Fortschritte der Physik / Progress of Physics: Band 24, Heft 11 1976 [Reprint 2021 ed.] 9783112520567, 9783112520550

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 11 • 1976 • B A N D 24

A K A D E M I E

- V E R L A G

EVP 10. 31728

M

•

B E R L I N

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Zeitschrift „Fortschritte der Physik'* Herausgeber: Prof. Dr. Frank Kaxchlubii. Prof. Dr. Artur Lüschc, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert ltoinpc. ¡tu Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4; Fernruf: 2 2 0 0 4 4 1 : Telex-Nr. 114420; Postscheekkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-XJnivcrsitüt zu Berlin, D D R - 104 Berlin, Hessische StraQe 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Cesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossictzky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines J a h r e s bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 120,— M). Preis je H e f t 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses H e f t e s : 1027/24/11. (C) 1976 by Akademie-Verlag Berlin. Printcd in the German Democratic Republic.

Fortschritte der Physik 24, 6 1 9 - 6 3 2 (1976)

Quantum Theory of Path-Dependent Field Operator Based on Characteristics of Displacement Operators TAKAO

Department of Physics,

OKABAYASHI

University of Tokyo, Tokyo,

Japan

Abstract The path-dependent operator formalism of quantum electrodynamics proposed by Mandelstam is reformulated through quantum field theory based on characteristics of displacement operators in Minkowski space. It is shown that total energy- and total angular-momentum operators can generate inhomogeneous Lorentz transformations on any local operator including path-independent bilinear forms constructed of path-dependent electron operator W(x, P), but that generators for W{x, P) itself are only their W(x, P)-dependent parts. Such an unfamiliar feature is characteristic of the path-dependent operator formalism. The present approach possesses unique merits in making the logic of the formalism transparent" as described in the following: i) Quantum electrodynamics can be formulated but for the help of potential operator even as a tool for calculation up to a final step, ii) Some restriction, which can be used to discuss propriety of gauge conditions, can be figured out. iii) By introducing a path-rearrangement operator, we can keep infinite variety of space-like pathes with the same end point throughout our formulation as they stand, iv) Several points which must be modified in the presence of magnetic monopole are closed up.

1. Introduction Since an early stage of development in quantum electrodynamics, it has been well recognized that the non-integrability of phase of electron wave function is intimately connected with the existence of electromagnetic field [1, 2] and that gauge invariant operators in quantum electrodynamics are electromagnetic field strength Fki and a X

path-dependent electron operator of the form W(x, P) = exp [—ie J dzkAk{z)~\ ip(x). In p

1962 Mandelstam proposed a formalism of quantum electrodynamics where electromagnetic potential Ak(x) appears only as a tool for calculation, and tried to express all physically meaningful relations only in terms of Fkl{x) and W(x,P) [3]. This formalism has been utilized for studies especially on magnetic monopole [4, 5], and on non-abelian gauge fields [6] and gravitation field [7] in addition to quantum electrodynamics itself [6, S]. The basic idea of the formalism is very attractive, and is expected to shed some light on the problem of magnetic monopole [2, 5]. However, there remains something veiled in foundation of the formalism, and some people raised questions to what extent quantum electrodynamics can be described only through gauge invariant operators, or whether the formalism is completely equivalent to the conventional theory. 47

Zeitschrift „Fortschritte der Physik", Heft 11

620

TAKAO OKABAYASHI

The first question will be rather natural, if two kinds of important elements in quantum field theory, wave equations and fundamental commutation relations, cannot be derived but for the help of potential variable. This kind of obscurity can be removed when we apply the quantum field theory based on characteristics of displacement operators in Minkowski space to the present problem. In this article, starting not from Lagrangian density operator but from axiomatically settled energy-momentum tensor density operator, we shall derive all fundamental commutation relations and wave equations through characteristics of displacement operators. Then, contrary to the method devised by P E I E R L S [ 9 ] we can proceed with the manipulation without specifying the path-dependence of electron operator. In other words, we need not use potential variable even as a tool for calculation. That is one of merits of our approach. Naturally, in order to make the theory definite, we have to fix the path-dependence of electron operator with the aid of potential variable as a final task. The formalism based on characteristics of displacement operators is not so beautiful as the Lagrangian or the canonical formalism indeed. But it is effectual when there is no well-established way to quantize a newly introduced physical object, or when we entertain a doubt in applying the conventional quantization method to a system. The formalism is always serviceable, because in any Lorentz covariant field theory generators of displacement in Minkowski space should be defined. Recently this method has been used to quantize non-abelian gauge fields in radiation gauge [10] and preferred fields in the non-linear realization of groups with a non-abelian subgroup [21]. From a completely new standpoint, Green utilized the characteristics of hamiltonian operator to find the para-statistics [12]. Needless to say, we are not pursuing such a new possibility that we have never known. One of intricacies of path-dependent operator formalism is in the fact that there is an infinite variety of space-like pathes with the same end point. Mandelstam's viewpoint is that there is only one independent path-dependent operator for each point since the pathdependence is known, and he used only a representative path for each point throughout his formalism. As the result, for instance, the commutator between conjugate pair of path-dependent operators is path-independent, more specifically it becomes a space-like delta function. His viewpoint is correct indeed, but the formalism will become more transparent if we can keep the infinite variety of space-like pathes with the same end point as they stand. That is possible in our method, as will be shown in Chap. 3, if a pathrearrangement operator is admitted to appear in the commutator as a factor to be multiplied into the delta function. In the present approach almost all parts of the pathdependent operator formalism can be constructed without referring to potential operator. The only exception is the path-rearrangement operator. In the manipulation explained above it is enough for us to impose some conditions on the operator. But we have to introduce potential operator, in order to show the existence of an operator such as to have properties which should be owned by the path-rearrangement operator. The second question mentioned in the beginning is twofold. The one is concerned in the freedom of gauge conditions, and the other in the local character of the pathdependent operator formalism. The present approach is advantageous also to discuss these questions. In the usual path-dependent operator formalism, while proceeding with their program, they always refer to the explicit form of W(x, P) given at the beginning, where no gauge condition is imposed on Ak(x). And then there is no place to discuss the propriety of a choice of gauge condition. For instance, in an argument on magnetic monopole Zumino interpreted the formalism as a kind of axial gauge formulation [5]. In our approach the path-dependence of W(x, P) is not specified up to a final step, where some restriction on gauge conditions comes in. All known gauge conditions appear to meet the qualification at least in quantum electrodynamics without magnetic

Quantum Theory of Path-Dependent Field Operator

621

monopole. Thus the path-dependent operator formalism seems to give the very same results with the conventional theory of quantum electrodynamics. As to the local character of our formalism, we should rather say it is almost local. All equations at each step of our manipulations can be translated into the ones in the conventional theory. However, an unfamiliar feature which is characteristic of the path-dependent operator formalism is disclosed in our approach. We shall find that total energy- and total angular-momentum operators play the role of generators of inhomogeneous Lorentz transformation group for any local operator including path-independent bilinear forms constructed of path-dependent operators, but that only their W(x, P)dependent parts generate elements of the group for *P(x,P). The physical meaning of our result is evident for angular-momentum operator. As for displacement operator, the naivest interpretation will be to attribute the peculiar property of W(x, P) to commutation relations among generators of inhomogeneous Lorentz transformation group. The fact is important, since that means there is no Lagrangian formalism which can reflect such an unaccustomed circumstance. Some people discussed about Green functions of path-dependent operator [6, S], but for a wonder properties of energy-momentum operator have never been analyzed as far as we are aware. In this article we shall study only quantum electrodynamics without magnetic monopole, since one of our purposes is to make the logic of the path-dependent operator formalism transparent. Physical meaning of each step of our manipulations is very clear, and, in the course of our study, few points which should be modified when magnetic monopole does exist are closed up. So occasionally we shall allude them. I n the next chapter all fundamental assumptions of our approach are put in order. We shall show in the first five subsections of Chap. 3 that quantum electrodynamics can be described without potential variable if we accept the existence of a path-rearrangement operator which should satisfy some conditions. The path-dependence of electron operator and then the explicit form of path-rearrangement operator are determined in the last part of Chap. 3. Few remarks on displacement operator and conditions imposed on the path are added in tKe last chapter.

2. Fundamental Assumptions In order to prevent confusion, first we should make the concept of path-dependent operator definite. In this article electron field interacting with electromagnetic field is represented by an operator Wa(x,P), which depends on a space-time point x and a space-like path P whose end point is x. (In the following, indices tx. and /? will be used for spinor indicis.) There is an infinite variety of space-like pathes with the same end point, and in any consistent scheme some prescription giving the change of \P(x, P) for any deformation of the path must be established. The prescription is not merely mathematical, but imposes a condition on the physical system concerned. For instance, in order to describe a system with magnetic monopole we must consider a singular pathdependent phase factor [13]. For quantum electrodynamics without magnetic monopole, it is enough to assume that the path can be continuously deformed through any amount of area, and that an operator connecting W(x, P) and xF(x, P') with each other for any two space-like pathes P and P' does exist. After having fixed the path-dependence of W(x, P), in Chap. 4 we shall discuss that assumption more definitely. Here it should be emphasized that, contrary to the usual path-dependent operator formalism, we are trying to fix the path-dependence through quantum field theory based on characteristics of displacement operators in Minkowski space, and that electromagnetic potential variable will be introduced only to specify the path-dependence explicitly at a final step. In the course of manipulations path-dependent operators, which depend on 47*

622

Takao Okabayashi

different pathes with the same end point, will appear. Another assumption on the pathdependence is that bilinear forms of ^(x, P) and its spinor adjoint *P{x, P) with the same path such as T(x, P) riJ(x, P) and ¥{x, P) rdkW{x, P) with any product F ofy-matrices are path-independent, and they will be simply written as (IP/1!?) (x) and (FFdklF) {x), respectively. The meaning of this assumption will be explained at the end of this chapter. The space-time derivative of 'Fix, P) is defined to be

8kW(x, P) = [V(x + Ax", P') - W(x, P)}/(Axk)\Ax^

(2.1)

as usual, where P' is obtained from P by extending its end point x through an infinitesimal amount Axk in the ¿-direction. It should be noticed that

8[k8t]W(x, P) da" = dxV(x, P) 4= 0,

(2.2)

where dxW(x, P) means a change of xF(x, P) for a shift of the path by an infinitesimal area dakl = —dalk at the end point. This is simply a result of the definition (2.1) [3]. Space-time indices k, I, m, etc., run from 0 to 3, and our metric in Minkowski space is

such that g00 = gm = — 1. Abbreviations A[kBt] = AkBt — AtBk and A • B = (1/2)

X (AB + BA) for arbitrary operators Ak, Bt, A and B will be frequently used. In the following we shall consider pathes on a flat space-like surface for simplicity. But generally we should not use natural coordinate system where the surface normal vector is in time-direction. Because we cannot define dQlP(x, P) through Eq. (2.1) in the system, although in spinor electrodynamics 8QP(x, P) can be regarded to be defined by wave equation. So we have to introduce complicated decompositions of elctromagnetic field strength Fki — —Fllc and y-matrices by utilizing projection operators —n k n l and 9kl — 9kl + nkn'j where nk is the surface normal vector with properties nknk = — 1 and n° > 0. Decompositions of various quantities are xk = nkrx + xk with tx — —ntxl and

xk = gk,xl, 8k = — nkd + %k with 8 = nk8k = 8j8r and bk = gkl8u Fkl = Fkl + n[kFt] with Fk = nlFkl and Fw = gkrgtsFrs, yk = nky + j>k with y = —n'yt and yk = gklyt, and finally akl = (1/2) \yk, y,] = n[kan + dkl with ak = nlakl and akl = gkTgtsars. Here yk

is defined by [yk, yt] = 2gkl and we shall use a convention where = —y and j V = yk. c Operators F(x, P) = VHx, P) STF and >F (x, P) = CFT{X, P) denote the spinor adjoint and the charge conjugate operators for [F(x, P). Hermitian operator stf = iy has properties = —srfyk and cr w W = and matrix C is defined by C+ = C L1 , GT T = —C and yk = — C ^ y i f i as usual. In any consistent field theory we should have generators of the inhomogeneous Lorentz transformation group, Pk and Jkh which satisfy commutation relations

[Pk, Pl] = 0,

-i[Pk, Jtm] = gk[mPtl

and

— k l > 4 » ] = 9mlkJl]n ~ 9n[k'h]m-

(2-3)

Furthermore, displacement operator Pk should have a property

i8kX{x) = [Pk, X(x)]

(2.4)

for any local operator %(x) including path-independent bilinear forms of W{x, P) mentioned

above. However, it is expected that Eq. (2.4) must be modified for path-dependent operators. Generators of homogeneous Lorentz transformations for F(x, P) will not be total angular-momentum operators but their lP(x, P)-dependent parts, since 'P(x, P) depends on electromagnetic field through its path-dependence. Then the second of commutation relations (2.3) will require that only lF(x, P)-dependent parts of Pk generate displacements in Minkowski space for W(x, P). (The other possible interpretation

Quantum Theory of Path-Dependent Field Operator

623

of such a modification will be discussed in Chap. 4.) Anyway we may take the effect into account in the form idkX{x, P) = [Pk, %(x, P)] + idkX(x, P)

(2.5)

for any path-dependent operator %(x, P). In other words, the last term is defined by this equation. Evidently dkW(x, P) should not be zero, since, owing to Eq. (2.3), dkW(x, P) = 0 means 8ik8l-iW{x, P) = 0 in contradiction with Eq. (2.2). Next it is assumed that generators with those properties can be expressed as Pk = f da,{x) T'k(x)

and

Jkl = f dam(x) x[kT^{x)

(2.6)

in terms of energy-momentum tensor density operator T'k(x) which is axiomatically settled in the following. Here da1 = nlda and da is the numerical measure of the surface element. We propose the following four axioms, which correspond to the ones usually used to fix Lagrangian density operator, i) Tlk is a hermitian second rank tensor density operator, ii) niT l k should involve only kinematically independent field operators, iii) Tlk is composed of two parts, T^lk and T(xf)lk. The first part is a bilinear form of Fmn. The second part is a path-independent bilinear form of E, E, dkS and dkS, but depends on 8kE and 8kE linearly at most. Here E(x, P) means W(x, P) or lPc(x, P). iv) Tlk is symmetric with respect to the charge conjugation. Then we have = - j

2Js{EyldkS

- ecEykals8sE - glk(2y*8sE + 2mSE)} + herm. conj.

(2.7)

and TWk = -F"» -Fkm + ^ ßg'k F™Fmn

(2.8)

with undetermined real parameters «, /? andTO.The overall factors in Eqs. (2.7) and (2.8) should be determined by referring to particle interpretation of electron and photon, but we put them equal to one in advance for simplicity. In Eq. (2.7) means the sum over the expressions for E = W and those for E = Wc. A few remarks on the axioms should be added here. First, the meaning of the second axiom is as follows: We can expect that wave equations for Fmn and are linear with respect to dkFmn and 8kxF, respectively. Therefore, 8Fmn and 8'F will be describable in terms of Frnn, W and their space-like derivatives, and then 8Fmn and d'F are not kinematically independent of Fmn and W. The second axiom is reasonable, since in our formalism all operator relations (i.e. fundamental commutation relations and wave equations) can be derived through relations (2.4) and (2.5). Owing to the axiom the coefficient of the g^-term in TM^ C an be fixed as it stands, and a term Eaksyl8sE, which is similar to the second term in Eq. (2.7), also can be excluded. Secondly, we did not consider any term which depends on both W and Fmn. This is because we are considering only minimal interaction which is expected to be involved in T k W. Third, T ^ \ k is not a symmetric tensor, but we can reform the expression (2.7) into symmetric one after having attained the wave equation and fixing a at a proper value. Fourth, we need not make any statement on the gauge invariance, since at the present stage electromagnetic potential variable does not explicitly appear. For a later convenience, we shall introduce new notations T and TA through the decomposition ntT'k = nkT + lk with T = —nl{n{Flk) and lk = gks(ntTls). More specifically we have W\

=

F»>.?km,

j M k = T H + ATMk

(2.9 a)

624

TAKAO OKABAYASHI

with M

= j

(1 + «) ([3», y ^ Y ] -

yf])

and AV*>t=

^

( 1~J P)

T(F) =

and TW = 1 ((1 + «)

(2.9b)

ybkmW\,

F m F m

+

T

(2,10a)

y ^ f ] - [b s T, m )

-

!F]).

(2.10b)

Corresponding to our decomposition, we shall write P t = nkgP + Pt. with = —n lPt = f da(x) T(x) and Pk = gk'Pt = f da(x) T k ( x ) . The meaning of and PW'k will be self-evident. Here we should make a remark on the requirement of path-independence of bilinear forms constructed of W{x, P): We are regarding Fmn as a local operator in the genuine sense. Then Eq. (2.4) means Pk and then T^ lk should be path-independet. Our final assumption is a set of "local" field conditions [Fki{x), Fm„(i/)] = 0,

(2.11)

[W{x, P), F m ,(»)] = 0,

(2.12)

and {Va(x,

P),W^(y,

P ' ) ) =

0,

(2.13)

which are required when x, y, P and P' are on the same space-like surface. (Hereafter all commutation relations should be understood as the ones for such a configuration.) Condition (2.11) is really a local field condition, but the others are not so simple. In the next chapter we shall see that condition (2.12) should not be imposed on a system with magnetic monopole. Moreover, W{x, P) is not a local operator in the genuine sense. 3. Operator Relations The purpose of this section is to show that all fundamental commutation relations, wave equations, undetermined parameter /S in Eq. (2.8) and the path-dependence of 'P(x, P) can be fixed by applying Eqs. (2.4) and (2.5) to Fmn(x) and W(x, P ) , respectively, and that the parameter « in Eq. (2.7) can be determined by studying the spinor character of W(x, P). A)

i8 k F mn = [P k , F m „]

Owing to "local" field conditions (2.11) and (2.12), we have ièkFmn(x) = j d a ( y ) [F s(y), Fmni^)] • Fks(y)Contrary to the cases when Eq. (2.4) is applied to potential variables [10,11] it is impossible to require that the right-hand side reproduces the left-hand side. In the present case the commutator should be fixed so that the resulting equation gives a tensor equation. Hence we have Vlfmn + ^[mF«]i = 0,

(3.1

Quantum Theory of Path-Dependent Field Operator

625

by imposing a commutation relation [Fs(y), F m , ( « ) ] = ig°iJn]

Sto-z).

(3.2)

Corresponding to our specification of coordinate system explained in Chapt. 2, here we have introduced a function o(y — x) which is defined by properties d(y — x) = d{x — y) and jda(y) 6(y — x) f(y) = f(x) for any operator / and any space-like surface passing through a point x. The partial integration formula for this function is ckd(y — x) f(x) = hHy — x) f(y) + Hy — $*/> as usual. Utilizing relation (3.2) just attained in addition to Eqs. (2.11) and (2.12), we have id Fmre(x) = 2) d[mFn]. Therefore, we should fix the parameter in Eq. (2.8) as ¿3 = 1, in order to get another tensor equation SFmn + dlmFn] = 0.

(3.3)

Equations (3.1) and (3.3) can be amalgamated into a four dimensional tensor equation ^

¿

= 0/

'

(3.4)

with the completely anti-symmetric tensor eklmn. It should be noticed that Eq. (3.4) means we have excluded the magnetic monopole, and that such a situation is due to "local" field condition (2.12) [14]. B)

idkFt = [Pk, F{\

First, analyzing the relation idFt = —\5P, Ft], we can easily derive relations SFt + A, Ft(x)] = i{dkFl — gk,drnFm). Here Eq. (3.9) for the case k = I should be understood as a definition of the time-like component of current. Equations (3.5) and (3.8) can be unified into a familiar form + ii = 0. C)

idk¥(z,

P) = [Pk, W(x, P ) ] + iikY(x,

.

'

(3.10)

P).

The space-like component of this relation can be used to find the commutation relation between T(y,P') and 'P(x, P). For that purpose, it is convenient to perform a partial integration in PMfc in advance so that 8kW{y, P') does not appear. Then, owing to "local" fiel dcondition (2.13), we have [P«*, !F„(a;, P)] = - a - 1 J da(y) {(?(?/, P') y)p, WJx, P)} X . okWf(y, P') with a-1 = (1 + «)/2, and this part can be used to reproduce the left-

626

TAKAO OKABAYASHI

hand side idkW(x, P). That is, we assume a commutation relation m y , P') v)ß, ¥a(x, P)} = -iaöj(y

- x) E(y, P';x, P).

(3.11)

and impose a condition b{y - x) E(y, P'; as, P) bkV{y, P') = d(y - x) 3kW(x, P)

(3.12)

on the path-rearrangement operator E(y, P'; x, P). Then we should have gk%W{x, P) = i[P(« t>

W(x, P)].

(3.13)

In order to settle commutation relation (3.11), we have made a partial integration in advance. But naturally the same result should come out without the procedure, and then operator E(y, P'; x, P) should satisfy another condition — (d(y - x) E(y, P'-x, P))

P') = dkd(y - x)

P) - b{y - x) bkW(x, P). (3.14)

Strictly speaking, the first term on the right-hand side should be multiplied by an undetermined constant. Judging from the property of delta function, we can take the value used in Eq. (3.14). The consistency of the value will be checked in subsection F. Moreover, by definition (3.11), we have E(y,P';x,P)i

= E(x,P-,y,P').

In the time-like component of the equation for n"dkW(x, P) =

(3.15)

P), it is preferable for us to put W(x, P)]

(3.16)

by referring to the form (3.13) and taking the Lorentz transformation property of d^ix, P) into account. Then we get a wave equation (;ykdk + am) W{x, P) = 0

(3.17)

with a help of an additional condition on E(y, P'; x, P) d(y - x) E(y, P'; x, P) W(y, P') = h[y-x)

W(x, P ) .

(3.18)

Equations (3.13) and (3.16) tell us that Eq. (2.5) essentially means idkW(x, P) = [ P H , W(x, P)],

(3.19)

as was expected in Chapt. 2. Such an unfamiliar form is natural in some sense, because PkM involves the minimal interaction. At the end of the next subsection we shall explain that local bilinear forms constructed of lF(x, P) obey the rule (2.4). The pathrearrangement operator E(y, P'; x, P), which satisfies conditions (3.12), (3.14), (3.15) and (3.18), will be explicitly given after having settled the path-dependence of ¥{x, P). D) Mandelstam function In order to manipulate Eqs. (3.7) and (3.9), we must determine the commutation relation between ^(x, P) and Fk(y). Taking account of the spinor character of W(x, P), we may write it in the form [W(x, P), Fk(y)] = Xk(y; x, P) W{x, P ) . (3.20)

Quantum Theory of Path-Dependent Field Operator

627

It is easy to observe that Xk (y; x, P) commutes with all independent operators Fmn , Fm , W(z, P') and