the vacuum state in quantum field theory

IL NUOVO CIMENTO VOL. XXIX, N. 1 1 o Luglio 1963

The Vacuum State in Quantum FieId Theory (*).

H. J. BORCHERS (**), R. I-IAAG and B. SCHROEIr (%*)

Department o] Physics, University o] Illinois - Urbana, Ill.

(ricevuto il 3 Gennaio 1963)

S u m m a r y . - - We wish to show that in a local quantum field theory which describes zero-mass particles the existence of a vacuum state is neither derivable from nor contradicted by the field equations and commutation relations which define the theory. It is an independent postulate which can be added for convenience or left off without changing the essential physical content of the theory.

1 . - I n t r o d u ~ l o n .

In all studies of quan tum field theory it is assumed that there exists a dis-

t inct state (the vacuum state) which is invariant under all translations and

Lorentz transformations. One might be inclined to question whether this is

reasonable assumption if the theory describes, amon~ other things, particles

of zero rest mass. From the physical point of view, one might argue tha t in

order to distinguish the vacuum from a state with one or several zero-mass

particles with very low momenta one would need a Geiger counter which is

moved with almost the velocity of light. Therefore a sharp distinction between

the vacuum and the neighboring low lying states seems experimentally impos-

sible. Mathematical ly one finds already in the simple case of a free Klein-

Gordon field a phenomenon tha t corresponds in some way to the above remark:

I f the mass is different from zero, then the Hilbert space of physical states is

uniquely characterized by the commutat ion relations and equations of motion

of the field operators and by the requirement tha t no negative energies shall

occur. This uniqueness s tatement fails, however, if the mass is zero. In tha t

case one can find many inequivalent solutions; i.e. the theory is not deter-

(') This research was supported in part by the National Science Foundation. ('*) Present address: Physics Department, New York University, New York. ('.') Present address: Institut fiir theoretische Physik der Univerit~t, Hamburg.

THE V A C U U M STATE IN Q U A N T U M F I E L D T H E O R Y 149

mined by the commuta t ion relations, equations of motion and the requirde

posit ivity of the energy (Section 2). One possible solution is, of course, again

Fock space. This is the only one which has a vacuum state. The others do not have any translationally invariant vectors. I t turns out, however, tha t

this ambigui ty is not very serious. While the various solutions differ in their

global behavior, they are essentially equivalent in their predictions for local

experiments. In Section 3 we shall s tudy this phenomenon in a more general

context. The discussion there seems to indicate tha t the assumption of a vacuum

state is always permissible: if a local field theory does not have a vacuum state

then it is possible to construct a theory with vacuum state from it. This

construction is unique and its physical meaning is simple. Consider as an

example the manifold of all states with charge one in electrodynamics. This

space of states with charge one is t ransformed into itself by the algebra of

observables and it clearly contains no normalizable state which is invariant

under translations. Hence it gives us an example of a local field theory without

vacuum. Bu t if we pick out an arbi t rary state vector T and apply to it a trans-

lation by a sufficiently large distance a, then we get a state which is practically

equivalent to the vacuum as far as measurements of local quantities are con-

cerned. The expectation value of any observable which can be measured

within the walls of the laboratory, taken in the state T a = exp [ - - i P a ] T is

almost equal to the vacuum expectation value of the observable if a is large

enough. In tha t way all physical statements pertaining to states of charge

zero can be obtained if we know the physics of the states with charge one.

The result of our discussion in Section 3 can then be summarized by saying

that the example just mentioned seems to represent the typical situation of a local field theory without vacuum.

2. - Example: system of noninteracting bosons.

We start with the commutat ion relations for the creation and destruction operators

(1) [a(k), at(k')] = 5 ( k - k') ;

and the Hamil tonian

(2)

where o~ is a function of k.

(3)

[a(k), a(k')] = [at(k), a+(k')] = 0

H -=j~ a*(k) a(k) dak,

The linear momentum is ~.dven by

P =fka+(k) a(k) d~k .

150 It . J . BORCHERS, R. HAAG a n d B. SCHROER

I f we pu t

(~) ~ = v / ~ + m 2 ,

the eqs. (1) and (2) are equivalent to the commuta t ion relations and field equa-

t ion of a Klein-Gordon field. Specifically, one defines the field operators in the usual way b y

t f dSk (5) A ( x ) = (2u)- (a(k) exp [ i ( k x -- wt)] -~ at(k) exp [-- i ( k x -- cot)]) 2-~

Then eqs. (1)-(4) are replaced by

(6) ([] - t o g a = 0 ,

(7) [A(x) , A(y)] ---- i d ( x - y ) ,

and the requi rement t ha t a space- t ime t rans la t ion b y the a-vector a is represented in the t t i lber t space b y a un i t a ry opera tor U(a) which t ransforms the field according to

(8) U ( a ) A ( x ) U-l(a) : A ( x - ~ a) .

I f we take the form (2) for the Hami l ton ian at its face value, then we con-

clude t ha t H is a posit ive opera tor ; no negat ive energies can appear . This is, however, a somewhat t r i cky point since (2) can only be regarded as a heuristic

definition of H which allows a certain amoun t of f reedom in its rigorous inter- pre ta t ion (*).

(') Various investigations of the uniqueness of the solution to the scheme (1). (2), (3) exist. An incomplete but representative list is given in refs. (1-4). In the work of ARAKI (which refers to more complicated Hamiltonians) two additional conditions were used, namely the positivity of the energy and the existence of a normalizable vacuum state (discrete eigenstate of H). SHAL~ and SEGAL abandoned the positivity condition but kept the assumption of a normalizable invariant state. Here we shall do the converse, i.e., keep only the positivity condition. I t was pointed out to us by B. ZUMINO that the essential content of our first theorem is contained already in Friedrich's book (ref. (i)). Since the terminology and proof technique is somewhat different there we have kept this theorem nevertheless in the present paper.

(1) K. 0. FRIEDRICHS: Mathematical Aspects o/ the Quantum Theory o] Fields (New York, 1953).

(2) H. ARAKI: Journ. Math. Phys., l , 492 (1960). (3) D. SHALE: Thesis University of Chicago, 1961. (4) I. E. SEGAL: The characterization o] the physical vacuum, to be published.

T H E V A C U U M STATE IN Q U A N T U M F I E L D T H E O R Y 151

We want to exhibi t the difference between the cases m :/: 0 and m = 0. Actual ly our a rgument does not at all rely on the relat ivist ic invar iance of

the theol T. So we could just as well take for ~o another nonnegat ive funct ion

of k t han t ha t given in eq. (4). The re levant dist inction for our purpose is whether to a t ta ins the value zero or whether it is separa ted f rom zero b y a finite gap. For simplici ty of lagnuage we shall, however , st ick to eq. (4).

The s t andard representa t ion (Fock representat ion) of the scheme defined b y eqs. (1), (2), (3) is obta ined if there exists a (normalizable) vec tor in Hi lber t

space, say To, which is annihi lated b y all the a(]e):

(9) a(k)~lo=O for all k ; (To, To) = 1 .

Applying the creation operators at(k) repea ted ly to this vector To one gene-

ra tes the Hi lber t space. The scalar product between any two vectors can be

worked out b y means of the commuta t ion relations (1) and eq. (9). I t also

follows f rom (3) t ha t To has l inear m o m e n t u m zero, i.e., it is invar ian t under t ranslat ions in space. This la t ter p roper ty we shall take always as the defi-

nit ion of the v a c u u m state. The question now is whether the existence of a vector To satisfying (9)

can a l ready be derived f rom (1) and (2) (and the posi t iv i ty of H) or whether

there are inequivalent a l ternat ives to the Fock representat ion. I t is easy to

see t ha t for m=/=0 (9) follows indeed f rom (1) and (2), bu t t ha t this is not the case for m = 0. The posi t iv i ty of H gives us the restr ict ion for the energy

spec t rum

E o ~ < E < c~ ; Eo>~0.

I r respec t ive of the s t ructure of the spec t rum (whether Eo is a point eigenvalue or the lower end of a cont inuum) we can choose an arbi t rar i ly small number and find a normalizable s ta te T which has exac t ly zero probabi l i ty for an

energy outside the in terval

E o < E < Eo + s .

I f one applies a destruct ion operator

a(/) ~-/a(k) /(k) d3k ,

on this s tate then one gets according to (1), (2) a s ta te whose energy is re-

str icted to the interval

(10)

152 H . J . BORCHERS, R. HAAG and B. SCHROER

where I k l i s the smallest value of I k t i n the support of the funct ion J. I f m s 0 we need only choose s < m and find tha t T is annihi lated b y all a(J), since then the inequalities (10) become contradictory. Hence T satisfies eq. (9). I f m = 0 we can only conclude t ha t kg is annihi lated by those operators a(]) for which the suppor t of the funct ion ] s tays away from the origin by a distance greater than s. Although we m a y choose e as small as we like we cannot

obta in any information about the l imit e = 0 b y this method.

One has, however, the following theorem:

Theorem I. - The commuta t ion relations and field equations of a free Bose

field with zero mass allow an infinite va r i e ty of inequivalent irreducible solu-

tions, all wi th nonnegat ive energy. Examples of solutions for which the Hami l - tonian (2) and the hnear m o m e n t u m (3) have purely continuous spec t rum

(no normMizable vacuum state exists) are obtainable f rom the Fock repre-

sentat ion in the following way. Let b+(k), b(k) be a system of creation and destruct ion operators in Fock

space and ](k) a numerical funct ion which has a singulari ty at the origin such

t h a t

(11) f []]2d3lc= c~ but f ]k] I]]2dak < c~.

P u t

(12) a(k) = b(k) - - J(k) ; a+(k) = bt(k) - - / * ( k ) .

Two such solutions are uni tar i ly equivalent if the difference between the two

functions J is square integrable. Le t us add one side r emark about Lorentz invariance. The defining equa-

tions of the theory are manifes t ly Lorentz invar ian t in the form (5), (6), (7).

B u t in none of the solutions given by (5), (12) with / satisfying (11) can we

find a un i t a ry operator U(A) representing the Lorentz t ransformat ion A such

tha t

U(A)A(x) U-I(A) = A(Ax) .

We have here, therefore, a (rather trivial) example of <~ breaking of a sym- m e t r y ~), a phenomenon frequencly occttring in sys tems with infinitely m a n y

degrees of f reedom: The invariance propert ies of the basic equations ar no

longer found in the Hi lber t space of s tates corresponding to a solution of type

{12) unless ] is square integrable. Le t us now call an operator <~ quasilocal ~> if it is such a functional of the


field opera tors A(x) t h a t on ly space- t ime points x in some finite region are

involved (*). Then we have

Theorem I I . - Let T be a n y s ta te and Q any <~ quasilocal ope ra to r )>; then

(13) l im (T~ [(2]T~> = Eo(Q)

exists, is i ndependen t of T and independen t of the choice of representa t ion ,

i.e., it is the same for ] = 0 and for a r b i t r a r y ] sa t is fying ( l l ) .

I n eq. (13) we have abbrev ia t ed

(14) T x = e x p [ - - i P . x ] T .

I f we t ake ] = 0 in (8), we come back to the usual F o c k rep resen ta t ion

of a(k), at(k) . Then Eo(Q) is the v a c u u m expec ta t ion va lue of Q. Therefore

Theorem I I tells us t h a t in a n y represen ta t ion we can find s ta tes which are

p rac t i ca l ly indis t inguishable f rom the v a c u u m state of the F o e k represen ta t ion

as long as we res t r ic t ourselves to quasiloeal onbservables . The inequivalence

of the different represen ta t ions can therefore show itself only in global meas-

u rements and is thus of l i t t le physica l interest .

This gives a s o m e w h a t unexpec t ed answer to the ques t ion raised in the

i n t roduc t i on : is the a s sumpt ion of a v a c u u m s ta te reasonable or no t? We

see t h a t in the present example the a s sumpt ion of a v a c u u m is i ndependen t

f rom the equa t ions which define the theory . We can adop t it or reject it

w i t hou t ge t t i ng a con t rad ic t ion and w i t hou t even changing the re levant physical predic t ions of the theory .

To prove the theorems let us use the fol lowing n o t a t i o n : I f L is a func t ion

of k and K = K ( k l , k2) a func t ion of two a r g u m e n t s (kernel) then we wri te

(15) (L1, L2) = f L l ( k ) L 2 ( k ) d3k; (L, K L ) = r E ( k 1 ) K ( k t , k2)L(k2) d3],'~ d3k2 .

(') Actually the following theorem and similar ones are valid for a much broader class of operators. For instance, the operator

Q = f / ( x I . . . . . xn)A(xl) .... 4 (xn) ddxl ... ddx,~ ,

would be called quasilocal according to the definition above if the function ] vanishes exactly outside of some finite space-time region. But theorem II holds already if ] vanishes asymptotically for large x, sufficiently rapidly to make f](x 1 .. . . . x,) (14xl ... d~xn finite. The term ((asymptotically local )) has been suggested for such an operator but this is perhaps not a very fortunate piece of semantics.

154 n . J . BORCHERS, R. HAAG and s. SCHROER

Le t ]0) be the Fock v a c u u m and b(k ) , b~(k) the o rd inary destruct ion and creat ion operators. In other words, we have

(16) b ( k ) 10) = 0

for all k (*).

I t is known tha t the states

(17) I h> ---- exp [(h, bt)] [0>

belong to Fock space provided t ha t h is a square integrable function. The

scwlar product between two such states is

(18) ( h ' l h ) ~- exp [(h'*, h)] .

I t is also known tha t these states provide a complete basis (nonorthogonal, of course) in Fock space. By this we mean t h a t any vector can be approxi-

m a t e d to an a rb i t ra ry degree of precision by a linear combinat ion of a finite n u m b e r of s tates ]h). Therefore, in order to define a un i ta ry operator in Foek

space i t is sufficient to define it oa the vectors ]h) and to ensure t h a t i t con-

serves the scalar products and ha, s an inverse. We shall do this for the t ime-

t rans la t ion operators

(19) U(t) = exp [ iHt ] ,

with H given b y (2) and (12). To mot iva te the definition of U(t) which will be given below (eq. (26)), we shall use the following formal identities:

e -~ F ( b , b~)e a -~ F ( b ' , b ' t) , (20)

with

b ' (k ) = e -~b(k) e";

I f A is a l inear form in b and bt:

(21)

t h in

(2~')

b t ' ( k ) -= e - ~ b t ( k ) e ~ .

A ~ - ( L I , b ) ~ (s t)

e-~b(k) e A = b(k) + L~(k); e-Abe(k) e ~ ---- b t (k) - - L l ( k ) .

(') For clarity let us emphasize again that the (( physical vacuum i) To, if it exists, would be characterized by the property P~o=0, where P is given by (3) and (12). The ((Fock vacuum,~ 10> is characterized by (16). Obviously ~Oor ]0> unless ](k)=0.

(26)

wi th

(26')

T H E VACUU.~[ STATE IN Q U A N T U M F I E L D T H E O R Y 155

If A is a bi l inear fo rm

(22) A = (b t, K b ) ,

t h e n

(22') e-A(L, b)e A = (L, eKb); e-A(L, bt)e ~ = (b*, e-KL) .

Final ly , we need the fo rmula

(23) exp [(L1, b) § (L2, b*)] = e x p [ l ( L x , L2)] exp[(L2, br exp [(L, , b)] .

This col lect ion of formulas is now used as follows. F i rs t one sees t h a t p u t t i n g L 2 ( k ) = - - / ( k ) , L d k ) = ]*(k) in (21), (21') we have the t r ans fo rma t ion f rom the sys tem b, b* to a, a* as defined by (12). Therefore , according to (20)

(24) U(t) = exp [iHt] = exp [ - - ( ]* , b) + (/, br exp [i(b*, o~b)t].

�9 exp[( /* , b) - - (/, b~)] (*). W e work out

U(t) Ih> = U(t) exp[ (h , b*)] ]0>,

b y using (22) to shift the exponen t i a l of (b*, cob) unt i l i t s tands immed ia t e ly in f ront of [0> where i t can be omi t ted . Thus we get

(25) U(t)]h} = exp [ - - (]:~, b) + (/, bt)] �9

�9 exp [(]*, exp [ - - icot]b) - - (b r exp [icot]])] exp [(b*, exp [icot] h)] ]0>.

Decompos ing the l inear exponen t i a l according to (23) and shif t ing the des- t ruc t ion pa r t s to the r ight we get f inal ly

U(t) i h> =- exp IN] ]](1 - - exp [iwt]) -? exp [icot] h>,

N = - - (/*, (1 - - exp[io, t ] ) ( / - - h)) .

Since for any t the func t ion 1 - - e x p [ i c o t ] has a zero of first order a t

I k ] = 0, i t is seen t h a t all quant i t i es appear ing on the r igh t -hand side of (26)

are well defined and finite, p rov ided t h a t ~/[ k[](k) is square in tegrable (which was assumed in Theo rem I). I t is immed ia t e ly checked wi th the help of (18) t h a t U(t) as defined b y (26) leaves the scalar p roduc t s unchanged and t h a t i t

(') Here r is, of course, the kernel ~(k~, k2)= [kl [6(kl--k2).

1 5 6 H . g . B O R C H E R S , R. HAAG and g . SCHROER

has an inverse, namely U(--t) . Therefore it is unitary. We also check

(27) U(t~) U(t2) = U(t~ 4- t~) ; U(O) =- 1 .

Therefore we can write

(28) U(t) = exp [iHt]

and definite s a self-adjoint operator H. That this operator coincides

with the naive definition (2) on a dense set of states is verified by differen-

t iation of (26).

In the same way we define the linear momentum operators. We only have

to replace hot in (26) by - - i k x to get the uni tary operators representing a

space translation

(29) U(x) = exp [-- i Px] .

Note tha t the condition of square integrability of the function v / i k ] / ( k ) is

also necessary for this purpose. Therefore, even if we take a different energy-

momentum relation for a single particle, for instance, if we put

(30) (o = k ~

we can still not use functions ] in (21) which are more singular. I f we wanted

only to define the t tamil tonian it would be sufficient to have v /m/ square

integrable. But if we also want the linear momenta we must have v~lkt/

square integrable as well. The last par t of Theorem I which remains to be proved is the posit ivity

of H. This is most easily tested by allowing t to become complex in (26).

I n particular, we c~n replace t by i~ and let the real number v tend towards 4- c~.

I f the norm of U(i~) lh} remains bounded in this limit for all [h} then the

spectrum of H contains no negative Values. According to (26) and (28) we find

(31) ]tU(iv)lh>[[~=exp[f[]h]~--L/--h]'(1--exp[--2~v] ]dSk I .

I t ] itself is not square integrable then as v --> 4- c~ the exponent approaches

- - c ~ so tha t

(32) l i m II u(i~)lh>a]2 = 0 for all h. T--~co

This shows tha t the energy spectrum is positive and tha t there is no discrete

eigenstate with zero energy. If / were square integrable, then the vector I I>


would be a discrete eigenstate of H to zero eigenvalue and we would have the s tandard representation of the system a(k), a+(k).

Concerning Theorem II , let us calculate, for instance

(33) <h[U(x) exp [(L~, a +) ~- (L1, a)] U-l(x) lh) =

= exp [ - - ((/* -- h*), exp[ - - i kx]L2) - - (L,, exp [i kx] (/ - - h)) +

+ (h*, h) + �89 L1)].

This corresponds to the choice of

Q = exp [L2, a +] exp [L1, a]

in (13). If Q is quasilocal then the functions L~(k) and Ll(k) can have no worse singularity than C.Ik[ -�89 We see this from (5), since

with

A(x)](x) d'x = (g, a) ~- (h, a+),

g(k) - ?(k, l kl) h(k) - ? ( - k, - I k l ) 24k! ' 21kl

If / has a finite support in x-space then f is regular in k-space. I f L has a [k[ - t singularity the functions ] .L and a ]ortiori k . L are

still absolutely integrable at the origin in k-space so tha t by the Riemann- Lebesgue Lemma

f [ ] [ exp [ - ikx] Ll.~(k) d3k and [ h ]exp [-- ikx]L~.~(k) dak,

vanish in the limit Ix[--> co. Hence the r ight-hand side of (33) has the limit exp[(h*, h)§189 L1)]. Dividing by the norm square of the vector [h> we see then tha t in the notation of (13)

(34) Eo(Q) : exp [�89 for Q -- exp [(L2, a+)+ (L1, a)]

which is indeed independent of the state [h> and of the function ] which char- acterizes the representation. We note tha t

(35) Eo(exp [(L2, a +) + (/51, a)]) = <0 ]exp [(L~, b +) -t- (L1, b)] 10> �9

158 H . J . BORCHERS, R. tIAAG a n d B. SCHROER

Fronl (34) we can deduce the E0-value of any product of factors a, a + by 7r

tionM differentiation with respect to the functions L2 and L~. Thus (34) is

sufficient to determine Eo(Q) for any quasilocM operator Q. I f Q = F ( a , a ~)

then we see h 'om (35) tha t

(36) Eo(F(e*, a*)) = 4.0 IF(b, b ~) 10>.

3 . - G e n e r a l c a s e .

We assume tha t we are given a quan tum theory with the following features:

1) A translat ion in space-time by the vector a --~ (a, ~) is represented by

the un i ta ry operator U(a) in the Hi lber t space of physical states. We wri te

(37) U(a) -- exp [ - - i ( P a - - H v ) ] ,

~nd call P the operator of linear momen tum, H the energy operator.

2) There exists an algebra ~ of << quasi-locM operators ~> which contains essentially all observables.

This second assumpt ion needs some explanation. In tu i t ive ly speaking, the t e rm << quasi-locM observable >> shall mean tha t the quan t i ty can be measured within some finite region of space-time. I t does not ma t t e r how large this region is as long as it is not infinitely extended. Mathemat ica l ly there are two a l te rnat ive ways to arrive a t a definition of ~ . The first one connects ~ wi th

the field operators of a local field theory. In such a theory one m a y associate with every region B of space-t ime an algebra of observables R B as explained in earlier papers (5). Then we m a y define ~ as the union of all those R 8 for which the region B is finitely extended (compact). In other words, Q E ~

means t h a t there exists a compac t region B such tha t Q ~ R B. The other a l te rna t ive is to define ~ abs t rac t ly by its re levant properties. We shall need

the following:

i) ~ is an algebra. This means tha t if Q and Q' belong to ~ and if

is ~ complex number then Q*, aQ, Q + Q ' , Q.Q' also belong to ~ . For sim-

plicity we shall take ~ to consist only of bounded operators a l though this

restr ict ion can have only very little relevance to our problem and should be

regarded merely as a technical device of el iminating pathologies which other-

wise would need an extensive discussion.

(~) R. HAAG: Colloque I~d. sur les probl~mes math. de la th~orie q~tatttique des champs Lille 1957, ed. by CNRS (Paris, 1959); R. HAAG and B. SCHROEZ: Jourl~. Math. Phys., 3, 248 (1962).

T I I E V A C U U M S T A T E I N Q U A N T U M F I E L D T H E O R Y 159

ii) ~ is stable unde r finite t ranslat ions: F rom Q 6 ~ it follows tha~

Q(x) ~ ~ where

(38) Q(x) = U(x) Q U-~(x) .

iii) For ever pair Q, Q' f rom ~ we have

(39) lira [Q(x, ~), Q'] o . Ixl~--> co

iv) The weak closure of ~ tzives the algebra of all bounded operators

in the Hi lber t space. This s t a tement was mean t above b y the phrase tha t

shall contain essential ly all observables.

I f ~ is defined concretely in the first ment ioned way, then it is immedia t e ly clear tha t all the propert ies listed above are realized. In the case of super-

selection rule ~ f has, of course, to be in te rpre ted as a single coherent subspace. We state now the following theorem.

Theorem I I I . - For any quasilocal Q we can find a numerical function Fr (x)

such tha t the difference Q(x)- -F~(x) converges weakly towards zero as Ix]-->cr The funct ion Fq(x) has the p roper ty

(40) ~ ( x + a) - F~(x) ~ 0 as l* I ~ ~ o .

Before proving this theorem let us comment on its simple physical meaning. An equivalent formulat ion would be

Theorem I I I ' . - For any two quasiloeal operators Q1 and Q~ and any s ta te T we have

(41) <TIQiQ~(x)[T > <T[Q, IT> <T!Q2(x)!T > -~ <T, T> as ] x l ~ .

In other words, two measurements in far separated regions are s ta t is t ical ly independent . I t is t r ivial to derive I I I ' f rom I I I . In the opposite direction

we have to make use of iv) which tells us t ha t by the appl icat ion of a quasi-

local operator on T we can get a rb i t rar i ly close to any vector r (*).

Therefore (41) can also be wr i t ten

(42) <r IQ(x) IT> ~ <q~' T> <T,~Q(x)[T> = F~(x). <r T>

(*) It is ~ well known theorem that for an angebra of bounded operators tile we-~k closm'e ~nd the strong closure are identical.

160 H . J . B O R C H E R 8 , R . H A A G and B . S C H R O E R

with

(42') F . ( x ) - <TIQ(x) IT> <T, T>

Interchanging T and q~, replacing Q by Q* ~nd taking the complex conjugate we find

and hence

(43)

<~lQ(x) IT> --+ F~(x) <~, ~>

/%(x) - - Fo(x) -+ 0 .

This fact, tha t the r ight-hand side of (42') becomes independent of the state T in the limit of large Ix I, is the content of Theorem III . In particular, it implies also (40), since

F~,(x + a) = F~, (x) with ~ ---- U - I ( a ) T .

Equat ion (40), unfortunately, is not quite strong enough to imply tha t F(x) approaches a constant as l xl--~ oo. The alternative possibility is tha t F(x) oscillates with a period increasing to infinity as [x[--> oo. This possibility is very pathological from the physical point of view and we want to exclude it by the

A s s u m p t i o n : lim Fq(x) exists.

Then Theorem I I I can be sharpened to

Theorem I V . - For any quasilocM operator Q the sequence Q(x, 3) converges weakly towards a multiple of the ident i ty as Ix[--> oo:

(44) weak l im Q(x 3) = ,~(Q).I . Ixl--+r

We have gained, thereby, a positive linear form over the algebra ~ , which is normalized and translationally invariant:

(45) ,~(~QI§247 2(Q*Q)>~0; 4 ( 1 ) = 1 ; ,~(Q(x))=,~(Q).

Theorem IV is the generalization of Theorem II . We can use the positive linear form ~ to construct a representation of the algebra ~ and of the translation group in a Hilbert space :If ' by means of the Gelfand construction. The translational invariance of ~ implies tha t there is a translationalIy inva- r iant state (vacuum) in 9~'.

T I I E V A C U U M S T A T E IN Q U A N T U M F I E L D T H E O R Y ] (~ l

We shall now prove Theorem i I I , or ra ther eq. (42). The relat ion of this equat ion to the other s t a tements has a l ready been discussed. We pick an a rb i t r a ry pair of s ta tes q~ a | ld T and a quasilocal operator Q. Le t P be the

project ion opera tor on T. Due to iv) we can find a self-adjoint quasilocal

opera tor Q' such tha t

where e is as small as we want . Then we have

<r = < ~ I Q ( x ) P t T > = <~ [(2(x) (2' I~> + ,+~ =

where

and

5~(x) -+ 0 as x -+ o0 .

The quanti t ies ~ and ~3 are independent of x since the norm of the operator

Q(x) is the same as tha t of (2 and they can be made arb i t rar i ly small by a suitable choice of (2'. Hence we have

< r IQ(x)1~> --+ <pqs]Q(x)IT> = (q~' T>(TJQ(x)IT> <T, T>

which is eq. (42).

There are two respects in which the present s tudy is incomplete. First , we feel t ha t the assumpt ion preceding Theorem I V mus t be derivable f rom the usual postulates of q u a n t u m field theory. Secondly, one would like to show tha t no physical informat ion is lost if one replaces thc originally given Hi lber t space J/f, which had no v a c u u m state in it by the space 5/f' which is constructed with the help of the linear form 2 and which has a v a c u u m state.

Tha t this mus t be the case is s t rongly suggested by the intui t ive discussion

and, of course, b y the consideration of special examples like t h a t in Section 2. Bu t we have not ye t found a general proof of this feature.

We would like to t hank H. ARAKI for some helpful suggestions concerning the proofs of the theorems in Section 2.

11 ~ I I Nuovo Cimento.

162 H . J . BORCHERS~ R. H A A G and B. SCHROER

N o t e a d d e d i n p r o f .

D. KASTL]~R has observed that the unsatisfactory features of our discussion in Section 3 can be overcome. Instead of making the assumption which precedes theorem IV, one can prove the following:

There is at least one sequence of points xn with ]x~ I--> ~ for n-+ c~ such that lira /~q(xn) exists for all operators Q in the algebra 2. n---~r

From this and theorem I I I it follows that one can construct at least one representation which has a translationaUy invariant state (vacuum). There remains therefore only the question of the uniqueness of the vacuum.

Concerning the (( physical equivalence ~> of two representations which are not equivalent by a uni tary transformation D . KASTLER found the following criterion.

All representations of ~ are physically equivalent if and only if the algebra is simple.

These matters will be discussed in some detail in a separate paper.

RIASSUNTO (*)

Si espone un'analisi della produzione di particelle strane nelle collisioni di protoni di 24.5 GeV/e con protoni (energia totale nel s.c.m.=6.72 GeV). Essa si basa su 50000 fotografie prese con la camera a bolle d'idrogeno da 30 cm del CERN. Si fa il eonfronto con un esperimento sulle interazioni u--p nella stessa camera a bolle. I bassi impulsi trasversali t rovati precedentemente rieevono conferma: solo gli iperoni positivi hanno impulsi trasversali elevati. Si di~nno le sezioni d'urto parziali.

(*) Traduzione a eura della Redazionr

the vacuum state in quantum field theory

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