Asymptotic quantum field theory

Asymptotic quantum field theory

ANNALS OF PHYSICS: 42, 176-493 (1967) Asymptotic Physics Department, Quantum Syracuse Field Uniaersify, Theory* Syracuse, New York In ...

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ANNALS

OF

PHYSICS:

42, 176-493

(1967)

Asymptotic

Physics

Department,

Quantum

Syracuse

Field

Uniaersify,

Theory*

Syracuse,

New

York

In this work, we have formulated a quantum field theory with t’he following features: (1) it is free of divergences and also free of the unphysical procedure of renormalization; (2) the interaction can be specified in our dynamical equation; (3) it gives consistent (renormalized) results for a renormalizable int,eraction; and (4) it is also applicable to a nonrenormalizable interaction. Our formulation assumes the Bogoliubov causality condition, t,he strong unitarity condition, the asymptotic condition, t,he completeness of the in-field (or out-field) and Lorentz covariance. lhe off- as well as on-mass-shell values play an important role in our formulation and they are treated carefully by employing the operator derivatives wit,h respect to the in-field which satisfies the weak (or ordinary) free-field equation. A dynamical equation which determines all S-matrix elements is developed. The specification of interactions is made in t)his equation. It is then shown that finite perturbative solutions exist to any order and that the scattering mat,ris elements can be obtained by imposing suitable boundary conditions. No restrictions on the types of interactions appear in our formrllation and, therefore, the same procedure can be carried out for renormalizable as well as for non-renormalizable interactions. For renormalizable interactions, in particular, our dynamical equat ion redllces to the one prcviollsly proposed by Pugh. I. INTRODUCTION l
sir~e

t,he

beginnings

of

asynlptotic

cluantum

field

theory

there

has

the off mass-shell values of the S-matrix. The importance of the off mass-shell values can be seen inmediately from the fact that the current operat,or is defined to be &!!*(~S/&C) which ran be determined only if the on- and off-mass-shell values of the S-operator are known. It is even yuite possible that all the diffioulties in field theory aan be resolved if one treats the off mass-shell values correctly. Recently several formulations in t,his direction have been proposed. In particular, Pugh (I), (2) made ass?wzptions which led t’o the continuation of the S-operator off the mass

been

the

c~ontinued

question

of how

one

can

cleternline

* Based on part of the Ph.D. thesis (Syracuse University, 1966) by the author. t Part,ial support, of this work by the National Science Foundation is gratefully acknowledged. 1 Present, address: Ijepartment of Physics, University of Toronto, Toronto, Canada. 476

,48YMPT‘OTIi

QUAXTUM

FIELD

477

1HEORY

shell in terms of t#he cp-products rather than any other products. The results of t’his assumption seems promisin.g: it, leads to corrert answers, at least, in pert,urbative approximations. This point of view is adopted in the present’ work, but, an at,tempt is made to improve the assumpt,ions and make these as natmural as possible. Thr on and oft-mass-shell values are treat)ecl carefully by employing the technicluc of the operator derivatives (3) wit,11 respect, to the in-field. In the present work, a field t,heory is formulated based on the assumption of the Hogoliuhov causality condition (4) (instead of microcausality), as well as the usual assumptions of asymptotic field theory. The operator derivative is so defined that it always commutes with coordinates differentiations; thus the operator derivative can hc performed without ambiguity. The idempotcnt cjperators PC:‘, P’ft’ and Bc”), which arc the generalization of Z’A , Pn , and B in a previous work (I ), (z?), are introduced. In terms of those projection operators an integral trcluation for t,he S-operat)or is derived which, for the case of renormalizahle interactions, is consistent with the mc in that previous formulat,ion. The specitkation of int,eractions is shown to be possible by this equation, mithout requiring restrictions to rcnormalizable inter:ietions (4). Finite perturbative snlutions to any order are then sktow~~ to exist. We shall begin with definitions and assumptions in Sections II and III. In Section IT’ a field theory is then formulated. The perturbative solutions are discussed in Section 1’. Finally, a 11101’e detailed study of the present) formulation for renormalizwhle interactions is given in Sect~ion VI. In particular, the expressions of the interpolating field :tntl the S-operator are derived.

1.

l’he IU (or Out)-Field The in (or out)-field

ui, (:t.) [IF aout (s)] satisfies, as usual, the free-field equation Ii a (.r ) = 0

where K = u -

(11.1)

m2, and the free-field commut~ation relation [I,

a(rz)]

=

-i

[email protected],

-

~2).

(11.2)

For simplieitjy WC shall only consider the neutral scalar fields. The X operator given by [for the equality “ % “, see the next section] Gut (X) g S* s*x

Uin

g xx*~

s j 0) = j 0). 2. The Interpolutiag

Field A (x)

is

(X)69,

(11.3)

1,

(11.4) (11.5)

47s

CHEN

As usual the interpolating field is given by IL4 (x) = j(x)

(11.0)

and the current j (2) is defined as (11.7) 6

6 -=6X

III.

-

GUin(X)

ASSUMPTIONS

Throughout this Chapter WCshall make various assumptions which we discuss in the present section. 1. Completeness

of the in (or out )-Field

We assume t,hat any operator F can be expanded in terms of in (or out )fields as follows F 2 ng -$ /f(xy.

ax,) :al. . .a,:

dzl..

.cZz,

(111.1)

where Ui E U(X;) andf&

E CLi,(Xi) (or U,,t(ICi))

. . - m,) are the coefficient functions which are obviously symmetric with

respect to the arguments ~1 . . . zn . The notion of the strong equality 5 in (111.1) states that the coefficient functions f&l . +* zn) are defined on and off the massshell such that the operator derivatives (3) of F are given by

for any 7~2 0. The coefficient functions, therefore, can be expressedby f (21. . .5,)

6°F = / 6x1. **6x, > 0 \

(111.3)

We shall always denote the in-field by a (x) and all the derivatives are taken with respect t,o t,he in-fields unless otherwise noted. The weak (or ordinary) equality will be denoted by = . We note that an operator derivative of a weak equation may no longer bc an equality. One can easily see the following useful relation from (IIJ), (111.1) and (111.2).

ASYMPTOTIC

QUASTUM

FIELD

THEOIlY

479

It, is important to note that the way we define bhe operator derivat,ive naturally implies that, t,he coordinate differentiat’ion and the operator derivative c*ommutc. Thus, in particular, we have

and

Therefore, Klnl = 0 but this equation cannot, be a skong equation.

Our theory is assumed to be Lorcntz covariant. In particular, we assumethat the interpolating field transforms in t,he same way as the in-field, i.e., it transforms as a scalar in the present formulation. :I. Bogoliubov C’ausality (4) We shall assumethat t,he current given in (11.7) satisfiesthe following causality condition : (1113)

The asymptotic condition (5) reyuires that, t,he limits :!$. (a j A*(t) 1\k) = (a j A,,, 1q) in

(111.6)

hold. In (111.6) / +), / *) are any vcetors in the domain of the operators, and n,(t)

z i 1 cl”x A(a) 7jJx)

4SO

(‘HEX

whcrc ,fa (x) is a c~onlplct~cset of orthonorrnal solutjions of t,he Iilci~l-( ;ord()l~ cquat’ion. The general expression of the interpolating field satisfying the field cquatioll (11.6) and the asymptotic condition (111.6) can be given by A(x)

2 a(x) - [ A,(x

- y) J(y)

rly

(III.;

j

where J(x)

3 j(x)

-Ku(s)

= is* g More preckly,

( 1113)

- Ku(z).

the intjerpolat,ing field is given by the weak equation A(z)

= a(x) - / A,(z

- y) j(y)

cly

011.9)

and the strong equations

sA(z) s = 8(x - 2,) 6x1

s =5. Strong Unitarity

AR(Z

s

-

y> (F

1

-K6(y-xl))

dy

(111.10) s

A&

&i(Y)

- y) 6s

cly. 1

(6)

The S-operat,or is asmunedto satisfy the st’rong unitarity condit,ion s*s 5 &ys” 2 1.

(111.11)

The immediate consequence of (111.11) is t’he following important relation which follows directly from the definition (11.7): (111.12) IV.

FORMULATION

Let us consider the quantity dn)(x, y)=

&,

s AA(XQ . . . z,)A~(yyl

. ..y.) (IV.1)

ASYMPTOTIC

QUASTUM

FIELD

481

THEORY

where A,,s(.ml

. . . .r,)

= A.&x

- x1) AA,&1

-

x2) . .+ Aa,s(r,-,

- s,).

Recalling t,he Hogoliubov causality c*oudition, we see inmcdiat~ely t,hat the integrauds can contribut)e only at x0 = x1’ = . . . =~l.,’ = y” z y10 zz . . = yn”. Rut, sir1c.e A.\ or As vanishes at equal time, wc c*ouclude that, Acn) (x, Y) vanishes

Sjl

unless - for equal time is of the form 6.1.2 (a,“) “I (a,“) It2 [6 (Xl” - .r?“)s(.rl

, J?)],

111 + nz 2 4n

where 9 (x1 x2) is au arbit,rary distribut,ion. [This can hc seen by performing t,he integrations by parts wit)h respect to x1 . x, , y1 . . Yn and making use of the propert,iesKAA(x) = KA&) = -S(J), (A.~(r)),o=o = (AR (.z))~Go = 0. In particular, as nl + 112= In - 1, the c~outributions to A(” (.c, Y) due to terms like (I&‘)“’ (&“)“2 [6 (xi0 - ~2”) S (~1, x2)] are of t,he form (O,, c~~A,(s - Y)] S’(s, Y) which vanishes since Ozu~3’A,(r

- Y) = do (0,, A.&(x - y)) - 6 (x - Y)A,(L

- Y) = 0.1

E’or convenience we shall denote this class of distributions by S:, in the following. It is quite obvious that, if m > n, then S:, 3 S:, . Thus, if bj1/6x2 E S:, , where N is t,he smallest,possiblenumber so that t’his relation still holds, then L1’n)(x, Y) 2 0 for all

‘782 N.

(IV.2)

Writing

II > 0

91> 0 =e z:y,

n = 0.’

(IV.3)

(IV.4)

4s

C!HE1\‘

Wt‘ an wwrite

(IV.2)

as (IV.5)

Together with the strong unitarity the following important equat’ion

c*ondition (III.ll),

i.e., (III.12),

we arrive at

n 2 N,

(1V.G)

where ] - p

s pp

+ pk”‘.

Iu krrns of t,he S-operatSor,this can be written as

l+Lluat,ion (IV&) or (IV.7) is our basic equation. WC uotc the following properties of B (n), pk”‘, nrld I$‘) :

B’“’ B’“’ & B’“’ >

(IV.8 j

B(“) pk”’ 5 0,

(IV.9)

B’“’ j>(d =’ R

m.10)

0,

for my ‘112 0. The proof of the above properties is given in the Appeudis. Making use of the properties (IV.8), (IV.S), and (IV.lO), the gcrmxl solution tmoIQ. (TVA) caanbe wrkten as Sj(x)/6y

2 b(")(cr, y) -

i Pk'([j(.r),

j(y)]),

II 2 N,

(I\‘.ll)

whcrc b’“’ (.r, y) is a solution of the homogeneousequation (1 - B’“‘)x(.r,

y) 2 0,

n 2 iv,

(n-.12)

and, hcrlce, is of the form (l), (2), (4) (&v

@,“P

P 6-O -

!?I Sk

Y)l,

Similarly, the general solution to (IV.7) is

111+ 112< Ah,

I1 2 AV’. (IV.13)

ISVMI’TOTI(’

QUASTUM

FIELI)

4S3

THEORY

whew t)““‘(.r,
SC)I,I’TIOSS

Before we try to solve I<:(l. (IV.11) or (I\l.l4), we shall first prow thca following important properties. For conveuiewe, we shall denote r’p’(F (.r, 21)) or I’g’(F(g, n)) by J’(n)(F(.r, g)) md the 14’ouricr tra?nsform of F(x, 11) by RP, !I). (a) If I)‘n)(F(~, y)) is finite for II = Q then it is finite for all II 2 /L,,. In fad, if P(p, p0 - a, q, 9’ + (Y) behaves like (a)’ for large N theu /l. > r/4. (h) If lJino’(F(x, y)) is finite then Z’(“)(F(.r, y)) - I’(‘rO’(F(.r, y)), II > ‘Q, is a single-point, distribution. T,et us prove the second property first. Considel Pk”‘(F(.r, y)) = (K,K,)”

e2,rG’“’ (n, y),

(V.1)

where

Since I~~“)(F(:~, u)) is finite, J’40”(F(.~, y)), II 2 1~0, is also finite from the proprrt,y (a). Thus we have /‘y’(F(.r,

y)) - zjk”-” (F(.r, y)) 2 (K&)“-

[ILK,

, S,,] G’“’ (.I-, g)

for n, 1~- 1 2 YL,,. Because of the O-fumtion, the left-hand side cm only be a single-point distribution; thus the number n can be reduced step by step until it reaches no and the difference between pp’(F(z, y)) and P(“‘)(F(x, y)) is a single-point, distribution. :1 similar proof holds for pg”‘. This proves the proprrty (b). Now we shall prove the property (a), T,et us wnsider

where G is the Fourier transform of G(:r, y). We see t’hat, 0 (n - g)G(x, y) is finite if G(p, p” - a, q, 9’ + ~2)falls off as l/(.y or faster. The Fourier tmnsform of G’“‘(x, g) of (V.1) can be easily obtained. n’oting that, Gln’(.r, r/) is :I convolution of x1 . . . x, and 1~1. . ’ yn , we have, with a

484

CHEN

G’“‘h

4) 2 [email protected]*(P))”

@R(dY

a4

a)

where 3, , AR the Fourier transform of A, , AR, respectively. Since a,(q, Q”+ CX) and iiR(p, p” - a) fall off as I/ (Y’,we conclude that, when i?(p, p” - LY,q, Q” + a) asymptot.ically behaves like (CX) ‘, the expression (V.l) is finite if 4n > Y. Kow we can easily conclude that Py’(F(s, no 5 n, is finite, since

(V.2) y))

is finite if P~“(F(.~,

u)),

n 2 no > r/4. Similarly one cm prove that Pp’(F (z, g)), for n 1 no , is finite, if I$“)(Ii (.r, z~)is finite. In fact, we have shown that if P(p, p” - LY,q, 9’ + CX)behaves like CYto t.he 7th power (7 finite) at, large 01,then one can always find no such that for all n 2 ‘~1~ , PCn)(F(r, y)) is finite. In particular, me can choose no to bc t,he minimum number satisfying (V.2). The importance of the property (a) can already be seen from (IV.ll) or (IV.14); if one choose n arbitrary large, all the terms involved arc meaningful finite quantities. 2. Perfurhative 2Mutions The integral equat,ion (IV.1 1) is most conveniently solved by a perturbative approximat,ion. Let us rewrite it as following

s’s =s (n) ___ P (x,y) 8x 6y

+ Pi?

(gs*g)+

P2$$s*(g),

(V.3)

where @‘)(.r, y) = Xb’(n’(~, y). The number n is chosen to be arbitrary large (but finite) so that Eq. (V.3) can be used for any t’ype of int’eraction. Expanding X and flcn’ in terms of a coupling constant’ g,

we have, for the lcth-order two-point operator,

(V.4)

ASYMPTOTIC

WC

note

the

that,,

because

quant,it,ies

to

any

of order

order,

n-point

once

operator

ntely.

Since

the

hence,

Eq.

(V.-l)

sp:1cc;

for

vertex

fun&on,

we must

be a single-point @i[l(r,

u)

In

way

t,his As

are

it

fore,

in

ICq.

int,roduce the

in the

as the

number

prescription

to

Consider

in general,

the

can

the

all

the

final

(V.3)

involves

operator strong

can hc equality,

order

follow

propert

1 should n-point

in

arbitrary,

we

such

except ~1) only

can

always

for

high

vanisher

:mtl

rnomcntun~

functions, &:;(.I.,

expression

I’j_)L’

determined

conditions

(1 ). Since

sol~rd the

inunc~di-

k’s of

bc

only

the has

to

choose energies.

uniquely.

subsection,

(V.4); for

/< >

the

boundary

is otherwise

bctwcen

value

all finite.

cncrgics

if

them

II is large

,&h”,’

any

arc

that

det enninctl

specific

to

large,

order

lurgc

previous

I$.

specific

following

which,

of

the

calculations.

hc

tlif?erencc

the

Although

require

the

bc

imposed

for

so that can

the

solutions

order

and

shown

t,hen

final

(V.4)

6’S/&r61~

as long

the

may

vanish

distribution

was

finite,

t’hr

of I$.

two-point, of

functions to

1~) in each

satisfy

example,

n-point

to any

side

the

4S5

THEORY

I3ec:tus:c

is chosen

/Iii,‘(:r,

results

1~. Thus

the

solutions

II and

final

than

~1) is spccificd.

aud, our

FIELD

1, t,he right-hand

2

lower

/3C1,(.r,

~1 in

that I-’ R“” assure The numbers that

of 8”’

QUASTUM

enough,

brmuse

P”‘)(F(.r,

y)),

is a single-point its

of the

for

TV =

/11 , ~2 ,

distribution.

specific

vxluc

houndnry

There-

is irrelevant

condition

to

one

has

to

a11 II. of II is irrelrvnnt dcterminc

to

II in

order

our

final

to

simplify

results,

we

our

shall

use

procedure

of

qua111 ity

he written

in tjcrrns

of normal

products

a(k’(x,y) 2 c cp (x,y), where ai (k)(X, Since of

CI~~)(.L.,

CX~~‘(:IZ, y)

bution

y)

y)

cm

can

t#o IQ.

3

i

/ jl”‘(

bc calculated

be observed.

(V.4)

due

the

ILL is the smallest

smallest,

number

in We

to &“‘(:r,

I Jk”l’ where

X, y, I1

(cry

each

shall

order

(I(., y) ) + which

: Cl] . ’ * Uj: of

always

?I) is of the

number which

* ’ ’ -2.;)

CZXl * ’ ’ CZLri .

approximation,

choose

t,he

‘~1 such

that,

(V.5) behavior

the

contri-

form

Pa”l’

(cp

(.T, y)),

assures

t,he

finiteness

of t,his

quantity

i.e.

satisfies IL > Yi/4

when

the

coefficient,

fur&ion

of

CI!~)(X:,

;v)

behaves

like

([I’)‘~

at

high

energy.

4%

CHEN

To illustrate our procedure, let us consider the cp3-interaction which was also considered recently by Pugh (1). We specify piln))(x, y) to be i6 (.r - y)a (y), so that, the first-order Noting that

two-point

operator

/Q+“\ \6z/o-

is 6’S’“/kSy

L i&(x

-

y)a(g).

_ () ’

we have

s(l) 2 ; 1 dx x”(x): and XP s i : dYMY> 6y=2

:.

Hence the quantit.y

can be calculated:

2*)(x,y) z -!/4 x2*(y)::ayr>: =s -1/‘4

:a’(x)a*(y):

+ ;A+(?/ - x) :u(x)c~(~):

+>d A+’ (j/ - 1.). The second-order two-point operator is then 8*&s(*) s p;;;[x,y) __ zx

6xSy

+

P:“‘(-,li

:a’(x)a’(y):

+$#+“(y

- x)] + Pa”’ /-f/4

+?4+%

- Y)]

2 P&J Y>- ?i

:a’(x)$(y):

+

:a2(x)u2(y):

+ @“&A+

+ P6”“‘($~A”+ (x - y) ) + P$‘(iA+(:r

iA+(y

-

+ iA+(x

(y -

s)

:a(x)a(y):

- y) :a(zr>a(y):

~2.): a(z)

- y) :u(z)u (y) : )

+ Pk”” (>$A+’ (y - x) ). Now we note that n+(p) has asymptotic behavior (p”)-’ and A+“(p) behaves

ASYMPTOTIC

like (p”)~“. Thus we choose 1~1= 0 6g

2 p&,1/)

+ Pa”(;/$A+“(y

- $d :a’(x)a”(~): -

z))

+ P:‘(dA+(.e

QUANTUM

FIELD

and

1, i.e.,

a~ =

+ @(iA+(y

4157

THEORY

- r)

- y) :a(z)a(y):)

:a(z)a(y):) + Pk”(>4A+“(n: -

yj).

Koting 1:hat 1’p’ = B,, , Pa”’ = BrU, we have g

2 &,(x,y)

- x

:a”(z)a’(y):

+ iA,(:c - y) :a(~)&):

+ d2)(a,y)

(V.6)

where UJ(*)(X,y) is t,he renormalized 2nd-order propagator and is given by

WC can continue our procedure t,o calculate t,he S-operator to higher-order approximations in this way. Thus we have shown that, our dynamical ccluation (IV.14) gives us finite solutions to arbitrarily large orders of approximations. And, in fact, this is t,rue independent, of the type of interactions. VI. RENORMALIZABLE

INTERACTIOKS

F’or all renormalizable intcrackions &jjl/8x2 c &‘; thus

This implies immediately that

or

As 11== 1, Eq. (VI.l) or IZq. (VI.2) is precisely the basic equation of Pugh’s formulation (1)) (2). We recall that there this equation was derived from st,rong unitaritjy and the dynamical axiom, i.e., i[6A(.z)/6y] 2 K&t,, [A(x), 11(1~)],while in the present formulat’ion it is a consequence of strong unitarity and Rogoliubov

4%

(‘HES

causality. In order to compare t,hesc two formulations fmthcr, let us invest,igate t,he commutator [A (R.), ,I (y)] in some detail. Wc note from Eq. (111.12) that [A (.T), -1 (my)]satisfies the ecpt’ion sj( x) G(Y) z i 6y- - 2,62:.

K,K,[A(r),A(y)] The general solut,ion to I$.

(IV.31

(VI.3) can he seen easily as

where x(x, y) satisfies KIKzx(.r,

y) 2 0 and

x (R.,y) 2 - x(y, -2.).

F‘rom the fact that, when there is no interaction, A 5 a, j 2 Ka we have [a(:c),a(y>l

2 x(z,y>

+ ,i / AA(y - yl) AR(X - x1) K,, 6(x1 - 1~1)~ZI dy,, - i

s

AA(Z - x1) AR(y - y,) K,,6(x1 - ye) dzl dy,.

The integrals can be easily calculated; thus t,ogether with the commutation relation (11.2), we conclude t,hat x(x, !I) 2 0. Thus we have [A(x),

A(y)]

dXl dy1

A i / AA(~ - 1~1)AR(.~ - 51) +)

-is

4j(Xl>

L&(X: - xl) AR(~ - yl) 61~

(VI.5)

7

dxl dyl

This immediately implies &,[A(z),

A(y)]

A - i h(‘)(q y) (VI.6) &i(Xl> + i / AA(Y - VI) &x(x - ~1) 6y 1

ax1

dy1

.4SYhfPTOTIC

QU.4STUM

FIELI)

Here A”’ (.r, y) is given by 1Sq. (IV.1 ). I;rom Ikls. t,he import,ant equalities

439

THEORY

(111.10) and (VI.6)

WC obt’ain

(VI.7)

and i*+

2 P&I&e,,

[A(z),

-4(y)]

+ iK,K,h”’

(qy).

(V1.S)

ns Eve have discussed before, the I3ogoliubov causalit,y condit)ion demands that A(‘) (x, TJ) can contribute only at’ .r = y, i.e., it can only be an operat’or-valued single-point, distribution [a linear comhinntion of 6 (-1. - ?/) and derivatives of 6 (x - ?];I]. For renormalizable

imeractions

in particular

A(” (x, y ) 2 0, i.e.,

i[SL4 (.i.)/6y] 2 K/3,,, [.4 (.r), 24 (y)l.

(VI.9)

Thus we have shown that in the cast of renormalizable interactions our assumptions imply the dynamical axiom of l’ugh’s formulation (1). Since Eq. (VI.9) is not valid for nonrenormalizable interactions, it, is natural that Pugh’s formulation is not8 applicable to nonrcnormaliznhlc irucractions. I’roni Eq. (VI.9), one can easily derive the following expressions for A (.r) and s. 6”AC.d 6X:1’ . .6x, 2 K 1’. .K,

R(x;x1..

.;r,J,

‘n 2 0 ‘I1 2 0;

i.e., VI.12) (VI.13) IIere R

r&

1’ arc the usual retarded and posit,ively timc-ordcred products, re-

spectively. The functions r and 7 are given by r(x;.rl

**. xn) = “)

7 (x1 * . . x,) = <7’ta41 ... L4,)>0

Thus we see that for rcnormuliznblc interactions, the expressions ~Jht~~irled ill (7) and (5) for A (a) mcl S are valid on and off the nlass-shell (with respect, to all the, argomcnt~s of coefficient functions). It is alsc) jnteresting to note that our c~pression of the X-operat)or is different from that of (1) in which the S-operator is expressed in terms of cp-products rather than ?‘-products of the interpolating fields. One can easily see that this diffc~rc~n~earises because of tliffcrcnt definitions of the operat,or derivatives. In the nonrenormalizable WSC, Eqs. (VI.12) and (VI.13) arc valid only weakly. This can be seen from the fact that the contribution due to K,h”’ (.r, !I) of (VI.7) to the expressions of A (.r) and S vanishes weakly.

Assuming the axioms of Bogoliubov causality, strong unitarity, the asymptotic condition, the completeness of the in-field and Lorentz covnriance, we have nrrived at the following dynamical equation for the two-point operator:

Because of the strong equality in this equation, all the TL-point operators, and hence all the n-point, functions, can be det,ermined from this equation provided WC have t,he specification of interactions and boundary conditions. The operator derivative plays an importjant role in our formulation. It is our main tool for treating the on and off mass-shell values. In c*ontrast to previous formulations (I), (4), WC have defined the operator derivatives with respect to an iI)-field which satisfies only t,he weak free-field equation. The advantage of the weak free field equation rather than t,hc strong free field equation for the in-field is t,hat our operator derivative always commutes with the coordinate difftrent’iaCons; t’hus our operator derivatives c~ur he performed wit,hout ambiguities and many computations become simpler. In fact, this definition of t,he operator derivative and the axiom of Bogoliubov c*nusality permit us to ~nakc our formulat,ion applicable to renormalizablc as well as nonrenormalizable interactions. But the assumption of strong unitarity bec*omes necessary, although one can still show [analogously to earlier work (S)] that for renormalizable interactions, the strong unitarit’y condition follows from the other assumptions. The specification of interactions corresponds to giving flCn’ (x, u) in our dynamcal equation. But pen’ (2, y) has to be of t,he form

otherwise it, is arbitrary; t’hus the specification of interactions, renormalizable as well as non-renornializable, is always possible as long as WC choose rz suffic~iently large.

ASYMPTOTIC

QUANTUM

FIELD

THEORY

491

This formulation of quantum field t’heory is free of divergences and does not. require renormalizat$ion. Because of the properties of PC”), all the integrals involved in our dynamical equation are convergent, and finite perturbative solutions to any order can be obtained in :I natural way for renormalizable a& nonrenormadizable interactions. (We omit here t’he question of exist)ence of solutions to infinitely high orders). The final unique solutions of the equation is then arrived at by imposing suitable b0undar.y conditions. lcor rSenormalizable int,era&ions, the present formulation reduces to Pugh’s formulation which yields finit’c results in agreement with the results of renormalization theory. Iqor nonrenormalixable interactions, the present work only showed that finite perturbative solutions to arbitrary high order exist. Furt,her work involving explicit, calculations now becomes essential. We expect to devot,e future puhlic&ons to this equation. ACKNOWLEDGMENT The author wishes to express his sincere gratefulness nnmerous helpful suggestions, continuous encouragement manuscript. He is also indebted to Dr. V. George and discussions.

to Professor F. Itohrlich for his and the critical reading of the Mr. J. G. Wray for many valuable

APPENDIX

WC>shall show the following

properties :

P (PF B'"' UP

ByI’.pF(.c,

(2, y)) 2 B(“)F(IC.,y), F(2,

y))

(11.1)

2 0,

(A.21

y)) 2 0

(A.3)

for IL 1 0, where PA(“) and 1’~~~)are defined in (IV.3) and (IV.4) and 13’“’ is given by 1 - B’“’ s &(I’) + pel(n’.

(A.41

For n = 0, Eqs. (A.l)-(A.3) are immediate since B’“’ = 0. For 11= 1, they were proved by Pugh (1). For n = 2, let us consider

492

CHEN

Performing the integration term vanishes because of ~y’(~j;L’F(rn,

y))

A,

by parts with respect to .I’~ we hnvcl (sinc.c> the> surface , An func*tions)

2 -(K,K,)‘e,,

/ AA!?/ - y~)hA(y~ - yyz)Aa(~ - n)

x F(xJ, ) y4) (lx1 (1X.l da3 fJ (hi>. Similarly, we can perform the integrations by parts with We can easily arrive at, t#he following expression

x j AA(y - ys)AA(y/3 - y&,(x

- Q)AR(~z.~ - zr)

The last term vanishes for F t 8;. Similarly

one can obtain

X s AA(y - y~)A~(yl

- YS)&(X

X (Rz2KV2)20z2y2A~(~~ X AR(~J~ Integrate

respect to s1 , y2 and y1 .

ydF(~.t,

~4)

- TI)AR(G

23)A~(x3

ft

-

xr)Aa(yJ

- ~2) -

~3)

(dzidy,).

by parts with respect to x2 , y2 , .q , yl. Since all the surface terms vanish

ASYMPTOTIC

Similarly

QUANTUhI

FIELD

49:-i

THEORY

one obtains

This proves (Ll)(h.3) for n = 2. In a similar nlanner, one ~‘a11 easily show that t,hey hold for arbitrary (but finite) 71. We would like to point, out that only 23“” is an idempotent operator for :rll h’. The idernpotency properties of Pp’ and Pg' do not, hold, neither does the orthogto belong to S:, . onnl pro#pert,y between Pp' and I$” unless F is restrict4 ~~ECEIVICI):

1. R. E. PUGH, 335 (1963).

4. 5. 6’. 7.

8.

J.

Malh.

I’h,!ts.

6, 750 (1965).

Also

we

R. 1’. I’u(:II,

.I w.

Ph,tp.

(S.Y.)

23,

E. PUGH, J. d~alh. Phi/a. 7, 376 (lYti6). F. ROHRLICH, J. ;Tfalh. Ph!/s. 5, 321 (196-k). Also F’. liotr~tr~rc*rr .ANI) hr. WIIAER, ./. :l/rr/h. Ph,y.r. 7, 482, (1966). T. W. CHEN, F. ROHRLK.H, .INU M. WILNER, J. Much. f’h,,ys. 7, 1365 (1966). II. LEHMANN, It. SYMANZIR, AND W. ZIMMERMANN, Xuovo Cincenlo 1, 1425 (1955). Also fI. LmnwNN, It. SYMANZIK ASU W. ZIMMERMANN, :\~uozw (‘inwrc(o 2, 125 (1955). M. M~JRASKIN AND K. NISHIJIMA, Phys. Rev. 122, 331 (1961). V. GIASER, 11. LEHMANN, W. ZIMMERMANN, ,\‘rrovo Civwdo 6, 1122 (1957). T. W. CHEN, ,~?tozw Ciwnfo 45, ,533 (1966).

2. It. 5.

July 6, 1966