On a construction of the maximal additive field of v. Neumann algebras in quantum field theory

Vol. 4 (1973)

REPORTS

ON MATHEMATICAL

No. 3

PHYSICS

ON A CONSTRUCTION OF THE MAXIMAL ADDITIVE FIELD OF v. NEUMANN ALGEBRAS IN QUANTUM FIELD THEORY

B. PAWLIK and P. SADOWSKI Department

of Mathematical

Methods

of Physics, Warsaw University,

(Received

Warsaw, Poland

May 15, 1972)

The construction presented below shows how an algebra without the additive property (called shortly a non-additive algebra) can be reduced to an additive algebra. The algebra obtained this way is the largest additive algebra contained in the original nonadditive algebra. Additivity is one of the features postulated for the observable algebra in Haag’s field theory (algebras of this type, belonging to a special class, are called by Haag kinematical ones). From another way it is well known that there are many algebras without such property (e.g. the algebra of fermions).

1. Introduction

Let us consider the algebraic formulation of the quantum field theory introduced in [2]. To any open subset 0, of the Minkowski space M a v. Neumann algebra of bounded operators a(O) acting in the Hilbert space H is assigned. An algebra &( .) is called additive if and only if for any family (O,),,. of open subsets of the Minkowski space, &‘(u 0,) is the smallest v. Neumann algebra containing all agebras &(c!?~). More exOL actly,

where (e)’ denotes the commutant of the set. We ‘assume that the algebras JZZ( .) have the following property:

If Oc&, then d(ol,)c.d((02)

for any open subsets U1, 0, of M.

2. Main theorem Let 0 be an open subset of the Minkowski space and let {U#},En be any open covering of 0. Let us introduce the following operation acting in the set of all v. Neumann algebras (U d (O,)}“. 5F& (0) : = n W&en

a

224

B. PAWLIK

and P. SADOWSKI

For any open Oc M we have THEOREM

additiveoF;d(.)=d(.).

d(.)

1.

Proofi =- This results simply from the definition G For any covering (~9~)~E,,, of 0

of F.

but

Assume

that a(

of v. Neumann Let

,) is the non-additive

algebra.

We define the following

transfinite

sequence

algebras: -c4, (0): =9-d(0)

and for any non-limit

ordinal

CC -c4,+r(0):=~~02,(O).

If 1 is limit ordinal,

then dA(0):=

Notice

that for any ordinal

The next theorem THEOREM

2.

n&=(o). a-zL

CEand p

asserts the existence

of the minimal

element _.of the sequence.

There exists such index S that for any y>6

~a(-)=~ey(*) Proof:

Assume

the sequence

is not stationary.

Notice

that for anyrset

CI and index tl

1?+&(8))=c,

where nt( .) denotes

the cardinal

number

or the power of the set and c continuum.

{AI> cd, m ({W)in

(1)

(0) cB

Because

(H),

(da (0)) < llt (B

03) ,

iit({U})=nt(B(H))=c, where B(H) denotes for CC
where K, denotes

the set of all bounded

operators

the power of the set of natural

acting in the Hilbert

numbers.

space H. Thus,

MAXIMAL

We can assume without

ADDITIVE

FIELD

OF V. NEUMANN

loss of generality

in (JZ?, (8) -dp

ALGEBRAS

225

that

(0)) = k ,

k-natural

number.

Note that o-first

limit ordinal 11t(d((O)-d,,(O))=K,X,=K(),

but

The last equation is contradictory to (1). Q.E.D. It is seen from Theorem 1 that d&(O) is additive. seen to be the largest one.

The algebra

obtained

this way is

The authors are very grateful to Professor K. Maurin for his grand encouragement in this work and to Doctors K. Napiorkowski and S. Woronowicz for their valuable discussions. REFERENCES [2] Haag, R., and B. Schroer, J. Math. Physics 3 (1962), 248. [l] Araki, H., Helvetica Physica Acta 36 (1963), 132.