ANNALS
OF
PHYSICS:
42, 176493
(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 infield (or outfield) and Lorentz covariance. lhe off as well as onmassshell values play an important role in our formulation and they are treated carefully by employing the operator derivatives wit,h respect to the infield which satisfies the weak (or ordinary) freefield equation. A dynamical equation which determines all Smatrix 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 nonrenormalizable 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 massshell values of the Smatrix. The importance of the off massshell 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 offmassshell values of the Soperator are known. It is even yuite possible that all the diffioulties in field theory aan be resolved if one treats the off massshell 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 Soperator 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 cpproducts 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 oftmassshell values are treat)ecl carefully by employing the technicluc of the operator derivatives (3) wit,11 respect, to the infield. 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 Soperat)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 Soperator are derived.
1.
l’he IU (or Out)Field The in (or out)field
ui, (:t.) [IF aout (s)] satisfies, as usual, the freefield equation Ii a (.r ) = 0
where K = u 
(11.1)
m2, and the freefield 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 infield by a (x) and all the derivatives are taken with respect t,o t,he infields 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 infield, 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(yxl))
dy
(111.10) s
A&
&i(Y)
 y) 6s
cly. 1
(6)
The Soperat,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 SoperatSor,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 6O 
!?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 singlepoint, 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 Ofumtion, the lefthand side cm only be a singlepoint 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 singlepoint, 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 lcthorder twopoint operator,
(V.4)
ASYMPTOTIC
WC
note
the
that,,
because
quant,it,ies
to
any
of order
order,
npoint
once
operator
ntely.
Since
the
hence,
Eq.
(V.l)
sp:1cc;
for
vertex
fun&on,
we must
be a singlepoint @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 npoint
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$.
specific
following
which,
of
the
calculations.
hc
tlif?erencc
the
Although
require
the
bc
imposed
for
so that can
the
solutions
order
and
shown
t,hen
final
(V.4)
6’S/&r61~
as long
the
may
vanish
distribution
was
finite,
t’hr
of I$.
twopoint, of
functions to
1~) in each
satisfy
example,
npoint
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 righthand
2
lower
/3C1,(.r,
~1 in
that I’ R“” assure The numbers that
of 8”’
QUASTUM
enough,
brmuse
P”‘)(F(.r,
y)),
is a singlepoint 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 cp3interaction which was also considered recently by Pugh (1). We specify piln))(x, y) to be i6 (.r  y)a (y), so that, the firstorder Noting that
twopoint
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 secondorder twopoint 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
 $d :a’(x)a”(~): 
z))
+ P:‘(dA+(.e
QUANTUM
FIELD
and
1, i.e.,
a~ =
+ @(iA+(y
4157
THEORY
 r)
 y) :a(z)a(y):)
:a(z)a(y):) + Pk”(>4A+“(n: 
yj).
Koting 1:hat 1’p’ = B,, , Pa”’ = BrU, we have g
2 &,(x,y)
 x
:a”(z)a’(y):
+ iA,(:c  y) :a(~)&):
+ d2)(a,y)
(V.6)
where UJ(*)(X,y) is t,he renormalized 2ndorder propagator and is given by
WC can continue our procedure t,o calculate t,he Soperator to higherorder approximations in this way. Thus we have shown that, our dynamical ccluation (IV.14) gives us finite solutions to arbitrarily large orders of approximations. And, in fact, this is t,rue independent, of the type of interactions. VI. RENORMALIZABLE
INTERACTIOKS
F’or all renormalizable intcrackions &jjl/8x2 c &‘; thus
This implies immediately that
or
As 11== 1, Eq. (VI.l) or IZq. (VI.2) is precisely the basic equation of Pugh’s formulation (1)) (2). We recall that there this equation was derived from st,rong unitaritjy and the dynamical axiom, i.e., i[6A(.z)/6y] 2 K&t,, [A(x), 11(1~)],while in the present formulat’ion it is a consequence of strong unitarity and Rogoliubov
4%
(‘HES
causality. In order to compare t,hesc two formulations fmthcr, let us invest,igate t,he commutator [A (R.), ,I (y)] in some detail. Wc note from Eq. (111.12) that [A (.T), 1 (my)]satisfies the ecpt’ion sj( x) G(Y) z i 6y  2,62:.
K,K,[A(r),A(y)] The general solut,ion to I$.
(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’orvalued singlepoint, 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 timcordcred 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 nlassshell (with respect, to all the, argomcnt~s of coefficient functions). It is alsc) jnteresting to note that our c~pression of the Xoperat)or is different from that of (1) in which the Soperator is expressed in terms of cpproducts 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 infield and Lorentz covnriance, we have nrrived at the following dynamical equation for the twopoint operator:
Because of the strong equality in this equation, all the TLpoint operators, and hence all the npoint, 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 massshell 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 freefield equation. The advantage of the weak free field equation rather than t,hc strong free field equation for the infield 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 nonrenornializable, 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 (196k). 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