Field quantization and wave particle duality

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  • Field quantization and wave particle duality

    M. Cini

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

    Universitaa La Sapienza, Rome, Italy

    Received 6 November 2002

    Abstract

    We owe to Pascual Jordan the rst formulation of a theory of quantized elds, in the

    framework of Heisenbergs 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 Jordans program leads to a formulation of quantum eld theory in terms ofensemble averages of the elds dynamical variables, in which no reference at all is made to theSchroodinger wave functions of rst 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 eld) which objectively exists in ordinary three-

    dimensional space. This theory, in which the elds statistical properties are represented byWigner pseudoprobabilities deduced without any reference to Schroodinger wave functions,is based on two postulates. The rst one is the requirement of invariance under canonical

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

    elds state. This invariance leads to an uncertainty relation for the conjugated variables ofthe elds 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 satised only if the

    systems physical variables are represented by noncommuting numbers. In this way what isgenerally 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) 8395

    www.elsevier.com/locate/aop

    E-mail address: marcello.cini@roma1.infn.it.

    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 rst formulation of a theory of quantized elds,

    in the framework of Heisenbergs matrix mechanics. No wave function existed atthat time, because Schroodingers paper had not been published yet. For this reasonJordans 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 Einsteins relations E hm and p 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 dierent nature of the twowaves was later claried by Borns 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 Schroodinger amplitudes, which had been in the mean-time accepted as the basic tool for the rst quantization of mechanics. One had to

    interpret single particle Schroodinger waves as classical dynamical variables, whichwere then quantized according to the rules of noncommuting quantum canonicalvariables, and to introduce successively Schroodinger probability waves dened in amany particle systems abstract space in order to compute the transition probabilitiesbetween its dierent 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 eld. Only at the end of his

    paper the connection with the classical waves of this eld 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 diraction and interference of single particles as properties of

    probability waves one should simply view them as primary properties of the eld

    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 dened in an abstract multi-di-

    mensional space, existing perhaps only in the observers mind. From Jordanspoint of view, instead, the wavelike behaviour of any elds state with any numberof discrete quanta simply reects 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-

    dans program leads to a formulation of quantum eld theory which eliminates allthe problems raised during many decades of debates on the paradoxical aspects of

    84 M. Cini / Annals of Physics 305 (2003) 8395

  • 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 eld implies a probabilistic description of its

    properties from the beginning. Therefore the states of the eld 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

    elds variables and of their correlations in any given state of the eld should appear.To this purpose an extension to eld 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-

    kenas 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

    Schroodinger equation, of its wave function. This means that Jordans program can-not be implemented by simply extending to the states of a eld in this procedure.

    In order to eliminate this drawback it is necessary to derive directly from rstprinciples, without having to recur to the previous determination of Schroodingerwave functions, the Wigner function of the elds state, as a function in the phasespaces 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 elds the same two a

    priori requirementsuncertainty principle and energy quantizationintroduced 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 Schroodinger wave functions.The paper is organized as follows. In Section 2 the statistical description of the

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

    3 the quantization of the free elds oscillators is formalized. In Section 4 the prop-erties of some standard ensembles of elds quanta are discussed. In Section 5 theemission 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 eld satisfying the uncertainty

    principle

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

    one-dimensional vibrating string of length l, whose motion is represented by the dis-placement ux; t given by

    ux; t Xr

    qrt sinxrx; xr pr=l; r 1; 2; . . . 1

    M. Cini / Annals of Physics 305 (2003) 8395 85

  • The Hamiltonian is

    H 1=2Z l0

    dxou=ot2 ou=ox2 1=2Xr

    2=lp2r l=2x2rq2r ; 2

    where

    pr l=2oqr=ot: 3Eq. (2) expresses the conventional superposition of normal modes as an assembly

    of harmonic oscillators of dierent 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; ar expressed in terms of each waves amplitude N

    1=2r

    and phase hr by means of

    qr h=lxr1=2ar ar ; pr 1=2ihlxr1=2ar ar ; 4

    ar N1=2r expihr=h; ar N1=2r expihr=h: 5The constant h has the dimension of an action and is introduced here to make

    ar; ar a dimensional. It will turn out, after quantization, to be equal to Plancks con-

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

    H Xr

    hxrarar Xr

    hxrNr: 6

    We introduce now a statistical ensemble for each radiation oscillator r dened bythe constraint that the intensity Nrq; p has with certainty a given value vmr. The dis-tribution function in phase space Pmqr; pr of each ensemble will be given by (the suf-x r is omitted in the following in order to simplify notation):

    Pmq; p hdq qdp pim 2ph2

    Z Zdxdk ei=hkqxphei=hkqxpim; 7

    where h im represents the average over the variables p; q with the constraint thatNq; p has the value m. The function hei=hkqxpim of k; x is the characteristic functionof the ensemble m introduced by Moyal in [5] and we denote it by Cmk; x.

    Since the average of any eld variable Mq; p in the ensemble m can be expressedin terms of the characteristic function Cmk; x, it will be sucient to write and solvethe equations satised by this function for any given ensemble in order to obtain all

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

    hN2im m2; 8where

    m hNim Z Z

    dkdxnk; xCmk; x; 9

    where nk; x is the double Fourier transform of the function Nq; p.

    86 M. Cini / Annals of Physics 305 (2003) 8395

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

    ei=hkqxpei=hk0qx0p ei=hkk0qxx0p 10

    that in order to satisfy Eq. (8) the function Cmk; x must satisfy the relationZ Zdy dhnh k; y xCmh; y mCmk; x: 11

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

    values m of N and the corresponding eigenfunctions Cmk; x. Its solutions can be im-mediately obtained from its inverse double Fourier transform. In terms of Nq; pand of Pmq; p Eq. (11) shows to be no longer an integral equation but a simple al-gebraic functional equation:

    Nq; pPmq; p mPmq; p; 12which has the solutions

    m Nq; p; 13

    Pmq; p fmhq; pdNq; p m 14with f h an arbitrary function of the phase variable h conjugated to N . This arbi-trariness reects the fact that theremay be an innity of ensembles inwhich the variable

    Nm has the value m and h any probability distribution. Eq. (13) implies that, given acouple 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 N fN;MgPB; 15where fN;MgPB is the Poisson Bracket of N with any arbitrary variable M, namelythat

    hfN;MgPBim 0: 16It is easy to see that only if Pm does not depend on h, namely if oPm=oh 0, Eq.

    (16) holds. This means that only if f constant the required invariance holds. In thisparticular 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 functionsatises, in addition to (11), also the equationZ Z

    dy dhnh k; y xky hxCmh; y 0 for all k; x: 17

    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 fullled 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) 8395 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 satises Eqs. (11) and (17) we select only the ensembles in which the

    uncertainty principle is satised.

    3. The quantization of the eld oscillators

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

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

    trum of the quantum variable N^N of each eld 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 modied 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 eigenvaluesof N^N, should not be confused with the sux r, introduced previously to distinguish thedierent radiation oscillators). Eq. (11) should be replaced therefore byZ Z

    dy dhnh k; y xgky hxCmh; y mnCmk; x; 18

    whereCmk; x is now the quantum characteristic function of the ensemble with N^N mn.The functional dependence on k; x and h; y of the nonseparable part of the kernelgky hx is justied [6] by dimensional requirements and by the limits for k x 0and k h, y x. It should be stressed that gky hxmust be universal, namely that itshould 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 nk; x remains the same double Fourier transformof the classical variable Nq; p. We shall see in a moment that for more general vari-ablesMq; p also the expressionmk; x should be modie...