IL NUOV0 CIMENT0 VOL. XXX.I, N. 5 1 o M~rzo 1963

On the SeIi-Adjointness of Field Operators (*).

H. J . BORCHERS and W. ZIMMERMAN~ ~

Coura~t Institute o] Mathematical Sciences, New York University - New York. N. Y.

(ricevuto il 21 Settembre 1963)

S u m m a r y . - - A new postulate of quantum field theory is formulated which implies: 1) the smeared out field operators are (essentially) self- adjoint; 2) the associated spectral projection operators commute for spacelike distances. These results allow to construct local rings of field operators in the sense of Hang and Araki.

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

Tile concep t of local r ings of bounde d opera tors was original ly in t roduced

by HAAG (1) aS a special tool wi th in the f r amework of q u a n t u m field theory .

R e c e n t l y ARAKI (2) developed tiffs concep t to an independen t t h e o r y which

has lost i ts d i rec t connec t ion to the usual q u a n t m n field t h e o r y (3). Since bo th

theories , the W i g h t m a n , as well as the A r a k i - H a a g theory , i n t end to describe

re la t iv is t ic q u a n t u m t h e o r y one expects t h a t b o t h theories give only dif- fe rent aspects of one and the same object . Bu t up to now there is no th ing

k n o w n which n m y suppor t t im hope t h a t bo th theories are real ly closely re-

luted to each o ther excep t in the t r iv ia l c~se of free fields where e v e r y t h i n g

can be computed .

(') This paper is supported by tlle Sloan Foundation's grant for mathematical physics.

(1) R. HAAG and B. SeHICOEmr Journ. Math. Phys., 3, 248 (1962); R. HAAO: Colloque International sur les Probl~mes Mathgmatiques sur la Th~orie Quantique des Champs (Lille, 1957).

(2) H. ARAKI: Lecture Notes (Zurich, 1962). See also M. GUENIN and B. MISRA: On the yon Neumann Algebras Generated by Field Operators, preprint (Geneva, 1963).

(s) A. S. WItmTMAN: Phys. Rev., 101, 8601 (1950) and Probl~mes Mathdmatiques de la Thdorie Quantique des Champs, Lecture notes (Paris).

1048 H . J . BORCHERS and w. ZIMMERMANN

Unfor tuna te ly it is not even known whether one c~n associate local rings to every Wigh tman theory. Haag ' s original way of construct ing them makes

use of the assumptions tha t every symmet r ic smeared-out field operator is ~utomatic,~lly self-adjoint and tha t the spectral resolution of two different smeared out field operators commute if these operators commute on some

common dense domain. I t is known tha t both assumptions are independent of each other and t ha t they ~re not au tomat ica l ly fulfilled (4).

In this note we want to give a condition under which H~mg's procedure

can be carried through. I f for every tes t funct ion I with compact support

the series

~=o ~!

has a nonzero radius of convergence then the local rings exist. This condition is interest ing because it coincides with the condition under which a functionnl formulat ion of quan tum field theory is possible. Whether this condition has also some physical meaning is not known, but , mathemat ica l ly , one h~s to expect some trouble if i t is not fulfilled. This will be demonstruted ~t an

example which ~dso shows tha t our above condition cannot be improved with-

out making fur ther assumpt ions than the usual ones. Section 2 conta, ins a detailed discussion of condition (1). Some mathem~tic~d

definitions and theorems f rom the theory of unbounded operators are collected

in Sect. 3. In Sect. 4 we prove tha t the field operators are self-adjoint and tha t their spectral resolutions commute for spaeelike distances ('~).

2. - G e n e r a l p o s t u l a t e s .

The f rame of our invest igat ions will be the set of postulates usually us- sumed in Wigh tman ' s formal~tion of quan tum field theory (3). We will not

depend in our proofs on the number of fields and their behavior under inhomo-

geneous Lorentz t ransformat ions so tha t we cun restr ic t ourselves to the case

of a single scalar and neutral field (a charged field and its hermit iun conjugate

can always be considered as equivalent to two real fields).

Ins t ead of the improper field operator A(x ) we will always use the smeared-

out opera tor

A (/) = / d x / ( x ) A ( x ) , J

(a) See, e.g., M. GUENIN and B. MISRA: De la Permutabilitd des Operat~urs non bornds, preprint (Geneva, 1963).

(5) For the special case of generalized free fields the self-adjointness of field operators was proved by R. JOST: Lecture Notes (Princeton, 1963).

ON THE S E L F - A D J O I N T N E S S OF FIELD OPERATORS 1049

where /(.r) is a real test function with compact support , i.e., /(x) is arbi t rar i ly

often differentiable and vanishes outside a bounded region. The opera,tots A(/) are defined on a dense linear subset D of Hi lber t space. Of the A(J) we need require only

a) tr~mslation invariance,

b) the spec t rum condition,

c) the existence of a vacllllni state,

d) cycli(.ity of tire v a c u u m state,

e) local cominuta t iv i ty ,

f) hermiti(.ity.

Postulates d), e) and J) will be of p~rt icular interest for this work. We therefore s tate t hem explici t ly

d) The domain D of tire operators A(J) is tire l inear span of the vectors

~r 2 t (/1)~(2 , A ( / , ) A (]2)~r . . . .

D is dense in Hi lber t space .

e) A(f) A(g)q~ = A(g) A(f)qb , q~ ~ D

for all vectors in D, if the sut)ports of ] and g are spacelike separated.

/) (~, A(/)W) = (A( / )~ , ~)

for all vectors (/) an(1 ~P in D.

For the formulat ion of m( addit ional postulate we need the concept of an analyt ic ve(,tor. In the mathemati( .M l i terature (6) a vector ~b in Hi lber t space is ealled m, analyt ic vector of an operator A if the power series

n=O n [

has a nonvanishing radius of convergence (il'[I denotes the norm of ~ vector) .

We now add to a)-f) the new postula te

g) tlle v a c u u m state be an analy t ic vector for every operator A(J).

(6) E. NELSON: Ann. Math., 70, 572 (1959).

1 0 5 0 H . J . B O R C I t E R S a n d W . Z I M M E R M A N N

(2)

Expl ic i t ly : The power series

~: IIA(/PQIt ~=o n !

has ~ nonvanishing radius of convergence, for every tes t funct ion ] wi th com-

pac t support .

We now prove: 51ecessary and sufficient for g) is the condition tha t for

each ] there exists a constant al such t ha t

f a x 1 ... dxn](xl ) ... ](x~) (A (x l ) ... A(x.))o = ] (A(])'~)o l < n !a'~ (3)

(<)o denotes the v a c u u m expecta t ion value). I t should be emphasized t ha t

the value al m a y depend on ] in an a rb i t r a ry manner and need not be bounded

as ~ funct ion of ]. For the proof of this s t a tement we observe theft (3) implies t ha t the ex-

pansion (2) is major ized by

n! a~z"

(because lid(/)" .(-2 II = v / ~ o < ~ / ~ n ) ! a~) with radius of convergence

1 r = ~ = / = 0 .

On the other hand, if eondition (2) holds the sequenee

~ IIA(I):' ~211 n!

must be bounded for n-->c~. Hence there exists a constant c such tha t

Thus we obtain

IIA(/PQII < n! c~.

<A(f ) '%< IlA(/)n~]l < n! cn.

This completes the proof of the equivalence of g) and (3). In the following we discuss some simple consequences of the postula te g).

Condition (3) implies the convergence of the expansion

0 ~=0 ~ - ! *

ON T I l E S E L F - A D J O I N T N E S S OF F I E L D OPERATORS 1051

This ullows to define the generating functional of the Wightman functions

by (4). Conversely, if the function W{z]} is analytic in z at z = 0 , condi-

tion (3) follows. We have thus proved: Necessary and sufficient for g) is that

the generating functional (4) of the Wightman functions exists and is analytic in the scale parameter z at the origin.

We will now show that condition (3) is equivalent to a corresponding con-

dition for the t runcated functions Wt~(x 1... x~) introduced by HAAG (7). Ac-

cording to SYMANZIK (s) the relation between W and Wt~ is conveniently de-

scribed by

(5) W{zl} = exp [ W~{z/} - - 1],

where Wt~{]} is the generating functiomfl of the t runcated functions. Equa-

tion (5) implies: If W{z]} is defined and analytic at z = 0, also W~{z]} is anal-

yt ic at z= -0 and vice versa. This proves the equivalence of (3) with the condition

( 6 ) fldx, dxn/(Xl) ..o /(Xn) ~ T t r ( X 1 . . . Xn) ... ~ n I.b l~ . !

(6) is trivially satisfied for ~ free field (or a generalized free field in the sense

of GREENBERG (9)) since in this ('~se the t runcated Wightman functions vanish

identically for n ~ 2. Hence postulate g) holds for a free field.

Lit t le is known about models of locally interacting fields. We quote a result obtained by SY~[ANZ[K (10) which concerns local invariant field theories

in the Euclidenn four-dimensional space. In this case SY~A~ZII( proved that

the functional (4) is an entire function of z provided the model is sufficiently

regularized and confined to a finite space-time box.

3. - M a t h e m a t i c a l pre l iminar ie s .

In this Section we will discuss the mathematical problems which arise from

the fact tha t the field operators A(]) are unbounded. For the convenience of the render we collect here some definitions and theorems of the theory of Hilbert space illustrated by examples in quantum field theory (1,). The ma-

terial will be used later for the proof of Haag 's assumptions.

(7) (s) (~)

Nuovo (10)

R. HAAG: Phys. Rev., 112, 669 (1958). K. SY~IANZIK: Zeits. ]. Natur]ors., 9 a, 809 (1954). 0. W. GREENBERG: Anr~. Phys., 10, 158 (1961); H. L. LICHT and J. C. TOLL: Cimento, 21, 346 (1961). K. SYMANZIK: Con]erenee on Functional Integrations (Boston, 1963).

(11) Except for Nelson's theorem (loe. eit. 6) all notations and results about unbounded operators are standard and can be'found in every textbook on Hilbert space. See e.g. the books by ACHIESER and GLASS,~IANN, RIESZ an4 NAGV, or STONE.

1052 H . J . BORCHERS a n d w . ZIMMERMANN

An operator A defined on a dense subset D~ of Hi lber t space is called synl- metr ic (i.e, hernlitian) if

(7) (~b, A T ) = (Aqb, T ) , for r T c D A ,

holds for all vectors ~b and kP in D , . Postula te ]) states tha t the operator A(/) of a neut ra l field with real test function is symmetr ic in the sense of (7).

S y m m e t r y should be well dist inguished f rom self-adjointness which we de- fine now. Let A be an operator with dense domain D~ (not necessarily sym-

metric)~ then we define the adjoint operator A* as the operator with the lar- gest domain D a. sat isfying

(8) (A*q~, T) = (qb, A T ) for q)~D~., T c D a.

This operator is uniquely defined, provided D a is dense. called self-adjoint if

(9) A ~ A* , in par t icular D a ~- Da. .

An oper'~t0r A is

A self-adjoint operator is always symmetr ic . The converse is not true, bu t the following can be said: I f A is symmetr ic the relation

Aq~ = A * # , q ~ D a ,

holds for all vectors of the domain DA, but D~. as defined by (8) m a y be larger than D A.

According to the cyclieity postulate d) the operators A(]) are defined on

the domMn D. D has the p roper ty of being invar iant under A(])

~b E D implies A(]) ~b ~ D .

Since ttle field operators A(]) are in general unbounded and symmet r i c they

cannot be defined everywhere in Hi lber t space. There are, however, m a n y

ways to extend the original domain of definition to a larger subset of Hi lbe r t

space. We call an operator B with domain D B an extension of the opera tor A with domain Da (writ ten as A c B ) i f

(10) D A c D~

and

A~b = Bq9 , # ~ D a ,

for all vectors ~b in D~. One example of an extension is the adjoint of a

symmet r i c operator (7).

ON T I [ E S E L F - A D J O 1 N T N E S S OF F I E L D OPERATOB, S 1053

]n many cases one can find for an unbounded operator an extension by

<( contimfity )>. This leads to a class of oi)erators whi('h are known as closed

operators. An operator A is ('Mled closed if the relations

imply

T ~ DA, lim T,~ = ~P, l i m A T , , - - q)

T c D ~ , A T - - q~.

If an operator has closed extension then there exists a minimal closed exten-

sion. This is called the closure of A and is denoted by A. One obtains A by

taking all convergent sequences T ~ D , for which also A T . converges. Then

we define

A T = q) = l i m A T . n---> r

The existence of a closed extension guarantees that the definition AT----q}

is independent of the sequence T~ we have chosen.

For tile existence of a closure we have the following criterion due to yon

NEUStANN: Let A be a densely defined operator. Then A has a closure A if

and only if the adjoint A* has a dense domain of definition. In this case the

equation _d= (A*)* holds.

Since the operators A(]) are symmetric the adjoint A*(]) is an extension

of A(J) and densely defined. Hence we have

L e m m a 1. The operators A(J) have a closure A(J).

So far we have considered extensions of operators in Hilbert space. Some-

times it is also useful to study conversely restrictions of a given operator. Let

A be an operator with a dense domain D a of definition in Hilbert space. Let

D ' be a linear subset of Da, dense in Hilbert space

D / c D A ,

Then we define the restriction B of the operator A to the domain D' by

B~b = Aq5 for q ~ D ' ,

and D B = D' as domain of .definition.

As an example in field theory we take an arbi t rary open set G in Minkowski

space and define D o as the linear span of all vectors

(11) f2, A(g).Q , . . . , A(gl) ... A ( g . ) ~ ... ,

1054 H . J . BORCUERS a n d w . ZIMMERMANN

where the functions g~ vanish outside the set G

(11') Supp (g,) c G .

A theorem due to REEH and SCHLIEDER (12) tells us tha t the domain D o

is dense in Hi lber t space. We m a y therefore restr ict an a rb i t r a ry operator A(]) to the domain D, and denote the restr ict ion by A(])~.

Concerning the relation between the operators A(]) and A(])~ we will now prove the following theorem:

Theorem 1. A(]) and A(f)a have the same closures

A(I) =A(])~.

This s t a tement is not self-evident and is not a consequence of the fact t ha t

D o is dense in D. The result assures tha t no impor t an t informat ion gets lost by res t r ic t ing A(]) to the smaller domain Du.

Proo]: A(]) is an extension of A(/)~. F r o m this i t follows ~4(])~A(I)G. We have now to show tha t the relat ion A ( ] ) c A ( / ) a holds. Bu t equivalent to this last relation is A*(])~A*(])a. :Now let T~DA.(])a and g~(x) be test func- t ions with support contained in G. Then we have

02) (A*(])aT , A(gl).. . A(gn)[2) = (~, A(]) A(g,) ... A(g~)~)

for all g with Supp (gj(x))cG. Or equivalent ly

(12') (A*(/)~T, A(x~). . .A(xn)~) ~-- (~, A(/)A(xl) . . . A(x , )~) for xjeG.

F r o m the spec t rum condition it follows tha t both sides are boundary values of analy t ic functions in the variables xl, x2--x l , ..., xn--x._~ which are holo- morphic in the region I m x ~ e V ~, I m ( x 2 - - x ~ ) E V +, ..., I m ( x n - - x n _ , ) e V + ( V + = forward light cone). A s tandard a rgument on analy t ic funct ions implies

tha t bo th sides of (12) are identical as analy t ic functions and hence so are

thei r boundary values. This implies tha t {12') holds for all real values of x.

Hence gl, ..., g~ in (12) m a y be replaced by a rb i t ra ry test functions with com-

pac t support . By definition of the adjoint we see now tha t A*(]) must be

an extension of A*(g)~. This implies Theorem 1.

We end this Section with some fur ther known results f rom the theory of unbounded operators.

(12) H. REEH and S. SCHLIEDER: N•OVO Cimento, 22, 1051 (1961).

ON T h E S E L F - A D J O I N T N E S S OF F I E L D OP]~;RATORS 1 0 5 5

L e m m a 2. The closure of a symmetric operator A with dense domain D~

is self-adoint if and only if for any nonreal number ;t the set (A - - ;t)D A is dense

in Hilbert space.

The following theorem (6) states a sufficient condition for self-adjointness

making use of the concept of analytic vectors ,~s defined in Section 2.

Nelson 's theorem. If A is a symmetric operator with dense domain of

definition D~ and if there exists a dense subset D ~ of analytic vectors

for A then :4 is self-adjoint.

4. - Se l f -adjointness of f ie ld operators.

Throughout this Section we assume the validity of the postulates a)-g).

We now turn to the proof of the main theorem.

Theorem 2. The closure A(/) of the fiehl operator A(]) is self-ndjoint if ]

is an arbi t rary real test function with compact support.

Proof. According to Nelson's theorem it suffices to show that A(/) has

dense set of analytic vectors. Let F be the support of ] and choose arbitrari ly

an open set G in Minkowski space which is spacelike separated from F. We

form the domain D~; which is defined as the linear span of the vectors

(]3) ~ = A(gl) ... A(g~)g2 , q - O, 1, 2, ...,

where gl, ..., gq have support contained in G. D a is dense in Hilbert space

(REEg-SCHLI~:DER). If tile vectors (13) are analyt ic for A(]) then this is also

true for every vector in D a and Nelson's theorem can be applied.

We will consider now an arbi t rary vector ~ of the fornl (13) and prove

that it is indeed an analytic vector of A(]). Using local eonmmtat iv i ty we

obtain the inequali ty

[ ] A " ( I ) A ( g , ) ... A(g~)-Q]I~ = (A~ '~ ( I )D , ~ t ' ) <

< ~/<A~'(/)>o II ~11 < V(4n)~ 4 = II~/I

where

= A(gq) ... A(gl) A(g,) ... A (gr

(because of (3)),

Hence the power series

Z n

1056

is nmjorized by

If. J . BORCHERS &n~ W. ZIMMERMANN

~-o n] ~

which tins the radius of convergence

Hence ~b is an analyt ic vector. vectors and A(f) is self-adjoint.

Theorem 2 permits to (~ diagonMize >> the operators A(]). of the spectral theorem we have the spectral decomposition

1 r=~-aar ~ 0 .

A(]), therefore, has a dense set of analyt ic

As consequence

If tile supports of the test functions J and g are spacelike separated, local com-

muta t iv i ty states tha t

(14)

for all vectors of D. A(g) we have the

A(]) A(g) q5 = A(g) A(g) q~ , ~b~D,

Concerning the commuta t iv i ty of the closures )1(]) und

Theorem 3. If f and g are test functions with spucelike separated sup- ports, the spectral projections of A(/) and :4Tgi commute with each other

(15) E(/, #) E(g, 4) = E(g, ,~) E(f, It).

t'roo/:

i) Since A(]i and A(g) are self-adjoint the resolvents

R(], #) = (A(I) - - It)-1, Imit ~ 0,

R ( g , 4) = ( A ( g ) - - 4) -1 , Im 4 ~ 0 ,

exist and are bounded for Ira It#=0, In l4~ :0 . Sufficient for (15) is tha t the

resolvents commute

(16) R(], It) R(g, 4) --= R(g, 4) R(f, it), I m # v a 0 , I m 4 r

+co

--co

ON T H E S E L F - A D J O I N T N E S S O F F I E L D O P E R A T O R S 1 0 5 7

For all vectors r in D we have

(A(g) - - 2 ) ( A ( f ) - - /~)qb = (A( / ) - - # ) ( A ( g ) - - ~)q5

because of (14). This implies

R ( / , ~) R(g, ~ )T = R(g, ~) Rq , ~ ) T

for all vectors T which can be wri t ten in the form

~J = (A( f ) - - /~ ) (d (g) - - ~)q~ , r e P .

Hence the resolvents commute for all vectors • of the domain

(17) Q =- (A(]) - - # ) ( A ( g ) - - ~ ) P .

I f (2 is dense (16) follows f rom (14) (since the resolvents are bounded) and the

theorem is proved.

ii) In order to prove tha t Q is dense we consider the linear span D a of

a.ll vectors

~Q, A(g,)~Q, A(gl) A ( g 2 ) 9 , . . . ,

where the supports of the g~ are contained in the support G of g. We form the domains

D ' = (A(g) - - ~)D~ ,

D " = (A(]) - - # ) ( A ( g ) - - ~)D~ = (A(]) - - f f ) D ' .

D ~ c D implies D " c Q. Therefore, if D" is dense in Hi lber t space then also is Q.

iii) D ' is dense.

Let A(g)a be the restr ic t ion of A(g) to Da, then A(g),~----A(g) (Theorem 1), hence A(g)a is self-adjoint (Theorem 2) and

D ' = (A(g) - - 2 ) P a~- ( d ( g ) o - - ~ ) P a

is dense in Hi lber t space (Lemma 2).

Final ly:

iv) D" is dense.

Consider

%

D"~- (A(])--/~)D'.

68 - I l Nuovo Cimento.

1 0 5 8 I t . J . BORCHERS and W. ZIMMS, RMANN

D' consists of all vectors

(A(g) - - ~)~b, ~ e D a.

Since G is spacelike separated from the support F of 1, every vector ~5~ D a is an analytic vector of A(]) (see proof of Theorem 2). How D ' = (A(g)--2)Dqc D o.

Hence all vectors of D ' are analytic vectors for A(]). Let now C be the re-

striction of A(]) to D' . Since C is symmetr ic and has a dense set D ' of anal-

yt ic vectors, the closure C of C must be self-adjoint (Nelson's theorem).

Hence

D"= (A(]) - - / t ) D ' = (C - - #)D'

is dense in Hilbert space by Lemma 2. This completes the proof of the

theorem.

Theorems 2 and 3 allow one to construct local rings of bounded operators.

To do this we take for every open domain G all operators A(]) with support f

contained in G, and their spectral resolutions E(/, 2). We define the local r ing

Ra as the yon Neumann algebra generated by all E(], ~). Theorem 3 gua-

rantees: If G and /v are spaeelike separated then the rings Rq and R e com-

mute with each other.

5. - E x a m p l e .

When we drop the assumption tha t the vacuum state is an analyt ic vector

for A(]) then we cannot draw the conclusion from the last Section. I n this

case we only know that A(]) is for real ] a symmetr ic operator. We want to give a very simple example to show tha t our condition for

the self-adjointness cannot be improved without making fur ther assmaptions.

To this end we assume tha t the representation of the Lorentz-group is

trivial. In this case every vector is a vacuum state. In such a theory there

exists only one operator and A(])----A(dx](x). Since one state f2 has to be

a cyclic state the Hilbert space is spanned by the vectors f2, A~2, A:f2, .... Since we assume that A(x) = A is a real field we have that A is a symmetr ic

operator.

I f now the sum Z n

has a finite radius of convergence then (A~}o = O(Cnn'O, but there exists an

example (13) of a symmetric operator with a cyclic vector Q which is not self-

(13) H. HAMBURGER: Math. Zeits., 4, 186 (1919).

ON THE S E L F - A D J O I N T N E S S OF F I E L D OPERATORS 1059

ndjoint and sutisfies the condition

with s arbi t rar i ly small.

(A'9o = O(n "(~+~))

The authors would like to thank Prof. K. O. FRIEDRICHS for reading the manuscr ip t and m a n y valuable comments . One of us (H. J . B.) would like to t hank Prof. B. Zu)IINO for his kind hospi ta l i ty and the U. S. Educa~tional Commission for a Fulbr ight t ravel grant .

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

Si formula un nuovo postulato della teoria quantist ica dei campi che implic~ che: 1) gli operatori di campo distesi sono (essenzialmente) autoaggiunti ; 2) gli operatori di proiezione spettral i associ~ti commut~no per distanze spaziali. Questi r isul tat i per- mettono di costruire anelli locali di operatori di campo nel senso di Haag ed Araki.

(*) T r a d u z l o n e a cura della Redaz ione .