IL NUOVO CIMENTO VOL. XXIX, N. 3 lO Agosto 1963

Structure of Field Quantities and S-Operator.

A. VISCONT~ and J. CA~O~A

.Faeultd des Sciences - Marseille

(ricevuto il 22 Febbraio 1963)

R 6 s u m 6 . - - Etude de la structure de l'opdrateur S e t des grandeurs de champ & partir de la fonctionnelle generatrice des valeurs moyennes dar t s le vide des produits chronologiquement ordonn4s de grandeurs de champ.

1 . - I n t r o d u c t i o n .

Our main task in this paper is to show how b y a coherent use of some mathemat ica l techniques known in Physics as functional techniques one can obtain in ~ form which is bo th elegant and neat several basic relations of Quan tum Field Theory. The compactness of the notations brings into light some logical connections which are more or less deeply hidden in the usual formulation.

We give in Section 1 the fundamental concepts and introduce some useful notat ions; in Section 2 and in Appendices I and IT, we define and s tudy care- fully the properties of an operator (called the mapping operator) which will

play a fundamental rble in this formulation. Section 3 is devoted to the comment of some essential results in Quan tum Field Theory. In Section 4,

we discuss the distinction between kinematical and dynamical considerations,

while in Section 5, we examine the logical connection between the formulat ion

we have been s tudying and the action integral. We hope to use in a for thcoming paper this formalism in order to inves-

t igate the form of possible interactions between fields. 1. - We star t from the set of assumptions which form the basis of L.S.Z.

approach of Quantum Field Theory (1) and consider for simplicity the

(1) H. LEHMAN, K. SYMANZIK and W. ZIMMERMAN: Nuovo Cimento, 1, 205 (1955).

STRUCTURE OF FIELD QUANTITIES AND S-OPERIkTOR 743

case of a scalar hermit ian field. We therefore have two free fields ~=(x) and

~o.t(x) (operator-valued distributions in a linear space E) and their creation and annihilation operators (a~(p), ao.t(p) defined through an or thonormal system (in the sense of the Klein-Gordon scalar product) fp(x) =~/+(x) and ]*p(x) -- ]~(x). We assume the existence of an uni ta ry operator S such tha t :

(1.1) q~o.~(x) = S -~ ~Ax) S

and of an interpolat ing field r

(L2a)

(1.2b)

q~(x) = q~=(x) -Fro(x--2)A (x -- 2) Yl~ ~(~) d42,

r = ~o.~(x) - f (1 -O(x - 2)) A (x - 2) ~ ~)(2) d,~ ,

The asymptot ic condition as a weak limit of the interpolating field is also assumed. As a fur ther hypothesis, we shall admit the existence of a state~ stable with respect to the S operator. This is the (~ vacuum state ~ such tha t :

(1.3) <o !~,o(z)IO> = <Ol~o.~(x)IO> = <OlO(x)IO> = o .

Fur thermore , we shall define normal ordered products of the ~vin and ~out fields with the know n proper ty :

(1.4) [ :~ . (x l ) ... ~ . ( z~) : , ~Ax)] =

= - i ~ :~,.(x~) ... ~,~(xj_~) A ( x - - x~) ~Axj+~) ... ~ A x . ) ' . J=l

We are going to consider functionals of a source J(x)~ where J is a scalar-valued function (e-number) with compact support. All our functionals are supposed to be Volterru-type, i.e. developpable in terms of J :

(L5) A[J] = ~ ~ A,,(21 ... 2n)J(21).*. J(2n),

where integrations have to be performed on each variable twice repeated and there is no restrict ion involved in supposing the A~ symmetric with respect to their integrat ion arguments. We shall therefore consider only symmetrical kernels.

744 a . VISCONTI &Ild g. CARI~IONA

The functionals themselves curt be scalar-valued or operator-vulued func- tionals in the E-space and will be called c-number or operator functionals. As an example of an operator functional, we m a y consider (~):

(1.6)

where Y is the t ime-ordering symbol. This is the well-known operator-gene- rat ing functional of t ime-ordered products of field quantities. The derivat ive 5/SJ(~) ~Dj(~) will be algebraically defined as follows:

(1.7) Dj(xl) ... D.,(xm) f ](~ ... ~ ) J (~) ... J($~) ~--

n ( n - - 1) , . . (U - - m) f l (x 1 .. .Xm, e I .~ ~m_n)J(~l) . . . J ( s e) . . . .

0

if m < n ,

if m > n ,

provided ] is symmetric with respect

~1 ... ~n (~). We therefore huve:

to all its integrat ion arguments

2. - The mapping operator ~ j .

Our main tool will be an operator ~2j which will map the space of the

c-number functionals of quan tum field theory into the E-space (in which the

operators ~tn and ~ou~ are defined), in other terms it t ransforms the scalar rune:

t ional a[J] in an operator functional A[J ] :

(2.1) A [ J ] = [ 2 j a [ J ] .

(~) J. SO,WIgGleR: Prec. Nat. Acad. Sci., 37, 452 (1951). (3) The operation Dj(x) i supposed to commute with Z in (1.5) even when

summation interests an infinite number of terms, we may also remark that:

[D,(~), Dj(~')] = 0.

the

STI~UCTUI~E OF FIELD QUANTITIES AND S-OI~ERATOR 745

The operator Dj or mapping operator will be defined as follows:

(2.2b) ~j(@) ~ 2r Dj(~)

and we ma y note tha t it operates on a c-number functional in two different ways, first through the derivat ion Dj(~) and secondly through the int roduct ion of factors buil t up by the field quantities ~i,(xj) (4). As an immediate conse- quence of eq. (1.4) one has the impor tan t relation:

o(y)l = (2.3) [~,

which is equivalent to the set of equations (5):

where ] + ~ ]~(~) and ]~(x)= ]~(x)(3). B y the introduct ion of a~t(p ) operators, we obtain another set Of equations equivalent to (2.3). The former equa- tions are fundamenta l for the proof of our main theorem which consists of two parts: the direct one and its converse.

Theorem A. Each operator-valued functional of the form:

(2.5) A[J] : ~ja[J],

where a[J] is a c-number functional Satisfies the following relation:

i (2.6) [ A [ J ] , ~ ( y ) ] = z~ (y - - ~) ~(~)A[o'].

(4) Suppose tha~/~[J] is a Volterra type functional end h(x) a well behaved function. One may prove"

1 l~[J(x) § = ~ ~ f h(~l) ... h(~_,,)DA~l) ...DAZ_,)-F[JJ = exp [ f h(~)DAS)] ~[J] .

We are led, therefore, to write symbollically

: ~2[J]F[J] : : :F[J(x) § ~.(x)g(~]: .

(5) Equation (2.4) results from (2.3) through the completeness relation:

f d3k (x)f~(x') : A+(x--x~). i

746 A. VISCONTI and J. CARI~IONA

Theorem B. Provided tha t the entire space E is spanned by the ortho- n

normal set of vectors (l/~/~.) ]-I a~(pj) ]0} (no bound states !), t ha t the operator- f=1

valued functional A[J] maps E onto itself and satisfies eq. (2.6), then A[J] is of the form:

A[J] = [2:;0 IA[J] ]0}.

The proof of part A is straightforward, we first note tha t since:

(2.7) (0X~9 , a[J]]0) = a[J]

for a v-number functional a[J], we have:

(2.8) a[J] = (0 ]A[J] [0};

we then apply both sides of formula (2.3) to the c-number (0]A[J] I0} in order to obtain (2.6).

The proof of the converse of part A (i.e. part B) requires some more deli- cate considerations and will be given in Appendix I .

As corollaries of the former theorems, we may list the following results:

a) A*[J] is of the form:

(2.9) A*[J] = [2:(0 [A*[J][O}

provided tha t A[J] satisfies theorem B (6).

b) Dj (X1) . . .D: (X , )A[J] is of the form:

(2.10) D:(X,) ... D j ( X , ) A [ J ] = Y2j(O ]D:(X1) ... D j ( X , ) A [ J ] 10}

provided A[J] satisfies Theorem B.

c) I f A~[J] and A2[J] satisfy Theorem B, then:

(2.11) A1 [J] A2 [J] = 9 j ( 0 I A~ [J] A2 [J] [0).

The proof of corollaries a), b) are straightforward, the proof of corollary c) requires the use of simple equalities on commutators.

(6) One remarks that for a hermitian field ~i~(x), one has, taking into account the hermieity of :~l~(xl) ... ~i~(x,,) :

( Q[J]a[J])* = t~[J]a*[J] .

S T R U C T U R E OF F I E L D Q U A N T I T I E S AND ~ - O P E R A T O R 747

3. - S operator, Haag's equation, Wick's theorem and unitarity type conditions.

We now introduce the expansion (1.5) of the functional A [ J ] into eq. (2.6) which is replaced by the infinite set of recurrent equations:

1) [ A . (x~ .. . x . ) , ~v,=(y)] = i ( A (y-- ~) ~t~A~+~ ( ~, x l . . . x . ) , (3. d

which are of the (~ reduction formulas )) type. Equat ion (2.6) is equivalent to the set (3.1) and conversely each set A~(x~ ... x v ) o f operator-valued distribu- tions with x~... x. as arguments generate the operator functional A [ J ] such tha t :

(3.2) A [J] =- ~gj<01A [J] 10>.

The first of the A , , namely Ao can be chosen arbitrarily since it defines A~(Xl) through the relation:

(3.3) [Ao, ~,.(y)] : i I A ( y -- ~ ) ~ A ~ ( ~ ) . 3

Now we have two remarkable operators in our theory, the S operator and the field itself: the choice of A0 as S operator generates the ST(x1 ... x . ) pro- ducts, the choice of Ao as q)(X) gives Haag's equation and generates the re- tarded products.

Choose Ao=--S, eq. (3.2) can be writ ten as:

(3.4) ~v,~(y) -- ~Vout(y) = i r a (y -- ~) J~f~ S -~ AI (~) , J

on the other hand, the asymptot ic condition as expressed by eqs. (1.2a) and (1.2b) gives:

qJln(y) -- q~ou~(y) = --JlA (y -- ~) Y ~ q~(~) , (3.5)

we therefore m a y define:

(3.6) A~(x~) -~ -~ i n qS(x~) .

This is the second term of the well known set of distributions (7):

~ T @ ( x , ) ... ~ ( x , ) } = S T ( x 1 ... x , )

(7) We remind the reader that:

(1)erm)

T(Xl... xn) : ~ 0 ( X l - - X2) . . . O(Xn-l--Xn) ~ ( X l ) . . . ~ ( X n ) �9

748 A. viscoN~i and J. CAR1VIONA

which satisfies the reduction formulas and generates the S~[J] operator as defined by (1.6). A direct application of Theorem B, permits to introduce the operator (s):

(3.7) S [J] ~ $~ [ j ] = /2 j (O 1~' [J] I o~ ~ ~., u [ J ] ,

where u[J] is the vacuum expectat ion value of ~[J], i.e. u[J] is the c-num- ber-generating functional o~ the vacuum expectat ion vulues of t ime-ordered products, which we shall call propagators. At the limit J = O, one gets the well known expansion of the S operator:

(3.8) (- i).f. The other choice of A0, Ao----qi(X), where X is a given space-time point generates the re ta rded commutators (5) R(X, xl ... x~). The shortest way of proving this point is to consider the generat ing operator-functional of R pro- ducts (1~ : , ~ [ j ; X] = - - i ~/* [J] Dj(X) ~ [ J J and remark tha t both of its fac- tors ( S ~ [ J l ) * and S D j ( X ) ~ [ J ] s~tisfy Theorem B (corolluries a) and b)), therefore in vir tue of corollary c):

(3.9) [ J ; X] = f)j(O I ~ [ J ; X] [O}.

Taking again the limit J = O, one gets Haag 's equations (11):

(3.10) ~ ( x ) = ~ n! 3"

We do not want to enter into a full discussion of these results, which are well known (1~). Nevertheless, it must be noted tha t there are still some ques- tions which should be cleared up in connection with eqs. (2.6) or the set (3.1).

I t is, for instance, impor tan t to know, if under physically acceptable condi-

tions, there are none other solutions than the ones we just mentioned. We

(s) Equations of the type (3.7) have been given by L.S.Z. I in Appendix. (9) We remind the reader that

R ( X , x 1 ... Xn) : (__g)n ~ O(X_Xl ) ...O(xn_l__xn ) [ . . . [ ~ ( X ) , Qp(xi) ] .... ~0(Xn) ] . l)erm

(lo) H. LEHMAN, K. SYMANZIK and W. ZIMM]~R~ANN - NUOVO Cimento 2, 319 (1957). (11) R. HAAG: Dan. Mat. Fys. Medd. 29, 319 (1955). (12) V. GLASER, H. LEHMANN and W. ZI~]~R~A~N: Nuovo Cimento, 5, 1122 (1957).

STRUCTURE OF FIELD Q~/ANTITIES AND S-OPERATOR 749

shall defer the answer to this question to Appendix I I , where we shall t ry to point out all the subtleties, which one comes accross when t rying to discuss

these equations (1~). Before s tudying uni ta ry type conditions, let us establish a formula, which

will be essential for the next paragraph, and which gives the functional form

of Wick's theorem as a by-product . Consider a free scalar field, 9(x), then S[J] reduces to ~o[J] , generating

operator-functional of t ime-ordered products, since in this case S = I . We therefore have:

(3.11) ego[J] = D+Uo[J],

where, as well known, uo [J] is the c-number-generating functional for the free field propagators,

Expanding ~Q+ and taking into account the equation of do:

5C, Ao(x - - x') = - - 2i O(x - - x') ,

some simple algebra shows tha t one can bring ~0[J ] into the form:

(3.13) ~o[J]=[2 . ,uo[J]~-uo[J] :exp[ i f J (~)q~ i . (~ ) ] "

and this formula is nothing else than Wick's formula, since it expresses the operator-generating functional ~0[J ] of the t ime-ordered products in terms of normal ly ordered products.

Le t us now examine the uni ta r i ty type conditions, we first note tha t the insertion of the or thonormal system which spans the space E (and has been

(1~) We list another formula which is interesting. Consider the functional of two sources ](x) and J(x):

Its expansion with respect to j(x), gives the multiple commutators of S[J] with the field %n(x), i.e. the operator

(-- it~'[... [S[J], ~0i=(x~/] ..- ~0io(x~)],

is the n-th coefficient, as in (1.5), of the Volterra expansion of the former generating functionM.

7 5 0 A. VISCONTI and Z. CAR~OlqA

given in Theorem B), ~s an in te rmedia te sys tem gives the following formula:

(3.1~) (O]Y2:~:a[J~ J'JlO} -=exp [i f A($--~')~:(~)~:,(~')] a[J~ J'] ,

where a[J, J'] is a c-number functional. I f one t~kes: a[J, J'] ~ u*[J]u[J'], then one obtains:

(3.15) (O I S*[J]S[J'] l O} = exp [i f ~J(~ -- ~') ~:(~) ~j,(~')] u*[J]u[J'] ,

and this is the generat ing funct ional for all un i t a r i t y - type relat ions (14). F o r

instance, t ak ing J (x )~ - J ' ( x ) - -= 0, one gets a necessary condition for the uni-

ra r i ty of the S operator , t ak ing der ivat ives with respect of J or J ' (or bo th of

them) and pu t t ing J = J ' = 0, one gets the (A) sys tem of equations of L.S.Z.

and all their possible ' generalizations.

One m a y wri te down analogous relat ions for the re ta rded commuta to r s ; we defer the e laborate s tudy of these questions to a for thcoming pape r and

content ourselves b y mak ing jus t a few comments . One of the possible extensions of q u a n t u m field theory lies in the s tudy

of all possible solutions of (3.15) in its funct ional form or in t e rms of its Vol-

te r ra coefficients. I t is useless to point out t h a t a global s tudy of (3.15) is, for the t ime being, an except ional ly difficult task, ~nd tha t on ly some kind of

approx imat ion scheme c~n possibly br ing any t~ngible results. One m~y, a t least ~t first sight, propose two kinds of approximat ions : the first one, similar

to ~ T~mm-Daneoff method, cuts down the expansion ~t a given number of

funct ional derivatives. I n this w~y, one gets results valid for the n- th order, n being the number of funct ional derivat ives ~/~J(xj) which h~ve been intro-

duced (15). Another way, is to in t roduce into the theory ~ dynamica l opera tor which

represents the effects of the in te rac t ion and to discuss as far as one c~n go

the possible forms of this operator .

(14) We note that under the assumption of the unitarity of the S operator:

(0 tS*[J ] S[J'] 10} = (0 ] ~ o2/[J'] 10) ~ .

(15) This kind of approximation method has been used by H. U~EZAW). and A. VISCO~TI: 2~uov~o Cimento, t, 1079 (1955). Perturbation methods have been used by K NISHIJIMA: Phys. l~ev., 119, 435 (1960). Functional methods have been used by K. SYMANZIK: Journ. Math. Phys., i, 249 (1960) and H. M. FRIED: Journ. Math. Phys., 6, 1107 (1962). This last paper came to our knowledge af ter this work was performed.

STRUCTURE OF FIELD QUANTITI]~S AND S-OPERATOI~ 75t

Although, as formerly said, we shall postpone the study of these questions to a forthcoming paper, we are going in the next paragraphs to consider in some details this last line of thought.

4. - The d y n a m i o a l l a w .

A dynamical constraint imposed on a free field can be expressed by the very simple property that the propagators of the new field differ from those of a free field. We shall express this relationship by the introduction of an interaction differential operator W[(1/i)Dj] which connects the generating c-number functional u0[J] of the propagators of the free field with the ge- nerating c-number functional u[J] of the propagators of the actual field, i.e. we are going to postulate that:

(4.1)

where W[j] is a Volterra functional of the c-number source j(x):

(4.2) 1 W, . . . W[j] = ~ # f .(~1 ~.) j($1)..d(~.)

and u0[J] is given by (3.12). The S[J] operator takes the form

(4.3)

But the operator Y2juo[J] has been already calculated in (3.11) and (3.13), we therefore have:

(4.4) S[J] W D+ ~o ('n> [J] ---- ~ Wn(~l... ~n)Dj(~l)... Dj(~n)~o(ln)[g].

Going to the limit J----0 in the former expression, we have:

(4.5)

One has thus a particularly simple connection between the operator S and the interaction differential operator W[j]: one gets S by replacing in W the

752 A. VISCONTI a n d $. CAI~IONA

c-number source j(x) by the field quantity %.(x). Suppose now th a t the field theory under consideration is a hamiltonimn theory. I t is well known tha t :

therefore if we introduce the functional ~/~.[j] such tha t :

(4.7) wU] = e x p [ iz, o,[j]],

the operator S[J] takes the form:

(4.8) S[J] = exp i~/,~t Dj ~0 [~j =

For an hamil tonian theory we shall take:

: exp : u 0 [ J ] .

(4.9) 5~',~t[q~,~] = f ~ , ~ t ( x ) ddx,

in agreement with (4.6). We are now coming back to the question we have been asking at the end

of last paragraph: what kind of restrictions shall we impose on W[j]? I t is first of all clear, tha t since di.~[j] is a real number, W[j] must be a phase, bu t one can hope to get some addit ional restrictions b y the consideration of uni tar i ty type conditions as given by (3.15). This problem will be studied

later on.

5. - The a c t i o n in tegra l .

Until now the mathematics we have been using can be considered as rela-

t ively well established. We want now to shift our considerations to more moving grounds and develop a formalism a t t rac t ive because of its physical

content even if it is less well founded from a pure mathemat ical point of view. Le t Y2 be a certain set of e-number scalar functions ~(x) and ~[q0] a

c-number functional defined for each ~(x)~ Q, and suppose t]2at there exists a linear operation denoted by f g ~ mapping the functional /~[~] onto a real

or complex number which depends on ~2 a n d / ~ bu t not on the choice of ~(x). We assume lur thermore this operation to be invar iant with respect of the

STRUCTURE OF FIELD QUANTITIES AND S - O P E R A T O R 7 5 3

translations of the argument ~(x) of F I l l such tha t :

(5.1) f F[~ + a] ~m =fF[m] 2~,

where the funct ion q~(x)+a(x)E Y2. Such an operation will be called invariant or junctional integration (~6).

We want now to show tha t if the functional:

where ~r is the c-number action of a c-number free scalar field, belongs to the /~-funet ionals class (i.e. if the functional integrat ion can be defined for such a functional), then the c-number generating functional uo[J] of the propagator of free fields can be brought into the form:

(5.2) u0[J] = ~ @~ exp i J(~) ~(~) + zr

~e o

where 5Vo is a normalizing factor such tha t u 0 [ O ] = l and ~9o is the set of all functions ~(x) such tha t the integral

f~(~) d~(~ _ ~,) ~(~')

(where d~ 1 is the reciprocal of ~ ) , is well defined. In order to prove (5.2), we first bring, b y a familiar technique, the action

d0[~] into the following form:

(5.3)

performing then the translat ion of the integrat ion variable:

~(x)= ~'(x) +~ A~(x-~)J(~),

(is) Properties of such a formal operation have been studied by K. SYlgANZIK: Zeits. /. Natur/orseh., 9, 809 (1954). See also A. VlSCONTI and H. UMEZAWA: Compt. Rend. Acad. Sei., 252, 1910 (1961). A. BIALYNICxI-BIRULA: Journ. Math. Phys., 3, 1094 (1962).

4:8 - I I N u o v o C i m e n t o .

7 5 4 A. VISCONTI ~ n d J . CARMONA

and finally taking into account the definition of the normalizing f~ctor N0~

we get without further difficulties the expression (3.12) of uoEJ] (1~). Let us

now consider the form (4.1) of u[J] with W given by (4.7):

(5.~)

+~r

where N ' is a normalizing factor (is) such tha t u [ 0 ] = l .

I f furthermore D+ commutes with ( ~ , then u[J] can be wr i t ten as. ~ 9c

(5.5) u[J] = ~ ~ exp i J($) q~(~) + ~9[~] + ~,~E~] �9

�9 Y 2 0

In other terms, we just showed tha t the generating :functional of the pro-

pagators is the Fourier t ransform of exp [i(d0[~]+~/in~[q~])]. I n part icular

for theories without derivative coupling, one can replace d 0 + d i n t by the

total action of the system.

We now question the existence and meaning of u [ J ] : unfortunately, it is

known tha t without renormalization, u[J] has no possible meaning. The

mathematical reason is straightforward, the application of a differential ope-

rator exp [i~/,,~[(1/i)D+]] to a well defined functional of J : Uo[J] given ex-

plicitely by (3.12) or expressed by the integral of a funct ional of two func-

tions ~(x) and J(x) is not generally a well defined funct ional (19). Due to the

imprecision of its basic concepts, this formalism does not seem fit for a dis-

cussion of the main properties of acceptable ~'int[~], but the results of ~ . ~

~.s obtained by methods suggested in the former paragraph can be of use for

a more precise formulation of the foundat ions of the theory of the action in-

tegral.

(17) One may also remark that we do not, in fact, require in (5.2) the existence of the functional integral but only its averaged value with the normalizing factor N.

(is) The introduction of N' means that we fixed an arbitrary phase in the definition of S, namely the one introduced by the sum of all diseonnected diagrams.

(19) This is true for a function of a parameter ~ defined by an integral on a function of two arguments x and ,~ and remains true a ]ortiori for funetionals.

S T R U C T U R E OF F I E L D Q U A N T I T I E S AND ~-OPER2~TOR 7 5 5

A P P E N D I X I

Lemma. - If, for a n y g iven in t ege r 1 and for a n y a r b i t r a r y choice of t he supe r sc r ip t s ~ of one has :

(I.l) <ol [... [A, ~(k~)] ... ,a~ (k,)] 10> = 0 ,

then , for all sets of in tegers m, n and s and a n y k ind of c o m b i n a t i o n of super- sc r ip t s + or - - , one has too :

(2.2) (01 a,+(p,) ... a,~(p~) [... [A, a~(k~)] . . . , a~(k~)] aT,,(q~).., aE(q~) I 0> ---- 0 .

Us ing the c o m p a c t n o t a t i o n :

(2.3) + + +

�9 .. ain(kl)J, . . . , <OIr a,.(p,,~)[...[A, a~(k,)Ja~(q~)...aV.(q~)]O> --= +

(a~=(p ; ] . . . . m ) [ A , a~(k; 1 .. n)]a~(q; 1 ... s ) } ,

we w a n t to p r o v e (2.2) for m = 0 and n~ s a r b i t r a r y b y m e a n s of r ecur rence me thods . I n d e e d (2.2) reduces to (I.1) for re=O, s=O and n a r b i t r a r y . Suppose n o w (2.2) to be t rue for m~-O, s~-r and n a r b i t r a r y , one sees:

(2.4) <[A, a~(ki; 1 ... n)] a~(q; 1 ... r + 1)> =

=- ( [A , a~(k; 1... n)], aV~(q~)]aE(q; 2, . . . , r § 1)> = 0 ,

i.e. (1.2) r e m a i n s t rue for s = r + l . W e thus p r o v e d t h a t (I .2) is t r ue for m ~ 0 , s a n d n a r b i t r a r y .

Then , b y r ecu r rence on m, one can show in the s a m e w a y t h a t the l e m m a is also va l id for a n y choice of m, n and s:

W e now m a y p r o v e the t h e o r e m

Theorem. - P r o v i d e d t h a t the o r t h o n o r m a l s y s t e m Z

1 (2.5) k, 1 . . . n>~o :- ~ aT~(k~) ... a~(k~) i 0 } ,

spans the en t i r e space E , a n y g iven o p e r a t o r A , which m a p s E on to i tself , and verif ies for a n y g iven n and a n y choice of t he supe r sc r ip t s ~ the fo l lowing r e l a t i o n

(2.6) <A, a~(k; 1 ... n)]) = 0 ,

is i den t i ca l l y 0. This is a d i rec t consequence of the f o r m e r l e m m a , since all i ts m a t r i x e l emen t s

w i th r e spec t to Z are 0.

'~56 A. VISCONTI and Z. CARMONA

In par t icular , if two operators A and B ver i fy :

(I.7) ([A, a~(k; 1 ... n)]} = ([B,a~(k; 1 ... n)]} ,

one has A = B . This is the case for the operators A(J) and B(J)----f2(J)(O IA(J)10} when

A(J) verifies eq. (2.6): then A[J] and B[J] are identical.

APPENDIX II

We want now to discuss under wha t physical ly plausible assumpt ions cer ta in types of solutions of the reduct ion formulas (3.1) are univoeal ly deter- + mined . Through the in t roduct ion of annihi la t ion operators ajn(k), the sy s t em (3.1) can be wr i t t en as follows:

(I i .~) [ A~(x l . . . x~), ai+.(k)] = f ?Z(x) (x, xl ... x.)

W e first t r ans form the integro-differential sys tem (iI .1) into an algebraical s y s t e m of equat ions which will be solved step b y step.

Taking into account the a s y m p t o t i c condit ion of L.S.Z., one finds the following re la t ion :

( I I .2)

w i th :

[A~(x~ ... x.), a[~(k)] = ~- i f ddx]~(x)J~f~A~+~(x,

A'~+~(x,x~ x~) ~ 0 for xO>A, . . . . [ EA~(x, x~... Xn) , ~(X)] f o r

X 1 . . . X n )

X o < - - B ,

A and B being two a rb i t ra r i ly large posi t ive numbers . We can also obta in a fo rmula s imilar to (Ii .1) for a~(k), then considering

b o t h formulas , we have :

(rod) [Ao(~:.. ~o), %o(y)] = ( d ~ ~(y - x)S~, ~'~+~(~, ~, ... xn). g

Compar ing ( I I .Q with (3.1) we get

(H.5) An+l(x, xl... Xn) : (-~ i)A'.+l(x, x~ ... x~),

(20) We remark that in addition to the dependence with respect to the arguments zl "" x~, all the operators can also depend on some other parameters x (or several ]oarameters).

S T R U C T U R E O F F I E L D Q U A N T I T I E S A N D ~ - O P E R A T O R 757

up to a solution of the equat ion

f A(y -- x)J['~u(x) ---- O,

for the unknown funct ion u(x). I n the f rame of the physical ly acceptable a s sumpt ions (as l isted below)

for the A~(x l . . . x~) , the solution u(x) is identical ly 0. Le t us now list the conditions on the A~ operators : they can be d iv ide4

into two classes:

a) Algebrical conditions. - We first note t ha t Am is a symmet r i ca l functiorL of its a rguments xl ... xn. For x~177 we r e m a r k t ha t A~+~(...) can be ob ta ine4 by r ight or left mul t ip l i ca t ion of A~ by ~(x) (up to a numer i ea l f ac to r :Li),. therefore i t is na tu ra l to assume A~ to be of the form

q~ (1)erm)

(II .6) A.(x~. . .x~) : (-~i) ~ ~ ~ Ek(xo .. I~=0

x~,) q(x,,)... ~(x,~)Ao ~(x~+~) ... q~(x,,) ,

where El0 is a funct ion tak ing the value 0, =~1. If , a s in Section 3, Ao is taker~ to be ~(x)~ then the expression (II.6) becomes:

n ( p e r m )

(H.7) A~(x; x~...x~) = ( + i ) n ~ ~ ( x ; % . . . x ~ , ) .

�9 ~(x,~)... ~(x,,)q~(x) ~(x~+,)...~(xa,).

b) Physical conditions. - We assume the invar iance of the A~,s w i th respect of the res t r ic ted homogenous Lorentz group, then b y considerat ions which are classical for R products , i t is easy to see t ha t E~ can depend only on the sign of the t ime componen t of the difference of two consecutive argumeuts~ Hence, we shall admi t t ha t EI~ will be bui l t up only wi th expressions l ike

O(x~+~- x~) or O(x~- x~+~) (~).

The me thod which we are going to use for the de te rmina t ion of the coeffi- cients E1c is again a recurrence method. Since i t requires some lenghty b u t very easy caculations, we shall jus t s ta te expl ic i t ly the calculations for t he

(51) In spite of these restrictions, the class of solutions we get in this way is still quite large since it contains all the Polkinghorne products. Proc. Cambr. Phil. Soc., 53, 260 (1957) . Furthermore, we note that as a consequence of their structure the difference of two coefficients An(x ~ ... xn) and Al.(x~ ... Xn) which both satisfy the former conditions, cannot be a solution different from zero of the equation f d (y - -x) J/f~ ~(x) = 0, as we already stated.

7 5 8 A. VISCONTI a n d 5. CAI~IVIONA

simple ease A~(x, x O. F r o m (II.5) and (II .3), i t follows

tO if - - i A ~ ( x , x O = E o ( x , x O q ~ ( x ) ~ ( x l ) + E~(x , x~)~(xOq~(x)= | IF(x),

thus :

0 if x ~ A E ~ 1 if x ~

0 if x~ > A o E~(x ,x~) -~ - - 1 if x l < ~ - - B

tile former equali t ies have the following consequences:

( i i . ~ o )

w t h

x~ > A ,

T(x~) if x ~

Eo(x , x~) = O(x - - x~) , E l ( x , x l ) = - - O(x - x l ) �9

(II.11) B n 4 _ ~ ( x ~ x 1 . . . Xn) = t ~ ( x ) B n ( x l " '" x n ) i f X o ~" A , (B~(z~ . . . x,~)~(x) i f Xo < - - B .

The same algebraical and physical restr ic t ions as for the former case, lead to following solutions

(I1.I2) B ~ ( x ~ . . . x ,~)=i ~ T , , ( x z . . . x~) ,

p rov ided one takes B o = l .

F ina l ly

( I I S ) Al(x, x l ) = ( + i ) [ ~ ( x ) , ~(~,1)~ = - - R(x, x~).

and one general ly finds

(II .9) A,~(x; x~ . . . x,~) = (-- 1)"R(x; x~ ... x~) .

Consider now the case when the field theory under considerat ion admi t s an S opera tor ; the sys tem (II.1) re tains i ts former solutions bu t has in addi t ion solut ions of the form:

A ~ ( x 1 . . . x~) = S B ~ ( x l . .x~) ,

as we saw in Sect ion 3o By the same methods , one finds

.. a i . (k ) ] i ( . . . x , , ) , [ S B . ( x l . x~), + --- d4x]*(x)i.V~SB'~+I(x, xi J

STRUCTURE OF FIELD QUANTITIES AND ~ OPERATOI~ 7 5 9

In conclusion let us show as an appl ica t ion of the uniquenes of the R-pro- ducts as solutions of (3.1), t h a t the operator

-- i ~ * [ J ] Dj(x) ~ [ J ]

is the i r genera t ing funct ional . Indeed the expansions of its factors introduce the coefficients E ~ ( x 1 . . . x~o) which sat isfy the MgebrMcM and physical conditions us t hey were s ta ted , fu r the rmore the f i r s t t e rm of i ts Vol ter ra expansion, the one corresponding to J = 0 , is q~(x).

R I A S S U N T O (*)

Si studia la struttur~ dell'oper~tore S e delle grandezze di campo partendo dal funzionale generatore dei vMori medi nel vuoto dei prodotti cronologicamente or- dinati delle grandezze di ca.mpo.

(*) T r a d u z i o n e a cura della Redazlone.