for

physical quantities obtained with the approach introduced here coincide with those

obtained with the methods of the standard theory, while the mathematical entities

which have no direct statistical significance have disappeared.

4. The statistical ensembles of field quanta

It is convenient to switch now from the quantum variables qq and pp to the variables aa

and aa� by means of the quantum equivalent of Eq. (4), and from k; x and h; y; b; b� and

c; c� by means of the same relations. In order to write down Eqs. (18) and (21) in terms

of these variables we have to work out the expressions of nðk; xÞ and gðky � hxÞ;f ðky � hxÞ in terms of the new variables. From Eq. (6) one finds

nðb; b�Þ ¼Z Z

daadaa�aa�aae�baa�þb� aa ¼ 2ðd=dbÞðd=db�Þdðb � b�Þdðb þ b�Þ ð27Þ

and

gðky � hxÞ ¼ cos½ðbc� � b�yÞ=2i�; f ðkx� hyÞ¼ ð2=�hÞ sin½ðbc� � b�yÞ=2i�: ð28Þ

Eqs. (18) and (21) become thereforeZ Z

dcdc� nðb � c; b� � c�Þgðbc� � b�cÞCmðc; c�Þ

¼ ðbb�=4ÞCnðb; b�Þ � ðd=dbÞðd=db�ÞCmðb; b�Þ ¼ mCmðb; b�Þ;Z Zdcdc� nðb � c; b� � c�Þf ðbc� � b�cÞCmðc; c�Þ

¼ bðd=dbÞCmðb; b�Þ � b�ðd=db�ÞCmðb; b�Þ ¼ 0:

ð29Þ

These equations can be easily solved to give the eigenvalues mn of the quantum

variable NN and their characteristic functions Cnðb; b�Þ:mn ¼ nþ ð1=2Þ; ð30Þ

Cnðb; b�Þ ¼ e�ðbb�=2ÞXn

k¼0

ckðbb�Þk; ckþ1 ¼ ckðk � nÞ=ðk þ 1Þ2; c0 ¼ 1: ð31Þ

Eq. (30) can be interpreted by saying that each radiation oscillator has n quanta of

energy �hx with a zero energy ð1=2Þ�hx when there are no quanta present. This result


(30) is expected, but remarkable, because it has been obtained by solving the integral

equations derived, without ad hoc assumptions, from those for the characteristic

function of a statistical ensemble of classical statistical mechanics (in which the in-

tensities of the radiation oscillators are given and their phase is completely undeter-

mined), modified in order to satisfy the requirement that these intensities shouldhave only discrete values. It should be stressed again that Schr€oodinger waves are

not needed in this derivation.

The state of the field in which each radiation oscillator has a given number nr ofquanta is therefore represented by the characteristic function

CEða1; a2; . . .Þ ¼Yr

Cnrðar; a�r Þ; E ¼

Xr

�hxr½nr þ ð1=2Þ�: ð32Þ

By means of (32) the expectation value of any quantum field variable

Mðqq1; pp1; qq2; pp2; . . .Þ can be obtained by means of

hMMin1;n2;... ¼Yr

Z Zdbr db�

r Cnrðbr; b�r Þmðb1; b

�1; b2; b

�2; . . .Þ; ð33Þ

where mðb1; b�1; b2; b

�2; . . .Þ has to be worked out, as already indicated in Eq. (26), by

inverting the definition

Mðaa1; aa�1; aa2; aa�2; . . .Þ ¼Yr

Z Zdbr db�

r mðb1; b�1; b2; b

�2; . . .Þebra

�r�b�r ar ð34Þ

and taking into account the commutation relations of the variables aar; aa�r .

It is also easy with this formalism to cope with the field�s coherent states. Eqs. (18)

and (21) with the change of variables from k; x to b; b� allow immediately to work

out the characteristic variables Caðb; b�Þ of each radiation oscillator�s ensemble in

a coherent state, defined as a state in which the variables aa and aa� are defined to have

given classical values a and a� with the minimum Heisenberg uncertainty. In fact,

from Eq. (4) we obtain the double Fourier transforms Cðb; b�Þ and Cðb; b�Þ of aaand aa� in the form

Cðb; b�Þ ¼ e�ðbrb�r =2Þðd=db�ÞdðbÞdðb�Þ;C�ðb; b�Þ ¼ �e�ðbrb�r =2Þðd=dbÞdðbÞdðb�Þ:

ð35Þ

The eigenvalue equation (27) with nðb; b�Þ replaced by Cðb; b�Þ and C�ðb; b�Þ yields

immediately

Ca;a� ðbr; b�r Þ ¼ C0;0ðbr; b

�r Þeab��ba� ; ð36Þ

where the characteristic function of the vacuum C0;0ðbr; b�r Þ (corresponding to

a; a� ¼ 0) is given by Eq. (31)

C0;0ðbr; b�r Þ ¼ e�ðbrb�r =2Þ: ð37Þ

One easily checks that (33) and (34) give the correct values for all the moments of

statistical distribution for the number of photons in the coherent ensemble. In fact, ifwe define the number of photons nn ¼ NN� ð1=2Þ we have


hnnia;a� ¼ aa� ð38Þ

and

hnn2ia;a� ¼ ðaa�Þ2 þ aa�: ð39Þ

This is the conventional result. By the way, Eq. (39) was found by Jordan to confirm

Einstein�s result of the energy fluctuations of the radiation energy in cavity. Higher

moments can be similarly worked out.

5. Emission and absorption of photons

Our last task is to reproduce with our formalism the results obtained by Dirac in

his seminal paper on the foundations of quantum electrodynamics. As recalled in the

introduction, absorption and emission probabilities of photons by an atom are de-

rived in that work by treating the photons as bosons which can be created and de-

stroyed in the evolution of a state described by a wave function wðN1;N2; . . . ;NrÞ in

the Fock space of the numbers of photons Nr generated by the Hamiltonian

HH ¼ HHs þ HHf þ HHint ¼ HHs þXr

�hxraa�r aar þ

Xr

ðVraar þ V �r aa

�r Þ: ð40Þ

In (40) HHs is the Hamiltonian of the source and Vr; V �r are functions of its vari-

ables. The ‘‘second quantization’’ creation and destruction operators aa�r ; aar are ex-pressed in terms of operators NNr and HHr by the same relations (5) introduced

before for the amplitudes of the field�s normal modes. It turns out that the absorp-

tion rate is proportional to Nr and the emission rate to Nr þ 1 (Einstein�s laws).

We will start with the same Hamiltonian (40), where the variables aa�r ; aar are now

our field�s quantum variables. Since we will be interested only in the field�s behaviourwe will not specify further the source�s variables. In our formalism, of course, we do

not have wave functions or state vectors, and we deal only with expectation values of

the field variables. We have to use only Eqs. (18), (20), (21), and (24). We chose toevaluate the rate of variation of the number of photons for a given mode, assuming,

for simplicity, that only n1 is initially equal to a given number n, all the other ones

being zero. The initial characteristic function of the field�s state CE0ða1; a2; . . .Þ will

be therefore given by (32) with n1 ¼ n and all the other nr equal to zero.

The rate of change RRi of Ni will be given by (we omit the index)

RR ¼ ðd=dtÞNN ¼ fNN; HHintgQPB ¼ ði=�hÞð�V aaþ V �aa�Þ ð41Þ

while its expectation value will be

hRRii ¼Z Z

dada� rða; a�ÞC1ða; a�Þ; ð42Þ

where rða; a�Þ is the representative

rða; a�Þ ¼ �eðaa�=2½vðd=da�Þ þ v�ðd=daÞ�dðbÞdðb�Þ ð43Þ


of RRi in the space a; a� (we omit all the explicit treatment of the source�s variables by

indicating their functions by v; v�) and C1ða; a�Þ is the solution of the eigenvalue

equation (18) in the eigenstate of the total Hamiltonian with total energy E to the

first order in the perturbation. We obtain

C1ða; a�Þ ¼ ½ð1=2Þða� � aÞ þ ðd=da � d=da�Þ�½vþC0nþ1ða; a�Þ

þ v�C0n�1ða; a�Þ�; ð44Þ

where C0nþ1 and C

0n�1 are the characteristic functions of the free fieldwith nþ 1 and n� 1

photons, respectively, and vþ; v� are the source functions with energy denominators

containing ��hx, respectively (see later). Introducing (43) and (44) in (42), one gets

hRRi ¼Z Z

dada� f½vvþðnþ 1ÞC0nþ1 þ vv�nC0

n�1� � ½� � ��g ¼ 0; ð45Þ

where the second square bracket contains the same terms of the first one. This result

is not surprising, because in a stationary state the rate of variation of the number of

photons is in the average zero. In fact the contributions of emissions balance exactly

those of absorptions. The two terms in the first square bracket represent, respec-

tively, the contribution to the rate of change in the number of photons from emissionand absorption of one photon by the source. They are proportional, as they should

be, to nþ 1 and n, respectively (see Appendix A.1).

6. Conclusions

The main result of this paper is to show that it is possible to derive directly from

first principles, without having to recur to the previous determination of Schr€oodingerwave functions, the Wigner function of the state of a quantum field, as a function in

the phase spaces of its normal modes. In this formulation the wave particle duality of

the field�s quanta is no longer a puzzling phenomenon. This result confirms the cor-

rectness of Jordan�s point of view, namely that the wavelike behaviour of the field�squanta simply reflects its property of being a physical nonlocal entity which exists

objectively in ordinary three-dimensional space.

The two requirements—uncertainty principle and discreteness of the field�s inten-

sity—imposed to the classical field lead to the two Eqs. (18) and (21) whose solutionsyield directly the quantum characteristic functions of the states of each mode, which

turn out to be the double Fourier transforms of their Wigner functions.

In the derivation of these equations one discovers that the field variables cannot

be represented by ordinary numbers but should be represented by means of noncom-

muting mathematical objects. The approach presented in this paper reverses there-

fore the usual formulation of quantum theory which starts by postulating that the

physical variables of any system should be represented by mathematical objects (op-

erators) which do not obey the commutation property of multiplication for ordinarynumbers. We have shown here instead that this new mathematical property is a nec-

essary consequence of two physical postulates: the uncertainty principle and the

existence of quanta.


With the direct construction of the Wigner functions of the states of quantum

fields, the de Broglie–Schr€oodinger waves are thus eliminated from the formulation

of quantum field theory. This means that, once that their nature of mathematical

auxiliary tools has been recognized, the endless discussions about their queer phys-

ical properties, such as the nature of long distance EPR correlations between two ormore particles or the meaning of the superposition of macroscopic states, become

meaningless as those of the queer properties of the aether after its elimination de-

clared by the theory of relativity.

This deduction from first principles of the Wigner–Feynman pseudoprobabilities

may help in increasing the consensus currently acquired in some domains of physics

such as quantum optics [7] (leading even to a proposal for their experimental determi-

nation [8]) on their use as the most adequate representation of the random character of

quantumphenomena. Finally, the direct deduction ofWigner functions fromfirst prin-ciples solves a puzzling unanswered questionwhich has beenworrying all the beginners

approaching the study of our fundamental theory ofmatter, all along its 75 years of life,

namely ‘‘Why should one take the modulus square of a wave amplitude in order to ob-

tain the corresponding probability?’’ We can now say that there is no longer need of an

answer, because there is no longer need to ask the question.

Appendix A

The result (A.1) should be compared with the calculation of the same quantity in

the Heisenberg representation of standard Quantum Mechanics. It is given by

hRRii ¼ hWjRRijWi; ðA:1Þwhere the state jWi has to be expanded in eigenstates of the free Hamiltonian

HHs þ HHf of energy E ¼ �þP

rðnr þ ð1=2ÞÞ�hxr:

jWi ¼ j�; n; 0; 0; . . .i þXðmÞ

c1ðmÞ j�0;m1;m2; . . .i: ðA:2Þ

The first order coefficients c1ðmÞ are easily derived from the eigenvalue equation of

the total Hamiltonian (40). It turns out that they are

c1�0 ;nþ1;0;0;... ¼ ðDEs þ �hw1Þ�1ðnþ 1Þ1=2h�jV �j�0i;c1�0 ;n�1;0;0;... ¼ ðDEs � �hw1Þ�1ðnÞ1=2h�jV j�0i;

ðA:3Þ

where the first one comes from the stimulated plus spontaneous emission and the

second one from the absorption of a photon in the first mode. For the other modes,

where photons are initially absent, only the contribution of spontaneous emission

will appear. The energy variation of the source ð�� 0Þ is denoted by DEs and h�jV j�0iis the matrix element of V between the two states of the source. From (A.2) and (A.3)

one obtains the result

hRRi¼ ði=�hÞf½ðnþ1ÞhV �ihV iðDEsþ�hx1Þ�1þþnhV �ihV iðDEs��hx1Þ�1�� ½. . .�g¼ 0:

ðA:4Þ


The terms ½� � �� exactly cancel the first ones, as we expect. The two terms in the first

square bracket, as we found in (45), give the standard rates of change from stimu-

lated and spontaneous emission and absorption, respectively, when the proper ex-

pressions for the source term are introduced and the energy denominator is

integrated over the continuum of the source states to yield the energy conservationd function. Eqs. (A.4) and (45) give the same result.

References

[1] M. Born, W. Heisenberg, P. Jordan, Z. Phys. 35 (1926) 557.

[2] P.A.M. Dirac, Proc. Roy. Soc. A 114 (1927) 243.

[3] E.P. Wigner, Phys. Rev. 40 (1932) 749.

[4] R.P. Feynman, in: B.J. Hiley, F.D. Peats (Eds.), Quantum Implications, Routledge & Kegan Paul,

London, 1987, p. 235.

[5] Moyal, Math. Proc. Cambridge Philos. Soc. 45 (1949) 99.

[6] M. Cini, Ann. Phys. 273 (1999) 99.

[7] Physics Today, cover April 1998; D. Leibfried, T. Pfau, C. Monroe, Phys. Today 51 (4) (1998) 22.

[8] L.G. Lutterbach, L. Davidovich, PRL 78 (1997) 2547.


field quantization and wave particle duality

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