Download - 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
E-mail address: [email protected].
0003-4916/03/$ - see front matter � 2003 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0003-4916(03)00042-3
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.
86 M. Cini / Annals of Physics 305 (2003) 83–95
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-
M. Cini / Annals of Physics 305 (2003) 83–95 87
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-
88 M. Cini / Annals of Physics 305 (2003) 83–95
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.
M. Cini / Annals of Physics 305 (2003) 83–95 89
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
90 M. Cini / Annals of Physics 305 (2003) 83–95
(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
M. Cini / Annals of Physics 305 (2003) 83–95 91
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
92 M. Cini / Annals of Physics 305 (2003) 83–95
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.
M. Cini / Annals of Physics 305 (2003) 83–95 93
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
94 M. Cini / Annals of Physics 305 (2003) 83–95
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.
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M. Cini / Annals of Physics 305 (2003) 83–95 95