field operators asc∞ functions in spacelike directions

1L NUOVO CIMENTO VOL. XXXII I , N. 6 16 Settembre 1964

Field Operators as C ~ Functions in Spacelike Directions.

t I . J . BoRcm~RS

Ins t i tu t des Hautes Etudes Scieut i / iques - Bures-sto'- Yvet te

(ricevuto il 13 Apfile 1964)

S u m m a r y . - - I t will be shown that it is only necessary to test the field operators in timelike direction, fA(x ~ x) / (x~ ~ exists 8s unbounded operator and is infinitely often differentiable in spacelike direction.

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

I t is well known in q u a n t u m field theory t ha t the field operators A ( x ) them- selves are not operators in a Hi lber t space bu t t ha t one has to smear out these

objects in order to get meaningful operators . There exists, however, no

physical in tui t ion telling us which kind of test-functions we have to choose.

The necessi ty of formula t ing local e o m m u t a t i v i t y forces us to choose a set

of tes t - funct ions containing a dense subset of functions with compact support .

Which space we ac tual ly choose is only a m a t t e r of convenience. Mathemat ica l

s implici ty suggests then to take the spaces ~ or S~ of Schwartz ' tes t -funct ions (1).

The space 5r is mos t ly t aken to have a more symmet r i c formula t ion between

x-space and momentum-space . In this note we s ta r t also f rom the choice ~ or 5P as space of tes t functions.

B u t we want to show tha t i t is sufficient to smear the field operators in the

t ime co-ordin~te alone in order to get mcimingful operators. Or more pre-

cisely the expression

(Aq~) (x) =f A(xo, x) (xo) d ,o

(1) L. SCHWARTZ: Thdorie des Dis tr ibut ions (Paris, 1950).

i

F I E L D O P E R A T O R S AS C ~ F U N C T I O N S I N S P A C E L I K E D I R E C T I O N S 1601

is defined as an unbounded opera tor for ~0(Xo)~=~ respect ively 5 ~ if fA(x)](x)ddx was originally defined for all tes t - funct ions from 2 respect ively 3 ' an4 has

the propert ies:

I) There exists a dense domain Do of definition wi th

(A~v)(x)Do c Do �9

I I ) (Aq))(x) is infinite often differentiable wi th respect to x.

I I I ) For every ~o E Do

II(A~) (x)~vll amt ~" ~" (A?)(x)~ (~Xl),~,'" (~x~)~

are bounded in x.

Ill Sect. 2 we give a review of the ma thema t i ca l tools we need in the proofs.

I n Sect. 3 we prove the above s t a t emen t for the case t ha t ~A(x)f(x)ddx is

defined for all test-funct ions in ~ .

In order to give a proof in the case where j'A(x)](x)d~x is only defined

for test-functions in ~ we have first to prove t ha t W i g h t m a n functions are

boundary values of analyt ic functions. This is done in Sect. 4.

Final ly we t r ea t the general ease in Sect. 5.

Our proofs in Sect. 4 are inspired b y a me thod due to KUNZE and STEIN (2)

concerning a special ease of Har togs tbeorem for analyt ic functions of real

variables. There exists an unpubl ished proof b y ZERNER (a) giving a result

similar to our L e m m a 9. Bu t the technique used in Zerner 's proof is of quite different nature .

2. - P o s t u l a t e s and m a t h e m a t i c a l pre l iminar ies .

The f rame of our inves t igat ion will be the set of postula tes usually assumed

in W i g h t m a n ' s formula t ion of q u a n t u m field theory (% We will not depend

in our proofs on the num ber of fields and their behaviour under homogeneous

Lorentz t ransformat ions , so t ha t we can res t r ic t ourselves to the ease of one

single field. For this field we require

(2) R. A. KUNZE and E. M. STEIN: zll~. Journ. Math., 83, 723 (1961). Lemma 21. (3) M. Z]~RNER: Semb~ar Marseille, unpublished (1960). 0) A. S. WmIIT)IAN: Phys. Rev., i01, 860 (1956); and Probl~mes math&tbatiques

de la thdorie quatdique des chaml~s , Lecture notes (Paris, 1957).

1602 i[. J. BOIiCnl~R8

a) translation invariance,

b) spectrum condition,

c) the existence of a vacuum state ~2,

d) cyclicity of the vacuum state,

e) local eommuta t iv i ty ,

/) existence of the Wigll tman functions as distributions.

We also will assume tha t the smeared-out field operators

Aft) =f/(x)A(x) d~c

are defined in a Hi lber t space with positive metric.

We now consider some results f rom the theory of vector-valued functions. Le t V be a topological vector space and V' it 's dual space. Be ~0(x)e V

for all x (x e R "~) then we say ~0(x) is a vector-valued function. ~(x) is called

differentiable if l im o (l/h)(q~(x + he) -- cf (x)) converges in the strong topology of V. The funct ion ~0(x) is called bounded when the set U,,~(x) is a bounded set

x in V. Wi th these notat ions we can define

E R

1) E(V) the set of all infi,lite often differentiable functions wi th values in V.

2) 0~(V) the set of all functions from E(V) which are, together with their derivatives, polynomial bounded.

3) B(V) the set of functions from O~(V) which are bounded and which have botmde4 derivatives.

An equivalent character izat ion of these spaces is given by SCHWARTZ (5).

Lemma 1: A funct ion ~(x) with values in V is in E(V), O~(V), B(V) if and only if for every e e V'<q)(x), e) is an element f rom E, 0~, B.

For our purpose we are in teres ted in two special cases:

I) How can we characterize functions from 0~x(~'~) which are at the same t ime elements f rom ~f'(x, y).

Since we know tha t our funct ion T(x, y) is from OM~(6~'~) we can form the

(5) L. SC~iWARTZ: Jourt~. d'Analyse Mathd,~atique (Jerusalem), 4, 88 (1954). Proposition 4.

F I E L D O P E R A T O R S AS C r176 F U N C T I O N S I N S P A C E L I K E D I R E C T I O N S 1603

convolution product wi th any funct ion from 5 f and lind tha t

T(x, y ) , ~0(y)

is from 0 , . for every fixed y. On the other hand i t is from 0 ,y for every

fixed x. Taking now into account tha t T(x, y) is f rom 5P'(x, y) we find

T(x, y ) , of(y) ~ 0~. .

Taking now the Fourier- t ransformed versio~ o[ i t we get:

Lemma 2: An clement T(x, y) h'om ocE'(x, y) is an element from O~(~ 'y) exact ly if for every test-function ~tq)E~(q) the product

~(p, q) ~(q)

is an element from Oe(p, q). T(p, q) denotes the Fourier- t ransform of l'(x, y).

I I ) We are in teres ted in functions from E~(~'~). Our special question

here is: Under what condition converges a sequence l'~(x, y) ~ E~.~ to an element in E~(~:). To decide this we nee4 first a topology in E~(~'~). To this end we define: a sequence Tn(x, y) converges to zero if for any compact sub-

set C in R~ the sequences

D~' Tn(x, y)

converge to zero in ~ uniformly in x ~ C. Since a family of elements Tn in ~'~ converges to zero if and only if for

any ~ v e ~ the set of numbers (Tn, ~} converges to zero (6) we have also: Tn(x, y) converge to zero in E~(~'~) if for any ~ (y )~6 ~ the sequence

<T,(x, y), q~(y))

converges to zero in E. Since now E(~ ' ) is complete we have in general.

Lemma 3. T,~(x, y) converges in E~(~'~) if for any ~v(y) c ~ (Tn(x, y) ~0(y)} converges in E~.

E ( _6J;t ~ On the other hand T(x, y)~_~ ,_y , can be considered as a linear map of I " t ! E'~ onto ~ . SCttWARTZ (~) has shown tha t E~(~) is in fact algebraic and

topological isomorphic to the space c~f(E:, ~ ) of linear maps of E~ onto ~ .

This gives us the possibility for a solution of our original problem.

(6) L. SCIIWARTZ: LOC. cit (1), Chap. III, Th6or6mes IX and XIII. (:) L. SCItWAI~TZ: Loc. cir. (5), Proposition 24.

1604 H.J. B O R C H E R S

Lemma 4. Let T,(x, y)eE:,: then l imT,(x , y) converges in E~(~:) if for

any r the sequence

<~(x) T,~(x, y)> converges in ~ : .

Proo/: Since for every ~b(x)+ E~' the sequence <r T,(x, y)> converges

in ~i~) we have a linear map T(x, y) of E~' onto ~ : and we need only to show

tha t this m a p is cont immus. To this end take any bounded set B c ~ then

an open neighbourhoo4 of the origin in ~ is characterized by (~be~:[ 1 <~b, ~>[-

�9 < M F ~ e B}. To show tha t 1'@, y) is continuous we ln~ve to show theft the set

(1) {O(x) e E:II<<~(~) T(x, y)> r / < M v~r e~}

contains an open subset. Bu t for every fixed ~b(x) the set of numbers

(<<q,(~,) T(xy)> ~(y)> Iq~ e B}

is bounded. T h a t means

{<<., T(~., ,j)> v(~)> [,p e ~}

is a bounded set in E (s) which is equivalent to the s t a t ement t ha t (1) is

open q.e.d.

3. - A (/) is defined for test funct ions in 9 ".

I n this Section we want to prove the s t a t emen t announced in the intro-

duct ion for the special ease t h a t the Wi gh t man functions are t empered distri-

butions. The ma in p rope r ty we use to get the result is the suppor t of the

W i g h t m a n funct ions in momentum-space . This leads us to s tudy distr ibut ions

wi th suppor t in a cone.

Lemma 5. Le t C be a proper closed cone in R "+1 wi th apex a t the origin

so t ha t x o = 0 n C = 0 and C c Xo ~> 0. Le t T be a t empered dis t r ibut ion with

suppor t in C and let (p(xo)~,~. Then T(xo, x)q~(xo) is an element f rom 0 C.

This means T(xo, x)q~(Xo) is a s t rongly decreasing dis tr ibut ion in all variables.

Proo/: Since C is a closed cone and has wi th the hyperplane xo = 0 only

the origin in common we can find d > 0 so t ha t

(2) x o > dixl if (Xo, x) eC.

(s) L. SCIIWARTZ: Loc. cir. (1), Chap. I I I , Sect. 7.

FIELD OPERATORS AS C c~ FUNCTIONS IN SPACELIKE DIRECTIONS 1605

This means especially t ha t there exists an open neighbourhood 2V(C) of C

and s 0 so t h a t

(3) (1 + x~ + x~) " ---- bn(xo, x) ( ] + (1 + dO')x~) ~ if (xo, x) eN(C) ,

where b,,(xo, x) is f rom B (the space of C ~ functions which are bomlded and which h:~ve bounded deriw~tions).

To show tha t T(x~ x)7,(xo ) is f rom 0 C we have to check tha t for every n

the expression (1 + x o + x2) " T(xo, x) q,(x,,) is a bounded distr ibution. Bu t f rom

a representa t ion theorem of t empered dis t r ibut ions we know tha t T(xo, x)

can be wr i t ten as

(4) T(xo, x) = (1 -1- 'a?" o -[- x2)mB(xo, x ) ,

where B(xo, x) is a bounded dis t r ibut ion. (3) together wi th (4) gives

2 nTm (t + x~ + x~)"T(Xo, X)q~(Xo) - ( l + X o + X ) B(xo, x)q:~(Xo) - -

-~- (1 + (1 + a'2)X2o)"+mg(xo)b.,+.,(xo, x) 'B(xo, x)

is a bounded dis t r ibut ion q.e.d.

Corollary. Let ~V(~ 1 . . . ~n) = (~r A l ( x 0 ) . . .An(d~n)~) , ~i ~ - - x * - - x i - 1 be a Wight-

man function. Assume the spec t rum condit ion is fulfilled and W(_~I... Sn) exists

as t empered distr ibution. Le t M be ally or thogonal n • t ransformat ion mapp ing o o ~I "'" ~n o n t o ~1 = ( 1 / V ~ ) ( ~ - ~ g ~ - . . . ~ ~ ~]2. . . ~ n .

Then ]~'(~]i...~]n, ~i.- '~n) is an element f rom 0~(.9~...~,, ~ . . .~ , ) (Sz~) .

Proo]: This is s imply a consequence of the spec t rum condit ion which implies t ha t the Fourier t rans form of 1V(~,... ~n) has a suppor t contained in

a proper cone. L e m m a 5 implies then t ha t the condition of L e m m a 2 is fulfilled. This gives the result.

We are now able to prove the main result of this Section.

Theorem 1. Let A(.r) 1)e a field fulfilling the nsu~d requirements except

local commutt~Civity and let the Wi gh t m an functions be defined as tempered

distr ibutions. Then for every tes t funct ion 9(Xo) f rom 5 P the expression

<A, 9> (x) =f A(x)W(Xo)

is defined as an unbounded opera tor in such manner :

a) There exists a dense domain Do of definition.

b) (A, 9)(x)Do=Do.

1606 H.J. BORCHER, S

c) V T~Do the vector <A, q> (x)T is polynomial bounded and infinite often 4ifferentiable with respect to x. M1 derivatives of (A, ~v>(x)T are bounded polynomials. Or (A, q~)(x)Te O~(x)(~).

If moreover this field is a local field then e) can be sharpened to

c') <A, (DF(x)W~_B(x)(.)F) V T e Do.

Proof: To prove a), b) :rod e) it is sufficient to show tha t the Wightman functions ($2, A(x~)...A(.r~)f2) are elements of the sp:~ce OM(Xl , X2... X,,)" �9 (Sf'(X~ X~ But this is an in,mediate consequence of the preceding corollary. The cyclicity postulate guarantees the existence of the domain Do with the properties a) and b). Explicit ley Do is defined as the linear span of the vectors

{~, <A, q)>(x)~, ..., <A, Q91)>(x1)... <.4, 0')n>(Xn)~(~., .�9

The posit ivi ty requ;rement implies tha t for any vector T of Do the function

<A, q,>(x)Tz OM(x)(~).

If we now assume local commuta t iv i ty we use a result of _A_RAKY, HEPP and ]~UELLE (9) which says: I f A(x) is a local field ~n4 ff all Wightman functions are tempered and if the spectrum conditions hold then the expressions

(5) ~XI.1 nl ... ~X~ In (~9 A(,~O Xl ) A(X o, X~t) ~f~) q31(,~,01) )pO 0 ..

are bounded distributions in (x~... x,) for ~v~(x ~ ~hf. These arguments c~m also be applied here to get the stronger result. I f A(x) is a local field and if the Wightman-functions exist as tempered

distributions and if the spectrum condition holds then for %(x~)~hf the expression (5) is an element from B(xl...xn).

This implies then the s ta tement c') q.e.d.

4. - Analytic properties of Wightman functions.

In this Section we want pro prove tha t Wightman functions are boundary values of analytic functions wi thout assuming the temperedness of these

functions.

Lemma 6. Let 2i~>0, ~ = 1 , i = ] . . . n an4 y/>~0.

(9) H. ARAKI, K. Hl~Pe and D. RUELLE: Helv. Phys. Aeta, 35, 164 (1962), Sect. 4.

.2

F I E L D O P E R A T O R S AS C r176 F U N C T I O N S IN S P A C E L I K E D I R E C T I O N S 1607

Then there exists a par t i t ion o~r ...p,,), j = O, 1 .... n of the uni t

so t ha t

exp [ - X ~ ~ O~, e x p [ - - y~1:5(1 _L s)]

= 1 ~ 2 , . . . , n .

Proof: Let us define e,~ 0~,

a t =

1 p ~ > - - l ~

0 p ~ < - - 2

and ~ o = H a s , then 7oexp [ - - ~ 2 , y , p J ~OM.

Since ~. 2, = 1 ) the regions

form for ~ > 0 an open covering of the complement of p ~ > - - l , i = 1 . . . ~ . n

Therefore we can find ~ wi th suppor t =jc G~ and = ~ 0.~ and ~ ~ = 1 - =o~ Since now ~=1

D " exp [ - - E'~iY,Pi] e x p [ - - (1 + e)y~p~]

w e h a v e

j~, exp [ - - (i + e) y~ p]

e:,p [-- y.

as required.

L c m m a 7. Let ](x~.. .x.) be a t empered distr ibution. tes t funct ion ff~(xi ... x~_~, xj+~ ... x~) ~ ~<f~_~ ; j = 1, 2 ... n.

Assume for every

(/, ~}(x~) has an analy t ic extension into 0 < I m z j < 1; j = ] ... n such t ha t

f(x~.., xr x~+ iyj. . , xn) is t empered in xl.. . x~. Then there exists an analyt ic

funct ion F(zl ... z,) holomorphie in I m z~ > 0, ~ I m z ~ < ] which is t empered

for every fixed yl. . . y~. This funct ion has boundary values in the sense of

distr ibutions and

]im /~(zl ... z~) ---- l(xl ... xn) Im zj =0

holds in the sense of distr ibutions.

Proof: Let I (Pl . . .P , ) be the Fourier t ransform of ]. l e 5 ~'.

1608 H.J. BORCIIERS

Since ](xl... x~-l, x~§ x~) is tempered we have

(6) ](Pl ... P~) exp [ ~ y~p~] e5 ~' for 0 < y~< 1.

Let now 2~> O, ~ ks = 1 and y~< 1 then we can find e >0 so tha t (1 +e)y~<l. Let ~o... ~ be the p~rtition of the uni t described in Lemma 6 then we have

,,xp [ - Z ~:,/,~,~]/(px ... ~,~) = ~o exp [ - ~: ~,.~/~t'~] l(~'~ ... p,,) +

--I- J=l '~ ~j ~'xpeXp[--[-- !I-;ps(I~'~"Y~P~]Jr- el i e:(p [ - - yjp;(l -~ e) ] t ( t ) l ... p~) ~ of,

according to Lemma 6 and (6). But this is equivalent to the s ta tement of the lemma.

Corollary. Let ](xl...x~) be a bounded function I](xl. . .x~)l< ] and also I](x~...xj ~, x~§ . . .x~)lK1 then F(zl.. .z~) is bounded by 1.

Proo]: Let I o~ I> 1 then (o~--/7(z~... z.)) -1 fulfills the condition of Lemma 8, and is therefore analytic in ~ I m z j < l , I m z j > 0 . So F does not take the value ~o. That means IF(z1... Zn) ]<1.

Lemma 8. Denote by C + - c<~ < e K + co the domain which is formed by intersection of the region Im z > 0 and the circle with center i~ and radius r - - ~ / 1 § ~ . Assume we have a function ](Xl...x~) conti~mous an4 bounded by 1 in the region Ix j /K] , j = l , 2 . . . n . For fixed (x~...xi 1, xj+~...x~) the function ] has an analytic continuation ](xx . . . x j+iy . . . x~) into the region C + which is botmded by 1. Then there exists a function /7(zt...z~) holo-

~ J

morphic in the region F defined below. /7 is bounded by 1 and /7 has boundary values lim/7(z~.., z~) which coincide with ](x~... x,,) as distribution in Ixj ] <: 1. F is defined as

F = U P(~.. .~, ,) 2 j ~ o

and

with F(~I "'" ~ ) = {21 '~ Zn IZJ e '+

Qj(XI ... ; t~)= {tg ( ~ ( 1 - - ~ j ) + ;t~ aretg ~ ) } -1 .

The arctg has to be chosen between 0 and +~) .

Proo]: We reduce Lemma 8 to Lemma 7 and the Corollary with aid of the t ransformation

exp [qoi] -- 1 Z _ _

exp [~] + 1

F I E L D O P E R A T O R S AS C r176 F U N C T I O N S IN SPAC~iLIKE D I R E C T I O N S 1609

This t ransformation maps the cube l x j l < 1 onto the whole space 1~ 1<r and the region C*ej onto the strip 0 .< ~ < = - - a r c t g 1/~j.

By Lemma 2 g(~l... ~ ) = / ( z ~ . . . z~) has an analytic cont inuat ion in ~ into

the convex hull, which is bounded by 1. Calculating back this region to the

z-variables we get Lemma 9.

L e m m a 9. Let L be a real nondegenerated linear tr~msformation and u = L x . Denote z = x + i y and w = u + i v . Assume the regions Im (za) > 0 and Im (w~) > 0 - Im (Lz)] > 0 intersect in an open tube with basis C which is a

cone with apex at the origin

Let G1 be the region l x ~ - - a ~ l < l ~ - - : l . . . n

and

G2 the region lug-- bj 1~ 1 j = 1. . . n

and assume G~ and G2 have an open intersection. Let f ( x l . . . x , ) be a function bounded and continuous i n G1. Assume

](x, ... x , ) has an ~nalytic cont inuat ion in the variables z~--aj into the region /71.

Let g(u~... ~,) be a funct ion bounded and continuous in G2. Assume g(u~.. , u~) has an analyt ic cont inuat ion in the variables w~--bj into the re-

gion /"2. If we have f (x~. . , x ~ ) ~ g(u~.. , u~) in GI n G2 then there exists an analytic

funct ion /7(Zl... z~) holomorphic in the union /"1 u ]"2 which coincides with

](z~... z,) in F~ and with g(wl . . , w , ) in F2.

Proof : F rom our assumption follows the existence of a region Go = = {xllx j - c j l ~ d >0} which is contained in G1 ~ G2. The intersectma of /'1 and F.~ is "flso nonempty an4 i t exists a constant e ~ 0 so tha t

is contained in FI n F~. Consider the function

h(Zl ... z,~) = f(z~ ... z,,) - - g(wl ... w.) ,

with D as its domain of definition. F rom our assumptions follows tha t

h(zl ... zn) has boundary values and

lim h(z l . . , z . ) = 0 . Im z~-~.O

Now the function h(51 ... z~)= J~(zl ... z,,) is again an analyt ic funct ion which is

: 0 2 - I1 Nuovo Cimento.

1610 H. J ' . BORCI{.ERS

holomorphic in D. We have

lira k(zx.., z~) : lim h(z 1.. . Zn) = 0 . lm zl--~O ImZ 1 -->0

Im (Z) ~ - - ~ Im(Z) ~ O

F r om the (( edge of the wedge ,) theorem (,o) follows tha t h(z, ... z,) has an analyt ic extension into the region D u / ) u {smM1 neighbourhood of

Ix~- ~, I< ~/~}. Since h is zero on real points it must be identically zero. This proves the

lemma.

Theorem 2. Let the spectrum of the representat ion of the t ranslat ion group U(t0 be contained in the forward light cone V*. Assume A(x) is a trans- lat ion-invariant field and ~2 a state which is inv~riant under this representat ion. If the expectat ion values

(~2, A(xo) ... A(x,).O) = I V ( ~ 1 . . . ~ n ) , ~ i ~ X i - - ' / ) i --1 ,

exist as distributions, then there exist functions F ( ~ . . . $ . )ho lomorphic in the

tube T + = {~ I Ira ~ V +) which have boundary values in the sense of distribu-

tions such tha t

Lira F ( ~ ... ~,) = W($~ ... ~,) Im~j--*O

Proo/: Let /o . . . /he ~ and A(L)=fA(x )L(x )dx , then equivalent to the sta- t ement of the theorem is t ha t the functions

(~, Afro){(~ , )A( / , )~(~) ... ~z(}~)A(}~)Q) = ~i(}~ ... ~ )

are boundary values of analyt ic functions which are holomorph in T +. We d

choose now d linear independent vectors J J - ,7+ ap . . ~d ~ ~ and write ~j = ~ ~jL~.

F rom the spectrum condit ion follows tha t L~

can be analyt ic cont inued in each variable 2~z separately into the region

I m A j L ~ 0 . In the regi (n 12~LI<A is the funct ion G I ( ~ ) bounded and we

can apply Lemma 8. Tha t means there exists a funct ion G 1 (AjL) holomorphic in some domain F which has G~(2~) as boundary values. I f we take a sequence Am-->co and apply Lemma 9 we see tha t there exists a funct ion GI(~jL ) holomorphic in i m A j ~ > 0 which has (~(2~L) as bound~ry values.

(10) H. J. BREMERlVIAN, R. OEHM:E and J. G. TAYLOR: Phys. Rev., 109, 2178 (1958).

F I E L D O P E R A T O R S AS C ca F U N C T I O N S I N S P A C E L I K E D I R E C T I O N S 1611

By taking all possible choices of ~ and application of Lemma 9 we find tha t there exists a funct ion F~ (~. . . ~ ) holomorphie in the tube T + which has F~(~I ... ~,~) as boundary value. This proves the theorem.

5. - T h e g e n e r a l c a s e .

In this Section we want to prove the analogue of Theorem 1 for the case

where the smeared-out field operators are only defined for test-functions with compact support. The result is

Theorem 3. Let A(x) be ~ field fulfilling the usual requirement except

local commuta t iv i ty and assume the Wightman functions cxist~s distributions. Then for every test-funct ion ~(x ~ from ~ the expression

defines an unbounded operator in such a manner tha t

a) there exists a dense domain Do of definition,

b) (Acf}(x)DoCDo,

c) VTeD o the vector ( A ~ } ( x ) T is from E~) (~ ) .

Is, moreover, this field ~ local field then c) can be sharpened to

c') VT~Do; (Aq~}(x)T~B(,)(z/f).

The proof of this theorem is essentially based on the well-known (1~).

Lemma 10. An analyt ic funct ion ](zl... z~) holomorphic in the product of the half-planes Im zj > 0 has boundary values in the sense of distributions exact ly if for every cube [x~ 1< M exists an integer N and a constant C so tha t

I](z~ ... z~)]< [(Im z~)(Im z~) ... (Ira z~)]-~C

holds for I x~ I< M, 0 < Im z~< 1.

F r om Lemma 10 we deduce now the following

Lemma 11. Let C be an open convex cone with apex at the origin and

assume the cone IYjI<~Yo is a subcone of C. Le t ~(Zo...zn) be a function holomorphic in the tube (Yo... Y~) ~ C and assume lim F(z o... zn) = ](xo... xn) exists as distr ibution then ](x0, xl . . .x~)is an element f rom E(x~..,n)(~,):

(11) H. G. TILLMANN: Math. Zeitschr., 59 (1953/54), Sect. 4.

1612 H.J . BO]~CHERS

Proo]: According to L e m m a 10 there exists a subcone, say ]y. I< (~/2)yo,

in which F(z~ Zn) is bounded by

6g [F( o O < y o < l , rxol, Ex l<i,

where d denotes the Eucl idean length of the vector (Yo... Yn). We can now

majorize the length of the vector (Yo...Y~) by the magn i tude of Yo itself so we get

'F(Zo z~) l< y ~ C ' for [y~[< ~ �9 .. : , :Jo, o < y o < 1, I~ol , I ~ 1 < M

I f we now consider especially y~=y~ . . . . y ~ = O then F(xo+iyo , x~. . .x . ) is clearly an e lement of E( .... , ..... ).

To prove the L e m m a we make now use o[ L e m m a 4. We consider the

expression

(8) ( qS(xl.., x~) F(x o -k iyo, x~ ... x~) ) for ~b e E ' .

Bu t f rom a representa t ion theorem of dis t r ibut ions wi th compact suppor t

we know the following representa t ion (~2)

(9) ~b(x, . . . Xn) : ~ ~,~nl . . . ~Xnm" Qgm(f 1 . . . X.n) ,

where q~(xl.., x,) are bounded continuous functions wi th

supp ~o(x~ ... x~) c supp r x~) -k s .

Inse r t ing (9) into (8) we get

(r ... x~)F(xo 4- iyo, xl ... x,,)) =

~f ~m~ ~n, = ~ ( - 1) ........ d~i ... d~cf,,,(x~ ... x~) ~xr~... c~-~ F(x . + iyo, x~ ... x , ) .

Since F(Xo§ Xl...x,~) is analyt ic in [z~.l< (~/2)yo we find b y means of Cauchy~s formula and (7)

[(q~(Xl...x.),F(Xo-~iyo, x~...x.))]~ ] ~/)m(~ql "~ 2n) I d'/~l ~ ~ d~Cn ? Y 0 y o / g e l .

(12) L. SCiiWAIr Loe. cir. (1), Chap. I I I , Th~or~rae XXVI.

FIELD OPERATORS AS C c~ FUNCTIONS IN SPACELIKE DIRECTIONS 1 6 1 3

This p roves (~b(xl... xn) F ( x o § xl . . . x,~)) has b o u n d a r y va lues as dis t r i -

bu t ions accord ing to L e m m a 4.

Corollary. L e t IV(~I... ~n) = ( ~ , Ao(xo) . . . . tn(Xn)~), ~ = X~--Xi--1 be a Wigh t -

m a n func t ion . Assume the s p e c t r u m condi t ion is fulfilled a n d W(~I,. . Sn) ex sts

as d i s t r ibu t ion . L e t M be a n y o r thogona l ~ x n t r a n s f o r m a t i o n m a p p i n g

1 o ~o ... $o on to ~, = (?~ (~1 ~ § ~ § ... § ~ , )~2 ... ~], �9

Then I V ( ~ I . . . ~ , r/~...~n) is au e l emen t f rom E(~l~...~/n , ~1...~,,)(~'~11).

Proo]: This is s i m p l y a consequence of T h e o r e m '2 and L e m m a l l .

Proo/ el Theorem 3. T h e p reced ing corol la ry impl ies t h a t all W i g h t m a n

func t ions (~ , A(Xl ) . . .A(x~)~) are e l emen t s of t he space E<~, ..... ) (~ ' (x ~176 The cycl ic i ty p o s t u l a t e g u a r a n t e e s the ex i s tence of a dense 4 o m a i n which

is def ined as the l inear span of the vec to r s

{~9, <A~> (x)f2, <A%>(x~) <A%)(x~)~9, ... l% e ~ } .

This set fulfills condi t ion~ b) and c). I f we now a s s u m e local c o m m u t a t i v i t y we use aga in the resu l t s of . ~ A K I ~

HEPP and RUELLE (za) i.e. : I f A(x) is a local field and if all W i g h t m a n func t ions

ex is t as d i s t r i bu t ions and if the s p e c t r u m cond i t ion holds t h e n the express ions

~ m z ~rnn [" ,0

(10) ~xm, ... ~xmnJd~c.1 ... dxn~ (~r ,~(Xl) . . . A(x,)~2)~I(X 0) ... ~n(~n 0)

are b o u n d e d d i s t r i bu t ions in x~... x , for ~c~(x~)~ . These a r g u m e n t s can also

be appl ied to ge t the s t ronge r resu l t t h a t (10) is an e l emen t f r o m B(x . . .$) . This impl ies s t a t e m e n t c' q.e.d.

The a u t h o r would l ike to t h a n k Dr. L. MOTCHANE foI his k ind hosp i ta l i ty .

(in) l,oc. cit. (9), Sect. 4.

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

Si dimostra che, per verificare gli opcratori di campo nella direzione temporale, b necessario soltanto che fA(x ~ x)](x)d~ ~ esista come operatore illimitato e ehe sin differenziabile iniinite volte nella direzione spaziale.

(*) Traduzione a cura della Redazione.

field operators asc∞ functions in spacelike directions

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