PHYSICAL REVIEW D VOLUME 8, NUMBER 8 15 OCTOBER 1973

Commutativity and Causal Independence

B. De Facio* Ames Laboratory-USAEC and Department of Physics, Iowa State University, Ames, Iowa 50010

Donald Curtis ~ a ~ l o r ' Department of Mathematics, University of Missouri-Columbia, Missouri 65201

(Received 21 June 1973)

We show that the original question concerning the relation between causal independence and local commutativity by Haag and Kastler remains open. Ekstein has given an infinite-dimensional reducible example which shows that causal independence neither implies local commutativity nor necessarily fonows from local commutativity, and we supply a finite-dimensional example of this interesting fact.

I. INTRODUCTION

One of the "axioms" o r better assumptions of local algebraic quantum field theory for a scalar field i s local commutativity, which means that for R,, R, regions (open sets with compact clo- sure) in Minkowski space which a r e spacelike with respect to each other the C* algebras %@,) and %(R,) commute elementwise. Haag and Kastlerl raised the following question on the algebraic structure of local commutativity: " For causally disjoint regions R, and R, a r e the algebras %(R,) and %(R,) essentially uncoupled?" Two regions R, and R, a r e causally disjoint if they a r e space- like with respect to one another, and local inde- pendence ,~~ o r statistical independence4 for two algebras % (R,), E(R,) (with R, and R, spacelike) means that for partial states w, on %(R,) and w, on %(R,) there exists a state w which extends w, and w,. Ekstein5 used the t e rm causal indepen- dence to refer to the normal state w which extends the normal states w,, w, on spacelike algebras.

Notation

Let 8 be a C* algebra (see ~ i x m i e r ' for the axioms of a C* algebra); B, and %, a r e sub-C* algebras of 91 on regions R,, R,. Bh is the set of al l Hermitian elements of %, and 91' i s the set of al l positive elements of a. When we consider %+ a s a cone in gh we let int(ge) denote the interior of 8' (its norm closure i s 91'). 3n i s a linear sub- space of %. The identity elements a r e written a s 1, (i = 1, 2) and the zero elements a r e written a s 0, (i = 1, 2) for the subalgebras 91, (i = 1 ,2 ) . Let X denote a complex Hilbert space with inner pro- duct (. , .). R, XR, means that regions R, and R, a r e spacelike. R,#R, means that, for a l l x E R,, and N being a suitably chosen neighborhood, one has @+N)XR,. Let a, b , c, a,, b,, and c, denote complex numbers; h, (z = 1, . . . , 6), e, and k vec-

tors in X ; w, and w, states of 8 ; and w,, w,, and w, states of XI, a,, % + .

Definitions

(i) The subalgebras %,, a, of % a r e said to be statistically independent or locally independent if, given states w, of a, and w, of %,, there exists some state w of &I that extends w, and w,.

(ii) The subalgebras a , , 8 , a r e said to be caus- ally independent if, given normal states w, on al and w, on 8,, there exists a normal state w on 8 which extends o, and w,.

(iii) The regions R, and R, a r e said to be supev- sflacelike if R, #R,.

(iv) The subalgebras 8, and %, a r e said to satisfy the Schliedev condition if for nonzero x E 8, and nonzero y E %, the product x y is nonzero.

Existing Results

(i) If R , # R, then %, and U , satisfy the Schlieder condition. Moreover, %, and %, a r e statistically independent . 3

(ii) Two algebras %, and 8, which commute ele- mentwise a r e statistically independent if and only if they satisfy the Schlieder ~ o n d i t i o n . ~

(iii) If the two algebras El and %, a r e infinite- dimensional algebras and constructed a s in Appen- dix C of Ref. 5 then, a s Ekstein showed, "causal independence neither implies nor i s implied by local c~mmutat iv i ty ."~ (Hereafter we refer to this statement a s the Ekstein statement.) In its exact form,5 there a r e two ways to evade Ekstein's ex- ample since it is an infinite dimensional and r e - ducible observable algebra. This i s that either/or finite dimensional o r irreducible observable alge- bras might fail to satisfy the Ekstein statement. In Sec. I1 we present an even simpler example of the Ekstein statement and in Sec. 111 we briefly comment on the present status of Haag and Kast- ler's original question.

B . D E F A C I O AND D O N A L D C U R T I S T A Y L O R

11. A FINITE-DIRIENSIONAL EXAMPLE OF THE EKSTEIN STATEMENT

Fi r s t of a l l one should note that a necessary con- dition for causal independence (local independence) of two algebras %, and 'kl, i s that 91, n a, i s non- trivial and has dimension one. Therefore com- mutativity of 91, and %, does not imply causal inde- pendence (local independence).

Next: we will construct a pair of finite-dimen- sional noncommuting algebras which a r e statisti- cally independent. This will complete the Ekstein statement.

Let X denote a complex Hilbert space with dim(%) a 6, let hl-h, be orthonormal vectors in X, define

and

and let P , Q denote projection operators onto the subspaces generated by @,, h,, e ) and (Iz,, h,, k ) , respectively. Let I denote the identity operator on all of X, let 91 = Q3 (X) be the C* algebra of inter - est where B(*) stands for the set of all linear bounded operators on (*) , let 91, denote the C* algebra generated by I and P, let 91, denote the C* algebra generated by I and Q, and let 31t denote the linear subspace of % generated by I, P, and Q. Note however that 311 is not a subalgebra unless either P I Q or P 2 Q (neither of which holds for this example).

Proposition 1. The algebras S1 and 91, do not commute.

Proof.

consequently

[P, Q ] - (h,) = c,e - c4k *O

Q.E.D. Definition. For each complex number a , b, c

define the linear functional w, by the expression

Proposition 2. The algebras a, and 91, a r e statistically independent.

Proof. Let w, of 91, and w2 of 91, denote states Choose the operator t

so that t has the form

where a, b, c a r e yea: numbers. Fo r each i = 1 , 2 , . . . , 6 o n e has

which by Eq. (5) i s

By successively using h,, h, , h,, and lz, in Eq. (7) one obtains

a s relations among the rea l numbers a, b, c . By considering al l four allowed cases from Eq. (8) a s applied to w,(t), we will show that w,(t) is positive on 37. n a ' . One has a 3 0 for al l cases in Eq. (8), and for b 3 0 and c a 0 we get

For b S 0 and c 0, we get

because w,(P) 6 1 and b 0 implies bwl(P) 2 b. For b 0 and c 3 0,

For b 3 0 and c 0 we get

Therefore w,(t) is positive on 3n n 91'. Now7 there exists a continuous linear functional w, on 91' which satisfies w,(x)> 0 on int(91') o r w,(x) 0 on %+. Now for each x in % define

where x = x, + ix2 i s the unique decomposition of x into Hermitian parts . Clearly w, is a state of 91 and

so that w, extends w, and w,. Hence, the algebras 91, and 8, a r e statistically independent. Q.E.D.

In Proposition 1 the finite-dimensional algebras a, and %, were shown to be noncommutative, and in Proposition 2 they were proven to be statisti- caliy independent. Therefore, we have presented a pair of finite-dimensional noncommuting algebras which a r e statistically independent, and the infi- nite -dimensional part of Ekstein's example was not a necessary requirement. Furthermore, the algebras used here a re irreducible so that Prop- osition 2 also closes that avenue of escape from the Ekstein statement.

C O M M U T A T I V I T Y AND C A U S A L I N D E P E N D E N C E 2731

111. CONCLUSIONS

We wish to observe that it is not known whether cazrsally disjoi~zt regions satisfy the Schlieder con- dition. If causally disjoint regions could be shown to satisfy tlre Schlieder condition (or not to) that would a.nswer the original question "yes" (or "no"). Since Roos assumed commutativity, his results have nothing to do with Ekstein's example o r the present example. Roos's theorem i s that the Schlieder condition together with commutativity can hold if and only if one has local independence. Neverkheless Ekstein has shown that commutativity neither implies nor follows f rom causal inciepen- dence, so the original question remains open. And the present work shows that the Elzstein s t a t e - ment also holds for finite -dimensional, i rreduc -

ible observable algebras and their normal states. This i s in agreement with Avishai and Eks te iq8 and shows that Ekstein5 was doing considerably more than tidying up the logical s tructure of relativistic theories.

ACKNOWLEDGMENTS

The authors wish to thank H . Ekstein for point- ing out that the cri t icism of Ref. 5 in a n ear l ie r preprint version of this note was unfounded, and R. Haag for pointing out a misstatement on caus- ally disjoint regions. One of us (B.D.F) thanks C. L. Hammer for useful conversations and the Ames Laboratory-USAEC for its hospitality during summer 1973.

*Summer Faculty Member. Permanent address : Department of Phys ics , University of Missouri-Co- lumbia, Columbia, Missouri 65201.

TPresent a d d r e s s : Department of Mathematics, Montana State Universi ty, Bozemaii, Montana, 5971.5.

'R. Haag and D. Kasxler, J. Math. Phys. 5, 848 (1964). 's. Schlieder, Coinmun. Math. Phys. 1, 265 (1965). 3 ~ . Schlieder, Commun. Math. Phys. 2, 216 (1969). '1-1. Roos, Commun. Math. Phys. 2, 238 (1970).

5 ~ . Ekstein, Phys. Rev. 184, 1315 (1969); Phys. Rev. D 1, 1851(E) (1970).

6 ~ . Dixmier, Les C %-algebras et leurs re$vesentations, 2nd edition (Gauthier-Villars, P a r i s , 1969).

'A. P . Robertson and W. J. Robertson, Topological Vector Spaces (Cambridge Univ. P r e s s , New York, 1966), p. 31 , Proposition 6.

8 ~ . Avishai and 11. Ekstein, Found. Phys. 2, 257 (1972).

P H Y S I C A L R E V I E W D V O L U M E 8 , N U M B E R 8 1 5 O C T O B E R 1 9 7 3

Current Uncertainty in the Ratio of Active-to-Passive Gravitational Mass

Daniel Morrison and Henry A. Hill University of Arizona, Tucson, Arizona 85721

(Received 8 August 1972)

The sensitivity of Kreuzer's experiment to a difference in active-to-passive mass ratios, hq/q, is calculated on the basis of design specifications. This provides an independent means for determining the experiment sensitivity not subject to the criticisms directed toward the original evaluation. The sensitivity is found to be (Aq/q)/S(O) = 1.9 X where S (0) is the signal recorded by the apparatus due to nonzero Aq/q. This result is in substantial agreement with the original results.

Gilvarry and Mullerl in a recent letter have criticized Mreuzer 'sZ conclusion from his 1966 experiment that the ratio of active-to-passive gravitational mass differs by no more than 5 X

between teflon (76% fluorine by weight), and a mixture of CHzBr, and CHC1: CC1, (74% bromine by weight). By refitting selected data points with a third-degree polynomial, they conclude that the uncertainty in the f i r s t -order te rm must be so la rge a s to invalidate i t s use in determining the sensitivity of the apparatus. The experimenter 's

f i r s t - degree fit produced a much smaller un- certainty in this te rm, and hence in his apparatus sensitivity. This note, by examining the e x ~ e r i - mental design, shows that the sensitivity inherent to this experiment must be approximately that claimed by the experimenter.

The analysis consists of two parts . F i rs t , the anomalous gravitational torque on the torsion balance due to a nonzero difference in mass rat ios is calculated on the basis of a model of the gravi- tational geometry of the apparatus. Equal and