On the independence of local algebras

REPORTS ON MATERMATICAL

Vol. 3 (1972)

No. 1

PRYSJCS

ON THE INDEPENDENCE OF LOCAL ALGEBBAS KAZIMIERZ NAPI~RKOWSKI Department

of Mathematical

Methods

of Physics,

(Received

University

of Warsaw,

Warsaw, Poland

June 15, 1971)

The correction between statistical independence of von Neumann algebras and their commutation is studied. An example of statistically independent but not commuting von Neumann algebras is given. It is shown that a stronger condition, similar to the condition of statistical independence, implies commutation of the von Neumann algebras.

Introduction The physical idea of independence of observations performed in space-like separated regions was formulated by Haag and Kastler in the language of local algebras of observables [l]. In this theory we assign a C*-algebra J& (0) to each region 0 of Minkowski space. We assume that the algebras JZZ(0) are subalgebras of a fixed C*-algebra d. States of the physical system are represented by positive linear functionals Q, on d (also called states), normed by the condition Q (1) = 1, where 2 is the unity of ~4. A positive normed linear functional on z$ (0) is called a pczrfiul sfate on 0. Two algebras d (0,) and d (0,) are called independent if for any two partial states p1 and q)2 on 0, and 02, respectively, there exists a state q on ~4 such that Pl&@l)= P1

and

&zng(02) = P2 -

The postulate of independence of observations can be formulated by assuming that if O1 and O)zare space-like separated regions then algebras .& (0,) and d (0,) are independent. As we shall see, the postulate above does not imply local commutativity (commutation of JXZ(0,) and & (O,)), which is usually assumed in quantum field theory. The aim of the present paper is to show that if one assumes independence and some additional conditions of a similar type, then local commutativity can be proved. The physical sense of the additional assumptions is, however, not clear. Let d be a C*-algebra of operators in a Hilbert space Hand let .G/, , .d2 be v. Neumann algebras di cd, i= 1,2. In the following we shall consider only states represented by density matrices, i.e. functionals of the form &4)=Tr @A), where p E L(H), pa0 and Trp=l. 1331

K. NAPIdRKOWSKl

34

DEFINITION. Algebras d, and dz are called independent if for any pair of states pl and q2 on zaI, and d,, respectively, there exists a state v, on d such that p’jdI = q1 and PI_&= 92. The following example shows that independent v. Neumann algebras not always commute.

EXAMPLE. Let H= C6, let d be the algebra of all matrices 6 x 6 and J&‘~(i= 1,2) be the algebras generated by I and Ei, where

E= 1

-100006 000000 001000 000000 000010 000000

E,=

-1 0 0 0 0 o000000 000000 000100 0000~~ oooo~+

The operators El and E2 do not commute, hence d, and &, do not commute. To prove independence, we notice that a state vi on ,c4, is uniquely determined by pi(Ei) and that 0 < pi(Ei) < 1. For any pair of real numbers a,, i= 1,2, such that O
operator

E independent

of an algebra L%if L%and the algebra generated by

INDEPENDENCE

Let li be eigenvalues and

of plo and let e, be corresponding Tr(E,

35

OF LOCAL ALGEBRAS

eigenvectors.

Then C’li= I

~10)~ C&(ei [El ei)= 1,

Tr(E,p,,)=

1, ;liZO (1)

CIZi(eiI E,e,)=O. I

(2)

Let H,, be the subspace of H spanned by the vectors ei for which il,#O. Because (et ( E,eJ
(2) iand the condition

get E2H10 = (0). The same way we obtain

hi >O we see that (non-trivial)

Ai#O implies

subspaces

(ei 1Ezei)=O.

This way we

HI 1, Ho, and Ho0 with the properties:

orthogonal (e.g. HooIHol because The subspaces Hoe, Ho,, Hlo, H,, are mutually Ho, cEzH and Hoo_L E,H). Let H’=Hoo@Hol~H1o@H,,~H. We consider the family of all subspaces Hi of H which have a decomposition H~=HOOaOHOlaOH1OaOH1la, where Hooa, HOI,, Hloa conditions as Ho,, Ho,, Hlo, and HII. Applying Zorn’s and Hlla satisfy analogous lemma we get a maximal subspace H” with this property. Let E” be the projection onto H”. It is easy to check that the operators El, E2 commute with E” and that 9J1 lEPfHand 9JzJs,a are independent. We shall show that at least one of the operators Elltr-E8tjH or Ezlcr-Er~~His trivial (0 or 1). If both are not trivial then E” is independent of 39r and gB,. This follows from the algebraic independence of pi and the algebra generated by E” (see [2], [3]). This way E” fulfills the condition of the definition of complete independence and we conclude that Using the same method as before we find a g31/(1-E”)rf and ~ZIo--E,J)H are independent. non-zero subspace H”’ c (H”)I having the decomposition H”’ = H;;d@ Hii@ H;‘; 0 H;;‘, which contradicts the maximality of H”. Thus one of the operators El or Ez must be 0 or I on (H”)I and the algebras 9Y1 and .c???~ commute on (H”)‘-. Since they commute also on H”, we have g1 c $8; and&, cd;. REFERENCES [l] Haag, R., and D. Kastler, J. Math. Phys. 5 (1969), 848. [2] Ross, H., Commun. Math. Phys. 16 (1970), 238. [3] Schlieder, S., ibid. 13 (1969). 216.