Finite model of four-fermion interactions with indefinite metric

ANNALS

OF PHYSICS:

64,

532-572 (1971)

Finite

Model of Four-Fermion Interactions with Indefinite Metric*+ JAGANNATHAN

GOMATAM

Physics Department, Syracuse University, Syracuse, New York Received July 27, 1970

The scattering operator theory for fermions is developed in the framework of an indefinite metric Hilbert space starting with the second power Dirac equation. It is based on Lorentz invariance, strong pseudounitarity and strong Bogoliubov causality. These assumptions, together with operator derivatives and generalized convolution operators lead to a strong scattering operator equation in the indefinite metric Hilbert space. There is enough latitude in the homogeneous solutions of the scattering operator equation for the specification of interaction. The second-order two- and four-point functions are explicitly computed for the four-fermion interactions. The scattering operator theory does not involve renormalization in the Feynman-Dyson sense. There are no ultraviolet divergences encountered and hence no momentum cut-off needed. The vacuum diagrams are divergent in perturbation theory and call for a space-time cutoff. Arguments are presented to show that all the higher order functions lead to convergent integrals, The present approach, though mathematically satisfactory, is at least so far not fully amenable to physical interpretation because of the indefinite metric.

1. INTRODUCTION

The development of quantum field theory from its early Lagrangian formulation to the Tomonaga-Schwinger and Feynman-Dyson quantum electrodynamics is an instance of physical intuition supplementing and sometimes surpassing the demands of mathematical rigor. The success of quantum electrodynamics in explaining and predicting the experimental phenomena is the highlight of the growth of quantum field theory. This engendered the need for developing a mathematically satisfactory formalism of quantum field theory, in general, and quantum electrodynamics, in particular, at a level suited to perform calculations of interaction dynamics. This stage of development was marked by the emergence of asymptotic [l] and axiomatic [2] quantum field theories in the 1950’s. The * Based in part on a dissertation submitted in partial fulfillment for the requirements for the Ph.D. degree at Syracuse University, 1969. t Work supported in part by the National Science Foundation.

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former emphasized that only physically observable quantities should enter into the formalism while the latter stressed the role of rigorous mathematics in a precise formulation of the theory. A formulation that was free from divergent quantities and suitable for computation was developed by Pugh [3] in 1963. The theory contained a basic postulate concerning the behavior of the scattering matrix elements off the mass-shell and successfully reproduced the results of renormalized quantum electrodynamics without introducing the renormalization scheme. Later the work [4] of Chen, Rohrlich, and Wilner demonstrated that the dynamical postulate of Pugh could be deduced from the usual assumptions of the theory coupled with the strong Bogoliubov causality condition. The result was a theory [5] which had the salient features of the asymptotic quantum field theory (LSZ) and all the computational advantage of the Pugh [3] formulation, however, without Pugh’s assumption about the behavior of the S-matrix elements off the mass-shell. A finite formulation for nonrenormalizable scalar interaction was first proposed by Chen [6]. With the aid of the generalized convolution operators1 P$ and Pi:: and the conventional free field equations, the formalism [6,7] produces finite results in a given order of perturbation. Of course, the framework in Chen’s approach [6] is the Hilbert space with a positive definite metric. However, it suffers from the defect that as the order of perturbation increases the parameter N, to be chosen to make the integrals convergent also increases. A direct generalization of Chen’s [6] formalism to spin-4 case would share this defect. Therefore, we propose a formulation in which the generalized convolution operators are employed in conjunction with field equations of appropriate higher order.2 The Green functions corresponding to higher order field equations have higher order poles in the PO-plane. As an elementary power counting calculation (Appendix B) shows such theories could be made to be either completely convergent or at least renormalizable by a judicious choice of the order of the field equation. The employment of the generalized convolution operator eliminates the need for a mathematically ill-defined renormalization procedure. However, the presence of 1 More explicitly, the generalized convolution

operators are

P’;‘(x, y) E (K&N 0, Aa”-“(x

- 5) A’[-“(

P;,“‘(x, y) = (K~K~)Ni3,, A’$-‘-“(x - 5) A’;-“(y

y - ?), - ?),

where K~=

&-mm”

and

d&&:‘(x)

= (-)N-‘/(2~)4

JERcAjd*pe’ps/(pe + mr)N

z Higher order field equations as a means of eliminating divergences in quantum field theory has been first considered in the frame work of Feynman-Dyson theory by Pais and Uhlenbeck, see Refs. [S, 91. Our formalism differs in that we seek to replace Feynman-Dyson approach by the use of convolution operators.

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higher order poles in Green’s functions occasion the need for a Hilbert space with indefinite metric. The notion of indefinite metric and its possible relevance to physical problems were considered by Dirac [IO], Pauli [I I], and Heisenberg [12]. When the metric is indefinite the norm of a state vector could be positive, zero or negative. The mathematical properties of such linear vector spaces have been outlined in the literature [13] and their physical implications studied [14]. In this work we develop a finite formalism for interaction among four fermions in the four-dimensional Minkowski manifold. We consider interactions of the type &L,(x) $ux> kc4 h3w 9 where I+&(X) is the spinor field satisfying the higher order Dirac equation

=0. (Y”&+my#(x) Our delineation, based on Pugh-Rohrlich-Chen formalism is certainly different from those of Sudarshan [ 151 and Lee and Wick [ 161 which are based on FeynmanDyson theory. The approach outlined here, through the employment of generalized convolution operators secures the advantages intrinsic to their algebra in contradistinction to the step functions which are the elemental units of time ordering so often used in a Feynman-Dyson approach. Another important feature of the investigation presented here is that the theory is set up in the framework of the scattering operator formalism [17] but in an indefinite metric Hilbert space. The interpolating field and the attendant asymptotic conditions which present physical as well as mathematical difficulties [18] in the context of indefinite metric are eschewed. However, no attempt is made to answer mathematical questions of existence or to salvage the indefinite metric theories from conventional difficulties that beset them. The justification for the study lies in the fact that indefinite metric field theories provide a natural framework for a cut-off free, local and covariant formulation of nonrenormalizable interactions in contrast to an arbitrary cut-off dependant approach using Hilbert space with positive-definite metric. A clean physical interpretation of indefinite metric field theories will have to await the corresponding elucidation of the mathematical properties of linear vector spaces with indefinite metric. In Section 2, the postulates and definitions of the self-interacting fermion theory with indefinite metric are stated. The generalized fermion convolution operators are introduced in Section 3 and the basic equation for the scattering operator is derived. The two- and four-point functions are computed in second order in Section 4. The calculation is mathematically well defined and no cut-off is employed. In Section 5 a concise discussion of the theory is presented. The Appendices at the end contain calculations germane to the presentation, the most notable among them being the Kallen-Lehmann-Umezawa-Kamefuchi spectral representation

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for the products of Green functions that are mathematically well defined. Of course, these spectral representations are evaluated in the frame of indefinite metric. 2. POSTULATES AND DEFINITIONS

A. Lorentz Invariance The theory is invariant under the inhomogeneous Lorentz group. When field quantities are constructed explicitly, additional requirements such as gauge invariance of the first kind may be imposed. B. Free Fields [ 191 (a) There exist four-component free spinor fields which satisfy a higher order Dirac equation with a mass parameter. These free fields satisfy anticommutation relations. More explicitly

P(41N Jiw = 09 $w4w QIN = 0,

(2.la) (2.lb) (2.2a) (2.2b)

where D(X) = (y@& The commutator

+ in)

function S(“-l)(x S’N-l’(X)

and

iyU s

a

- m)

- y) is defined by

= (y &

- rn), A’N-l’(X),

where dcN-l)(x) are Green’s functions that satisfy K3s1vA(~v-1)(x) = 0. The product of the Dirac operator in Eqs. (2.1) is the ordinary matrix product. The second quantization of the higher order Dirac equation is carried out in Appendix A. The free fields are required to be generalized functions belonging to Y’. When suitably smeared these fields act as operators on a linear vector space Z with a nondegenerate indefinite metric. (b) There exists a vacuum vector @,, in S-P which is characterized by $(-j(j) @,, = 0 and $(-) (f) @,, = 0 for all fC Y and polynomials of free fields acting on @,, produce vectors that again belong to SF. Also Q(GO, @,J > 0, where Q(@, @) is the inner product in s?.

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C. Completeness of Free Fielak and Operator Derivatives The free fields are complete in 2. This implies that an operator on S? has the strong representation (explained below)

x : $(x1>.‘. $(&>$(Yl)**-$(yd:*

(2.4)

The quantity &(x1) = $( x 1) is a free spinor field. The spinor indices have been omitted for simplicity but could be easily recovered by associating a spinor index with each xi and yi . To be able to define the operator derivative [20] of F, each term of (2.4) should contain either even or odd numbers of spinor fields; in other words, each operator term on the right side of the expansion should transform according to some definite representation of the Lorentz group. The C-number coefficients

which transform like tensors (spinors) for even (odd) numbers of operator factors are totally antisymmetric under the permutations of the coupled set [;;a]. If F is a multispinor operator, then the unsaturated indices and the unintegrated arguments will be associated with the coefficient function f. Apart from the above mentioned symmetry conditions, the coefficient functions are assumed to be tempered distributions belonging to Y’ and are also restricted by the requirement that F is an operator on X. The presence of free fields in the expansion restricts the momenta in the Fourier transform of &(x1 ..* x7yl -** ul) to the mass-shell. The knowledge of MXl * . . xr Yl .*- yr) on the mass-shell is thus sufficient to determine the scalar operator on the mass-shell. The operator derivative of F with respect to &,l(zl) is defined by [20]

The notations

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537

will be employed throughout the text. Now in (2.5) the momentum variable corresponding to z1 is not restricted to the mass-shell. This situation will be henceforth referred to as z1 being off the mass-shell. Thus, the knowledge of the first operator derivative of F is equivalent to the knowledge of allf,,+, with one argument in each off the mass-shell. It is easy to see that the n-th derivative will relieve n-arguments in the coefficients of the mass-shell restriction. Thus the knowledge of all the coefficient functions off the mass-shell is equivalent to the knowledge of F and all its derivatives. This calls for a distinction between strong and weak equations with respect to the operator derivative. An equation which remains valid after arbitrary number of operator differentiations on both sides is a strong equation. Otherwise it is a weak equation. The notation g will be used for weak equality and z for strong equality. Following Chen [6], the free field equations will be taken as weak equations. However, the functional independence of # and $ calls for a special qualification.

KxX)lN ?&9 “=”0,

(2.6a)

W[(x)

(2.6b)

aIN 2 0.

Here 2 means that the equality is weak respect to #. However, Eq. (2.6a) is strong respect to 4. From the definition of the (m vanishing of the vacuum expectation value we arrive at:

under operator differentiation with under operator differentiation with + r)-th operator derivative and the (vev) of a nonempty Wick product (2.7)

D. The “in” and “out”

Fields and the Scattering Operator

There exists two sets of free fields, namely the “in” and “out” fields which satisfy the postulate Section 2B. This means that there exists a weakly pseudounitary operator S in %‘, called the scattering operator such that apt(x) : S%p(x) gFyt

s,

(2.8a)

: S@glr(x)in s,

(2.8b)

where So is the pseudo-Hermitian adjoint of S. However, we do not interpret the “in” and “out” fields as representing physical particles. E. The Strong Unitarity

Condition

We require the pseudounitarity

of the S operator to hold strongly. More explicitly se % s-1.

(2.9)

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GOMATAM

F. StabiIity of the Vacuum State QO and the States ~3~= #(+ND,, and 5, = $(+)00 It is assumed that SD, = Go (2.10) and sq = @I) (2.11a) sG1 = ?sl . (2.11b) G. Strong Pseudocausality

Condition

The strong Bogoliubov causality of the conventional hold here with appropriate modifications.

theory [17] is assumed to

seo5ir,(x - y), Support 6x(y)

(2.12)

where Q(x) = (iSo 2

, is@ g)

and

we adopt the name pseudocausality condition usual strong causality [17] condition.

x(y) = (#(y), m), in order to distinguish it from the

H. Physical Interpretation The above postulates should be supplemented by physical interpretation. A theory with indefinite metric leads to negative probabilities. There are attempts in the literature [14] to extract a subset of & spanned by physical state vectors. The physical space should have positive definite metric with S operator that is unitary, The question relevant to the present formalism is whether the Bogoliubov causality condition is satisfied in the physical space. But these problems will be investigated elsewhere.

3. THE GENERALIZED

CONVOLUTION

OPERATORS

The generalized convolution operators were first introduced by Chen [6] to give a finite formulation of nonrenormalizable interactions of scalar fields. The functions dh!W1’(~), out of which these convolutions were composed, emerged 3 In the conventional theories [4] the strong Bogoliubov causality implies microcausality for the interpolating field. A similar situation obtains in the case of higher order field equations if one defines an interpolating field through the S operator in an analogous manner, provided the interaction is such that SO(x)/Sx(v) C 94,-r where 9&-i contains all distributions of the type 8j$,$,8(x”-~),m+n<2N-l.

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naturally as Green’s functions of higher order Klein-Gordon equations in the works of Rohrlich [21] and Gorge and Rohrlich [22]. These ideas are easily extended to spinor case as is demonstrated in Appendix A. We take [D(x)]” I&x) “=” 0,

(3. la)

$0(x)

(3.1 b)

a]* 2 0

as free field equations. The various Green functions are given by $)(xy)

= [D(x)]* &‘(xy),

(3.2)

where S”)(Xy) = S’l’(x r r

v)

and

The dy’(xy) .SZdF’(x - y) are tabulated in Ref. [22]. We introduce the following generalized convolution operators:

L,%f(x, Y>= P(x) NY)Y ezvs AZdq d?M>

$‘(~rl)f(&

L&/lx,

Y)

i%Mx,

.Y>= 1 &i-d7,@(tx) ~i%W-(5, 4 &AW

rlk (3.34

= P(x) %~)l” @,,j 6 drl&?4 @b~rl)f(5, ~1, (3-j (2) (2) (2) (3.3c) I - LA = Lm + L4J-n

[%.I fk Y>= j- @ do @@‘x) ~~‘(~~)f(~~ (2)

(2)

(YYV,

(3.4a)

7) U(x>Q CYM”,

(3.4b)

12)

I - Pm1 = uL,l + ULI;

(3.4c)

Lkl f-(x, Y>= FWI” ox, J^4 4 $?(xE>f@~)~!?h~)[(~)~l~, (3Sa) L,~JJ(x, Y>= PWJ2 4, j d=$dq $?(xOf(5rl) ~~‘(~YMY)~~“, (3.56) (2)

(21

(2)

I - L41 = La,1 + Wzl.

(3.5c)

We adopted the convention that the spinor indices follow the same order as the letters of ~/IF argument. Note that the location of the subscript letters xy with respect to R indicates the location of the Dirac operators D(x) D(y) with respect to the rest of the factors in the convolution; whereas the order of the letters in

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GOMATAM

the subscript refers to the order in which the same letters occur in the step function. The meaning and power of similar convolution operators occurring in scalar theories [4, 61 has been well demonstrated. The various S functions occurring in these convolution operators are inhomogeneous advanced and retarded Green’s functions of the “squared” Dirac equations (3.1). The properties of these convolution operators are discussed in Appendix C. They satisfy the following orthogonal projection properties: (3.6a) (3.6b) Similar equations hold for other convolution operators. These convolution operators are distributors in Y(R*) and the class of distributionf2which form well defined products with these convolution operators is denot$ by 9”. T@ class of distributf;ns tha;s,form null convolution products with [,LQ and [,17,] are denoted by *A, and sR’, respectively. Their properties are discussed in Appendix C.

4. THE DYNAMICAL EQUATION FOR THE SCATTERING OPERATOR

In this section we will derive a new operator differential equation for the scattering operator. While the method of derivation is analogous to the scalar case [17], the underlying Hilbert space has a nondegenerate indefinite metric. We use the product rule [20] for operator differentiation and obtain (4.la) (4Sb) (2)

(2)

Convoluting the first of the above equations with [J7J and the second with [flu] and then using

we obtain

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Here we have assumed that various conv$nions occurring in (4.2) exist. This is guaranteed by requiring that S2S/(S~ Sy) E 3’. Later we will find that this is the case for four-fermion interactions. To be able to incorporate the Bogoliubov causality conditions (2.12) in (4.2) we must first formulate it in a particular fashion, viz. (4.3) The left side of the above equation is a convolution produc,t,of distributions. It exists on{;! for those interactions for which 6/&S U(y) E 9’. It is zero if 6/G 8(y) E sA’. For a detailed discussion of these ideas the reader is referred to Appendix C . Proceeding with the derivation of the S-operator equation, we see that because of (4.3), Eq. (4.2) becomes

(2) -{I - [&]} SQg Incorporating

the strong unitarity

=s [~J+gT~ condition

- $yEg.

(4.4)

(2.9), we rewrite (4.4) as

or

where (4.7) Here we note that [$?,I E sB’ should contain no more than the third derivative of 6(x0 - y”) as is shown in Appendix C. We will prove in the next section that this class of interactions include four fermion interactions. The adjoint S-operator equations could be derived by a similar procedure. (4.8a) (4.8b)

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GOMATAM

The following remarks are in order. We utilized the strong unitarity and strong Bogoliubov causality condition in deriving (4.6) and (4.8a,b). To make sure that the theory satisfies strong unitarity, we should consider only those solutions of (4.6) that satisfy strong unitarity. We would like to emphasize the fact that (4.6) is a new S-operator equation. We maintain that Chen’s [6] approach to nonrenormalizable interactions is not consistent with Hilbert space of positive definite metric. The field formalism developed by Rohrlich suggests that the Wick contraction function for normal ordered products of free fields in perturbation calculations should be SF)(x) rather than S*(x), as a straight forward spinor generalization of Chen’s [6] approach would have demanded.

5. PERTURBATIVE SOLUTION OF THE S-OPERATOR EQUATION It is well known that in a conventional Feynman-Dyson approach to the fourfermion interaction :$(a$ #(x) 4(x) Z/(X):, only the first-order perturbation calculation yields finite result. All higher order graphs correspond to divergent integrals. In Appendix B we have demonstrated that in the present formulation (N = 2, r = 4, s = 3) only the vacuum graphs and two- and four-point functions can be primitively divergent in the Feynman-Dyson sense. The vacuum divergence can be taken care of by a space-time cut-off which is removed after performing computations. The space-time cut-off also guarantees that the scattering operator of various orders in perturbation exists as a well-defined operator intelhe Hilbert space with indefinite metric. The use of the convolution operators 17’s eliminate the need for renormalization of the primitively divergent diagrams. In this section we calculate the two- and four-point functions in second-order perturbation and exhibit the finiteness of the computation at every stage. No ultraviolet cut-off is employed. We will obtain the solution of (4.6) in perturbation theory. Expanding fl and S in terms of the coupling constant h,

with (5.3)

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we obtain from (4.6)

(5.4) We define the two-point function to the k-th order through ,W(X,

y)

=

(p&"

From the perturbation expansions, it is clear that mcr)(Z, y) is expressed in terms of the S operator and its first derivatives to (k - I)-th order and lower orders. Now

~W, We consider the interaction is specified by S(I) 1 &

Y)

= wJ,ld”

(5.6)

.

:&(x) #&Y) #B(~) $Q(x): . The first-order

j” dx :$a(-4 VW) $&)

vertex4 (5.7)

hdrc): .

This implies that

Lo&yl(I) 2 &w -

Y)CL.

343(Y) iJa(Y>:

- :&Y(u) v47tY):17

(5.8)

hence

Next we consider the two-point

function to second order. From (5.3) and (5.4),

6Jyx,y) = ([&](2))*+ [JYy](+

Jp), - [J?,](F

Jg),.

(5.10)

Consider first, the second and third terms on the right side of (5.10). From (5.7), 4 To be mathematically precise the vertex part should be specified by S”’ = 1/Q 9” J d&x :&(x> ~.dx) s&W #&): f(x), wheref(x) is a c-number function with support in UP, and which in the end is allowed to tend to 1. The above form guarantees that S (l) is a well defined operator in Z. See Ref. [23].

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GOMATAM

the rules [20] for operator differentiation

Wl) Qh

qFsy,,=

>

and Wick’s theorems, we see that

-i[S”‘(xy)] f

p Tr[S?( vx) S”‘(xy)] + YY

+ i[SJl)(xy) Sl’)(yx) SJ’)(xy)] YY ’

(5.11a)

+ i[S’l’(xy)

(5.11b)

and

gqyx)

s2)(xy)]y,y .

In Appendix D the spectral representation for the products of $1) functions are obtained and then it is shown that the convolution~p,(5.11a,b) are well defined. Because of the presence of two [D(x)]~ operators in n’s, the above convolutions are proportional to (ip * y - m)” in momentum space. The ([J3n/]&o is restricted to sB’, but is arbitrary otherwise. But the stability of the state G5,, under S-operator transformation (Section 2F) would demand that the Fourier transform of cP(ZJJ) should vanish5 on the mass shell like (ip . y - m)“. But this implies that ([&?y](a))o should involve the factor (ip . y - PZ)~, which we know is not permissible since ([~j3y]~z))o E sB’. Thus we demand that

azSlJd0 = 0. This means that [&]~z, is a linear combination More explicitly the two point function is:

u(2)(%y) =

-

(2&

[D(x)]2

- 2nZ(K +

1

m)”

@2

dzfl”(mK)

:

(5.12) of :9(x> J&J): and :m

K2)4

&(X

[~z??~‘(VZK)

&(X

-

K)][(J’)(T12.

y,

-

y,

I,&): .

K)

(5.13)

Thus we have shown that the two point function can be computed without encountering divergences which is the case in the conventional theory of fourfermion interaction with positive definite metric. A brief comparison of the present formalism with N = 3 and the theory of leptons proposed by Sudarshan [15] can now be made. The propagators in both 6 This stipulation has a physical significance and consequence in theories with positive metric while the role of this postulate is a little vague in theories with indefinite metric. See the reference cited in Ref. [17]. Since the spirit of the presentation here is to keep as much contact as possible with positive definite metric theories we have retained this stability criterion.

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formalisms behave as l/p3 for large momentum. Consider, for example, the twopoint function in second order. By counting powers of the momentum variables, we see (Appendix B) that K = -1. Thus with the choice iV = 3, the interaction belongs to the class 9:. In Appendix D a sufficient condition for the equivalence of our formalism and that of Feynman and Dyson is derived: Terms like SSc,)/SX SS,,JSy must belong to the class s8’. Next we will compute the four point function”

to second order and show that it is finite, without using any cut-off whatsoever. First, we will derive the equation for the “four-point operator” S4S/(SZ SW 6X 6~1). Since (4.6) is a strong equation we can take its operator derivatives to obtain 64s sz SW 6X sy

-

s;yszp E - +& (ss”E sz

SY

- [j?*.

[*

6% ss SQx--+ sz SW sy

__ II 6‘3 sz sy

SSQ ss - ___SW 6.2 1

(se&

SSQ 63s -- ss --+sQ,~;~,~-g-~-~&) sy ( SF SW sx -

sy t- so

Z

62s

SSQ ss ( --sz 6% -1 P&)1.

(5.15)

We will solve (5.15) in perturbation expansion using the specifications (5.1), (5.2), and (5.3). Since each term on the right side of (5.15) is a product of three operators ABC say, a typical term in the perturbation expansion will have the structure Xz,$ A(s)B(~+.-,)C(,) . For the second order, the expansion is 40)B(2)C(~) + 4441)C(0) + 41)40)C(1) + 40)%G) . The term B is one of these S@, SS@/Sx, and S2S8/(Sx 64) to 0, I or 2nd order, where x and 4 denote “in” spinor fields. The terms A and C are either first or higher derivatives to 0 fi There are no three-point functions for the four-fermion interactions.

546

GOMATAM

or 1st order. With the specification (5.4), the only surviving terms have the structure A(1)B(,)C(1) . Thus in second-order perturbation, we obtain

When we take the vev of the above equation, by virtue of (5.7) products like (6S~,,/6X 63S(,)/(6Z 6w Sy)), yield terms proportional to (:&&&): :#( Y):)~ . But this is zero because there will be nonempty normal-ordered products left out after performing contractions. Thus, terms in which third-order operator derivatives occur vanish. Thus rdy,-, co, x, y) =

S4S(2) ( &if 6w 8X sy > 0 = ( +

[.&J

[(gg

- [,Yi$ [Restoring

82 62 *w

a),

-

(+$),

(2)

>

(&

%)J

+ (*a)J

the spinor indices and performing

+ ${ [J7,] Tr($!(xy)

L&1(2)

(5.17)

relevant calculations we see that

S!?( yx)) + [,EJ

Tr S!!‘(xy) Sl”( yx)}

x 6(x - 4 %Y - 8 hA0 (2)

- t{r&l SJl)(XY)o, Sf)(xY)KY + rJ‘t1 x 6(x - 2) S(w - y) (2)

+ a{[J$l S%Y>,, S%YL x 6(x - 2) S(w - y)

+ Lfizl

~Yxy)o,

~YXY)KJ

~YXYL

mxY>K,>

S%Y)K,

fmY&>

(2)

+ ~{[a&1 SJ”(XY),, SYYX),, + dzl x 6(x - w) S(y - 2).

(5.18)

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In Appendix D we have obtained the spectral representation for the convolution products of S functions occurring in (5.18) and shown that various convolutions in (5.18) lead to convergent integrals without invoking any cut-off whatsoever. The term
for all pi2 == -m2. This would require that (@/(SZ SW)[$~]~~,.,,,should be a linear combination S(x - z) S(w - y) and S(Z - JI) S(M,- x) which, in fact, is the case as is seenfrom (5.12). The two- and four-point functions in higher order do not pose problems of convergence as is seenfrom Appendix B. This is also true for higher point functions. Of course, an explicit demonstration of these facts call for involved computations.

6. DISCUSSION The scattering operator theory of the interacting spinor field developed here is based on the second-power free field Dirac equation, Lorentz covariance, strong unitarity, strong Bogoliubov causality, and other pertinent assumptionsof quantum field theory-all in the framework of an indefinite metric Hilbert space. That there are no divergences in the second-order two- and four-point functions for the four-fermion interaction is explicitly demonstrated. This is the chief feature of this work. We feel that earlier attempts to formulate nonrenormalizable theories using generalized convolution operators, but the usual contraction functions in the framework possitive definite Hilbert space is a particular realization of the projection to the physical space. This approach, while regains the obvious advantages of a physical Hilbert space, suffers from a rather serious defect of containing an arbitrarily large number of undeterminable parameters [7]. There are still unanswered questions. While the indefinite metric approach is mathematically satisfactory, it is not yet amenable to complete physical interpretation. The prospect of finding a projection of the indefinite metric Hilbert space to a physical space in realistic theories is not yet clear. Whether the theory can be local in that physical space is also still open to question.

APPENDIX SECOND QUANTIZATION

OF HIGHER

A: ORDER FREE DIRAC

EQUATION

The second quantization of higher order Dirac equation can be performed by a suitable generalization of the corresponding problem for the scalar field [21]. Throughout this section the equations are weak equations.

548

GOMATAM

1. Plane Wave Decomposition7 The free field equation is given by PWIN

K4

(A.la)

= 03

with the stipulation mX>IM

M
16-(x>f 0,

(A.lb)

to exclude the solutions of lower order equations. Substituting $6) -- t2&

s d*p F(P) eipx

64.2)

in (A. la) and (A. 1b), we obtain (A.3a)

tip . Y + mlN F(P) = 0, M < N.

(ip * Y + mY F(P) # 0,

(A.3b)

We introduce the operators [A*(p)]N

= [ m ;z

. y 1”

with the properties k&?)l”

[&p)ln

=

I%:(P)I~-~ ( Pz~~a ):

m Z n I=- 0,

= k’L(~)l”-~ ( pa4~~e )m,

(A.4a)

n 2 m > 0. (A.4b)

However, on the mass-shell (p” + mz)” y(p) = 0, V+(P>lN

+ [A (PY

= M+(P) + aP>l” =1-p2i-mz

2m2

- y

7=1

for

(;)

V+(P>lN=-

RUPW

N = 2

(A.9

in general.

64-6)

and L,(P)lZN

f kL(P)lN

’ The notation used here is adopted from Ref. [19], Appendix

A2.

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549

INTERACTIONS

From (A.3a) and (A.3b) it follows that T(P) = s-w2

+ ~“wuP)l”

where x(p) is an arbitrary function of

#(x) =

-(2;)2

Substituting

p.

atN-l)(p2

64.7)

(A.7) in (A.2) we obtain

+ m2w(P)lN

i 8P [@(PO) atN-YP2

+ 0(-p”)

X(P),

+ m”)[~L(p)]~

x(p)]

X(P) eiPz.

64.8)

Using the identity wJ-yp2

+

my

N-l =

lro

[&N-l-u’(”

"q&GJ

-PO)

+

(-l)N-1-v

~w-l--V(PO

+

w)],

(A.9

where 1 (N-l++)! “=y?(Njzq~ w = +(p”

+ my,

and using the product rule for differentiation

(F $y” = Nzi”

tV-(P7 p”)lN x(P,PO>e-i*“9j (”

- ,’ -

y)(~)N~‘~“~r

one finds

Po=TW

([A-@,

T-co)]~ x(p, +~)(-lixO)’

e*tiwzo.

(A.10) Thus

x [(--I)’

(&)“-‘-“-’

+ (g-l--

([L(p>lN

W+(P)I~

x(p)) eips

X(-P))

(A.1 1)

e-i-1.

We now define the plane wave spinors u and v through

[~+(P)lNU(P)

= 0,

r4(pN”4P)

f- 0%

M
(A.12a)

550

GOMATAM

and

PUPY 4P> = 0,

LUPP

a> + 0,

M
(A. 12b)

and set

and (A.13b) where u,(p) and v,(p) are solutions of (A.12a) and (A.l2b),

x [(- 1)’ (&)N-l+r + (a&+-

respectively. Thus

([8.rrmu]N/2 a,(p) u,(p)) eiPx ([8rrmco]N/2 b,@(p) v,(p)) ciPs].

(A. 14)

We note the presence of the time-dependent term (ix”)‘, for r > 0. The above decomposition is Lorentz invariant, because the decomposition (A.8) is. However, the meaning of the individual terms in the summation is not obvious. There is another approach to the problem of plane wave decomposition which is less cumbersome. We rewrite (A.la) as

w(XwwlN-l We immediately

N-4 = 0.

(A.15)

see that but for additive constants

[w41N-’ $44 = #oc4,

(A.16)

where D(x) #o(x) = 0. By virtue of (A. lb) we can immediately write down the most general solution of the inhomogeneous equation (A.16) as

W) = - j 4 S:N-2'(v)$43(Y),

(A.17)

where SjN-2)(xy) is th e inhomogeneous Green’s function, defined later, by Eq. (A.26). For reasons which will be clear presently, we will choose the inhomo-

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INTERACTIONS

551

geneous contour c,. Now using the plane wave decomposition obtain: $xX) = ____ ‘,;;’

#(f) = w 77

for z/~,(y), we

I R (p2 + m2)Np1 S(p’ + n?) L(p)

s 44ip . y - mY1

i(p) eiPJ

(A.18)

i dp R (p2 +1m2jN-l

&P2 + ~2)[~-(P)lN-1

UP)

%I?)“&). (A.19)

Setting 1

(A.20)

LUP)Y+‘,

T(P) = R (p” f my-1

we find L’-(P) A-(P)

= g$g

1 j dP (fj””

%P>~~PI + T(-P)

4(p)

~(-P).&P)

(A.211

v-(P) G?.(P)UPlAP)

T

V(P) C(P)&P)l.

(A.22)

HP) = ( 87TmQJPZ cc d,(P) UP),

(A.23a)

+ T(-P) Here we have set -UP)

A+(P) 3-P)

= (8 rmoY2

(A.23b)

c 6,@(P) 6,(P),

where A+(p) a, = 0 = A-(p)

6, )

r = A.

The representation (A.22) resembles the expression for the plane wave decompot sition for the ordinary Dirac equation excepting for the presence of the weighdistribution densities (2~71)~-~T(p) and (~FYz)~-~T( -p) and the test function f(p). 2. Green’s Functions

We define the homogeneous and inhomogeneous

Green’s functions by

F = homogeneous contour,

(A.24a)

r = inhomogeneous

(A.24b)

contour,

552

GOMATAM

respectively. From the scalar theory [21], we know K,,$‘-1)

= 0,

(A.25a)

= --6(x).

(A.25b)

Therefore, we obtain S;N-l)(x) = [D(x)]~ AiN-“(

(A.26)

where D(x) = (~8, - m). The A$?‘J(x) functions (the d:"-')(p) of Ref. [9] must be multiplied by (-)” to obtain the corresponding quantities used here) for various contours are listed in Ref. [20]. In particular, we have tJ(N-l)(p) c

=

(ip

’ Y

(P”

where giN--l)(~) is the Fourier conditions are now

transform

+

-

dN MN

(A.27) OIR’

of SAN-‘@). The usual quantization

{$Nx>,&9> = iS’N-l’(X - Y>,

(A.28)

{VW, w>> = 0 = Gm

(A.29)

x3>.

3. Operator Derivatives The results of Ref. [20] are valid here with the replacement of D(x) by [D(x)]~ and (x)Q by [(x) aIN. For example, we obtain 6F8 8 --i x = PWIN

C#), F.&-I

(A.30)

,

(A.3 1) 4. Two-point Function In order to demonstrate the indefinite metric of the associated metric space we will compute the two-point function explicitly:

(A.32) =I ’ s dx dyf(x) g(y)[D(x)]”

AIN-l’(x

- y).

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Here the test functions f and g belong to Y. Expressing the integral (A.32) in terms of their fourier transforms, we obtain

(0 I WI $(g>I 0) = ,,,,!$"

1>,j dp4 4 tip - Y - 4" &PO)

x ~(N-1)(P2 + Mw=((g + p) g”(q’ - p).

(A.33)

In the conventional theory with N = 1, the left side of (A.24) is the square of norm of the one-particle state. If we define the left side to be the square of norm of the “one-particle state” even when N # 1, we note that it is not positive definite, because of the fact that the derivative of 6(p2 + m2) is not a positive definite measure while the distribution 8(p2 + m”) is. Thus indefinite metric makes its appearance in theories with higher order field equations.

APPENDIX

B:

PRIMITIVE DIVERGENCES IN THEORIES WITH HIGHER ORDER FIELD EQUATIONS

In this appendix we will investigate the conditions under which the integrals associated with primitive diagrams, corresponding to theories with higher order Dirac equation converge, using elementary power counting methods. We will employ the following notation: Fi = number of internal fermion lines; Fe = number of external fermion lines;

n = number of corners in a diagram; r = number of lines meeting at a corner; s + 1 = the dimension of the Minkowski manifold of spacelike dimensions; Ly(p)

=

1 = Fermion propagation (ip . y + m)”

where s is the number

function of the theory.

We denote by M the integral associated with a primitive known [19] that one can write M=

j:dn

j;dy

.- j$“ds+‘k,

.,. &+lk I),

diagram.

It is well

(B.1)

where x, y... are Feynman auxiliary variables and p is the number of independent momenta with (s + 1) components that are associated with the internal lines of

554

GOMATAM

a diagram. Because of energy-momentum conservation at each corner and the overall energy-momentum conservation we obtain p = Fi - (n - 1).

At each corner there are r lines and each internal Therefore,

03.2)

line connects two corners.

nr = 2Fi + Fe .

(B.3)

According to Feynman rules we associate the propagator @‘$kJ with each internal line. Thus, .9/N is a multinomial of degree NFi in internal momentum variable. We now define the degree of divergence by k = (s + 1)~ - NF,.

(B.4)

Expressing p in terms of number of external lines Fe , with the aid of (B.3) we obtain k = n [-

(s + 1 - N) - (S + I)] - $ (S + 1 - N) + (s + 1). (B.5)

We will now investigate (B.5) in detail. We are now interested only in Lorentz invariant and gauge invariant interactions. This will restrict r to even numbers. Then from (B.3) it follows that Fe is always an even number. Also, for an integration over internal lines II 3 2. For a theory to be renormalizable, no new primitive divergences should appear at higher orders. This means that the coefficient of n in (B.5) should satisfy the condition ; (s + 1 - N) - (S + 1) < 0. Since we want to consider all multiparticle restriction

(B.6)

processes we should impose the

- $ (s + 1 - N) + (s + 1) < 0.

(B-7)

Case 1. Lets = 3 and N = 1. Then k = n($r - 4) - $F, + 4.

The condition

WV

(B.6) is obeyed if r 6 3. The only possible nontrivial choice is - QFB + 4. The vacuum diagrams corresponding to Fe = 0 are at worst (n = 2) quadratically divergent. All other diagrams yield convergent results. From (B.8) it is clear that the four fermion interaction in four dimensions is not renormalizable with N = 1. r = 2. Then k = -n

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Cuse2.

555

1NTERACTIONS

Lets=landN=l.Then k=n&2)-%+2.

(B.9)

The restriction r < 4 satisfies (B.6). When r = 2, k = -n - FJ2 + 2. While the vacuum diagram is at worst (n = 2) logarithmically divergent, all other diagrams are convergent. On the other hand, when r = 4, k = -Fe/2 + 2. Then the vacuum diagram is quadratically divergent. The other diagrams with F, = 2 and 4 produce linear and logarithmic divergence. Thus, we see that the four fermion interaction in two dimensions is renormalizable with N = 1. Case3.

Let s = 3 and r = 4. Then k = n[2(4 - N) - 41 - 2 (4 - N) + 4.

(B. 10)

The choice N = 2 satisfies (B.6) and we obtain k = -F,

(B.ll)

+ 4.

Corresponding to F, = 2 and 4, we obtain quadratic and logarithmic divergences. Note that (B. 11) is independent of the order of perturbation. Also for all F, > 4, k < 0. Thus, only the vacuum graph and two- and four-point functions are primitively divergent. The higher point functions yield convergent integrals. Case 4.

Let s = 3, r = 4, and iV = 3. Then k = -2n - 2 + 4.

While the vacuum diagram is logarithmically divergent all other diagrams are convergent. The choice N = 3 yields a propagator which behaves as l/p3 for large momentum. Case 5. Let s = 3 and N = 4. Then (B. 12)

k = -4n + 4.

This choice makes the theory completely convergent for any finite r.

APPENDIX

C:

PROPERTIES OF CONVOLUTION

OPERATORS

We will study in this Appendix, properties of some of the convolution operators introduced in Section 3. (2) The class of distributions 9’ with which the convolution operators form well-

556

GOMATAM

defined products is obtained as follows. Substituting transforms we obtain for (3.5a) [$v]f(x,

y) = w

the appropriate

Fourier

f dp dq eciszmiav) &,

&&+

k, ' y - m]" [@ + k) * y - ml2 - iE)[W92- (PO+ kl - i4”l”

+ k -$ + W 2 -

(q.

+

k.

_

4212

(C.la)

[(y)(7]2,

where k = (k,, , 0). Similarly one finds [,Ez]f(x,

y) = & X

1 dp dq eciDSwiQY) m &,

r'bk+ :'

*; - ml2 I@/ + 4 ' y - ml2 ir wD2- (p,, + k, + i~)~]”

s

x ji~+*,-6-!W 0,

2 -

(q.

+

k.

+

i42]2

(C.Ib)

[(y)a]2'

Now for the products (C.la) and (C. lb) to exist as well defined distributions, impose the following restrictions* onJ(p + k, -(q + k)): (0 Rip + 9, -(q

we

+ 9) - analytic around k = 0;

(ii)

I~((P + 4, --(q + k))’ --, 0 Ik I4

(iii)

(ipy + m)” [i(p + k) y -

for

k -+ 4 co uniformly

in p and q;

ml”

x ft’(p+ k -(q + k))R I(p + k;2+ m212 , t&v + ml2 [Kq + k) y - ml” x AP + k, -(q + k)) R I(q + k;2 + m2]2 , (ipy + 4” P(P + k) y - ml” X Jim + k -(q + k)) E (po + k,) W’[(p + k)z + m2], GQ + ml2 b(q 4 k) y - ml” X i(~ + k -(q + k)> E (a, + k,) 8(l’[(q + k)2 + m”],

should be well defined generalized functions ~9” in the variables p and q. * Through the existence of this class was tacitly assumed in earlier literature (Refs. [3,4]) a clear characterization was first given in Ref. [17].

FOUR-FERMION

The class oij(x, y) such thatfip denoted by S’. The subclasses (21 %’ = I faCx, ykf=A~ .I%P

+

k

+ k, -(q

+ k -(q

44

I ko

+

k))

551

INTERACTIONS

-

+ k)) satisfies the stipulations (i)-(iii)

analytic and

+ k)) o

for

ik,j+

a

in Imk,tO

I

I4

(El analytic and %’ = i .h(x, Y);~R(P + k -(q + 4) LO + k;(q + k)) --j o for j k, j -+ co in Imk, 4 0

>O

i

satisfy (2)

Ltn~lf4(x, Y>= 0, (2)

L/Fzlfi(~, v> = 0,

(C.24

(C.2b)

respectively. Considering (C-la), we easily see that because of the analyticity of fA(p + k, -(q + k)) and its asymptotic behavior as 1k, j4-s, T,I> 0, we can close the contour in Im k, < 0. Then the k” integral will vanish by residue theorem. p alss, reco;d that the class of distributions satisfying (4.7) is given by SB’ = FA’ n SR’. In coordinate space this means that [&,] is a distribution with point support and can contain utmost the third derivalue of 6(x0 - ~0). The other convolution operators could also be treated to similar calculations. We will next prove the assertions (3.6a) and (3.6b). Using the identity

where

558

GOMATAM

we can prove (3.6a) provided the surface terms vanish. That this is indeed the case can be seen as follows: The second term on the right of (C.3) is

The most direct way to demonstrate the vanishing of (C.4) is to reexpress it in terms of volume integral as follows: s

d4qPWI” LkP~‘CM - s d4q wi2

d%q) [@$I’ ~%~Y>KJW~~

~3cy~md%4

d%&i%12

d%.~~6%1~.

(C.5)

Next we will show that inside the integral of the second term we can write

~4+9%diwV = 4dd%4&FV~.

(C.6)

We first observe that, in the integrand,

= scl)(xO - $1 s2ju17) - 26(x0 - VP ao?3~)(xT) yO + - 2m[6(x0 - 74 So)

+

ecxO- $Jj s:)(~~)

Sn2

ecxO- 7’) $)(x7) Sn . ~1 + m28(x0 - 7’) So). (C.7)

The terms containing 6 functions and their derivatives But the product of 8,,8,, with 8(x0 - q”) and its support only at x0 = q” = y”. Bu t in this region terms containing 8(x0 - TO) and its derivatives do justified. Using [D(q)‘Jz Sjql)(rly) = -S(v - y) in the

in the second integral in (C.5), we obtain

have support only at x0 = ~0. derivatives has nonvanishing P(77y) = 0. Thus, in (C.7), not contribute. Thus (C.6) is first integral and

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INTERACTIONS

The support properties of (C.8) are in part controlled by those of e2y0az0U20YV= kdu~ . This implies that this product is non vanishing at u” = x0 = y”. But in this plane W(vy) = 0. A similar nontrivial calculation shows that the secondsurface term also vanishes. Then using the fact that S(j) are inhomogeneous Green’s functions and 8,,8,, = Bz, we arrive at (3.6a). The identity (C.3) can also be used to prove (3.6b).

APPENDIX LORENTZ

INVARIANT

D:

GREEN'S FUNCTIONS

AND THEIR PRODUCTS

1. This part is devoted to products of Green’s functions S, and d, in fourdimensional space-time. The following expressions are well known 1241:

CD.11

0+(x, 4 a+(-%b) = i j=0 +dAa, 6, K”) d+(x, K), where

- ta - b>2>1”” eLK2_ ca + by] dK2 &da, b, K”>= [tK2- Ca+ b)2HK2 16dK2

VW

and

where du&,

b,

K)

=

‘(K2

-

a21;nyrK;

4a2b211’2

[(b +

K)”

-

a”]

o[K2

-

(Q

+

b)2]

dK. (D.4)

It is easily checked that

S+(x - y, m) S-b - x> 4

dq eiq(2-r)8(-qo) 6(q2 + m2)(iq . y - m) f = [y . &4+(x - y, m)]2 - m2[Ll+(x - y, m)]” X

= Kl,

- 4m2)V+.(x - Y, m>Y,

(D.5)

where we have used (D.1) with a = b = m to obtain the last step. Also Tr[S+(x - y, m) S-(y - x, m)] = 2(0,

- 4m2)[d,(x

- y, m)12.

(D.6)

560

GOMATAM

2. In this part we will outline the calculations that led to the assertion that the two-point function in second order is finite for the four-fermion interactions (Section 5). We first note that SJl’(xy) S!f’(yx)

= [(cl,

+ m”) OJy(x - y, m)]” - 4m2[a,“&‘(x

= 2m2[-13,

- y, m)]”

+ 4m2][dJ1)(x - y, m)12 + [Ll+(x - y, m)]”

- 8m2Lp(x

- y, m) d+(x - y, m).

(D.7)

Therefore, the two-point function is given by w@(x, y) = - y {[EJ

S.f’(xy) Tr[SF’(xy> S
(2)

- [,Il,]

d?(xy) Tr[S!‘(xy)

5’:‘( yx)]).

Next we will express the right side of (D.7) in terms of d+(x - y, K) integrated over a suitable function. We will then express S~)(X - y, m) d+(x - y, K) in terms of S+(x - y, K’) and d+(x - y, K’) integrated over K’ with suitable weight functions. We also note the result

= d,(x

- y, a) Ll+(x - y, b) + 2a2 @‘(x

- 2a[y - a, ol”(x

- y, a) d+(x

- y, b)

- y, a)] Ll+(x - y, b).

V-9

We now proceed to perform the various calculations mentioned above:

Ll+(x,a) dJl’(x, b) = - & Settingp

1 dp dq 8(p”)

B(q”) 6(p2 + ~2) W(q2

+ b2) ei(p+g)o.

+ q = u and choosing the frame II = 0, we obtain

d+(x, U) A:‘(&

b)]o=o = &

s du eius 1 #p dp0 *cu;, &4

X

- u” + PO> @-%I3

where w, = (p2 + u2)lj2. After integrating obtains A+(% a> d%

a

A- &lb + dp”

po) - 240+ PO)

GJd2

I ’

over dpo and d3p = 47r 1p I2 d 1p I, one

b> = i ,,” dp,A,,,(u2, b2, K2) 4+(x, ‘d,

(D.9)

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561

INTERACTIONS

where

p(a2, b2, K”)

=I g

@K2- (a -t bj21icK2+ b2 _ a2j2

_ (K” + U2 - b2)[8a2K2 + (K2

with

b2,

p(U”,

K”)

=

{[K”

(K”

+

b”

-

2(K2

+

U2

-

b9)2]

(D

dK2

lo)

K”)

(a + b)2][fc2 - (a - b)2]}1’2. We note that

-

dpddc1,(u2, b2,

A similar computation

cl')2

-

b2 - u~)~~(u~, b2,

+

K”)

.2-t%

5

(D.ll)

.

yields

kt’(x,

ml]”

=

i s,r

dpA,l,A,,,(m2m2K2)

d,(x,

(D. 12)

K),

where dpd,ud,l,

(m2, m2,

K”)

= & X

--

2(38m2K2 - 96m4 K5(K2

o(K”

-

h2)

-

~K~)I

J&)3/2

dK2.

) (D.13)

Note that (D.14) Using (D.2), (D.9), and (D-12) we obtain $%Y)

~‘l’(YX)

= i j-w dps,l,s,l,(m2, m2, K”) A,@ - Y, K), 0

(D.15)

where (m2, m2, K’) dpsClts,l>

= 2m2(--tc2 + 4m2) dpd,l,dc~,(m2, m2, - 8m2 dpddtJm2, m2,

For large

K”)

+ dp,,(m”,

m2,

K”)

(D.16a)

K”).

K’,

dp

stl~s(Jm2, m2,

K2)

*

const. dK2

because of the presence of dpAd on the right side of (D.16a).

(D.16b)

562

GOMATAM

We next proceed to evaluate SF)(x - y, m) d+(x - y, m): [$&‘(x

- y, a)] d+(x - y, b) = i 1: da (u2, b2, tc2)S+(x - y, + i lrn da (a2, b2,

K2)

K)

Kd+(X

-

y,

(D.17)

K),

0

where da(a2, b2,

K2)

E

!‘(a”,

-!-

8.rr2

b2,

4K2(Kz

K”)

62 - a”)

+

P(a’, bZ, K”) + t (K” + a2 - b2)

~~

2 a2 - b2 +

+

1 1 K2 +2”,“e&(a”, b2, K2) j

-

~~

1 b2 - a2

+

“‘1 QK2 _ (a + b)2] d/c%. (D.18)

Therefore sl” A+@ - y, K) = i s,” das&m, - 2mi

K2,

K’)

m da(m2, I0

K2,

d+(X

-

y,

K’)

(D. 19)

K”) s+(X - J’, K’),

where dus(l)d(m, K2, K’) = dpbd(mz,

K2,

It is easily seen that

for

K’?

+

large

2m2

K

dpgAw(K2,

and

daswd(m,

K’

K’~)

-

+

COIlSt.

du(m2,

2mK’

with the restriction K’2)

K~,

m2,

K <

K2,

K’3.

K’,

d//2.

(D.20)

Using (D. 15) and (D.19), we obtain )S~‘(xy) Tr[SF’(xy) S?‘( yx)] = i Jrn dLW(m,

K’)

d+(x

0

-

y,

K’)

-

2mi lrn dz(l)(m!c’) s+(x - y,

K’),

(D.21)

0

where dz(l)(mK’)

= / dps(l)s(l)(m2, m2, K”) dos(l)d(m,

K~K’>

(D.22a)

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and d%‘l”(m~‘)

Similarly,

-e

d~s(,,s(&12,

Y)l’,

K”)

h(m2,

K2,

(D.22b)

K’2).

we obtain

&S(“(xy) Tr[Sl’)(xy) \ 55 dz”‘(m~‘) -4 0

St’< yx)] d-(X

-

y,

K’)

-

hi

li d,i?““(md) 0

S-(x

-

y,

That this is so could be easily checked by using (D.21) in conjunction relations S+(y - x, -m) = S-(x - y, m), sl”(y

- x, -m)

= --s’l’(x

- I’, m),

d+(y

- x, -m)

= -A-(x

- y, m),

dDl’(-f?l,

-K’)

=

d?“(m,

cfYW’(-m,

-K’)

=

&P’(m,

K’).

(D.23) with the

K’)

and K’).

The two-point function is therefore (2) 0+(X - Y, K'> - [,%I 0-b - Y, K')] (2) I”) ~')1[&] S+(x Y, K’) - [,%I s-(X - Y, K’))(D.24)

(21

cLP(x, y) = 3 j dD1 (m, K'>[[&,] - 6m A trivial calculation

drP"(m,

s

yields

(2) t,fl,l

)(

12) d+(X

-

Y,

K’)

-

[,%I

0-b

-

Y,

K’>

(D.25)

eLP’“-?/‘[(y)qz,

where w,’ = p2 + m2. After recognizing side of (D.25) becomes

the presence of 6(pz +

K”),

the right

(D.26)

564

GOMATAM

where S,(,

- y, K’) SE [(K/2 + my

+ 4rn2K’2 - 4m(m2 + K”) $1 &(x

- Y, K’).

Similarly, (2)

[flv]

(2) [&I s-6

s+(x - Y, K’) -

- Y> K’)

In arriving at (D.27) we have recognized the presence of 6(p2 + and used the result (@

’ y

-

m)”

(ip

’ y

-

K’)

=

(K’

+

m)”

(z$

’ y

in &(p,

~‘3

-

K’).

d,(X

-

K’)

(D.28)

Therefore the two-point function is rD(x)]2 -

b(K’

1

+

(m2

m)”

:

[dz’l’(m,

K’2)4

d,??l”(m,

K’)

K’)

&(X

-

y,

JJ, K’)

(D.29)

K’)][(JJ)a]“.

Next we will show that the K’-integration converges. From (D.22a) and the expression for dpsll,scl, and duscl,,, , we see that dz(l)(rn,

K’)

=

dK’2 x

m dK2 j&~~~(m2, s I?=0

i?&,(l7?,

K2,

K”)

F?Z2, K2)

6[K’2

-

(RI

8(K*

+

-

4??12) (D.30)

K)‘],

where and dcyo,,

=

8s~ljd6[K’2

-

(In

+

dK’2.

K)2]

We obtain d.WJ(m,

For

K’ ---f

co,

K’)

dK’2

=

( Id I-mP s 4ma

dK2

/+so)(m2,

m2,

K2)

&s(&Z,

K2,

K”).

have

we

&&~ dJP(m,

K’)

-+

dKr2

s lP=4ma

&2 = K12 &‘2 ,

(D.31)

because &ws(~)(m2,

m2,

~3

a

(const.)

and

8s(1)d(m, K2, ~‘3 K’e*m-Ke+w*K
FOUR-FERMION

Expressing d,(x - y, K’) in momentum in (D.29) converges because

I‘dfc-

Kr2

$

Also, it is easily seen that for large diP’(m,

565

INTERACTIONS

space, we see that dZ(l)(m, K’) integration

-$

=

j

$

<

(D.32)

CO.

K’,

di2 K’) -+ K’2

KLK’P ,,2,4m dK2 = dK’%, s 2

because

as is evident from (D.18). Thus diP’(m, because

f

dKf2$

K’)

integration

in (D.29) also converges

. + = j $$. < *.

(D.33) Q.E.D.

Next we will supply the necessary calculations that led to the assertion that the four-point function (Eq. 5.18) in second-order perturbation is finite. To this end we have to first obtain the spectral representation for products like

(+yxy) (;(yxy) KY OP

‘l’(+) SK, (XY) 3AYX).

and

We will first compute ‘l’(+) S,,, (xy) ‘,.$~‘(x~). Note the identity ‘l’(+) ‘l’(+) So, (XY) SK, (w> = &AKl,

+ m3 @(xy)12

- 2mf&A, + rSLJtKl,

+ m”) 4f’(~u)lt~,~J”(xy)l (D.34)

+ 4m2~~~r~,[a,d,“‘(xy)I[ap J”(w>l = LL~4m4[~s-l’(xv)lz + d+2(x~)+ 4m2d+(xy)&(xy)l

- 2mfSA, + rK”ys~,l[m”a,[dJ”(x~Y)12 + d+(v) %4%)1 + 4m2~~~r~~f~,~~‘(xy>I[ap~‘(xy)l. We have already obtained the “spectral”

representation

(D.35)

for LI+~, [4($12, d+dy),

566

GOMATAM

and A+a,A$‘. So the only product to be computed on the right side of (D.35) is the last term. This is computed as follows:

Letting p + q = u and choosing the frame in which u = 0, the integral on the right side becomes

- [+-I2 &y:, j du@(r-r)j d3pdp0[& x P&G - PJ [&F

S(0- p")+ &

&J - u” + PO) + &

PJ(oJ-PO)]

W’(OJ - u” + pu,].

(D.37)

Now

(D.38) Under d3p = / p I2 d 1p ] dsZ integration only the first and the last terms in (D.38) survive. With y”pyv E B,,,,s,, , (D.38) yields

I 1

I P I* k&,

I P I -$-

- r?‘yP 45-p”(uo - PO)\ d I P I.

(D.39)

Also the products 6(w - p”) &~)(cIJ - a0 + p”) and 2Yu - p”) 6(u - u” + p”) cancel each other under dp” integration. We therefore have

yg,y~,[a,dI”(xy)l[a~4I”(xy)l = ~~]2~due”.‘.-Y’~~p~2d~pIdpo[~S(w-po)S(w+po--o)

+

-

(ii)’

x [qEmploying

sy,

- p”) syw

+ pa - d)]

u” - PO)]. I P I2 ~.JK, - 47v?Yy;;YP0(

tD.@)

the identities

p”(u” - p”) &I

- p”) S(w + po - u”) = w%(w - p”) S(w + p” - u”)

FOUR-FERMION

567

INTERACTIONS

and pO(Q - p”) SCl)(, - p”) S(l)(, + po - u”) z w2syw

- p”) W(w + po - 11’))

-- S(w - p”) S(w + po - UO), we rewrite (D.40) as 4rr [&J

j du eiu(Z-9) j 1p 12d / p 1dpO[S(w - p”) 8Cw$- P” - 11°)

x \ IPI” - l 3(20)~ s,,s~, + syw

-

- p”) syw

w2 - (zo)6

Y?%iy

+

YXOYKo;’ ! (&,)4

\

+ po -

The d ] p 1integration is transformed to dw integration by making the substitutions jpj”=o2m2 and ) p ) dp = odw. Then (D.41) becomes 4~ [$-F]2

j dU eiu’L-y)

j dp” dw [8(w -

x p,s,, 1_02- m2)3’2+ rzw 3(2~)~

@WI4

(

~0) a(w + p” _ 1~0)

1 _ $!&-I

(w” - m2)*i

.

(D.42)

After performing dw-integration and boosting the rest frame u = 0 to some finite arbitrary value u, we obtain

&(m2,

K) _ s

1145 cK2 ;8y2)3’2

1

_ ; cK2 -K;m2)1’2 _ 4K2(K2

x

595/64/2-17

8(K”

-

4m2)

-

h2)1/2

\ 2K

(D.44a)

568

GOMATAM

and 3(K2

2m2

4W2)”

-

16~~

K5(KZ

Note that for large

K,

we

m2 4f?@

-

+

1

fl(K2

K4(K2

-

4m2)#

(K”

-

6mP)/

4m2)

-

(D&b)

have dCt(t?l’,

K)

--+

(D.45a)

$

and d/3(VZ2,K) -+ $.

(D.45b)

‘l’(+) (xy) ‘l’(-) In order to obtain the spectral representation for the product SK, S,, (Yx), we first note the identity (+) - (l’ SW (XY) ‘3-YYX~

= {4~“[@‘(xY)12

+ 4m2 &(xY) - Y3wm2

+ ~bcAY

d+(xY) + @+(xY)12) 6,$,,

%L$%Y)12

+ d+(XY) M%VJ)l (D.46)

- 4m2y~~,y~,ra,4J’)(xY)l[ap~)(xY)l. Since the spectral representation for all the spectral representation for ‘l’(+) SK, (xy) We first consider the combination of (2) I&Y

Tr

(1)

(1)

S+(XY)

S-W

Using the spectral representation i I dpswsdm2, =&

the products on the right side are known (l)( S,,; )(Yx) could be written down explicitly. terms: (2) -

W,l

Tr

(1)

‘1)

SXv>

S+(Yx).

P.47)

(D. 15) for the products of S-functions we obtain

Ma, Ke,t[i$]

d+(X - Y, K> - [&I d-(X - Y, K)> e d~sws’~)(m2, m2, K2)[@X)12 ‘$ 1 $ [(Y)Q12, I

(D.48)

where we have utilized (D.25) and (D.26). From (D.16b) we see that the dK2 integration is convergent because dtc2K4

I K2.K8 <*.

FOUR-FERMION

569

INTERACTIONS

(2’ ‘l’( ) (l’( ) The term [&IV] ?6+‘(xy) ‘l’(+) SK, (xy) + [,;%I S,; (xy) SK; (xy) leads to

(2, [&%I 0-h

1%) i&dKY J* dud(~),@, m2, K2){[a&l d+(x - Y, K> - 2MCiL + r3,,1 j Wm2, m2,K”) (2)

(2)

x {[J&]

ijuzA+(x

(2) x {[,~vlb’~>:,

+ 4m2i

s

y, K) - [,17,] a,zL(x

(?‘%Y d+(X - Y, K) -

dor(m2, K){[,~J

- 3’. ‘+

- y, K)} + 4m2i s d/3(m2, m2, fc2)

(2) k/~&’

- a):~ (7’ ’ %w d-(x

8 pSq Ll+(x - y, K) - [,Z,l

- Y, K)}

S,,S,, n-(x - y, K):, (D.49)

where dud(,jd(m2, m2,

K”)

=

4m4 d pA(l)ddm2, m2, ~“1 + dpAd(m2, m2, K~> (D.50a)

+ 4m2dpAddm2, m2, K”> and dA(m2, m2,

K’)

=

m2 dpd(l)d(1)(m2, m2,

K”)

+

du(m2, m2,

K~).

(D.51a)

We note that: dug(1)d(m2, m2,

K”)

h

(COnst.)

dA(m2, m2,

K”)

*%

5

dK2, .

A careful computation using explicit expression for I7 convolutions similar to (D.25) and (D.27) yields for (D.49) the expression i LL Gw4

I

- -$$I

dudwA(m2,m2,~3[Wl” dA(m2, m2,

&

K> cm2_- I’,K2j4

(DSOb) (DSlb) and results

KvP12

K~)[D(x)I~[ ($ T $i4 {[Ux - Y, 410pa,,

(D.52)

570

GOMATAN

Next we will show that all K integrations in (D.52) converge. From (D.50a) we immediately see that the first integral converges because

That the second integral converges in clear from (D.51a), because

The convergence of the third and fourth integral are guaranteed by (D.45a) because

Next we will consider the term

Since this term could be obtained from (D.49) by yzp, its convergence is guaranteed by that of (D.49) which has been proven just now. We will now focus our attention on the term

Using the identity (D.46) and the spectral representation occurring therein, we rewrite (D.53) as

for various products

This has the same structure as (D.49) the difference being only in the signature in front of K-integrals. Since we have explicitly demonstrated the convergence of (D.49), the convergence of (D.54) automatically follows.

FOUR-FERMION

Next we will investigate separation is valid:

571

INTERACTIONS

the sufficient condition

under which the following (N)

(A’)

(D.55)

L~,l f(x - u> = e,vf(x - r> + z (x - .I’). (Nl

yNhere each term on the right side is a well-defined distributionNandfE 4’. where 9, we recall, is the class of distributions which multiply [Ji’,]. Therefore the left side of (D.55) is well defined. Thus, it is enough to find out the further restriction off such that the product 0J(x - y) is well defined. Expressing 8,, and f(x - y) in momentum space, we see that the following further restriction on f is sufficient: h4

25%

(PO>“,

where

6 < 0.

(D.56)

We will denote this class of functions by g8’. If the interaction is such that 8SjS.Y S”(SS/S~) E Pa’, after a certain order of perturbation, the present formalism will yield the same result as that obtained by using Feynman rules. If we choose N = 3, we obtain a propagator that behaves as l/p3 for large momentum. By power counting (Appendix B, case 4) we see that the F.T. of S,!P)/GX rSP)/6y behaves as 1/p and therefore &V/8X SP)/6y E &‘. Thus, for the choice N = 3, the Feynman-Dyson formalism yields finite results [I 51 for four-fermion interactions.

ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Professor F. Rohrlich for his interest and encouragement during the course of this work and to Professor E. C. G. Sudarshan and Dr. W. J. Eachus for useful discussions.

REFERENCES 1. H. LEHMANN, K. SYMANZIK, AND W. ZIMMERMANN (LSZ), Nuovo Cimento 1 (1955), 205; 6 (1957), 319; V. GLASER, H. LEHMANN, AND ZIMMERMANN, 6 (1957), 1122. 2. For references on axiomatic formulation, see R. F. STREATER AND A. S. WIGHTMAN, “PCT,

Spin and Statistics and All That,” Benjamin, New York, 1964. 3. R. E. PUGH, Ann. Phys. (New York) 23 (1963), 335; J. Math. Phys. 7 (1966), 376. 4. T. W. CHEN, F. ROHRLICH, AND M. WILNER, J. Math. Phys. 7 (1966), 1365. 5. For discussion of the relationship of this theory to those of LSZ and Pugh see: F. ROHRLICH, “Perspectives in Modem Physics” (R. E. Marshak, Ed.), Interscience, New York, 1966. 6. T. W. CHEN, Ann. Phys. (New York) 42 (1967), 476. 7. J. G. WRAY, J. Math. Phys. 9 (1968), 537.

572 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

COMATAM

G. E. UHLENBECK, P/zp. Rev. 79 (1950), 145. H. UMEZAWA AND Y. TAKAHASHI, Progr. Theoref. Phys. 9 (1953), 501. P. A. M. DIRAC, Comm. Dublin Inst. Adv. Studies A (1943), No. 1. W. PAULI, Rev. Mod. Phys. 15 (1943), 175. W. HEISENBERG,Nucl. Phys. 4 (1957), 532. R. NEVANLINNA, Ann. Acud. Sci. Femz. Ser. A 1 (1952), 108, 113, 115; (1954), 163; (1956), 222. L. K. PANDIT, Nuovo Cimento Suppl. 11 (1959), 157. K. L. NAGY, “State Vector Spaces with Indefinite Metric in Quantum Field Theory,” P. Noordhoff Ltd., GrGningen, The Netherlands, 1966. E. C. G. SUDARSHAN,Nmvo Cimento 21 (1961), 7. T. D. LEE AND G. C. WICK, Nucl. Phys. B9 (1969), 209. F. ROHRLICH, Phys. Rev. 183 (1969), 1359. J. G. TAYLOR (preprint), Queen Mary College, London, April 1969. The Minkowski metric and the notation for invariant functions adopted from: J. M. JAUCH AND F. ROHRLICH, “Theory of Photons and Electrons,” Second printing, Addison Wesley, Reading, Mass., 1959. F. ROHRLICH, .I. Moth. Phys. 5 (1964), 324. F. ROHRLICH AND M. WILNER, 7 (1966), 482. F. ROHRLICH, Quantum field theory and generalized functions, Acta Phys. Austr. Suppl. 4 (1967), 228. V. GORGE AND F. ROHRLICH, J. Math. Phys. 8 (1967), 1748. A. PAGNAMENTA AND F. ROHRLICH, Phys. Rev. D1(1970), 1640. WALTER E. THIRRING, Principles of Quantum Electrodynamics, Appendix II, Academic Press, New York, 1958. J. GOMATAM AND F. ROHRLICH, J. Math. Phys. 10 (1969), 614.