Finite formulation of quantum field theory

ANNALS

OF

PHYSICS:

Finite

23, 335-373

(1963)

Formulation

of Quantum ROBETAT E.

Department

of Physics

rind Astronowly,

Field

Theory*

PUGH~$ University

of Iowa,

Iowa

City,

Iowa,

A fundamental integral equation to he satisfied by all quantum fields is proposed. It is shown that this equation determines all S-matrix elements. The axiom of microscopic causality is not assumed, and the asymptotic behavior of field operators follows from the theory. The boundary conditions imposed upon the S-matrix elements limit the ones. Within the framework types of interaction to the “renormalizable” of perturbation theory, it is proven that finite solutions exist to all orders. It is also shown how the general S-matrix elements may be determined. The formalism is extended to quantum electrodynamics where it is shown that the gauge invariance of the theory is a consequence of the unitarity of the S-matrix and the Lorentz condition. The latter is imposed only on the wave functions of the photons in tlie asymptotic regions, not on either the operators or the state vectors. Finally, it is shown how the concept of the ester& field arises naturally in the appropriate physical situations. I. INTRODUCTION

Since 195.3 several formulations of quantum field theory have been presented (I ) that do not involve renormalization. These theories are all based on at least the following three fundamental axioms: 1. Lore& invariance. 2. Asymptotic behavior. 3. Microscopic causality. Microscopic causality means that’ [A(z),

A(y)]

= 0

if

(Z - y)” > 0

for the “renormalized” field A (.r). It is felt by this author that there is no compelling reason to assume microscopic causality. While it is true that macroscopic causality has strong esperi* This work was supported in part by the National Science Foundation. t The author has submitted this paper in partial fulfillment for the requirements for the Ph.D. degree at the University of Iowa. $ Present address: Physics Department, IJniversity of Toronto, Canada. r It is also possible to demand t,hat the matrix elements of this commutator vanish at spacelike points, and the argument,s given helow apply equally well to this weaker form of microscopic causality. 335

336

PUGH

mental support and must therefore be a feature of any theory, microscopic causality is only a sufficient condition for macroscopic causality; not a necessary one. If the interacting fields satisfy the usual asymptotic conditions then the causality of the field operators to some extent follows from the Lorentz invariance of the time ordered product (2) of the fields which enters in the LSZ form of the S-matrix. The equal time commutator (or anticommutator) of field operators may, however, be extremely singular at the origin of the light cone and this singularity should not be ignored. There is some evidence both from perturbation calculations using the renorma!ized spinor fields and a!so from calculations (3) involving the spectral representation of Green’s functions that the equal time anticommutator of spinor fie!ds is the product of the position &function, 6(r1 - r2), and an infinite factor which is related to the wave function renormalization “constant.” One might then conclude that the equal time anticommutator of spinor fie!d operators is a physically meaning!ess expression since it involves physically meaningless (infinite) factors. When one considers that the S-matrix elements are not functionals of the field (anti-) commutators, one is further inclined to take the position that these (anti-) commutators have no physical meaning and should therefore not enter into the formulation of a quantum field theory. This point of view is adopted in the present work. In this paper, instead of assuming microscopic causality and the asymptotic conditions, an integral equation for the interacting fields is proposed. This equation plays the role of the field equations in the usual formulation but in addition, it implies the asymptotic behavior of the field operators. We begin with a formulation and discussion of the theory of an hermitian scalar field in interaction with itself. The form of the equations, and the theorems proven about the solutions of these equations, are easily generalieable to the case of several different fields in interaction with each other. Such a generalization is outlined for quantum electrodynamics, and the special problems associated with the photon field are discussed in some detail. The notation used here for scalar fields is the same as that used in the first paper by Lehmann, Symanzik, and Zimmermann (I), except that the homogeneous and inhomogeneous A-functions are defined as in Jauch and Rohrlich (4). When spinor fields are discussed, we use the differentiation operators

and a, which is the same as a except that the differentiation acts to the left. When many position four-vectors, yl , y, , . . . , yYn, z1 , z2 , . . . , 2%, are involved, the particular variable used in the differentiation is denoted by the corresponding numerical subscript on a or e: a, , a, , . . . a, , & , & , . . . e, , respectively. Spinor indices are usually omitted, but the matrix multiplication can be fol-

QCANTUM

337

FIELD THEORY

lowed by remembering that spinor indices follow the space-time variables in any product, e.g.,

We adopt. the following three assumptions:

The S-matrix elements must be Lorentz invariant. This does not necessarily imply that the basic equations need to be manifestly covariant. For example, we shall have occasion to use the time ordered product of field operators and the definition of such time ordered products is not covariant unless the field operators satisfy some weak form of microscopic causality. Also, the “smoothed out” interacting field operators A(t) depend on the spacelike surface over which they are smoothed out and are thus not manifestly covariant. The integral equations developed for the S-matrix e!ements will have a wide class of solutions and we shall admit only those solutions that are Lorentz invariant. B. C0MPLETEh-ESS It is assumed that there exist complete sets of in and out scattering states. This means that there are no stable bound states, or, more precisely stated, that we are discussing only those interactions that do not lead to bound states. The scattering states are constructed by use of the creation operator;’

where fU(z) (a = 1, 2, 3, . ..) is a complete set of positive frequency3 orthonormal solutions of the Klein-Gordon equation and Ai,, out(:r) are the free particle in and out fields defined by4

and [Ain, out(x), Ain, out(

= -iA(z

- I’)

(11.3)

2 We use A$, 0Ut asa creation operator and a&: OUt as an annihilation operator. 3 This is the customary terminology (f), but one must be wary of the notation of (4) d enote the creation which is also used here wherein Ai’,+’ (z) and al,‘(x) operator and destruction operator parts of d,,,(x), respectively, whereas in fact in the terminology of (I), A i$’ (CC) and B i;’ are the negative and positive frequency parts of A i,,(x), respectively. 4 Part of the definition of din “=6(x) is that, of the possible inequivalent represent,ations of (IIJ), we select the one that possesses a vacuum.

338

PUGH

The S-operator

is defined by the relations5 A,,,(a) sts

= S+Ai,,(z)S

(11.4)

= Ass’ = 1

(11.5) (11.6)

s I 0) = I 0) where ) 0) is the vacuum state of the operators Ai,, 0,t(~). these equations, the in and out states are related by

As a consequence of

* out = Stain C.

FIELD

EQUATIONS

It was shown by Lehmann, Symanzik, and Zimmermann “interacting field” operator A (2) satisfying the asymptotic lim (a, A”(t)*) t+*:m

(1) that, given an conditions

= (+, AzUh, in*)

(11.7)

the S-operator is completely determined. It is clear, therefore, that if the field A(z) is given in terms of the S-operator, a system of equations will result which should be sufficient to determine the S-operator. It is only necessary to ensure that A(z) satisfies the asymptotic conditions. In analogy with Eq. (11.4) we write6 A(X)

= S+(Ain(X)S)+

(11.8)

is a (positively) time ordered product.7 This product is where (Ain(Z) defined as follows: First imagine S to be written as a sum of positively time ordered products of the operators Ai, ; we then insert Ai, into each of these products in such a position that the result remains a time ordered product. It is shown in Appendix I that this equation for A (x) is completely “flexible,” i.e., S+(Ai,(z)S)+

= (S’Ain(Z))-S

= S(A~“+,(X)S~)-

= (SAo,+,(x))+S+

(11.9)

We make frequent use of these identities in the following sections. In particular it follows easily from (11.9) that A(z) is hermitian if Ai,, is hermitian. It follows easily from (11.8) that A(z) satisfies the asymptotic conditions. The expression (Am(t) S)+ depends on t only through the limits of integration 5 The symbol t indicates hermitian conjugation and is to be contradistinguished from * which is used later to indicate complex conjugation. the symbol 6 When the statee of the system satisfy subsidiary conditions, it will be possible to include other terms on the right side of (11.8) and still satisfy the asymptotic condition (11.7). Compare, for example, Eq. (11.8) with Eq. (X.42). 7 The subscript + indicates that the largest t,imes are to the left; we shall accordingly use the subscript - on time ordered products in which the largest times are to the right.

QVAh-TUM

FIELD

THEORY

and not through any operator dependence on t. If we writesVg

where W(L~ . . . x,) is a symmetric function of its variables that is suitably restricted by (11.5) and that can otherwise be considered unknown at this point, then

(iZin(x)S)+ =XAin(X) -ngq s .

d4< d4Xl . f * d4xnAR(x

-

(11.11)

~)co($I

. . * Xn)

:Ain(xl)

* *. Ain(

Thus

.w(&

X1)

. . . x,)

:Ain(Xl)

. . * Ain(

XI,

* * . xn)

:Ain(xl)

* * . Ain(

(11.12) J

d3xiA(x

-

~)~o~~(x)u(~,

-dt,21,

2,) :Ain(xl)

*.-

*** Ai,(

In the last step, use has been made of s

d3xA(x - f);c&(x)

(11.13)

= -fa(g)

This should be compared with the similar result

(11.14) -w(l,

XI)

.. *

2,) :Ai,(zl)

-. * Ai,(

from which we seeimmediately * That the S-operator can be written invariance and completeness. 9 The term n = 2 is considered absent,

in this from

form

follows

S at the outset.

from

the

axioms

of Lorentz

840

PUGH

!im (AZlt)S)+ t++m

= AGS

lim (A,F,(t)X)+ I---m

= SAb

(11.15)

a tltl thus

Having proven the asymptotic conditions we can make use of the LSZ result, viz., t,hat the asymptotic behavior (5) of A (x) implies that W(Xl ) . . . tn) = K, *. . &cp(R.1)

. . . z,)

(11.17)

where da The Ip-product p(A(sl)

rn> = (0 I cp(A(zd

. . . A(G)

is defined in analogy to Wick’s

Theorem:

... A(G))

I *”

= (A(G)

+ i C

‘..

) IO)

(11.18)

a(~))+

A,(z~ - xj)(A(zl)

*.. Ali ..* A? ... A(x,))+

pZtirs

(11.19)

. (A(s,)

. . . Ai . . . Aj -. . ilk . . . A, . . . A(x,))+

+ etc. (all possible contractions) Here the symbol Ai means that the operator A(z.~) is omitted. The S-matrix elements w(z1 . . . z,) can be evaluated by the following procedure. The interacting field operators A(z) are expressed in terms of the W’S by (11.8). When this expression for A(r) is substituted into (11.17) one obtains an infinite set of simultaneous integral equations for the w functions. It is the solution of this set of equations that is discussed in the following sections. III.

THE

INTEGRAL

EQUATION

IN

PERTURBATION

THEORY

In this section the form of the equations determining S-matrix elements will be examined in perturbation theory. We assume that S can be expanded in a power series in g : s = g (7” S(R) n=O’

(111.1)

QUASTT’M

FIELD

341

THEORY

and similarly A(x)

= gg”A’“‘(r), ,,=,I

9’o’(s)

IFrom Eqs. ( II.q5) and (11.8) it follows immediately

= _4:.n ir) _.

(111.2)

that

71-l

A’“‘(tr)

= [Ain(

where the retarded commutator [A;,C,.r’), Ai,(yj]rct.

S(n)]p~t. - ,g S”‘A’“-“‘(:r), of two field operators

z B(:c - 1~) [sin(T),

(111.3)

is defined as

ilin(!/)]

= -iA,t(.r

- u)

The integral equation determining a particular nth order matrix wy.cl ) . . . :r,,,) is obtained from the definition of w(~): w”‘~(,.Y~, . . . R.,) = Kl 1. . K, c (0 1&4’*“(.e~) cn)

. . . A(Ym’(ccm)) ) 0)

(111.4) element (111.5)

where c(g) is a sum over all nonnegative integer values of each qi consistent with the restriction xyZ1 yi = 71. This sum may be broken up as follows: cd(‘L)(.rl) . . . .r,,j = k’l . . .

ilin(:rm))

where the prime on restriction yj < n for A (%), as may be seen We collect all terms this collect,ion A’“’ (x1 terms, viz., x:;“)(xl

IO) \

(III.(jj

the second summation symbol indicates the additional all j. On the right side of (III.6)) w(~) occurs only in the from (111.3). All other terms involve W’S of lower order. in (III.6 j that involve w’s of order less than IL and call , . . . .r,) . This collection contains two intrinsically different

) . . :rm) 3 K, . . . K, c’(O (a)

j (o(A(yrl)

. . . L4(*~-~(,,J)l

0)

(111.7)

and

Thus X’“‘(zl

, . . . .r,,) = ,:7c) (II ) . . . 2,) + xi”’

(Xl ) . . . 2,)

C,III.D)

312

PUGH

Using this notation, uvd (.x7:1,. ..x.)

-

the integral

equation

d4& *.. d44;,B(Xl,

s

. ...2&.(1,

.Jy&) where,

according

-.. ,tm)

. . . fm) = P(x1,

(111.10) . . . x,)

to (111.3)) the second term on the left is

& . . . Km *$ (0 ( cP([Ain( The matrix element, it is found that” Ha;..

(111.6) takes the form

Gn;h,...

S(n)]ret.Ain(xl)

(III.ll),

is explicitIy

. * * Ai ’ * ’

Ain(

IO)

evaluated

in Appendix

(1Ir.1l) II. There

a)

The integral appearing in (111.11) (111.11) may be written concisely as

will

(1 - B)&p

be abbreviated

as BwCn’ so that

= A’“’

(111.13)

This is the integral equation that must be solved to obtain CO(%).Its solution is made especially simple by the fact that B is a projection operator, i.e., B2 = B

(111.14)

or s

,j4& . . . d4&nB(sl,

. . . xm ; 41, .** aM-5,

... hn;;yl,

= B(xl,

... Y,>

... z,;yl,

(111.15)

... y/,)

as is shown in Appendix III. Multiplying (111.13) from the left by B one sees that BX = 0

(111.16)

is a necessary condition for the existence of a solution to (111.13). That it is also a sufficient condition is clear from the fact that if it holds, then CO(~)= X’“’ is a particular solution of (111.13). It will be shown in the next section that (111.16) holds independent of any details of the interaction. The general solution of (111.13) is discussed in Sections V and VII, but it should be already clear, however, that CO(~)is determined by X (n) . We thus have a recursion relation by which the nth order w is determined by the w’s of order less than n. 10 The

prime

on the product

sign

indicates

that

the

term

i = X is to be omitted.

QUANTUM

IV.

FIELD

343

THEORY

EXISTENCE

PROOF

In the last section we established that RX = 0 is the necessary and sufficient condition for the existence of a solution of (111.13). In this section we shall prove that this condition is generally satisfied. It will first be proven that

from which

it follows

BX, = - h,

(IV.1)

Bh, = hb

(IV.2)

that

so that BX = B(X, + In order to prove (IV.l), s

d4EiA(x, -

xb)

=

(IV.3)

0

we first establish the formula

tL)KEi(O I (A(511 . . . A (&IL) I+ IO) = (01 (A($) -

(IV.4)

*.. Ai “0 A(&n))+Ain(xi)(O) (OIAout(zi)(A(h)

... Ai ..- A(tm))+lO)

This follows directly from the asymptotic conditions: We integrate the left side of (IV.41 by parts twice and only the boundary terms (&@ + + 00 and ti” + - a, ) survive. Consider BX,: BX, = -~g,...li,sd4E1...d4F,,, X=1

-{&(a - Edfj% .k’, . . . Km 5’ We apply (IV.4)

ZA - z<)A(zi

(0 1(p{Acql)(td

to each homogeneous function

A(ci

- ti) . . . A(4m%m) I IO> - li) in (IV.5)

and get (IV.6)

-(O 1iVj/l~P,:“(~~‘)

’ *. A~“(:~pl)A’**‘(ZX)Ain(26+?)

. . ’ Ain(zm,‘)f

10)

where (sl’, . f .rP’, lea+2 , . . . x,‘) is some arrangement of the variables (.rI , 1 . . -(‘h--l ) .h+1, . . . z,) and the sum Ccuml,. , is over all combinations (x1’, . . . -I+‘) that can be chosen from (.Q , . . J&--l , JA+1 , . . . z,). The N in (IV.6) is to indicate that the product of field operators is normally ordered insofar as no two free field operators may completely contract on one another. This normal ordering is a consequence of the p-ordering in (IV..?). Next, the product

344

PUGH

may be replaced by

because only derivatives of the 8 functions operators K1 . . . K, are applied in (IV.6). When it is noted that (Z (0 1 :A::t”(xl’)

. ’ ~:%“(2,1):

will enter when the Klein-Gordon

= (0 1 :A;~(x~‘)

.. * -4in(Xi):

Scnwnh) (IV.7)

we can write BX, = K1 . . . K, 2

#&I

c

5’

fi

comb.

k=l

i=l

.(O ( N(Ain(sl’)

O(zi’ - 2x1 jxc+2 &A

’ ”

- Xi’)

Ai,(Xp’)S’n--k’A’k’(XX)Ain(Za+O) * ’ ’ Ain

m =

K,

. . .

(IV.8)

K, C C (0 / cp{X’n--k’A’k’(2X)Ain(Z1) +* * h x=1

k=l ...

=-

1 [ O>

n-l

Ain(

IO)

hb

It3 is now trivial to prove (IV.2). BX,, = B(-BX,)

From (IV.1)

and (111.14)) we have

= -BB?X, = -BBh, =

hb.

(IV.9)

V. UNIQUENESS

The general solution, w(~), of (1 - B)w’“’ = A’“’

(V-1) is the sum of the particular solution, XCn),and the general solution, x, of the homogeneousequation (1 - B)x = 0. w.21 (n) is made unique by specifying boundary conditions that it must satisfy. These boundary conditions are usually given for the ‘lFourier transform,” p, of w(d which is defined by w

(2r;4ma4 d4p, . . . d4pm6(pl + . . . + Pm) w(21, . . . 2,) = ____ s +W(p, , . . . p,le

i(P*zl+~‘.+Pm4

(V.3)

QUANTTM

FIELD

345

THEORY

In stating these boundary conditions, one must consider two cases: (a) If w’l)(:rl . . . m,) is nonzero, then w(.T~ . . . 2,) it called a vertex function and must satisfy the boundary condition: iL)(pl . . . p,) approaches a polynomial in the energies, pl”, . . . pmn, for large energies. Since the higher order (n 2 3) parts of the vertex function vanish when all but one of the momenta are on the mass shell, the Bw term vanishes for n 2 2 and thus the higher order vertex parts are determined entirely by XCn)in (V.l) and no boundary conditions are applied. (b) Once we have decided upon what w-function to use as a vertex in the theory, all other &functions must vanish for large energies for otherwise more vertex-like terms would be introduced. These boundary conditions actually follow from an apparently weaker boundary condition: O(pl , . . . , pm) shall haue no essential singularity at injknitg. This result is a consequence of the structure of the homogeneous and inhomogeneous equations for the w-functions. It is found that if the vertex function used corresponds to a nonrenormalizable interaction in the conventional formulation of quantum field theory, then the w-functions necessarily have an essential singularity at infinity, while if the chosen vertex function corresponds to a renormalizable interaction it is always possible to eliminate the essential singularity by forcing each o(~’ (p, , . . , pm) (n 2 2) to vanish for large energies. The proof of the above statements is not given here since our methods of evaluating the w-functions are not sufficiently streamlined to permit a concise proof. These facts have, however, been checked by exhaustive calculations. VI.

The

solution

THE

HOMOGESEOCS

EQVrATION

of the equation (1 - B)x

is especially simple. equal times. Hence we expect that x is derivatives of such

= 0

One first notes from (111.12) that B is nonzero only at x must also have this property. From Lorentz invariance a product of d-dimensional S-functions or possibly covariant products. First, we establish that

x0(& , . . Em) = a(& - FIN.5 is a solution of (VI.l). Bx,, = -

(VI.2)

- [I) .. . S(& - fl)

Upon substitution,

2 k’l . . . K, 1 X=1

(VI.1 j

d4,tA,(

xx

-

()

fi’O(z,

-

ai)A(.r;

-

<)

i=1

(VI.3)

where we used

346

PUGH

Noting the identity

and using A,(z

- [) = - O(t -

x)A(z

-

E)

we have Bxo = - K1 +. . K, / c14t; fi A(zi 1 i=l =

K1

. . .

K,(-1)”

.$) -

(-1)”

fi

A,(&

i=l

- E) !

/ d4t fi A,(zi - E) i=l

which verifies that (VI.2) is a solution of (VI.1 ) . Let, us now consider other solutions of (VI.l). It has already been remarked that such solutions must be derivatives of a product of &functions. The form of the projection operator, B, restricts the number of time derivatives that can occur in a solution of (VI.l). We find that the set of solutions of (VI.l) is precisely that found by Lehmann, Symanzik, and Zimmermann (1) : the total number of derivatives with respect to any two time variables is always less than four. VII.

An arbitrary

EVALUATION

OF

nth order matrix

(1 - B)o’“’ where XCn) involves boundary condition Since

matrix elements that its “Fourier

that A’“’ may always

ELEMENTS

= A(=)

(VII.1)

of order less than 12. It also satisfies the transform,” hCn) vanish for large energies.

BX’“’ it follows

MATRIX

element w(~) satisfies the equation

be written

= 0, in the form

),‘“’ = ( 1 - B)x;“’

(VH.2)

solution, X, of with Xi”’ determined by XC*) up to the addition of an arbitrary the homogeneous equation (VI. 1) . If the vertex function corresponds to a renormalizable interaction then XC”) is asymptotically a polynomial in the energies in which the sum of the powers

QUANTUM

FIELD

347

THEORY

of any two energies is always less than four. If the vertex function corresponds to a nonrenormalizable interaction then this is not so. (One may, therefore, define “renormalizable” and “nonrenormalizable” vertex funct(ions by this criterion without reference to conventional field theory.) Thus with a “renormalizable” vertex function it is always possible to choose x in xb”’ = X(n) + x so that xp’ vanishes for large energies. In this way, Ah”’ is uniquely by X’“‘. A comparison of (VII.1) and (VII.2) indicates that Wh =

b

determined

(n)

(VII.3)

is the desired solution of (VII.l) satisfying the appropriate boundary conditions. Thus the boundary conditions imposed on w are sufficient to ensure a unique solution. As a simple example, we shall work out the second order propagator CP(E1 , x2): since W(G , x2) does not appear in the X-operator, the term Bw does not occur in the equation for W.Hence from (VII.l),

u(~)(s~ , x2) = X'2'(xl ,x2) = K1 Kz (0 I(A"'(.r:,)

A'"(z?))+

j 0)

(VII.4)

This result is true regardlessof the form of the interaction. For any selection of interaction (e.g., direct or gradient vertex function), (VII.4) can be shown to be the “renormalized” propagator. The nth order propagator is determined by all matrix elements of order less than ‘n, viz., n-1 &P(Xl

, ~2) = XC”‘&, ~2) = K1 K2 c (0 1(A'i'(xl) i=l

A'"-i'(xz))+

/ 0)

(VIM)

The calculation of the higher order vertex functions is also very simple because of the fact that these vertex functions vanish on the mass shell and thus the term Bw is absent from the equation for w. Hence,

VIII.

CPT

INVARIANCE

In this chapter it will be shown that the system of equations for the S-matrix elements is invariant under time reversal, charge conjugation, and parity reflection. The proof is given for the hermitian (pseudo) scalar field in interaction with itself, but the generalization of this proof to other fields is straightforward. The equations are invariant under the separate operations of time reversal,

348

PUGH

charge conjugation, and parity reflection, but here it will only be shown that they are invariant under the combined operation of all three transformations. It follows that the S-matrix elements have the same invariance properties as the interaction. The hermitian scalar field is unchanged by charge conjugation. But under space inversion

A?Ln,out(x) = Ain. out( -X, 20) while

under

time

(VIII.l)

reversal (VIII.2)

AL, out(x) = A,*,t, in(X, -20) where * indicates complex conjugation. of C, P, and T, the fields become

Therefore,

under the combined

(VIII.3)

AIn, out(x) = A,*,,, in(-X) The CPT inverted

S-operator

is defined by the relation

A&,(r) where

t indicates

hermitian

action

= s’+ A[,(s)S’

conjugation. From St = s*+ $p

(VIII.4) (VIII.3)

it then follows

that

(VIII.5)

and a can be chosen to be zero without loss of generality. Thus we can write down an explicit expression for S’ in terms of Al,(x). From (VIII.5) and (11.9) it follows that

xl = 2 ‘--I’* n-=0 f Replacing

/ dkX1 . . . d4xnW(x1, . . . x,) :A:(xI)

~i by -xi

s’ = 2 0” n-0 n!

and using (VIII.3)

1 d4q . . . d4x,u( -xl,

. . . Aj”,(len):

we have

. . . , -2,)

:A:,,(zl) . . . A:,t(z,)

= 2 0” n=o n!

(VIII.6)

/ d4Zl . . . &&u( -x1 7 . . . , -2,)

(VIII.7)

:

:Al,(z,) (VIII.8)

. . . A;,(L): The step from (VIII.7) to (VIII.8) was made by making use of the fact that S has the same form in terms of the in and out operators. This fact. is proven in Appendix I. (See (AI.6) and (AI.7).) Next it will be proven that

Qumwhl

FIELD

A*( -x)

To prove

(VIII.9)

(VIII.9)

= S’+(A;,(r)S’)+

so that we may define the CPT inverted A’(x)

349

THEORY

interacting

= S’+(A;,(z)X’)+

field operator as

= A*(-x)

(VIII.10)

we note first that ,4(x)

= S+(Ain(X)S)+

with S considered as a functional ordering; then

of the operators

A( -z)

= S+(Ail(

with S considered as a functional time ordering. Hence A*( -2)

-x)S)-

Using (11.10) the desired result Finally we shall prove that if

-r)S*)-

(VIII.12) Ai,( -- y) for the sake of the

= S’(A:,,(x)S”)-

(VIII.13)

follows.

= @I (A(a)

... A(GJ)+IO)

then 7(-x.1,

... , -4

= (0 I (Ah4

. . . A’(erJ)

(VIII.14)

+ IO)

This result is proven by first noting that 7 ( 51 ) . . . .c n) = c e(21 - .Q?)e(zz - 53) * .. tl(z,,-, pw Ill. .(O I ACal

- 2,) (VIII.15) . . . A(G)

/ 0)

so that 7(-x1,

...

) -. c n) =

c B(Zl - :c:2)e(zz - ss) . . . B(&--I perm. -(O I A(--2,)

- 2,)

... A(-x,)

(VIII.16) IO)

Since l(OIA(.rd by the definition (0 / A( -2,) for hermitian

... A(L)

/O))* = (0 / A*(a)

. . . A*(x,)

IO)

(VIII.17)

/ 0)

(VIII.18)

of the complex conjugate, . . . A(-s,)

operators.

10) = (0 ( A*( -.x1)

Thus (VIII.16)

I)

Ai,( y) for the sake of the time

of the operators

= S*+(A:n(

7(x1 ) . . . 4

(VIII.1

becomes

. . . A*(-r,)

350

PUGH

7(-x1, ... ) -zn) = (0 I c e(z1 - ZZM~Z - 23) perm. . . . e(x,-1 - x,)A*( -x1) . . * A*( -x,) 10) (vlll*lg) = (0 1(A’h) *** A’(d)+ IO> which proves (VIII. 14). Now w(-x1, ... , -2,) is formed from the functions r( -xl, . . . , -z,,) via Wick’s Theorem with the contraction functions A,( xi - xj) . Since Ad-x) o(-X:1)

= A,(x),

. . . ) -Z,)

(v111.20)

= K1 . . . K,(O 1&l’(Xl)

. . . A’(&))

j 0)

(VIII.21)

Since A’ depends on o’ = W(-2) via (VIII.10) just as A depends on w via (VIII.11)) the system of equations determining W’ is identical to the system of equations determining w. The boundary conditions are the same, so the solutions, being unique, are identical: w(-Xl,

‘.a,

-2,)

= w(x1, *.. ,x,)

(VIII.22)

provided that the same interaction (first order solutions) are used for w and w’. Therefore we have shown that the S-matrix elements are TCP invariant if the interactions are. More generally, the equations for w are invariant separately under T, C, and P. Hence the X-matrix elements have the same invariance properties as the interaction. IX. POSSIBLE

THEORIES:

SELECTION

RULES

In this section we shall generalize the basic vertex equation (1 - B)x = 0

(1X.1)

to the ease of particles with arbitrary spin and charge interacting with one another. The formulation of theories involving arbitrary particles is given by the generalization of (11.8) : cY(X) =

X+(ain(x>

S)+

(1X.2)

where ai,(Z) is the quantum field corresponding to the free particle of given spin and charge that is under consideration. Under Lorentz transformations ai, behaves like a scalar, spinor, vector, etc., if the particle has spin 0, $& 1, etc., respectively. The general vertex equation is of the form

4x,y,2)= sd4t dbd43-Hx, Y,2;t,9,i-ME, v,s-1

(1X.3)

QUANTVM

FIELD

351

THEORY

where 4x, Y, 2) = Kz k’, K (0 I P(4~MYMz)

1 I o>

(1X.4)

Here K, , K, , K, are the appropriate “Klein-Gordon” operators: 0 - me for scalars; (a + m) for spinors; and so forth. The integral operator B is formed according to (111.12) : B(x, y, x; t, ?I,(-) = iK, K, K, {[0(x - E)6(x - g)B(r - z) + ecy - MY

- de(~ - 4 + e(z - s>e(x - 2)eb - ~11

. [(Yin(Z),ah(t

(1X.5)

[Pin(Y),Bin(r1>1* [Yin(z>,7in(Ol*l

where hi, , @in, Yin , are the appropriate “adjoints” of ain , Pin , Yin : hermitian conjugate for scalar fields, adjoint spinors for spinor fields, etc. Since a;,(L), pi”(y), yin(Z), satisfy the free particle equations Kzzain(~) = K$in(Y)

= Ksyin(z) = 0,

(1X.6)

it follows from (1X.5) that B(s, y, z; E, 7, [) is proportional to a product of &functions, 6(x0 - y”)6(yo - x0), or possibly derivatives of such products. Hence by Lorentz invariance and (1X.3), w(z, y, z) is a product of B-functions: w(x, y, x) = L6(x - y)S(y - 2)

(1X.7)

where L is a covariant differential operator. The general (anti-) commutator of free fields of arbitrary spin is a differential operator, I,‘, operating on the distribution A (5)) [ain(

nin(t)]*

= L’A(s - 0.

(1X.8)

The first remark to be made is that if L’ contains time derivatives of higher order than the first then B is no longer a projection operator and Eq. (1X.4) no longer has solutions. It is therefore impossible, for example, to couple the charged vector boson in any way because its commutation relation

[A?,(x), AL(t)1 = i(6,, - (C4A/m”))A(z - 5)

(1X.9)

contains time derivatives of second order. In fact, it would appear that the only particles whose commutation rules permit a solution of (1X.3) are the hermitian scalar, charged scalar, spinor, and hermitian vector. It is possiblethat by use of subsidiary conditions one can also formulate commutation rules for neutral bosons of higher spin that will allow a solution of (1X.3). The solution of (1X.3) for the interaction of three scalar fields has already been discussedin Section VI. Of the three covariant solutions only the direct interaction (no derivatives) allowed a solution of the inhomogeneous equation (1 - B)w = x satisfying the boundary conditions.

352

PUGH

We now turn to the solution of (1X.1) for the case of spinor particles interacting with bosons (scalar or hermitian vector). Here the presence of the time derivative in the commutation rule

allows even fewer solutions of (1X.1) and (1X.3) than were allowed in the boson-boson case. The general solution of the homogeneous equation (1X.1) in this case is similar to the general solution in the boson-boson case; if we define a parameter & which for a boson variable is the number of time derivatives entering x and which for a fermion variable is one more than the number of time derivatives entering x then x is a solution of (1X.1) provided that

for all i and j. The vertex equation, (1X.3), for the spinor-scalar boson case has the following covariant solutions (written in terms of interaction Hamiltonians for convenience) : $A#, $rr~a,,A, $r,,&$~a,A, &.,r,,J/d,A, $r,,$4+,&4. Here rllv is either S,, or a,, . Of these, only $A$ leads to solutions of the inhomogeneous equation satisfying the boundary conditions and hence it is the only admissible interaction. For the spinor-vector boson interaction, (1X.3) has the solutions: $r,A,,fi, $r,,a&4,, $P,,r,v~Au, $r,,$d,A, , and $rpV&d,d,A~. Of these only &,,A,# will lead to solutions of the inhomogeneous equation that can satisfy the boundary condition. X.

QUANTUM

ELECTRODYNAMICS

We shall outline below the formulation of quantum electrodynamics and shall solve for all first and second order matrix elements as examples of the application of this formulation. The reason that quantum electrodynamics must be given special attention is that there are special problems associated with the photon field: definition of the vacuum, subsidiary condition, longitudinal and scalar photons. These problems are not new; they appear to a greater or lesserextent in almost all other formulations (6) of quantum electrodynamics. A. THE

ELECTRON

FIELD

The in and out electron spinor fields satisfy k-&in. {#in,

out(x) 3 (a + out(x),

gin,

..t(y)l

m)$in,

out(z) = 0

= iX(z

- v)

(X.1) (X-2)

The corresponding creation operators to be used for state vector construction are

QUANTVM

FIELD

353

THEORY

(X.3) a Xin.

out

E

d’Cia(~)Y4#in.

s

out(:C)

(X.4)

Here &,, out creates a free negaton with wave function V,(Z) and x3, out creates a free position with wave function iim(.r). The wave functions form a complete set of orthonormal positive frequency solutions of the free particle equations = 0

(X.5)

jia(.x)(e - VL) = 0

(X.6)

(a + m!%?(s)

The statement

of orthonormality

is (X.7) (X.8 1

while the statement

of completeness

is

g Xa(~)ji&) The S-operator

so that the fundamental

PHOTON

(X.1 1)

S++in(Z)S

field equation for the interacting GC2>

THE

(X.10)

is defined by the relation #out(JJ) =

B.

= is-(x - Y>.

=

field is (X.12)

S+(#in(~)S)+

FIELD

1. The Operators The in and out photon fields are defined by the equations &At,,

out(t)

= CL4Lo.t

ML, out(~1, Aln,..,~~/)l = and the S-operator

= 0,

J.4= 1, 2, 3, 4,

-iie,,D(x - y)

(X.13) (X.14)

is defined by AL(x)

We note that from (X.14),

= S+A:&)S

the operator

Ayn, out(.r) = -iAt”,

(X.15) Out(x) satisfies

354

PUGH

the commutation

relation

MPn,out(Z),Ak, .“dY)l = iD(z - y)

(X.16)

There are two possible interpretations of this commutation rule. The first is that if A!::&,(z), (k = 1, 2, 3), is a creation operator then AFi2\+d,,(s) is a destruction operator. Lorentz invariance then demands that the vacuum of the photon state have an indefinite number of scalar photons in it. This leads to some difficulties in the normalization of the vacuum state which we would prefer to avoid by adopting the second possible interpretation of (X.16). If AFiT,Ut(~), (k = 1, 2, 3), creates a state of positive norm, then A!iltd,,(x) creates a state of negative norm. This means that there are no scalar photons in the vacuum and normalization difficulties are thereby avoided. We shall see below that all physical states are of positive norm. The creation operator to be used for the construction of photon state vectors is given by (X.17) Here faP(z), (a = 1, 2, ... , 00), are a set of positive solutions of the free particle equation tlfa’(2)

frequency

orthonormal

(X.18)

= 0.

2. The Subsidiary Condition As it stands, Am, out creates a photon state that is a superposition verse, longitudinal, and scalar photon states; i.e., A:,

out ( 0) = K,

out = a~?~out + b~‘iL,41,ut+ c&F Out

of trans(X.19)

where a, b, and c are superposition amplitudes and @$,TOUt, @zOUt , @z, out are state vectors for transverse, longitudinal, and scalar photons, respectively. These states are normalized to unity: (acAut ) a,

out) = (e&t

, eout)

= -@e,

out ) @!, out) = b

(X.20)

and are orthogonal to each other. As a first step in eliminating any contribution to a physical process from the longitudinal and scalar parts of the state vector we demand that the norm of #I. out be equal to the norm of its transverse part. In view of (X.19) this means that ) b 1 = 1c 1 . W e can arrange that this is always so in a covariant fashion by imposing the restriction &fu”(x)

= 0

(X.21)

QUANTUM

FIELD

355

THEORY

which can easily be seen to lead to the result 0 = -c

(X.22)

since the longitudinal and scalar parts of fP” (2) are then equal in view of (X. 18). It will be shown that all states constructed with Al:, out with ffi”(z) restricted by (X.18) satisfy the condition a,A;i&?&)&,)out Let @if,),,t contain

= 0

(X.23)

n photons: &?,u,

(X.24)

= AZ, out +1 . A:: o,,*k,‘du,

Here \k!Z,‘&,, contains no photons: A!fIII. ..&c,‘dut Now we apply tt,AY~;-,ut(x)

(X.25,)

= 0

to (a!:,‘,,, :

&‘Gz”t(X>~~SLt

We note that the commutator la,A%,

in (X.26)

A,p,o,tl

=

s

vanishes because of (X.21) :

&&X&o&D-(y

- x) (X.27)

= -

s

d”y&sei(7&IL(y

- z) = 0

In the last step in (X.27) we have integrated by parts with respect to the space variables (V = 1, 2, 3) and have made use of the fact that D- satisfies (Vi - a,“)D-(z)

= 0

(X.28)

in order to transform the term involving two time derivatives (V = 4). Thus the operator A&, out creates only those photon states that satisfy subsidiary condition (X.23). Finally, we show that the commutator LG$,ut , A’!In,o”tl =

-i / d”y.fe*iy~~o.I~~y)

the

(X.29)

is determined only by the transverse parts of Ayn , out(x). This can easily be seen by inspecting the Fourier transform of (X.29). Writing J;Bh)

= (2&2

s

d’ke”““ca(~)s(k’)e(k)e~(~)

(X.30)

356

PUGH

as the most general solution of (X.18), kc

(X.29)

becomes

out , A?In, outl = / d”kc,*(k)cs(k)s(k”)e(k)e,(/c)e,(k)

Now the subsidiary

condition

(X.21) es(k)

if the 3-axis is chosen in the direction e,(k)e,(k)

(X.31)

implies that, for example, (X.32)

= edk) of k. Hence = e*(k)

(X.33)

+ fa2(k)

and no contribution is made to (X.29) by the scalar and longitudinal photons. This means that in matrix elements between two in states or two out states w,Lt

) e?outJ

the scalar and longitudinal photons make no contributions. This completes the proof that the scalar and longitudinal pletely absent from the kinematics of the theory.

photons

are com-

S. Weak Gauge Invariance The elimination of the scalar and longitudinal photons from the dynamical aspects of the theory is equivalent to the requirement that they make no contribution to the matrix element (&i

) d,B’>

We show that this is so by proving that such matrix elements remain unchanged if the wave function of any photon undergoes the gauge transformation .fP”(Z) -+.f!A”(z) rJha(Z)

+ &LA”(z) = 0,

(X.34) (X.35)

so that the matrix elements formed with an arbitrary wave function j,,a(z) are equal to those formed with an fPa(z) whose longitudinal and scalar parts are zero. It will now be shown that this gauge invariance of the S-matrix elements is a consequence of the completeness of the in-Hilbert space and the unitarity of the S-matrix, i.e., it is a consequence of a::: We note first that if, as follows

= 2 @$‘(a$? from

(X.36)

= 0

(X.37)

(X.23),

(@in”’ , &,A:&x)@,I,B’) then

) @,6::,‘,

QUANTUM

FIELD

357

THEORY

(X.38) because (X.38) can be expressed as a sum of matrix elements of the form of (X.37) by use of (X.36), and each term in the sum vanishes. The gauge transformation (X.3+ j then yields

This becomes

upon integration by parts on the space variables (p = 1, 2, 3), and by making use of (X.35) to transform the terms with k = 4. Thus if

then

,~5’~,,,~,+ Xc,,ca, + i

s

d”z~~‘(~)~o(&:,

a,Ak,(.~~jd?j

(X.41)

under the transformation (X.34). Hence the S-matrix element is invariant since the integral in (X.41) vanishes according to (X.38). This proves that weak gauge invariance is a consequence of the unitarity of the S-matrix. ,/t. Generalization

of the Fwldamental

Field

Equations

We are now in a position to write down the field equation for the interacting photon field, A’(t) . First it must be noted that becauseof (X.23)) the addition to A”(z) of any term of the form Ma,&A?n(z) doesnot affect the weak asymptotic convergence of the “smoothed out” interacting field operator A*(t) (defined in analogy with (X.18) with AL(z) replaced by A”(r)) to the operators AL, OUt since according to (X.23) all matrix elements of M~,,c?~A~~(x) vanish. Thus, as far as the asymptotic conditions are concerned, we may write A,(x)

= X+(A;n(~)s)+

+ llla,,&Ah(~j

(X.42)

as the basic field equation for the photon field. It is, in fact, convenient to include this M-term in the basic field equation because similar terms may arige in 2f(A~,,(z)s)+ and it will be desirable to eliminate them as they may violate our boundary conditions while having no physical significance. This complication is the price we pay for dealing with interacting field operators rather than the physically significant matrix elements. It is assumedthat M can be written in a power seriesin the coupling constant P, the leading term being proportional to e2.

358

PUGH

C. FORM OF THE X-OPERATOR The interacting fields satisfying the basic integral equations (X.12) and (X.42) satisfy the asymptotic conditions. Hence the form of the S-operator may be derived by the method of LSZ. We find

(X.43 )

where” 4x1 ) . . * zg) = K, . . . K,( a, + m) . . . (a, + m)(O j qJ{A”(Xl) .$(?A)

-** NY*Ma)

. * * &)I

In the S-operator the subscripts are multiplied as matrices, viz.,

I o>(eql - 4

are summed

pi

over

. . . Apqx,)

. . . (e, - f-4

while

the

spinor

(x.44) quantities

(X.45) Thus w transforms trices.

as a tensor of rank p and as a direct product

of q spinor ma-

D. FORM OF THE INTEGRAL EQUATION IN PERTURBATION THEORY Using the method function becomes u(n) (XI, . . . x,> -

s

of Section III,

d4& . . . d4HqB(xl,

the integral

equation

for an nth order w

. . . zq ; .$I, . . . j-c&r~(~)(&, . . . j-,)

(X.46)

= A@) (21) * . . 2,) where XCn) is that part of w(~) that depends only on w’s of order less than n, and B(x1 , . . . zg ; & , . . . l,) is found from (1X.5). The theorems proven for the scalar field are also valid in quantum electrodynamics. In particular it can be shown that B is a projection operator and that B),‘“’

=

(X.47)

0

If we write X(n) = A;“’ + xl”’ ‘I For

an explanation

of notation

see the last

paragraph

(X.48) in Section

I.

QUANTUM

FIELD

359

THEORY

where &“’ and A:,“’ has a Fourier transform solution of (X.46) is

= $’

which

(X.49)

vanishes

for large energies, then the

w(n) = hy) FJ.

lkwr

ORDER

The solutions

(X.50)

VERTEX

of the first order vertex equation

j

w(I)(~:, y, x) =

d2 dkq d4<~(~, y, z; E, 17,w)(t,

q, 0

(X.51)

were given in Section IX. There it was pointed out that the only solution that leads to higher order matrix elements that can satisfy the boundary conditions is the direct interaction: cP(x, y, x) = -iy,S(R: - y)6(y - 2). (X.52) I‘.

fhxosn

ORDER

ELECTRON

PROPAGATOR

Since w(y, z) does not appear in the S-operator, not occur in the equation for wC2’(y, z). Hence cP’(y,

2) = P(y,

From (X.52)

z) = 0, + a0

the term Bw in (X.46)

1 W1%ht1)(4)+

I o)(e,

- m).

does (X.53)

and (X.12) 4”‘(Y)

=

i

j

d4~~i,(?7>r’Ai,(?)Sa(a

-

(X.54)

Y>

(X.55) Hence wyy,

2) = -j

d47jd4r(az + m)~X,(z .[e(z - ?/h~+(l-

+ e(li/ - +d-(i-

- 77Uu.r

- {) - rm+Q-

d-iXIsAh

(X.56)

- 17)YX -

dl(e,

- n1)

Using the formulas S*(z>D*(z>

= -2-s 4(2~)~

d4peiP”8( -p2 . $qiy., cc

e(ztz>

= +$

m”)e(fp) (eg)

(X.57)

ikq

dk e s --m li =F ie

(X.58)

360

PUGH

and (X.59) we get, after a few changes of variables, if

(X.60)

3K”) which is the “renormalized” It should be noted that “renormalized” propagator viz., the “unrenormalized”

propagator. the “unrenormalized” propagator differs from the of (X..53) only in the position of the O-functions, propagator is

(0 I WJ(d& CT. SECOND ORDER PHOTON

- d(az

+ +tW~+

(X.61)

lo>

PROPAGATOX

As in the caseof the electron propagat,or, the Bw term does not appear in the equation for the photon propagator since ~(51, x2) does not appear in the S-operator. Hence, d2)(21

) x2)

=

x%1

) z2)

=

[7102{(0

I (AI”(dP(~2)),

+ A~‘“‘[a,axArn(z,)Ar,(z,)

+ Axr1hA&(r2)1

I O)}

(X.62)

From (X.*52) and (X.42) it follows that A:‘(z)

=

-i

/ 0)

=

/

d4&DR(x

-

E)

(X.63)

:$in(()rp#in(():

Hence, (0

/ (&!%d-d”(d

)+

.[O(~I

-

~2)

]

Tr

+ 0(x2

-

d%

d4tddx~

tdDR(%

-

-

IL.I)

{S-C&

b)r,S+(&

-

12hyJ

E2hYX+(E2 -

Using (X.58) and the formula Tr [S-(t)yPS+(O~YI

= &

.$2>

(X.64)

{X-(E2 Tr

-

/ d4PeiPee(-P2 - 4m2@(p)

,tl>r,ll

QUANTVM

(X%4)

FIELD

361

THEORY

becomes after a few manipulations:

The second term in (X.66)

may be evaluated

explicitly

to yield (X.67)

The second term in (X.62)

can also be evaluated easily. One gets

l~f’2’(o j [&ih~~n(.d~b(~-?) + i2~~:,(x1)a,dxA~,,(xz)]+ 10) (X.68) = -%nr”‘{a,a,D,(x, - x2) - Br*&4B(x1 - x*)1 We choose MC2’ so that the c$,& terms in (X.66)

and (X.68)

cancel. This gives (X.69)

Collect.ing the remaining

terms, (X.62)

becomes

which is the “renormalized” propagator. It should be noted that if the term (X.67) were to remain in w”)(z~ , z?), both gauge invariance and Lorentz invariance would be violated. Also, such a term could not be eliminated by the separation (X.48) because of the absence of the Bw term in the equation for the propagator. The result (X.69) has also been obtained by Nishijima (1) and by Rollnick, Stech, and Nunnemann (7’). Nishijima introduces M into the theory through the use of the commutation rules [Afn(x),

A&(y)]

= -i(6,,

+ zMa,a,)D(z

- 2/).

(X.71)

We choose not to do this because we like to keep At,(x) zero order in the electric charge, and also because the derivative terms lead to difficulties in the general formulation as pointed out in Section IX.

362

PUGH

Rollnick, Stech, and Nunnemann have shown that the operator A,,(x) must approach asymptotically AL , W(X) + Md,&Ak,(z). This is consistent with our choice of field equation (X.42) but it has no important consequencesin the formulation of the theory since we demand only weak convergence of I” to A,“,, OUtin the asymptotic regions: the vanishing of all matrix elements of &At,,(z) ensures this asymptotic behavior. H. SECOND ORDER MILLER

SCATTERING

For second order electron-electron scattering the function A”’ is x’2’(Yl , yz ) 21) 22) = (1 a+ m>(a2 + m)(O 1(4”‘(yl)~“‘(~*)~in(Xl)~in(Z2) + 4”‘(T/l)~in(y2)~‘l’(Zl)~in(Z2)

+ 4”‘(yl)~in(~2)~/in(Z1)~‘1’(Z2)

+ ~i*(yl)~“‘(Y2)~‘(l’(Xl)~in(x2)

+ ~i~(Yl)~“‘(~yZ)~i~(Xl)~“‘(~?)

+ ~in(~l)~in(~2)~‘1’(Zl)~‘1’(Z2)}+’

I o)(el

- m)(e,

- In)

(X.72)

Here the prime on the time ordering indicates that the in fields are not to be contracted on each other. This is necessary to ensure that the result is a pproduct rather than a time ordered product. This function must be separated into parts Xi*’ and hi”, according to (X.48). The simplest way of carrying out this separation is to pick out terms belonging to Xi”’ and throw them away; the remaining terms then constitute AA’). Terms belonging to Xi” are easily identified since they are nonzero only when all times are equal: yIo = yZo = zIo = 2,“. These terms include products of three Lfunctions of the times. It will now be shown that in all terms belonging to Xi2’, the first order fields are differentiated. For example, that part of (al + m)(a2 + m)

(X.73)

.(O I {~“‘(yl)~“‘(y2)~in(Zl)~in(ZP))

f’ I o)(e, - 4 (e2 - m!

that belongs to Xi2’ is (al + m)(32 + ml

-(O 1(iW~l)(el

(X.74) -

hW1J2)(e2

This can be proven as follows: Consider the a cp-product, the in fields cannot contract on the first order fields. Thus J/in(Zl) contracts simplicity consider only that part of the first

- m>lLin(Xl)#in(Z2)}+I I 0) first term in (X.72). Since this is each other; they contract only on on either $(l)( yl) or qC1)(~2). For term where #in(Zl) contracts with

QUAh-TUM

$‘“(yl)

(a,

and

+

z2) contracts

#in(

7r/J(a,

+

7n)(O

/ -

FIELD

with s(l) ( yz) . Writing

e(z1

-

yMz2

+

‘~i~(~l)~“)(Yl!~i*iZ,~~“‘C~)

L-1

363

THEORY

out this part of the first term:

2/2)e(y1

e(?/l

-

-

y2)

zl)e(G?

-

y2)e(l/l

-

Consider the various terms that arise when the differentiation yl is carried out. (a) If B(yl - y2) is differentiated the resulting only at equal times since the differentiations with respect to x1 and a &function. Hence these terms belong to Xi”. (b) If e(xl - yl) are differentiated we get terms like --ical

+ IWI,

+ do

1Qlo

- 23%

. ih(zl),

- ~4 oh

Z/2)

(x.75)

with respect to term is nonzero zq each produce and e(yl - zl)

- y2)

~“‘(Y1)Y4)~in(Z2)S/“‘(Y2)

+

etc. j o)(e, - 111)

which is cancelled by a term -i(a,

+

,~)(a~

+

711)(0

1 6(y1”

-

. {#“‘(G),

210)e(22

-

y2)

e(2,

~i~(~l)Y~)~i~(~*)~‘l’(~*)

L-d

-

7~~)

+ etc. 1o>(e, - m)

that comes from the fourth term in (X.72). These terms cancel because the equal time anticommutators of the spinor fields vanish in first order. Hence the y1 differentiation gives a nonzero contribution to Xi*) only when it acts on $“I( yl). This proves the contention (X.74). The terms in (X.72) contributing to Xi” are therefore easily calculated. Using iy4s(x)qxo)

= 6(r)

(X.76)

we get finally W’*)(yla, yzp, 217, 2*&) = -iD,(y1 *[E(Q

-

z/*Plz2

- 1/2)

- y1,rh’v

- S(z1 - y1)6(zn - y*)rrq]

(X.77)

which is the usual result. Here (Y, /3, 7, and 6 are spinor indices. It shoule be noted that the method used here to separate x,,“’ from x’“’ is not always applicable. After all terms manifestly proportional to products of three B-functions have been dropped, the remaining terms may still include parts that eliminated-a belong to A:~‘. In this case these parts must be subsequently process not unlike renormalization.

364

PUGH

I. SECOND ORDER COMPTON SCATTERING The function

X for second order Compton

scattering

is

x’2’(z1 ) J-2 ) y, 2) = i?lO~(a

+ m)(O 1 IA~‘(xl)Aj1’(x2)~in(y)~/in(X)

+A~“(xl)A;“(x2)~“‘(y)lC/i,(~)

+ Aj”(xl)A1,(1:2)~in(I/)~“)(X)

+A~~‘,(xl)A~“(x2)~“‘(y)~i,(~)

+ A~~(x2)AJ1)(s2)~i,(y)~“‘(z)

+A~~,(x~)A~(x~)~‘~‘(Y)~‘“(~))

+’ 1o>(e - ~4

(X.78)

The method of Section X, H may be applied here to separate Xc’ from A’~‘. As before, in the part belonging to Xi” the differential operators are applied directly to the first order field operators. The result is (J(2)

(Xl ) x2, y, 2) = i{@Xl - XhpS,(X - ybdy +

5(x2

-

- zhdc(z

/TTCA\

z2)

(LLIY) - y)rJ(y

- Xl)}

J. SUMMARY TO SECOND ORDER The interacting field operators can be evaluated to second order using (X.1 2) and (X.42) and the w’s evaluated above. The results are

A,(z)

= AL(z)

- ie /” d4EDR(x - t) :$in(t)rkin(t):

+ e2 / d4.$d4qDR(x - t)[:$in(E)YpSR(t

- ~)Y*Ain(~)Gin(~l):

+ :$in(q)Y.Ain(q)Sa(q

+ e2M’2’4ihA~n(z)

- E)Yp+in([):]

XI.CONCEPTOFTHE

(X.81) + ...

EXTERNALFIELD

In this section it will be shown how the above methods may be applied to problems usually referred to as external field problems. Since the concepts involved are much clearer in the case of quantum electrodynamics, we shall restrict our discussion to this subject.

QU.4STUM

iz.

FIELD

365

THEORY

FORMULATION

Assume that in addition to any particles undergoing scattering, the initial and final states contain a large number of charged particles that are, in a certain approximation, unaffected by whatever scattering process is taking place.” These unaffected particles will be referred to as the external particles and the st,ate of these part,icles mill be called the external state. The external charges are described by the state vectors @$’ and @iit . The fact that these particles are Lmaffected by the scattering is equivalent to saying @CO) In = a(O) out

(X1.1)

It is assumed that there are no photons in the external state: A~~-‘(x)~~,O’

= 0

(X1.2)

so that, @I? ) A:&)~::‘)

= 0.

(X1.3)

The in and out state vectors, (a!:’ and +$Y> , are constructed from the external state vectors, a!:’ and @iti. , by the application of the operators &, out , xt,, OUt, A,‘,, out defined by Eys. (X.3), (X.-l), and (X.17), respectively. ol eII at<+1 @!a’ 111,out G Pin. out ’ . ’ Pin, outXin, out ’ ’ xlPnlroutAC%t . . . Ai2 out@:! out (X1.1) The state vectors @i,4’,,,t represent the actual physical state of the system in the asymptotic regions. They include both the external particles and the particles undergoing appreciable scattering. The S-matrix elements and the S-operator are defined by S,,,(/3, 5% (a::;

, &“)

= (a::’

, X%4’)

(X1.5)

One may, in principle, undertake a complete reduction of (X1.5) to bring it into the usual form in terms of w functions. The interacting fields G(x) and A,(r) are to satisfy the fundamental integral equations (X.12) and (X.42). Thus, since both the S-matrix and the fie!d equations have the same form as in Section X, the i,ntwacting fields +(r) and A,(.r) are identical to those jm.md is 5. It is both conr:enient and desirable to introduce an auxiliary set of state vectors, \kj,T’. These state vectors are constructed simply by applying the creation operators, & , xl’n , and ilrn , to the physical vacuum state, / 0). Accompanying these auxiliary state vectors is an operator S’ which is to satisfy (&us ) &‘)

= (W,

ml:‘)

(X1.6)

Carrying out a partial reduction of the left side of (X1.6) so that it becomes L2 It might, be helpful to imagine a scattering process taking place between the plates of a capacitor. The charges on the plates are not affect.ed appreciably in the asymptotic region by, sap, an electron traveling through roughly parallel to the plat,es.

366

PUGH

the expectation value of an operator between the states +$i and @ii’ , we see that

with

w’(z1, ... 2,) = 01 ... q ,(& + m) ... (a, + m) .
. . . #(z,) )a&?) (e, - m) . . . (e, - m>

(X1.8)

i.e., w’ has the sameform as w except that the vacuum is replaced by a!,‘?. It is important to note that in contradistinction to the conventional formulation of external field problems, the intereacting fields used here satisfy the same equations, (X.12) and (X.42), as they did without the presence of the external particles, and they have the same form with and without the external particles. Thus in the calculation of the matrix elements w’ one merely inserts into the right side of (X1.8) the interacting field operators found by the methods of section X. B. ELEMENTARY

CALCULATIONS

In the following we shall evaluate w’ using Eqs. (X1.8), (X.80), and (X.81). For the first order vertex we find, of course, no effect due to the external states: u’(l)(x, y, 2) = w(l)(x, y, 2) = -iey,d(x

- y)6(x - 2)

(X1.9)

Also the matrix element for second order electron-electron scattering is unaffected by the presence of the external particles and one gets the same result as in Eq. (X.77). In the calculation of second order Compton scattering there are some ghostlike contributions from the external particles : cd’(2) (Xl , x2 7Y, 2) =Cd (2)(21, x2, y, x) - ieyd(x2 - yY)6(x2 - z)O~@Y(xd - ierJ(3-3 - y)6(xl

-

2)02@?)(22)

+ie26(xl - y)6(z2 - x)y,S(y, z)-yY (X1.10) + ie26(x1 -

z)6(x2

- yhS(y,

xh,

where s(y, x) = -i(&? and

7 :+in(Y)Sin(z):

+!I’)

(X1.11)

QUAiVTUM

C$“(X)

= ie

s

FIELD

d4.Q~(X

367

THEORY

- ()(@Z’,

:$jn(()rp#in(C;):

@ii’)

(X1.12)

The function a,(z) may be called the external jkld. When w’(~)(~1 , x2 , y, z) is substituted into (X1.7), it is found that only co@) (Xl ) x2 , y, Z) gives a nonzero contribution to S’. Hence the external state does not make any contributions to second order Compton scattering. The first instance of a nonzero contribution by the external particles to a physical matrix element is in the second order electron propagator. Here we find w’yy,

x)

(X1.13)

=CO(2)(y, 2) + e6(y - x)-r-a’“(z)

-

e’yhS(y, x)D,(y

-

X)YX

When w’@)( y, x) is substituted into the expressionfor S’ one finds the following contributions e

s

d4t :4in(F)r’a”‘(E)J/in(~):

(X1.14) -

e2

d4&

d4C;2

:$in(tl)YhS(El

-

tz)D~(E1

-

t2)h#in(t2):

s

The first term gives what may be called scattering from the external field while the second term gives the M#ller scattering of an electron from an electron in the external state. The external field approximation may be defined as the approximation wherein this second type of scattering is ignored. This is usually justifiable experimentally by arranging that the scattering electrons do not come close to the source of the external field. One may proceed in this way to calculate any physical matrix element in the presence of the external particles. XII.

CONCLUSIONS

It has been shown that the postulated relationship between the interacting field operators CZ(~) and the S-operator, cU(X) = S+(CXin(X)S)+

(X11.1)

together with boundary conditions, Lorentz invariance, and completeness uniquely determines the S-matrix elements, w(r.1 , . . . .r,). This formulation may be considered as intermediate between the Lagrangian formulation and the L&Z formulation. Lorentz invariance and completenessare common features of all formulations. In addition, in the Lagrangian formulation the interacting fields are required to satisfy field equations, K&(X)

= 6H/6n(x)

(X11.2)

368

PUGH

the interaction H plus asymptotic conditions determine 6’ immediately. Because there is no opportunity to apply boundary conditions to the S-matrix elements, they have certain nonphysical characteristics that can only be removed by the procedure of renormalization. The LSZ formulation, on the other hand, uses no fie!d equations and determines S-matrix elements by the use of boundary conditions plus the axioms of microscopic causality and asymptotic behavior. The disadvantage of the I,SZ formulation is that an integral equation must be solved at each stage of the calculation, and the solution of these integral equations does not appear to be particularly easy. In the formulation presented here, the basic equation, (XII.1 ), is a solution of the field equation (see Eq. (AI.3) ) K&i(X)

= iS+6X/GEi~(X)

(X11.3)

which satisfies the asymptotic conditions. As we have seen, (X11.1) leads to a set of integral equations for the S-matrix elements w(zl . . . 2,). In contradistinction to the LXZ integral equations, these equations are easily solved because of the fact that the kernel of the integral equation is a projection operator. The application of the boundary conditions is formally simple, but, in practice, appears to require detailed calculations. Perhaps the most satisfying feature of this theory is that only vertices corresponding to renormalieable interactions are allowed by the formalism. In most cases this restricts the form of interaction between arbitrary particles to such an extent that the interaction is uniquely predicted. In the case of the charged vector boson, no interaction is possible at all. As in other formulations the Fermi interaction cannot be introduced in a consistent fashion. This means that unstable particles can be treated by an in-out formulation of quantum field theory only in the approximation in which their instability is ignored. The proper treatment of unstable particles will probably come about in a theory in which many of the other properties of elementary particles (masses, etc.) are also explained. The solution of the equations for S-matrix elements by nonperturbative methods is not out of the question, although we are a long way from accomplishing such a program at present. The structureal simplicity of the system of equations presented here may be quite helpful in developing an exact solution. APPENDIX

In this appendix S’(Ai”(Z)X)+

I

we shall prove the relations

= (X+Ai,(x))-S

= h’(A,“,(~)s~)-

(11.9),

viz.,

= (SA,,,(s))+S+

(AI.l)

We first write iP(Ain(z)s)+

= Ain

+ s+[Ain(x),

Slret.

(AI.2)

QUAXTUM

FIELD

alid then note that the retarded derivative : S’(ilin(r)S)+

commutator

= ;lin(~)

THEORY

X(3

may be written

- i / d4yS’ &) - IlI~

.A,(L

as a functional

- y)

(AU)

Then since

G(XiS) _ al)

_ () = St- Is

(‘41.4)

6A in(u)

6-4in(Y)

we have s+(Ai,(z)S)+

= i4in(:L.) + i 1’ d4yAv(x = Ai,

- [Ai,(

- y) A) j In

‘3 (AI.6)

S’],,t S

= (Ai,(z)s+>-X which proves the first, part of (AI.l). In order to deal with the out operator parts of (AI.1 ) it is important to observe that the S-operator has the identical form when written in terms of either the in or out operators, i.e., if

sout E 2 0” wL=~ m!

/‘d4xl

. . . d4x,w(z1

, . . . , r, :A..,(.c,)

. . . il,:,,(.x,):

then Sin = Sout = AS

(-41.7)

That this is so can be seenas follows. Write Ai,(.ci) for each Ai,

= SA,“t(-ri)St

in Sin . Then since Sin = S, we have’” Sin = S = SS,“tS+

Multiplying

(AI.8)

on the left by S ’ and on the right by S we have Sout = S+LSS= S = Sin

13 It is seen that the normal product replacing each 9 ill the left and a factor il..,(.r,.): since

(111.8)

! 241.9)

:d ;,,(zrl) .4i,,(~m): = S :ilout(.iz) .. rl,,,t(.r.z): St by first expressing of A ill’s as a sum of time ordered products via Wick’s theorem and then of S can be taken to in each time ordered product by SA.., St. A factor of St can be taken t,o the right. The middle part recombines to :ilour(.tz~) to those formed by ilout’s. contractions formed by .4 >,,‘s are identical

370

PUGH

It will now be proven that S(A,“,(Z)S+)-

=

(AI.10)

(S+Ain(X))-8

We first note that (Ao”,(2$s+)~

= Ao”t(dS+

-

[Ao,t(z),

s+lret.

(AI.ll)

= AoddS++i / d4yAdz- Y)&

(AI.12)

In the above expressions S is considered as a functional of the A,,, operators Once the functional differentiation has been carried out, the remaining out operators may be written in terms of in operators via (11.4). Hence we get (AI.13) so that

(Ao,,(x)St)Multiplying

= Aout(x

i- 8% /

- Y> af

d4yAdz

In

b’

(AI.14)

on the left by X gives

~(~4,,,(x>~+>-

= SA,ddS+ = Ain

+ i / d4yAdz + i J d4y&(~

-

Y>

Y>

&

&S

In

In

(AI.15)

~‘3

= (X”Ain(z))-8 where we have used (AI.5). The proof that

X(A,,,(z)~+)-

(AI.lG)

= (SAm&))+X+

is similar to the proof leading to (AI.5). APPENDIX

II

In order to investigate the properties of the integral equation (III.lO), it is necessary to get an explicit expression for B(s , * * . , CL ; 4‘1, ’ . . , L). This is done by explicitly evaluating the matrix element (111.11). We note first that (AII.1) * (E,

El,

*.

.

, &)AB(x

-

E)

:Ain(El)

*. . Ain

:

QUAXTUM

FIELD

371

THEORY

so that (111.11) becomes:

(-q--l ^ 4 BWcn) = ---k-l ... Km (m _ 1)1 d [I ... d’&no ‘“‘(El , . . . ) fm) J (1~11.2) ‘(0 j ilin(.Cl)

’ ’ ’ i‘iin(Xh-1)

” ’ Ai . ’ izi”(,t,)

:Ain(,tl)

1 Ain(Xh+l)

’ f ’ Ai”

IO)

of the variables (x1 . . . z,) Here c,,p,,,, indicates a sum over all permutations and also a sum over all values of X. In the evaluation of (AII.2) the field operators Ai,( . . . , ‘Ix, . . . , Ain contract with the operators A in( (1)) . . . , dx , . . , A in( twL), and not with each other. Hence the position of the operators d in(Xl), . . . , LIx , . . , flin(lCm) relative to each other is immaterial; only the position of Ai, relative to the normal product is of any consequence. If iii,(.ri) is to the left of the normal product, me get a factor 6(si - q)[--~A+(L~ - (;‘)I (AII.3) where ,$&’ is any one of ((1 , . . . , nh , . . . , l*). On the other hand, if Ain lies to the right of the normal product, we get a factor B(.rx - q)[iA-(~~

- ti’)]

(Al1.l)

Combining these two factors we get the total contribution from the contractions of Ai, to be i6(ax - r;)A(zi - &‘) - iA+(ei - $;‘) (AII.5) Because A+ satisfies the homogeneous Klein-Gordon equation, the A+ term in (AII.5) makes no contribution to the expression (AII.2). Since 6~~~)($1 , . . . , lm) is a symmetric function of its variables, the sum over all possible permutations of the variables Ei’ produces a factor (m - 1) !. Hence (AII.2) becomes Bw’“’

= -Kl

. . . j-c,

I

d4& . . * d4& 2 A,(:ch - &) X=1

(AII.6) .@

B(rx - az)A(.ci - {,)u(~)({,

, .

, &A

Thus

\

(AII.7)

372

PUGH

APPENDIX

III

In this appendix it will be shown that the integral operator B(x~ , . . . , xn ; 5 , . . . , Em) is a projection operator. One first observes that R is a sum of operators, & ,

(AIII.1)

It will now be proven that the Bx’s are a set of independent projection operators:

Rx’ = Bx BxB,,

(-4111.2) if

= 0

X# a

(AIII.3)

These results are proven by direct calculation:

(AIII.4)

Integrate by parts with respect to Fj, (j # X), and only the boundary terms, ,&’ -+ - 00, will survive. Then using

s

d”a$j A(zj

-

lj)

k.

7

A(a$i -

yj)

= A(Xj

-

uj)

(AIII.5)

and KtiAdEx one gets (AIII.2). Bx B, = K1 . . . K,

- YA) = --6(b - YX)

(AIII.6)

To prove (AIII.S), s

d4$ . . . d4.&

(AIII.7)

one integrates by parts l&h respect to & and &” + + 00, survive. Using (MII.5) one gets

\vl-herr the double primes iudicate (A111.8) vanishes because of’

and thus \ve have proven I;inally, using (AIII.2) B” = which

proves

that

that i and j are equal

(AIII.3). and (MII.3)

2 2 &B, x=1 a=1

B is a projection

only

the

boundary

to neither

terms,

X 1101’ K. KOW

we have = 2 &“ x=1

=

2 & x=1

= B,

operator.

The :Lntllor wislles to express his sincere appreciation to Professor F. Rohrlich for his unfailing encouragement nnd helpful suggestions during the performance of t,his research. The :tut,hor llas benefited greatly from many convers;tt8ions with Professors M. Dresdrn. I’. K~LhIl, :tnd I). Kaplan. RECEIVED:

January

15, 1W-l

1. H. LEHMANN, IX. SYMASZIK, AKU FT. ZIMMERMANN, Xuovo C’inlento 1, 205 (1955); H. LEHMANN, I(. SYMANZIK, AXII W. ZIMMERMANX, .Vuooo CivLento 6,319 (1957); K. NISHIJIMA, Phys. Hev. 119,485 (1960); h. S. WIGHTMAN, Phi/s. Rev. 101,8GO (195G). 2. H. LEHMANN, N/tow C’imento 11, 342 (1954). 3. J,. I<. EVANS AND T. FULTON, Sut2eur Phys. 21, 492 ClXiO). 4. J. JAI:VH ANI) F. ROHRLICH, “Theory of Photons and Electrons.” Addison-Wesley, Reading, Mass., 1955. 5. J. HILGICVOORD, ,V’l’ud. Php. 15, 057 (1960). 6. E’. ROHRIXH, State University of Iowa Research Report, WI-62-15 (1962). 7. H. ROLLNIK, IS. STECH, AND E. NUNXERIASS, Z. Phqsik 159, 482 (1960).