- Email: [email protected]
The modal interpretation of algebraic quantum field theory
19 June 2000
Physics Letters A 271 Ž2000. 167–177 www.elsevier.nlrlocaterpla
The modal interpretation of algebraic quantum field theory Rob Clifton Department of Philosophy, 1001 Cathedral of Learning, UniÕersity of Pittsburgh, Pittsburgh, PA 15260, USA Received 6 March 2000; received in revised form 17 May 2000; accepted 19 May 2000 Communicated by P.R. Holland
Abstract We show that Dieks’ recent proposal for extending the modal interpretation to algebraic quantum field theory fails to yield a well-defined prescription for which observables in a local spacetime region possess definite values. On the other hand, we demonstrate that there is a well-defined and unique extension of the modal interpretation to the local algebras of quantum field theory Žwhich may, however, face a serious difficulty in connection with ergodic field states.. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 03.65.-w; 03.70; 11.10.-z; 01.70.q w Keywords: Modal interpretation; Von Neumann algebra; Quantum field theory; Ergodic state
1. Introduction It is well-known that standard quantum mechanics faces the problem of measurement. At the conclusion of a unitary measurement interaction, the joint state of the measured system and measuring apparatus will typically have the form of a superposition that entangles eigenstates of the observable measured with different eigenstates of the apparatus’ pointer observable. If we subscribe to the orthodox ‘eigenstate–eigenvalue link’, according to which an observable of a system can only possess a sharp value if the quantum state of the system is an eigenstate of that observable Ži.e., allows one to predict its value
E-mail address: [email protected] ŽR. Clifton..
with probability 1., then we are forbidden to assert that after the measurement the pointer actually points in some direction, even though we know from experience that quantum measurements yield determinate results. The orthodox response to this problem is to postulate a collapse of the joint statevector of the measured systemq apparatus into an eigenstate of the pointer observable, with the probability of collapse to any particular eigenstate taken to be the usual quantum probability for the corresponding measured value. The difficulties of this collapse postulate are well-known, and to some extent alleviated by wavefunction collapse theories that modify the standard unitary evolution so that in measurement-like scenarios collapse occurs naturally. The aim of the modal interpretation, however, is to resolve the measurement problem without modifying the standard formalism and, in particular, without postulating col-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 6 4 - 9
168
R. Cliftonr Physics Letters A 271 (2000) 167–177
lapse. The key idea is that there is nothing in the formalism that enforces acceptance of the orthodox interpretation of the formalism embodied in the eigenstate–eigenvalue link. Thus the modal interpretation posits a different method of determining from the quantum state of a system which of its observables possess sharp values – a method that Žit is claimed. is well-motivated, does not add anything to the standard formalism, and avoids the measurement problem. The modal interpretation is actually a family of closely related interpretations w2–7x. But almost all versions of the interpretation have in common that a system’s density operator determines which of its observables possess sharp values. We start by recalling how this works from a specifically algebraic point of view w8–10x, since our main interest is in how the interpretation fares with respect to algebraic quantum field theory.
2. The modal interpretation of nonrelativistic quantum theory At the most general level, we can consider a ‘universe’ comprised of a Žnonrelativistic. quantum system U, with finitely many degrees of freedom, represented by a Hilbert space HU s mi Hi . At any given time, U will occupy a pure vector state x g HU that determines a reduced density operator DS on the Hilbert space H S s mi g S Hi of any subsystem S : U. Let BŽ HS . denote the algebra of all bounded operators on H S , and for any single operator or family of operators T : BŽ HS ., let T X denote its commutant Ži.e., all operators on HS that commute with those in T .. Let PS denote the projection onto the range of DS , and consider the subalgebra of BŽ HS . given by the direct sum MS ' PSH B Ž HS . PSH qDSXX PS .
Ž 1.
The self-adjoint members of this algebra MS comprise the basic set of observables of S that the modal interpretation takes to have sharp values. Note that DSXX is none other than the set of bounded functions of DS , thus MS consists just of all operators that agree with a function of DS on its range. Leaving the double commutant out of the definition of MS would
have the unfortunate consequence that functions of observables with definite values would not also have definite values. Good physical arguments for choosing the definite-valued observables of S to be those in MS Žindependent of the motivation to solve the measurement problem. have been given in w8,10,13x, and the first proposition we prove in Section 4 below shall strengthen those arguments still further. For more insight into the structure of MS , consider two common special cases. First, when S is not entangled by x g HU with its environment S Žrepresented by H S s mi f S H i ., DS will itself be a pure state, induced by a unit vector y g HS . In this case, PS is simply the one-dimensional projection associated with y, and then MS consists of all operators that commute with that projection, or equivalently, that have y as an eigenvector. Thus the self-adjoint members of MS are exactly those that satisfy orthodoxy’s eigenstate–eigenvalue link, and the modal and orthodox interpretations coincide in this case. On the other hand, in the case where there is entanglement between S and S, forcing DS to be mixed – in particular, in the extremely mixed case PS s I Žwhere essentially every vector state y g H S is a component of the mixture. – then MS s DSXX , and MS consists of all bounded functions of DS . Yet, in this case, orthodoxy has nothing to say about the properties of S, because only multiples of the identity have values predictable with certainty in the state DS . Thus, when S is Schrodinger’s cat entangled with ¨ some potentially cat killing device S, we get the infamous measurement problem. At this point, it is necessary to make a distinction between two different ways modal interpreters employ the algebra MS to pick out definite-valued observables. In ‘non-atomic’ versions of the modal interpretation w3,11–13x, there is no preferred partition of the universe into subsystems. Any particular subsystem S : U is taken to have definite values for all the self-adjoint operators that lie in MS , and this of course applies whether or not DS is pure. The values of these observables are taken to be distributed according to the usual Born rule. Thus, the expectation of any observable X g MS is TrŽ DS X ., and the probability that X possesses some particular value x j is TrŽ DS P j ., where P j g MS is the corresponding eigenprojection of X. As in orthodox quantum theory with collapse, which precise value for X
R. Cliftonr Physics Letters A 271 (2000) 167–177
occurs on a given occasion Žfrom amongst those with nonzero probability in state DS . is not fixed by the interpretation. However, the occurrence of a value does not require that it be ‘measured’. And, unlike orthodox quantum theory, no miraculous collapse is needed to solve the measurement problem. Instead, after a typical unitary ‘measurement’ interaction between two parts of the universe – O the ‘measured’ system, and A the apparatus – decoherence induced by A’s coupling to the environment O j A will force the density operator DA of the apparatus to diagonalize in a basis extremely close to one which diagonalizes the pointer observable of A w14x. Thus, after the measurement, the definite-valued observables in MA will be such that the pointer points! By contrast, in ‘atomic’ versions of the modal interpretation Žw2x, w15–18x., one does not tell a separate story about the definite-valued observables for each subsystem S : U. Rather, S is taken to inherit properties from those of its atomic components, represented by the individual Hilbert spaces in mi g S Hi . In the approach favoured by Dieks w16x Žcf. w15x., each atomic system i possesses definite values for all the observables in Mi , as determined by the corresponding atomic density operator Di in accordance with Ž1.. The definite-valued observables of S itself are then built up by embedding each Mi in BŽ HS . Žvia tensoring it with the identity on H S ., and taking the von Neumann subalgebra of BŽ HS . generated by all these embeddings. The definite properties of S will therefore include all projections mi g S Pi jŽ i. that are tensor products of the spectral projections of the individual atomic density operators, and their joint probabilities are again taken to be given by the usual Born rule Trw DS Žmi g S Pi jŽ i. .x. In the presence of decoherence, the expectation is that atomic versions of the modal interpretation can yield essentially the same resolution of the measurement problem as do non-atomic versions w16,19x. In both versions of the modal interpretation, the observables with definite values must change over time as a function of the Žgenerally, non-unitary. evolution of the reduced density operators of the systems involved. In principle, this evolution can be determined from the Žunitary. Schrodinger evolution ¨ of the universal state vector x g HU w20x. But the evolution of the precise values of the definite-valued observables themselves is not determined by the
169
Schrodinger equation. Various more or less natural ¨ proposals have been made for ‘completing’ the modal interpretation with a dynamics for values Žw15,21– 23x.. Unfortunately, it has been shown that the most natural proposals for a dynamics, particularly in the case of atomic modal interpretations, must break Lorentz-invariance w24x. Most of Dieks’ recent Letter w1x is concerned to address this dynamics problem by appropriating ideas from the decoherent histories approach to quantum theory Žcf. w25x.. However, we shall focus here entirely on the viability of Dieks’ new proposal for picking out definite-valued observables of a relativistic quantum field that are associated with approximately point-sized regions of Minkowski spacetime Žw1x, Section 5..
3. Critique of Dieks’ proposal Dieks’ stated aim is to see if the modal interpretation can achieve sensible results in the context of quantum field theory. For this purpose, he adopts the formalism of algebraic quantum field theory because of its generality w26–28x. In the concrete ‘Haag– Araki’ approach, one supposes that a quantum field on Minkowski spacetime M will associate to each bounded open region O : M a von Neumann algebra RŽ O . of observables measurable in that region, where the collection RŽ O .:O : M 4 acts irreducibly on some fixed Hilbert space H. It is then natural to treat each open region O and associated algebra RŽ O . as a quantum system in its own right. Given any Žnormal. state r of the field Žwhere r is a state functional on BŽ H .., we can then ask which observables in RŽ O . are picked out as definite-valued by the restriction, r O , of the state r to RŽ O .. The difficulty for the modal interpretation Žthough Dieks himself does not put it this way. is that when O has nonempty spacelike complement OX , RŽ O . will typically be a type III factor that contains no nonzero finite projections Žw26x, Section 5.6; w28x, Section 17.2.. Because of this, RŽ O . cannot contain compact operators, like density operators, all of whose Žnon-null. spectral projections are finite-dimensional. As a result, there is no density operator in RŽ O . that can represent r O . Moreover, if we try to apply the standard modal prescription based on Eq.
170
R. Cliftonr Physics Letters A 271 (2000) 167–177
Ž1. to a density operator in BŽ H . that agrees with r O , there is no guarantee that the resulting set of observables will pick out a subalgebra of RŽ O ., and we will be left with nothing to say about which observables have definite values in O. The moral Dieks draws from this is that ‘‘We can therefore not take the open spacetime regions and their algebras as fundamental, if we want an interpretation in terms of Žmore or less. localized systems whose properties would specify an event’’ Žw1x, p. 322.. We shall see in the next section that this conclusion is overly pessimistic. In any case, Dieks’ strategy for dealing with the problem is to exploit the fact that, in most models of the axioms of algebraic quantum field theory, the local algebras associated with diamond shaped spacetime regions Ži.e., regions given by interior of the intersection of the causal future and past of two spacetime points. have the split property. The property is that for any two concentric diamond shaped regions er ,erq e : M, with radii r and r q e , there is a type I ‘interpolating’ factor Nrq e such that RŽer . ; Nrq e ; RŽerq e .. Now since Nrq e f BŽ H . for some Hilbert space H, we know there is always a unique density operator Drq e g BŽ H . that agrees with r on Nrq e , and therefore with re r. The proposal is, then, to take both r and e to be fixed small numbers and apply the prescription in Eq. Ž1. to this density operator Drq e , yielding a definite-valued subalgebra Mrq e : Nrq e . This, according to Dieks, should give an approximate indication of which observables have definite values at the common origin of the two diamonds in ‘‘the classical limiting situation in which classical field and particle concepts become approximately applicable’’ Žw1x, p. 323.. Thus, Dieks proposes to build up an atomic modal interpretation of the field as follows. Ži. We subdivide Žapproximately. the whole of spacetime M into a collection of non-overlapping diamond regions er , with some fixed small radius r. Žii. We choose some fixed small e and an interpolating factor Nrq e for each diamond, using its r-induced density operator Drq e and Eq. Ž1. to determine the definite-valued observables Mrq e ; RŽerq e . to be loosely associated with the origin. Žiii. Finally, we build up definite-valued observables associated with collections of diamonds, and define their joint probabilities in the usual way via Born’s rule Ždefining transition
probabilities between values of observables associated with timelike-separated diamonds using the familiar multi-time generalization of that rule employed in the decoherent histories approach.. As things stand, there is much arbitrariness in this proposal that enters into the stages Ži. and Žii.. Dieks himself recognizes the arbitrariness in the size of the partition of M chosen. He also acknowledges that this arbitrariness cannot be eliminated by passing to the limit r, e 0, because the intersection of the algebras associated with any collection of concentric diamonds converging to a point is always the trivial algebra C I w29x. Indeed, one would have thought that this undermines any attempt to formulate an atomic modal interpretation in this context, because it forces the choice of ‘atomic diamonds’ in the partition of M to be essentially arbitrary. Dieks appears to suggest that this arbitrariness will become unimportant in some classical limit of relativistic quantum field theory in which we should recover ‘‘the classical picture according to which field values are attached to spacetime points’’ Žw1x, p. 325.. But it is not sufficient for the success of a proposal for interpreting a relativistic quantum field theory that the interpretation give sensible results in the limit of classical relativistic Žor nonrelativistic. field theory. Indeed, the only relevant limit would appear to be the nonrelativistic limit; i.e., Galilean quantum field theory. But, there, one still needs to spatially smear ‘operator-valued’ fields at each point to obtain a well-defined algebra of observables in a spatial region w30x, so there will again be no natural choice to make for atomic spatial regions or algebras. There is also another, more troubling, degree of arbitrariness at step Žii. in the choice of the type I interpolating factor Nrq e about each origin point. For any fixed partition and fixed r, e ) 0, we can always sub-divide the interval Ž r,r q e . further, and then the split property implies the existence of a pair of interpolating type I factors satisfying
™
R Ž er . ; Nrq e r2 ; R Ž erq e r2 . ; Nrq e ; R Ž erq e . .
Ž 2.
The problem is that we now face a nontrivial choice deciding which of these factors’ r-induced density operators to use to pick out the definite-valued observables in the state r associated with the origin. If
R. Cliftonr Physics Letters A 271 (2000) 167–177
we pick Drq e g Nrq e , then by Ž1. all observables X g Nrq e that share the same spectral projections as Drq e will have definite values. However, no such X can lie in Mrq e r2 : Nrq e r2 , nor even in RŽerq e r2 .. The reason is that X ’s spectral projections are finite in Nrq e . So if those projections were also in the type III algebra RŽerq e r2 ., they would have to be infinite in RŽerq e r2 ., and therefore also infinite projections in Nrq e – which is impossible. Clearly we can sub-divide the interval Ž r,r q e . arbitrarily many times in this way and obtain a monotonically decreasing sequence of type I factors satisfying RŽerq e r2 nq 1 . ; Nrq e r2 n ; RŽerq e r2 n . that all interpolate between RŽer . and RŽerq e .. The sequence Nrq e r2 n 4`ns0 has no least member, and its greatest member, Nrq e , is arbitrary, because we could also further sub-divide the interval Ž r q er2,r q e . ad infinitum. Thus there is no natural choice of interpolating factor for picking out the observables definite at the origin, even if we restrict ourselves to a ‘nice’ decreasing sequence of interpolating factors of the form Nrq e r2 n 4`ns0 . On the other hand, it can actually be shown that RŽer . s F `ns0 Nrq e r2 n Žw27x, pp. 12-3; w28x, p. 426.. Furthermore, suppose Nrq e n 4`ns0 is any other decreasing type I sequence satisfying R Ž erq e nq 1 . ; Nrq e n ; R Ž erq e n . , e 0s e ,
e n ) e nq1 , lim e ns0.
Ž 3.
Then, since for any n there will be a sufficiently large nX such that Nrq e nX ; Nrq e r2 n Žand vice-versa., clearly RŽer . s F `ns0 Nrq e n, and this intersection will also be independent of e . It would seem, then, that the natural way to avoid choosing between the myriad type I factors that interpolate between RŽer . and RŽerq e . is to take the observables definite-valued at the origin to be those in the intersection F `ns 0 Mrq e n : RŽer . Žwhere, as before, Mrq e n is the modal subalgebra of Nrq e n determined via Ž1. by the density operator in Nrq e n that represents r .. Indeed, one way to understand Dieks’ proposal Žwhich may, however, not have been his original intention. is that when we take successively smaller values for e Žholding r fixed., and choose a type I interpolating factor at each stage, we should be getting progressively better approximations to the set of observables that are truly definite at the origin.
171
What better candidate for that set could there be than an intersection like F `ns 0 Mrq e n? Unfortunately, we have no guarantee that this intersection, unlike F `ns 0 Nrq e n itself, is independent of the particular sequence e n4 or its starting value e 0 s e . The reason any intersection of form F `ns 0 Nrq e n is so independent is because Nrq e n > Nrq e nq 1 for all n. But this does not imply Mrq e n > Mrq e nq 1. To take just a trivial example: when r is a pure state of Nrq e n that induces a mixed state on the proper subalgebra Nrq e nq 1of Nrq e n, Mrq e nq 1 will contain observables with dispersion in the state r , but Mrq e n will not. We conclude that there is little prospect of eliminating the arbitrariness in Dieks’ proposal and making it well-defined. One ought to look for another, intrinsic way to pick out the definite-valued observables in RŽer . that does not depend on special assumptions such as the split property.
4. The Modal interpretation for arbitrary von Neumann algebras There are two salient features of the algebra MS in Eq. Ž1. that make it an attractive set of definitevalued observables to modal interpreters. First, MS is locally determined by the quantum state DS of system S together with the structure of its algebra of observables. In particular, there is no need to add any additional structure to the standard formalism of quantum theory to pick out S’s properties. Second, the restriction of the state DS to the subalgebra MS is a mixture of dispersion-free states Žgiven by the density operators one obtains by renormalizing the Žnon-null. spectral projections of DS .. This second feature is what makes it possible to think of the observables in MS as possessing definite values distributed in accordance with standard Born rule statistics w9x. Let us see, then, whether we can generalize these two features to come up with a proposal for the definite-valued observables of a system described by an arbitrary von Neumann algebra R Žacting on some Hilbert space H. in an arbitrary state r of R. Generally, a state r of R will be a mixture of dispersion-free states on a subalgebra S : R just in
R. Cliftonr Physics Letters A 271 (2000) 167–177
172
case there is a probability measure mr on the space L of dispersion-free states of S such that
r Ž A. s
HLv Ž A . d m Ž l. , for all l
r
AgS ,
Ž 4.
where vlŽ A2 . s vlŽ A. 2 for all self-adjoint elements A g S . This somewhat cumbersome condition turns out to be equivalent Žw10x, Prop. 2.2Žii.. to simply requiring that )
r Ž w A, B x w A, B x . s 0 for all A, B g S .
Ž 5.
In particular, r can always be represented as a mixture of dispersion-free states on any Abelian subalgebra S : R. Conversely, if r is a faithful state of R, i.e., r maps no nonzero positive elements of R to zero, then the only subalgebras that allow r to be represented as a mixture of dispersion-free states are the Abelian ones. There is now an easy way to pick out a subalgebra S : R with this property, using only r and the algebraic operations available within R. Consider the following two mathematical objects explicitly defined in terms of R and r . First, the support projection of the state r in R, defined by Pr , R ' n P s P 2 s P ) g R : r Ž P . s 1 4 ,
Ž 6.
which is simply the smallest projection in R that the state r ‘makes true’. Second, there is the centralizer subalgebra of the state r in R, defined by Cr , R ' A g R : r Ž w A, B x . s 0 for all B g R 4 .
Ž 7. For any von Neumann algebra K, let Z Ž K . ' K l K X , the center algebra of K. Then it is reasonable for the modal interpreter to take as definite-valued all the observables that lie in the direct sum Z Ž Cr , R . Pr , R : R , S s Mr , R ' PrH, R R PrH, R qZ
Ž 8.
where the algebra in the first summand acts on the subspace PrH , R H and that of the second acts on Pr , R H. The state r is a mixture of dispersion-free states on Mr , R , by Ž5., because r maps all elements H of the form PrH , R R Pr , R to zero, and the product of the commutators of any two elements of Z Ž Cr , R . Pr , R also gets mapped to zero, for the trivial reason that Z Ž Cr , R . is Abelian.
The set Mr , R directly generalizes the algebra of Eq. Ž1. to the non-type I case where the algebra of observables of the system does not contain a density operator representative of the state r . Assuming the type I case, R f BŽ H . for some Hilbert space H, r is given by a density operator D on H, Pr , R is equivalent to the range projection of D, and Z Ž Cr , R . f Z Ž CD, B Ž H . .. So to show that Mr , R is isomorphic to the algebra of Eq. Ž1., it suffices to establish that Z Ž CD, B Ž H . . s DXX . It is easy to see that CD, B Ž H . s DX Žinvoking cyclicity and positivedefiniteness of the trace., thus Z Ž CD, B Ž H . . s DX l DXX . However, since DX always contains a maximal Abelian subalgebra of BŽ H . Žviz., that generated by the projections onto any complete orthonormal basis of eigenvectors for D ., we always have DXX : DX . Choosing Mr , R is certainly not the only way to pick a subalgebra S : R that is definable in terms of r and R and allows r to be represented as a mixture of dispersion-free states. There is the obvious orthodox alternative one can always consider, viz., the definite algebra of r in R, S s Or , R ' A g R : r Ž AB . s r Ž A . r Ž B . for all B g R 4 ,
Ž 9.
which coincides with the complex span of all self-adjoint members of R on which r is dispersion-free Žw10x, p. 2445.. Note, however, that we always have Or , R : Mr , R . Indeed the problem, as we have seen, is that the orthodox choice Or , R generally will contain far too few definite-valued observables to solve the measurement problem. For example, when r is faithful – and there will always be a norm dense set of states of R that are – we get just Or , R s C I. Thus it is natural for a modal interpreter to require that the choice of S : R be maximal. In the case where r is faithful, we now show that this singles out the choice S s Mr , R s Z Ž Cr , R . uniquely Žand we conjecture that a similar uniqueness result holds for the more general expression for Mr , R in Eq. Ž8., using the fact that an arbitrary state r always renormalizes to a faithful state on Pr , R R Pr , R ..
Proposition 1. Let R be a Õon Neumann algebra and r a faithful normal state of R with centralizer
R. Cliftonr Physics Letters A 271 (2000) 167–177
Cr , R : R. Then Z ( Cr , R ), the center of Cr , R , is the unique subalgebra S : R such that: 1. The restriction of r to S is a mixture of dispersion-free states. 2. S is definable solely in terms of r and the algebraic structure of R. 3. S is maximal with respect to properties 1. and 2. Proof. By 3., it suffices to show than any S : R satisfying 1. and 2. is contained in Z Ž Cr , R .. And for this, it suffices Žbecause von Neumann algebras are generated by their projections. to show that S, the subset of projections in S , is contained in Z Ž Cr , R .. Recall also that, as a consequence of 1. and the faithfulness of r , S must be Abelian. And in virtue of 2., any automorphism s : R R that preserves the state r in the sense that r ( s s r , must leave the set S Žnot necessarily pointwise. invariant, i.e., s Ž S . s S . S : CrX, R . Any unitary operator U g Cr , R defines an inner automorphism on R that leaves r invariant, therefore U S Uy1 s S . Since S is Abelian, wUPUy1 , P x s 0 for each P g S and all unitary U g Cr , R . By Lemma 4.2 of w10x Žwith V s CrXX, R s Cr , R ., this implies that P g CrX, R . S : Cr , R . Since r is faithful, there is a oneparameter group st :t g R 4 of automorphisms of R – the modular automorphism group of R determined by r Žw31x, Section 9.2. — leaving r invariant. Since Cr , R consists precisely of the fixed points of the modular group Žw31x, Prop. 9.2.14., it suffices to show that it leaves the individual elements of S fixed. For this, we use the fact that the modular group satisfies the KMS condition with respect to r : for each A, B g R, there is a complex-valued function f, bounded and continuous on the strip z g C:0 F Im z F 14 in the complex plane, and analytic on the interior of that strip, such that
™
f Ž t . s r Ž st Ž A . B . , f Ž t q i . s r Ž Bs t Ž A . . , t g R .
Ž 10 .
In fact, we shall need only one simple consequence of the KMS condition, viz., if f Ž t . s f Ž t q i . for all t g R, then f is constant Žw31x, p. 611..
173
Fix an arbitrary projection P g S. Since the modular automorphism group must leave S as a whole invariant, and S is Abelian, w st Ž P ., P Hx s 0 for all t g R. However, there exists a function f with the above properties such that f Ž t . s r Ž st Ž P . P H . s r Ž P H st Ž P . . sf Ž tqi. , tgR , so it follows that f is constant. In particular, since s 0 Ž P . P H s PP H s 0, f is identically zero, and r Ž st Ž P . P H. s 0 for all t g R. And since st Ž P . and P H are commuting projections, their product is a Žpositive. projection, so that the faithfulness of r requires that st Ž P . P H s 0, or equivalently st Ž P . s st Ž P . P, for all t g R. Running through the exact same argument, starting with P Hg S in place of P, yields st Ž P H. s st Ž P H. P H , or equivalently, st Ž P . P s P, for all t g R. Together with st Ž P . s st Ž P . P, this implies that st Ž P . s P for all t g R. QED.
The choice Mr , R s Z Ž Cr , R . has another feature that generalizes a natural consequence of the modal interpretation of nonrelativistic quantum theory. Suppose the universal state x g HU defines a faithful state r x on both BŽ HS . and BŽ HS .. This requires that dim HS s dim HS s n Žpossibly `., and, furthermore, that any Schmidt decomposition of the state vector x relative to the factorization HU s HS m HS takes the form n
xs
Ý c i Õi m wi , c i / 0 for all i s 1 to n,
Ž 11 .
is1
where the vectors Õi and wi are complete orthonormal bases in their respective spaces. As is wellknown, for each distinct eigenvalue l˜ j for DS , the span of the vectors Õi for which < c i < 2 s l˜ j coincides with the range of the l˜ j-eigenprojection of DS , and similarly for DS . Consequently, there is a natural bijective correspondence between the properties represented by the projections in the two sets MS s DSXX and MS s DSXX : any definite property S happens to possess is strictly correlated to a unique property of its environment S that occurs with the same fre-
174
R. Cliftonr Physics Letters A 271 (2000) 167–177
quency. More formally, for any P g MS , there is a unique P g MS satisfying
zero, since r x is faithful.. Letting A g Cx, R be the double of B in R, we get
Ž x , PPx . s Ž x , Px . s Ž x , Px . .
PBx y BPx s PAx y BPx s APx y PBx
Ž 12 .
To see this, note that any P g MS Žin this case, the set of all functions of DS . is a sum of spectral projections of DS . Let P g MS be the sum of the corresponding spectral projections of DS for the same eigenvalues. Then it is evident from the form of the state expansion in Ž11. that P has the property in Ž12., and no other projection in MS does. This has led some non-atomic modal interpreters, such as Kochen w11x, to interpret each property P of S, not as a property that S possesses absolutely, but only in relation to its environment S possessing the corresponding property P. For a general von Neumann factor R, Ž R j R X .XX s BŽ H . need not be isomorphic to the tensor product R m R X Žparticularly when R is type III, for then R m R X must be type III as well.. Therefore, there is no direct analogue of a Schmidt decomposition for a pure state x g H relative to the factorization Ž R j R X .XX of BŽ H .. Nevertheless, we show next that there is still the same strict correlation between definite properties in Mr x , R s Z Ž Cr x , R . and Mr x , R X s Z Ž Cr x , R X .. Proposition 2. Let R be a Õon Neumann algebra acting on a Hilbert space H, and suppose x g H induces a state r x that is faithful on both R and R X . Then for any projection P g Z Ž Cr x , R . , there is a unique projection P g Z Ž Cr x , R X . such that Ž x, PPx . s Ž x, Px . s Ž x, Px . . Proof. For any fixed A g R, call an element B g R X a double for A Žin state x . just in case Ax s Bx and A) x s B ) x. By an elementary application of modular theory, Werner Žw32x, Section 2. has shown that Cr x , R consists precisely of those elements of R with doubles in R X Žwith respect to x .. Moreover, the double of any element of R clearly has to be unique, by the faithfulness of r x on R X . Now it is easy to see Žagain using the faithfulness of r x . that the double of any projection P g Cr x , R is a projection P g Cr x , R X satisfying Ž12.. We claim that whenever P g Cx,X R , we have P g Cx,X R X . For this, it suffices to show P g Cx,X R implies that for arbitrary B g Cx, R X , w P, B x x s 0 Žand then w P, B x itself is
s APx y PAxs0,
Ž 13 .
as required. Finally, were there another projection P˜ g Z Ž Cr x , R X . satisfying Ž12., then by exploiting the ˜ .s fact that P is P ’s double in R X , we get Ž x, PPx ˜ .; or, equivalently, Ž x, Px . s Ž x, Px
˜ . s 0. Ž x , PP˜ H x . s Ž x , P H Px
Ž 14 .
Since Z Ž Cr x , R X . is Abelian, both PP˜ H and P H P˜ are Žpositive. projections in R. But as r x is faithful on R, Eqs. Ž14. entail that PP˜ H s P H P˜ s 0, which ˜ as required for uniquein turn implies that P s P, ness. QED. Let us return now to the problem of picking out a set of definite-valued observables localized in a diamond region with associated algebra RŽer .. Let r be any pure state of the field that induces a faithful state on RŽer . and its commutant; for example, r could be the vacuum or any one of the dense set of states of a field with bounded energy Žby the Reeh– Schlieder theorem – see w27x, Thm. 1.3.1.. By Proposition 1, the definite-valued observables in R Žer . are simply those in the subalgebra Z Ž Cr , R Že r . .. Note that this proposal yields observables all of which have an exact spacetime localization within the open set er and are picked out intrinsically by the local algebra RŽer . and the field state r . Contrary to Dieks’ pessimistic conclusion, we can take open spacetime regions as fundamental for determining the definite-valued observables. In fact, since this proposal works independent of the size of r, it could also be embraced by non-atomic modal interpreters not wishing to commit themselves to a particular partition of the field into subsystems Žor to thinking from the outset in terms of approximately point-localized field observables.. Finally, note that since the algebra of a diamond region er satisfies duality with respect to the algebra of its spacelike complement er X , i.e., RŽer .X s RŽer X . Žw26x, p. 145., Proposition 2 tells us that there is a natural bijective correspondence between the properties in Z Ž Cr , R Že . . and strictly correlated properties r
R. Cliftonr Physics Letters A 271 (2000) 167–177
in Z Ž Cr , R ŽeXr . . associated with the complement region.
5. A potential difficulty with ergodic states We have seen that there is, after all, a well-motivated and unambiguous prescription extending the standard modal interpretation of nonrelativistic quantum theory to the local algebras of quantum field theory. We also, now, have a natural standard of comparison with Galilean quantum field theory. At least in the case of free fields, it is possible to build up local algebras in M from spatially smeared ‘field algebras’ defined on spacelike hyperplanes in M. A diamond region corresponds to the domain of dependence of a spatial region in a hyperplane, and it can be shown that the algebra of that spatial region will also be type III and, indeed, coincide with its domain of dependence algebra Žw27x, Prop. 3.3.2, Thm. 3.3.4.. These type III spatial algebras in M, and the definite-valued observables therein, are what should be compared, in the nonrelativistic limit, to the corresponding equal time spatial algebras defined on simultaneity slices of Galilean spacetime. Unfortunately, since the algebras in the Galilean case are invariably type I Žw27x, p. 35., this limit is bound to be mathematically singular, and its physical characterization needs to be dealt with carefully. But this is a problem for any would-be interpreter of relativistic quantum field theory, not just modal interpreters. All we should require of them, at this stage, is that they be able to say something sensible in the relativistic case about the local observables with definite values Žwhich was, indeed, Dieks’ original goal.. However, as we now explain, it is not clear whether even this goal can be attained. If R f BŽ H . is type I, it possesses at most one faithful state r such that Z Ž Cr , R . s C I. This is easy to see, because if the density operator D g BŽ H . represents r , Z Ž Cr , R . f DXX , and DXX s C I implies that D itself must be a multiple of the identity. So when H is finite-dimensional, we must have D s Irdim H, the unique maximally mixed state, and in the infinite-dimensional case, no such density operator even exists. Elsewhere Dieks w33x has argued convincingly that there is no problem when a system, occupying a maximally mixed state, possesses only
175
trivial properties, because such states are rare and highly unstable under environmental decoherence Žcf. w2x, pp. 99-100.. However, the situation is quite different for the local algebras of algebraic quantum field theory. In all physically reasonable models of the axioms of the theory, every local algebra RŽ O . is isomorphic to the unique Žup to isomorphism. hyperfinite type III 1 factor Žw26x, Section 5.6; w28x, Section 17.2.. In that case, there is a novel way to obtain Z Ž Cr , R ŽO . . s C I, namely, when the state r of RŽ O . is an ergodic state w34,35x, i.e., r possesses a trivial centralizer in RŽ O .. ŽWere R a non-Abelian type I factor, this would be impossible, since DX s C I implies DXX s BŽ H . f R, which is patently false.. In fact, we have the following result. Proposition 3. If R is the hyperfinite type III 1 factor, there is a norm dense set of unit Õectors in the Hilbert space H on which R acts that induce faithful ergodic states on R (i.e., with triÕial centralizers in R). Proof: First recall the following facts provable from the axioms of algebraic quantum field theory: Ži. the vacuum state of a field on M has a trivial centralizer in the algebra of any Rindler wedge Žw28x, Section 16.1.1.; Žii. the vacuum is faithful for any wedge algebra Žby the Reeh–Schlieder theorem.; and Žiii. wedge algebras are hyperfinite type III 1 factors Žw28x, Ex. 16.2.14, pp. 426-7.. Since being faithful and having a trivial centralizer are isomorphic invariants, it follows that any instantiation R of the hyperfinite type III 1 factor possesses at least one faithful normal state r with trivial centralizer Ževen when R is the algebra of a bounded open region, like a diamond.. Now since R is type III, all its states are vector states Žcombine w36x, Cor. 2.9.28 with w31x, Thm. 7.2.3.; in particular, r s r x for some unit vector x g H. Furthermore, by the homogeneity of the state space of type III 1 factors Žw37x, Cor. 6., the set of all unit vectors of the form UU X x, with U g R and U X g R X unitary operators, lies dense in H. But clearly any such vector must again induce a faithful state on R with trivial centralizer. QED. Combining Propositions 1 and 3, there will be a whole host of states of any relativistic quantum field
176
R. Cliftonr Physics Letters A 271 (2000) 167–177
in which the modal interpreter is forced to assert that no nontrivial local observables have definite values! Note, however, that while the set of field states ergodic for any given type III 1 local algebra is always dense, this does not automatically imply that such states are typical or generic. Indeed, results of Summers and Werner Žw38x, particularly Cor. 2.4. imply that for any local diamond algebra RŽer ., there will also always be a dense set field states whose centralizers in RŽer . contain the hyperfinite type II 1 factor, and so will not be trivial. Still, the modal interpreter needs to provide some physical reason for neglecting the densely many field states that do yield trivial definite-valued observables locally. Obviously instability under decoherence is no longer relevant. Perhaps one could try to bypass Proposition 1 by exploiting extra structure not contained in the particular field state and local algebra to pick out the definite-valued observables in a region. For example, one might try to exploit the field’s total energymomentum operator, and, in particular, its generator of time evolution. In the context of the nonrelativistic modal interpretation, Bacciagaluppi et al. w20x Žcf. also w15x, p. 1181. have successfully invoked the analytic properties of the time evolution of the spectral projections of a system’s reduced density operator DS to avoid discontinuities that occur in the definite-valued set MS at moments of time where the multiplicity of the eigenvalues of DS changes. In particular, their methods yield a natural dynamical way, independent of instability considerations, to avoid the trivial definite-valued sets determined by maximally mixed density operators. So one might hope that these same dynamical methods could be extended to type III 1 algebras so as to yield a richer set of properties in a local region than Proposition 1 allows for ergodic states. In any case, modal interpreters need to do more work to show that their interpretation yields sensible local properties in quantum field theory Ževen before one considers, with Dieks, how to define Lorentz invariant decoherent histories of properties.. Acknowledgements The author wishes to thank Hans Halvorson Žfor supplying the argument immediately following Eq.
Ž2. and suggesting the use of the KMS condition in the proof of Proposition 1., and Reinhard Werner Žfor helpful correspondence about centralizers and inspiring Proposition 2..
References w1x D. Dieks, Phys. Lett. A 265 Ž2000. 317. w2x R. Healey, The Philosophy of Quantum Mechanics: An Interactive Interpretation, Cambridge Univ. Press, Cambridge, 1989. w3x B.C. van Fraassen, Quantum Mechanics: An Empiricist View, Clarendon Press, Oxford, 1991. w4x R. Clifton, Brit. J. Phil. Sci. 47 Ž1996. 371. w5x J. Bub, Interpreting the Quantum World, Cambridge Univ. Press, Cambridge, 1997. w6x R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998. w7x D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998. w8x J. Zimba, R. Clifton in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 69. w9x R. Clifton, in: J. Butterfield, C. Pagonis ŽEds.., From Physics to Philosophy, Cambridge Univ. Press, Cambridge, 1999 ŽChap. 2.. w10x H. Halvorson, R. Clifton, Int. J. Theor. Phys. 38 Ž1999. 2441. w11x S. Kochen, in: P.J. Lahti, P. Mittelstaedt ŽEds.., Symposium on the Foundations of Modern Physics, World Scientific, Singapore, p. 151. w12x R. Clifton, Brit. J. Phil. Sci. 46 Ž1995. 33. w13x R. Clifton, Ann. ŽNY. Acad. Sci. 755 Ž1995. 570. w14x G. Bacciagaluppi, M. Hemmo, Stud. Hist. Phil. Mod. Phys. 27B Ž1996. 239. w15x G. Bacciagaluppi, M. Dickson, Found. Phys. 29 Ž1999. 1165. w16x D. Dieks in: R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998, p. 144. w17x R. Spekkens, J. Sipe, Contingent Determinate Variables in Modal Interpretations of Quantum Mechanics Ž2000., preprint. w18x R. Spekkens, J. Sipe, quant-phr0003092. w19x P. Vermaas in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 103. w20x G. Bacciagaluppi, M. Donald, P. Vermaas, Helv. Phys. Act. 68 Ž1995. 679. w21x P. Vermaas, Stud. Hist. Phil. Mod. Phys. 27 Ž1996. 133. w22x M. Dickson, in: R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998, p. 160. w23x J. Ax, S. Kochen, quant-phr9905077.
R. Cliftonr Physics Letters A 271 (2000) 167–177 w24x M. Dickson, R. Clifton in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 9. w25x M. Hemmo, in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 253. w26x R. Haag, Local Quantum Physics, 2nd ed., Springer, Berlin, 1992. w27x S. Horuzhy, Introduction to Algebraic Quantum Field Theory, Kluwer, Dordrecht, 1988. w28x H. Baumgartel, M. Wollenberg, Causal Nets of Operator ¨ Algebras, Springer, Berlin, 1992. w29x A. Wightman, Ann. Inst. Henri Poincare´ 1 Ž1964. 403.
177
w30x M. Requardt, J. Phys. A: Math. Gen. 15 Ž1982. 3715. w31x R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, American Mathematical Society, R.I. Providence, 1997. w32x R. Werner, quant-phr9910077 w33x D. Dieks, Phys. Rev. A 49 Ž1994. 2290. w34x R. Herman, M. Takesaki, Comm. Math. Phys. 19 Ž1970. 142. w35x L. Barnett, Proc. Am. Math. Soc. 123 Ž1995. 543. w36x S. Sakai, C ) -Algebras and W ) -algebras, Springer, Berlin, 1971. w37x A. Connes, E. Størmer, J. Func. Anal. 26 Ž1978. 187. w38x S. Summers, R. Werner, Ann. Inst. Henri Poincare´ 49 Ž1988. 215.
Physics Letters A 271 Ž2000. 167–177 www.elsevier.nlrlocaterpla
The modal interpretation of algebraic quantum field theory Rob Clifton Department of Philosophy, 1001 Cathedral of Learning, UniÕersity of Pittsburgh, Pittsburgh, PA 15260, USA Received 6 March 2000; received in revised form 17 May 2000; accepted 19 May 2000 Communicated by P.R. Holland
Abstract We show that Dieks’ recent proposal for extending the modal interpretation to algebraic quantum field theory fails to yield a well-defined prescription for which observables in a local spacetime region possess definite values. On the other hand, we demonstrate that there is a well-defined and unique extension of the modal interpretation to the local algebras of quantum field theory Žwhich may, however, face a serious difficulty in connection with ergodic field states.. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 03.65.-w; 03.70; 11.10.-z; 01.70.q w Keywords: Modal interpretation; Von Neumann algebra; Quantum field theory; Ergodic state
1. Introduction It is well-known that standard quantum mechanics faces the problem of measurement. At the conclusion of a unitary measurement interaction, the joint state of the measured system and measuring apparatus will typically have the form of a superposition that entangles eigenstates of the observable measured with different eigenstates of the apparatus’ pointer observable. If we subscribe to the orthodox ‘eigenstate–eigenvalue link’, according to which an observable of a system can only possess a sharp value if the quantum state of the system is an eigenstate of that observable Ži.e., allows one to predict its value
E-mail address: [email protected] ŽR. Clifton..
with probability 1., then we are forbidden to assert that after the measurement the pointer actually points in some direction, even though we know from experience that quantum measurements yield determinate results. The orthodox response to this problem is to postulate a collapse of the joint statevector of the measured systemq apparatus into an eigenstate of the pointer observable, with the probability of collapse to any particular eigenstate taken to be the usual quantum probability for the corresponding measured value. The difficulties of this collapse postulate are well-known, and to some extent alleviated by wavefunction collapse theories that modify the standard unitary evolution so that in measurement-like scenarios collapse occurs naturally. The aim of the modal interpretation, however, is to resolve the measurement problem without modifying the standard formalism and, in particular, without postulating col-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 6 4 - 9
168
R. Cliftonr Physics Letters A 271 (2000) 167–177
lapse. The key idea is that there is nothing in the formalism that enforces acceptance of the orthodox interpretation of the formalism embodied in the eigenstate–eigenvalue link. Thus the modal interpretation posits a different method of determining from the quantum state of a system which of its observables possess sharp values – a method that Žit is claimed. is well-motivated, does not add anything to the standard formalism, and avoids the measurement problem. The modal interpretation is actually a family of closely related interpretations w2–7x. But almost all versions of the interpretation have in common that a system’s density operator determines which of its observables possess sharp values. We start by recalling how this works from a specifically algebraic point of view w8–10x, since our main interest is in how the interpretation fares with respect to algebraic quantum field theory.
2. The modal interpretation of nonrelativistic quantum theory At the most general level, we can consider a ‘universe’ comprised of a Žnonrelativistic. quantum system U, with finitely many degrees of freedom, represented by a Hilbert space HU s mi Hi . At any given time, U will occupy a pure vector state x g HU that determines a reduced density operator DS on the Hilbert space H S s mi g S Hi of any subsystem S : U. Let BŽ HS . denote the algebra of all bounded operators on H S , and for any single operator or family of operators T : BŽ HS ., let T X denote its commutant Ži.e., all operators on HS that commute with those in T .. Let PS denote the projection onto the range of DS , and consider the subalgebra of BŽ HS . given by the direct sum MS ' PSH B Ž HS . PSH qDSXX PS .
Ž 1.
The self-adjoint members of this algebra MS comprise the basic set of observables of S that the modal interpretation takes to have sharp values. Note that DSXX is none other than the set of bounded functions of DS , thus MS consists just of all operators that agree with a function of DS on its range. Leaving the double commutant out of the definition of MS would
have the unfortunate consequence that functions of observables with definite values would not also have definite values. Good physical arguments for choosing the definite-valued observables of S to be those in MS Žindependent of the motivation to solve the measurement problem. have been given in w8,10,13x, and the first proposition we prove in Section 4 below shall strengthen those arguments still further. For more insight into the structure of MS , consider two common special cases. First, when S is not entangled by x g HU with its environment S Žrepresented by H S s mi f S H i ., DS will itself be a pure state, induced by a unit vector y g HS . In this case, PS is simply the one-dimensional projection associated with y, and then MS consists of all operators that commute with that projection, or equivalently, that have y as an eigenvector. Thus the self-adjoint members of MS are exactly those that satisfy orthodoxy’s eigenstate–eigenvalue link, and the modal and orthodox interpretations coincide in this case. On the other hand, in the case where there is entanglement between S and S, forcing DS to be mixed – in particular, in the extremely mixed case PS s I Žwhere essentially every vector state y g H S is a component of the mixture. – then MS s DSXX , and MS consists of all bounded functions of DS . Yet, in this case, orthodoxy has nothing to say about the properties of S, because only multiples of the identity have values predictable with certainty in the state DS . Thus, when S is Schrodinger’s cat entangled with ¨ some potentially cat killing device S, we get the infamous measurement problem. At this point, it is necessary to make a distinction between two different ways modal interpreters employ the algebra MS to pick out definite-valued observables. In ‘non-atomic’ versions of the modal interpretation w3,11–13x, there is no preferred partition of the universe into subsystems. Any particular subsystem S : U is taken to have definite values for all the self-adjoint operators that lie in MS , and this of course applies whether or not DS is pure. The values of these observables are taken to be distributed according to the usual Born rule. Thus, the expectation of any observable X g MS is TrŽ DS X ., and the probability that X possesses some particular value x j is TrŽ DS P j ., where P j g MS is the corresponding eigenprojection of X. As in orthodox quantum theory with collapse, which precise value for X
R. Cliftonr Physics Letters A 271 (2000) 167–177
occurs on a given occasion Žfrom amongst those with nonzero probability in state DS . is not fixed by the interpretation. However, the occurrence of a value does not require that it be ‘measured’. And, unlike orthodox quantum theory, no miraculous collapse is needed to solve the measurement problem. Instead, after a typical unitary ‘measurement’ interaction between two parts of the universe – O the ‘measured’ system, and A the apparatus – decoherence induced by A’s coupling to the environment O j A will force the density operator DA of the apparatus to diagonalize in a basis extremely close to one which diagonalizes the pointer observable of A w14x. Thus, after the measurement, the definite-valued observables in MA will be such that the pointer points! By contrast, in ‘atomic’ versions of the modal interpretation Žw2x, w15–18x., one does not tell a separate story about the definite-valued observables for each subsystem S : U. Rather, S is taken to inherit properties from those of its atomic components, represented by the individual Hilbert spaces in mi g S Hi . In the approach favoured by Dieks w16x Žcf. w15x., each atomic system i possesses definite values for all the observables in Mi , as determined by the corresponding atomic density operator Di in accordance with Ž1.. The definite-valued observables of S itself are then built up by embedding each Mi in BŽ HS . Žvia tensoring it with the identity on H S ., and taking the von Neumann subalgebra of BŽ HS . generated by all these embeddings. The definite properties of S will therefore include all projections mi g S Pi jŽ i. that are tensor products of the spectral projections of the individual atomic density operators, and their joint probabilities are again taken to be given by the usual Born rule Trw DS Žmi g S Pi jŽ i. .x. In the presence of decoherence, the expectation is that atomic versions of the modal interpretation can yield essentially the same resolution of the measurement problem as do non-atomic versions w16,19x. In both versions of the modal interpretation, the observables with definite values must change over time as a function of the Žgenerally, non-unitary. evolution of the reduced density operators of the systems involved. In principle, this evolution can be determined from the Žunitary. Schrodinger evolution ¨ of the universal state vector x g HU w20x. But the evolution of the precise values of the definite-valued observables themselves is not determined by the
169
Schrodinger equation. Various more or less natural ¨ proposals have been made for ‘completing’ the modal interpretation with a dynamics for values Žw15,21– 23x.. Unfortunately, it has been shown that the most natural proposals for a dynamics, particularly in the case of atomic modal interpretations, must break Lorentz-invariance w24x. Most of Dieks’ recent Letter w1x is concerned to address this dynamics problem by appropriating ideas from the decoherent histories approach to quantum theory Žcf. w25x.. However, we shall focus here entirely on the viability of Dieks’ new proposal for picking out definite-valued observables of a relativistic quantum field that are associated with approximately point-sized regions of Minkowski spacetime Žw1x, Section 5..
3. Critique of Dieks’ proposal Dieks’ stated aim is to see if the modal interpretation can achieve sensible results in the context of quantum field theory. For this purpose, he adopts the formalism of algebraic quantum field theory because of its generality w26–28x. In the concrete ‘Haag– Araki’ approach, one supposes that a quantum field on Minkowski spacetime M will associate to each bounded open region O : M a von Neumann algebra RŽ O . of observables measurable in that region, where the collection RŽ O .:O : M 4 acts irreducibly on some fixed Hilbert space H. It is then natural to treat each open region O and associated algebra RŽ O . as a quantum system in its own right. Given any Žnormal. state r of the field Žwhere r is a state functional on BŽ H .., we can then ask which observables in RŽ O . are picked out as definite-valued by the restriction, r O , of the state r to RŽ O .. The difficulty for the modal interpretation Žthough Dieks himself does not put it this way. is that when O has nonempty spacelike complement OX , RŽ O . will typically be a type III factor that contains no nonzero finite projections Žw26x, Section 5.6; w28x, Section 17.2.. Because of this, RŽ O . cannot contain compact operators, like density operators, all of whose Žnon-null. spectral projections are finite-dimensional. As a result, there is no density operator in RŽ O . that can represent r O . Moreover, if we try to apply the standard modal prescription based on Eq.
170
R. Cliftonr Physics Letters A 271 (2000) 167–177
Ž1. to a density operator in BŽ H . that agrees with r O , there is no guarantee that the resulting set of observables will pick out a subalgebra of RŽ O ., and we will be left with nothing to say about which observables have definite values in O. The moral Dieks draws from this is that ‘‘We can therefore not take the open spacetime regions and their algebras as fundamental, if we want an interpretation in terms of Žmore or less. localized systems whose properties would specify an event’’ Žw1x, p. 322.. We shall see in the next section that this conclusion is overly pessimistic. In any case, Dieks’ strategy for dealing with the problem is to exploit the fact that, in most models of the axioms of algebraic quantum field theory, the local algebras associated with diamond shaped spacetime regions Ži.e., regions given by interior of the intersection of the causal future and past of two spacetime points. have the split property. The property is that for any two concentric diamond shaped regions er ,erq e : M, with radii r and r q e , there is a type I ‘interpolating’ factor Nrq e such that RŽer . ; Nrq e ; RŽerq e .. Now since Nrq e f BŽ H . for some Hilbert space H, we know there is always a unique density operator Drq e g BŽ H . that agrees with r on Nrq e , and therefore with re r. The proposal is, then, to take both r and e to be fixed small numbers and apply the prescription in Eq. Ž1. to this density operator Drq e , yielding a definite-valued subalgebra Mrq e : Nrq e . This, according to Dieks, should give an approximate indication of which observables have definite values at the common origin of the two diamonds in ‘‘the classical limiting situation in which classical field and particle concepts become approximately applicable’’ Žw1x, p. 323.. Thus, Dieks proposes to build up an atomic modal interpretation of the field as follows. Ži. We subdivide Žapproximately. the whole of spacetime M into a collection of non-overlapping diamond regions er , with some fixed small radius r. Žii. We choose some fixed small e and an interpolating factor Nrq e for each diamond, using its r-induced density operator Drq e and Eq. Ž1. to determine the definite-valued observables Mrq e ; RŽerq e . to be loosely associated with the origin. Žiii. Finally, we build up definite-valued observables associated with collections of diamonds, and define their joint probabilities in the usual way via Born’s rule Ždefining transition
probabilities between values of observables associated with timelike-separated diamonds using the familiar multi-time generalization of that rule employed in the decoherent histories approach.. As things stand, there is much arbitrariness in this proposal that enters into the stages Ži. and Žii.. Dieks himself recognizes the arbitrariness in the size of the partition of M chosen. He also acknowledges that this arbitrariness cannot be eliminated by passing to the limit r, e 0, because the intersection of the algebras associated with any collection of concentric diamonds converging to a point is always the trivial algebra C I w29x. Indeed, one would have thought that this undermines any attempt to formulate an atomic modal interpretation in this context, because it forces the choice of ‘atomic diamonds’ in the partition of M to be essentially arbitrary. Dieks appears to suggest that this arbitrariness will become unimportant in some classical limit of relativistic quantum field theory in which we should recover ‘‘the classical picture according to which field values are attached to spacetime points’’ Žw1x, p. 325.. But it is not sufficient for the success of a proposal for interpreting a relativistic quantum field theory that the interpretation give sensible results in the limit of classical relativistic Žor nonrelativistic. field theory. Indeed, the only relevant limit would appear to be the nonrelativistic limit; i.e., Galilean quantum field theory. But, there, one still needs to spatially smear ‘operator-valued’ fields at each point to obtain a well-defined algebra of observables in a spatial region w30x, so there will again be no natural choice to make for atomic spatial regions or algebras. There is also another, more troubling, degree of arbitrariness at step Žii. in the choice of the type I interpolating factor Nrq e about each origin point. For any fixed partition and fixed r, e ) 0, we can always sub-divide the interval Ž r,r q e . further, and then the split property implies the existence of a pair of interpolating type I factors satisfying
™
R Ž er . ; Nrq e r2 ; R Ž erq e r2 . ; Nrq e ; R Ž erq e . .
Ž 2.
The problem is that we now face a nontrivial choice deciding which of these factors’ r-induced density operators to use to pick out the definite-valued observables in the state r associated with the origin. If
R. Cliftonr Physics Letters A 271 (2000) 167–177
we pick Drq e g Nrq e , then by Ž1. all observables X g Nrq e that share the same spectral projections as Drq e will have definite values. However, no such X can lie in Mrq e r2 : Nrq e r2 , nor even in RŽerq e r2 .. The reason is that X ’s spectral projections are finite in Nrq e . So if those projections were also in the type III algebra RŽerq e r2 ., they would have to be infinite in RŽerq e r2 ., and therefore also infinite projections in Nrq e – which is impossible. Clearly we can sub-divide the interval Ž r,r q e . arbitrarily many times in this way and obtain a monotonically decreasing sequence of type I factors satisfying RŽerq e r2 nq 1 . ; Nrq e r2 n ; RŽerq e r2 n . that all interpolate between RŽer . and RŽerq e .. The sequence Nrq e r2 n 4`ns0 has no least member, and its greatest member, Nrq e , is arbitrary, because we could also further sub-divide the interval Ž r q er2,r q e . ad infinitum. Thus there is no natural choice of interpolating factor for picking out the observables definite at the origin, even if we restrict ourselves to a ‘nice’ decreasing sequence of interpolating factors of the form Nrq e r2 n 4`ns0 . On the other hand, it can actually be shown that RŽer . s F `ns0 Nrq e r2 n Žw27x, pp. 12-3; w28x, p. 426.. Furthermore, suppose Nrq e n 4`ns0 is any other decreasing type I sequence satisfying R Ž erq e nq 1 . ; Nrq e n ; R Ž erq e n . , e 0s e ,
e n ) e nq1 , lim e ns0.
Ž 3.
Then, since for any n there will be a sufficiently large nX such that Nrq e nX ; Nrq e r2 n Žand vice-versa., clearly RŽer . s F `ns0 Nrq e n, and this intersection will also be independent of e . It would seem, then, that the natural way to avoid choosing between the myriad type I factors that interpolate between RŽer . and RŽerq e . is to take the observables definite-valued at the origin to be those in the intersection F `ns 0 Mrq e n : RŽer . Žwhere, as before, Mrq e n is the modal subalgebra of Nrq e n determined via Ž1. by the density operator in Nrq e n that represents r .. Indeed, one way to understand Dieks’ proposal Žwhich may, however, not have been his original intention. is that when we take successively smaller values for e Žholding r fixed., and choose a type I interpolating factor at each stage, we should be getting progressively better approximations to the set of observables that are truly definite at the origin.
171
What better candidate for that set could there be than an intersection like F `ns 0 Mrq e n? Unfortunately, we have no guarantee that this intersection, unlike F `ns 0 Nrq e n itself, is independent of the particular sequence e n4 or its starting value e 0 s e . The reason any intersection of form F `ns 0 Nrq e n is so independent is because Nrq e n > Nrq e nq 1 for all n. But this does not imply Mrq e n > Mrq e nq 1. To take just a trivial example: when r is a pure state of Nrq e n that induces a mixed state on the proper subalgebra Nrq e nq 1of Nrq e n, Mrq e nq 1 will contain observables with dispersion in the state r , but Mrq e n will not. We conclude that there is little prospect of eliminating the arbitrariness in Dieks’ proposal and making it well-defined. One ought to look for another, intrinsic way to pick out the definite-valued observables in RŽer . that does not depend on special assumptions such as the split property.
4. The Modal interpretation for arbitrary von Neumann algebras There are two salient features of the algebra MS in Eq. Ž1. that make it an attractive set of definitevalued observables to modal interpreters. First, MS is locally determined by the quantum state DS of system S together with the structure of its algebra of observables. In particular, there is no need to add any additional structure to the standard formalism of quantum theory to pick out S’s properties. Second, the restriction of the state DS to the subalgebra MS is a mixture of dispersion-free states Žgiven by the density operators one obtains by renormalizing the Žnon-null. spectral projections of DS .. This second feature is what makes it possible to think of the observables in MS as possessing definite values distributed in accordance with standard Born rule statistics w9x. Let us see, then, whether we can generalize these two features to come up with a proposal for the definite-valued observables of a system described by an arbitrary von Neumann algebra R Žacting on some Hilbert space H. in an arbitrary state r of R. Generally, a state r of R will be a mixture of dispersion-free states on a subalgebra S : R just in
R. Cliftonr Physics Letters A 271 (2000) 167–177
172
case there is a probability measure mr on the space L of dispersion-free states of S such that
r Ž A. s
HLv Ž A . d m Ž l. , for all l
r
AgS ,
Ž 4.
where vlŽ A2 . s vlŽ A. 2 for all self-adjoint elements A g S . This somewhat cumbersome condition turns out to be equivalent Žw10x, Prop. 2.2Žii.. to simply requiring that )
r Ž w A, B x w A, B x . s 0 for all A, B g S .
Ž 5.
In particular, r can always be represented as a mixture of dispersion-free states on any Abelian subalgebra S : R. Conversely, if r is a faithful state of R, i.e., r maps no nonzero positive elements of R to zero, then the only subalgebras that allow r to be represented as a mixture of dispersion-free states are the Abelian ones. There is now an easy way to pick out a subalgebra S : R with this property, using only r and the algebraic operations available within R. Consider the following two mathematical objects explicitly defined in terms of R and r . First, the support projection of the state r in R, defined by Pr , R ' n P s P 2 s P ) g R : r Ž P . s 1 4 ,
Ž 6.
which is simply the smallest projection in R that the state r ‘makes true’. Second, there is the centralizer subalgebra of the state r in R, defined by Cr , R ' A g R : r Ž w A, B x . s 0 for all B g R 4 .
Ž 7. For any von Neumann algebra K, let Z Ž K . ' K l K X , the center algebra of K. Then it is reasonable for the modal interpreter to take as definite-valued all the observables that lie in the direct sum Z Ž Cr , R . Pr , R : R , S s Mr , R ' PrH, R R PrH, R qZ
Ž 8.
where the algebra in the first summand acts on the subspace PrH , R H and that of the second acts on Pr , R H. The state r is a mixture of dispersion-free states on Mr , R , by Ž5., because r maps all elements H of the form PrH , R R Pr , R to zero, and the product of the commutators of any two elements of Z Ž Cr , R . Pr , R also gets mapped to zero, for the trivial reason that Z Ž Cr , R . is Abelian.
The set Mr , R directly generalizes the algebra of Eq. Ž1. to the non-type I case where the algebra of observables of the system does not contain a density operator representative of the state r . Assuming the type I case, R f BŽ H . for some Hilbert space H, r is given by a density operator D on H, Pr , R is equivalent to the range projection of D, and Z Ž Cr , R . f Z Ž CD, B Ž H . .. So to show that Mr , R is isomorphic to the algebra of Eq. Ž1., it suffices to establish that Z Ž CD, B Ž H . . s DXX . It is easy to see that CD, B Ž H . s DX Žinvoking cyclicity and positivedefiniteness of the trace., thus Z Ž CD, B Ž H . . s DX l DXX . However, since DX always contains a maximal Abelian subalgebra of BŽ H . Žviz., that generated by the projections onto any complete orthonormal basis of eigenvectors for D ., we always have DXX : DX . Choosing Mr , R is certainly not the only way to pick a subalgebra S : R that is definable in terms of r and R and allows r to be represented as a mixture of dispersion-free states. There is the obvious orthodox alternative one can always consider, viz., the definite algebra of r in R, S s Or , R ' A g R : r Ž AB . s r Ž A . r Ž B . for all B g R 4 ,
Ž 9.
which coincides with the complex span of all self-adjoint members of R on which r is dispersion-free Žw10x, p. 2445.. Note, however, that we always have Or , R : Mr , R . Indeed the problem, as we have seen, is that the orthodox choice Or , R generally will contain far too few definite-valued observables to solve the measurement problem. For example, when r is faithful – and there will always be a norm dense set of states of R that are – we get just Or , R s C I. Thus it is natural for a modal interpreter to require that the choice of S : R be maximal. In the case where r is faithful, we now show that this singles out the choice S s Mr , R s Z Ž Cr , R . uniquely Žand we conjecture that a similar uniqueness result holds for the more general expression for Mr , R in Eq. Ž8., using the fact that an arbitrary state r always renormalizes to a faithful state on Pr , R R Pr , R ..
Proposition 1. Let R be a Õon Neumann algebra and r a faithful normal state of R with centralizer
R. Cliftonr Physics Letters A 271 (2000) 167–177
Cr , R : R. Then Z ( Cr , R ), the center of Cr , R , is the unique subalgebra S : R such that: 1. The restriction of r to S is a mixture of dispersion-free states. 2. S is definable solely in terms of r and the algebraic structure of R. 3. S is maximal with respect to properties 1. and 2. Proof. By 3., it suffices to show than any S : R satisfying 1. and 2. is contained in Z Ž Cr , R .. And for this, it suffices Žbecause von Neumann algebras are generated by their projections. to show that S, the subset of projections in S , is contained in Z Ž Cr , R .. Recall also that, as a consequence of 1. and the faithfulness of r , S must be Abelian. And in virtue of 2., any automorphism s : R R that preserves the state r in the sense that r ( s s r , must leave the set S Žnot necessarily pointwise. invariant, i.e., s Ž S . s S . S : CrX, R . Any unitary operator U g Cr , R defines an inner automorphism on R that leaves r invariant, therefore U S Uy1 s S . Since S is Abelian, wUPUy1 , P x s 0 for each P g S and all unitary U g Cr , R . By Lemma 4.2 of w10x Žwith V s CrXX, R s Cr , R ., this implies that P g CrX, R . S : Cr , R . Since r is faithful, there is a oneparameter group st :t g R 4 of automorphisms of R – the modular automorphism group of R determined by r Žw31x, Section 9.2. — leaving r invariant. Since Cr , R consists precisely of the fixed points of the modular group Žw31x, Prop. 9.2.14., it suffices to show that it leaves the individual elements of S fixed. For this, we use the fact that the modular group satisfies the KMS condition with respect to r : for each A, B g R, there is a complex-valued function f, bounded and continuous on the strip z g C:0 F Im z F 14 in the complex plane, and analytic on the interior of that strip, such that
™
f Ž t . s r Ž st Ž A . B . , f Ž t q i . s r Ž Bs t Ž A . . , t g R .
Ž 10 .
In fact, we shall need only one simple consequence of the KMS condition, viz., if f Ž t . s f Ž t q i . for all t g R, then f is constant Žw31x, p. 611..
173
Fix an arbitrary projection P g S. Since the modular automorphism group must leave S as a whole invariant, and S is Abelian, w st Ž P ., P Hx s 0 for all t g R. However, there exists a function f with the above properties such that f Ž t . s r Ž st Ž P . P H . s r Ž P H st Ž P . . sf Ž tqi. , tgR , so it follows that f is constant. In particular, since s 0 Ž P . P H s PP H s 0, f is identically zero, and r Ž st Ž P . P H. s 0 for all t g R. And since st Ž P . and P H are commuting projections, their product is a Žpositive. projection, so that the faithfulness of r requires that st Ž P . P H s 0, or equivalently st Ž P . s st Ž P . P, for all t g R. Running through the exact same argument, starting with P Hg S in place of P, yields st Ž P H. s st Ž P H. P H , or equivalently, st Ž P . P s P, for all t g R. Together with st Ž P . s st Ž P . P, this implies that st Ž P . s P for all t g R. QED.
The choice Mr , R s Z Ž Cr , R . has another feature that generalizes a natural consequence of the modal interpretation of nonrelativistic quantum theory. Suppose the universal state x g HU defines a faithful state r x on both BŽ HS . and BŽ HS .. This requires that dim HS s dim HS s n Žpossibly `., and, furthermore, that any Schmidt decomposition of the state vector x relative to the factorization HU s HS m HS takes the form n
xs
Ý c i Õi m wi , c i / 0 for all i s 1 to n,
Ž 11 .
is1
where the vectors Õi and wi are complete orthonormal bases in their respective spaces. As is wellknown, for each distinct eigenvalue l˜ j for DS , the span of the vectors Õi for which < c i < 2 s l˜ j coincides with the range of the l˜ j-eigenprojection of DS , and similarly for DS . Consequently, there is a natural bijective correspondence between the properties represented by the projections in the two sets MS s DSXX and MS s DSXX : any definite property S happens to possess is strictly correlated to a unique property of its environment S that occurs with the same fre-
174
R. Cliftonr Physics Letters A 271 (2000) 167–177
quency. More formally, for any P g MS , there is a unique P g MS satisfying
zero, since r x is faithful.. Letting A g Cx, R be the double of B in R, we get
Ž x , PPx . s Ž x , Px . s Ž x , Px . .
PBx y BPx s PAx y BPx s APx y PBx
Ž 12 .
To see this, note that any P g MS Žin this case, the set of all functions of DS . is a sum of spectral projections of DS . Let P g MS be the sum of the corresponding spectral projections of DS for the same eigenvalues. Then it is evident from the form of the state expansion in Ž11. that P has the property in Ž12., and no other projection in MS does. This has led some non-atomic modal interpreters, such as Kochen w11x, to interpret each property P of S, not as a property that S possesses absolutely, but only in relation to its environment S possessing the corresponding property P. For a general von Neumann factor R, Ž R j R X .XX s BŽ H . need not be isomorphic to the tensor product R m R X Žparticularly when R is type III, for then R m R X must be type III as well.. Therefore, there is no direct analogue of a Schmidt decomposition for a pure state x g H relative to the factorization Ž R j R X .XX of BŽ H .. Nevertheless, we show next that there is still the same strict correlation between definite properties in Mr x , R s Z Ž Cr x , R . and Mr x , R X s Z Ž Cr x , R X .. Proposition 2. Let R be a Õon Neumann algebra acting on a Hilbert space H, and suppose x g H induces a state r x that is faithful on both R and R X . Then for any projection P g Z Ž Cr x , R . , there is a unique projection P g Z Ž Cr x , R X . such that Ž x, PPx . s Ž x, Px . s Ž x, Px . . Proof. For any fixed A g R, call an element B g R X a double for A Žin state x . just in case Ax s Bx and A) x s B ) x. By an elementary application of modular theory, Werner Žw32x, Section 2. has shown that Cr x , R consists precisely of those elements of R with doubles in R X Žwith respect to x .. Moreover, the double of any element of R clearly has to be unique, by the faithfulness of r x on R X . Now it is easy to see Žagain using the faithfulness of r x . that the double of any projection P g Cr x , R is a projection P g Cr x , R X satisfying Ž12.. We claim that whenever P g Cx,X R , we have P g Cx,X R X . For this, it suffices to show P g Cx,X R implies that for arbitrary B g Cx, R X , w P, B x x s 0 Žand then w P, B x itself is
s APx y PAxs0,
Ž 13 .
as required. Finally, were there another projection P˜ g Z Ž Cr x , R X . satisfying Ž12., then by exploiting the ˜ .s fact that P is P ’s double in R X , we get Ž x, PPx ˜ .; or, equivalently, Ž x, Px . s Ž x, Px
˜ . s 0. Ž x , PP˜ H x . s Ž x , P H Px
Ž 14 .
Since Z Ž Cr x , R X . is Abelian, both PP˜ H and P H P˜ are Žpositive. projections in R. But as r x is faithful on R, Eqs. Ž14. entail that PP˜ H s P H P˜ s 0, which ˜ as required for uniquein turn implies that P s P, ness. QED. Let us return now to the problem of picking out a set of definite-valued observables localized in a diamond region with associated algebra RŽer .. Let r be any pure state of the field that induces a faithful state on RŽer . and its commutant; for example, r could be the vacuum or any one of the dense set of states of a field with bounded energy Žby the Reeh– Schlieder theorem – see w27x, Thm. 1.3.1.. By Proposition 1, the definite-valued observables in R Žer . are simply those in the subalgebra Z Ž Cr , R Že r . .. Note that this proposal yields observables all of which have an exact spacetime localization within the open set er and are picked out intrinsically by the local algebra RŽer . and the field state r . Contrary to Dieks’ pessimistic conclusion, we can take open spacetime regions as fundamental for determining the definite-valued observables. In fact, since this proposal works independent of the size of r, it could also be embraced by non-atomic modal interpreters not wishing to commit themselves to a particular partition of the field into subsystems Žor to thinking from the outset in terms of approximately point-localized field observables.. Finally, note that since the algebra of a diamond region er satisfies duality with respect to the algebra of its spacelike complement er X , i.e., RŽer .X s RŽer X . Žw26x, p. 145., Proposition 2 tells us that there is a natural bijective correspondence between the properties in Z Ž Cr , R Že . . and strictly correlated properties r
R. Cliftonr Physics Letters A 271 (2000) 167–177
in Z Ž Cr , R ŽeXr . . associated with the complement region.
5. A potential difficulty with ergodic states We have seen that there is, after all, a well-motivated and unambiguous prescription extending the standard modal interpretation of nonrelativistic quantum theory to the local algebras of quantum field theory. We also, now, have a natural standard of comparison with Galilean quantum field theory. At least in the case of free fields, it is possible to build up local algebras in M from spatially smeared ‘field algebras’ defined on spacelike hyperplanes in M. A diamond region corresponds to the domain of dependence of a spatial region in a hyperplane, and it can be shown that the algebra of that spatial region will also be type III and, indeed, coincide with its domain of dependence algebra Žw27x, Prop. 3.3.2, Thm. 3.3.4.. These type III spatial algebras in M, and the definite-valued observables therein, are what should be compared, in the nonrelativistic limit, to the corresponding equal time spatial algebras defined on simultaneity slices of Galilean spacetime. Unfortunately, since the algebras in the Galilean case are invariably type I Žw27x, p. 35., this limit is bound to be mathematically singular, and its physical characterization needs to be dealt with carefully. But this is a problem for any would-be interpreter of relativistic quantum field theory, not just modal interpreters. All we should require of them, at this stage, is that they be able to say something sensible in the relativistic case about the local observables with definite values Žwhich was, indeed, Dieks’ original goal.. However, as we now explain, it is not clear whether even this goal can be attained. If R f BŽ H . is type I, it possesses at most one faithful state r such that Z Ž Cr , R . s C I. This is easy to see, because if the density operator D g BŽ H . represents r , Z Ž Cr , R . f DXX , and DXX s C I implies that D itself must be a multiple of the identity. So when H is finite-dimensional, we must have D s Irdim H, the unique maximally mixed state, and in the infinite-dimensional case, no such density operator even exists. Elsewhere Dieks w33x has argued convincingly that there is no problem when a system, occupying a maximally mixed state, possesses only
175
trivial properties, because such states are rare and highly unstable under environmental decoherence Žcf. w2x, pp. 99-100.. However, the situation is quite different for the local algebras of algebraic quantum field theory. In all physically reasonable models of the axioms of the theory, every local algebra RŽ O . is isomorphic to the unique Žup to isomorphism. hyperfinite type III 1 factor Žw26x, Section 5.6; w28x, Section 17.2.. In that case, there is a novel way to obtain Z Ž Cr , R ŽO . . s C I, namely, when the state r of RŽ O . is an ergodic state w34,35x, i.e., r possesses a trivial centralizer in RŽ O .. ŽWere R a non-Abelian type I factor, this would be impossible, since DX s C I implies DXX s BŽ H . f R, which is patently false.. In fact, we have the following result. Proposition 3. If R is the hyperfinite type III 1 factor, there is a norm dense set of unit Õectors in the Hilbert space H on which R acts that induce faithful ergodic states on R (i.e., with triÕial centralizers in R). Proof: First recall the following facts provable from the axioms of algebraic quantum field theory: Ži. the vacuum state of a field on M has a trivial centralizer in the algebra of any Rindler wedge Žw28x, Section 16.1.1.; Žii. the vacuum is faithful for any wedge algebra Žby the Reeh–Schlieder theorem.; and Žiii. wedge algebras are hyperfinite type III 1 factors Žw28x, Ex. 16.2.14, pp. 426-7.. Since being faithful and having a trivial centralizer are isomorphic invariants, it follows that any instantiation R of the hyperfinite type III 1 factor possesses at least one faithful normal state r with trivial centralizer Ževen when R is the algebra of a bounded open region, like a diamond.. Now since R is type III, all its states are vector states Žcombine w36x, Cor. 2.9.28 with w31x, Thm. 7.2.3.; in particular, r s r x for some unit vector x g H. Furthermore, by the homogeneity of the state space of type III 1 factors Žw37x, Cor. 6., the set of all unit vectors of the form UU X x, with U g R and U X g R X unitary operators, lies dense in H. But clearly any such vector must again induce a faithful state on R with trivial centralizer. QED. Combining Propositions 1 and 3, there will be a whole host of states of any relativistic quantum field
176
R. Cliftonr Physics Letters A 271 (2000) 167–177
in which the modal interpreter is forced to assert that no nontrivial local observables have definite values! Note, however, that while the set of field states ergodic for any given type III 1 local algebra is always dense, this does not automatically imply that such states are typical or generic. Indeed, results of Summers and Werner Žw38x, particularly Cor. 2.4. imply that for any local diamond algebra RŽer ., there will also always be a dense set field states whose centralizers in RŽer . contain the hyperfinite type II 1 factor, and so will not be trivial. Still, the modal interpreter needs to provide some physical reason for neglecting the densely many field states that do yield trivial definite-valued observables locally. Obviously instability under decoherence is no longer relevant. Perhaps one could try to bypass Proposition 1 by exploiting extra structure not contained in the particular field state and local algebra to pick out the definite-valued observables in a region. For example, one might try to exploit the field’s total energymomentum operator, and, in particular, its generator of time evolution. In the context of the nonrelativistic modal interpretation, Bacciagaluppi et al. w20x Žcf. also w15x, p. 1181. have successfully invoked the analytic properties of the time evolution of the spectral projections of a system’s reduced density operator DS to avoid discontinuities that occur in the definite-valued set MS at moments of time where the multiplicity of the eigenvalues of DS changes. In particular, their methods yield a natural dynamical way, independent of instability considerations, to avoid the trivial definite-valued sets determined by maximally mixed density operators. So one might hope that these same dynamical methods could be extended to type III 1 algebras so as to yield a richer set of properties in a local region than Proposition 1 allows for ergodic states. In any case, modal interpreters need to do more work to show that their interpretation yields sensible local properties in quantum field theory Ževen before one considers, with Dieks, how to define Lorentz invariant decoherent histories of properties.. Acknowledgements The author wishes to thank Hans Halvorson Žfor supplying the argument immediately following Eq.
Ž2. and suggesting the use of the KMS condition in the proof of Proposition 1., and Reinhard Werner Žfor helpful correspondence about centralizers and inspiring Proposition 2..
References w1x D. Dieks, Phys. Lett. A 265 Ž2000. 317. w2x R. Healey, The Philosophy of Quantum Mechanics: An Interactive Interpretation, Cambridge Univ. Press, Cambridge, 1989. w3x B.C. van Fraassen, Quantum Mechanics: An Empiricist View, Clarendon Press, Oxford, 1991. w4x R. Clifton, Brit. J. Phil. Sci. 47 Ž1996. 371. w5x J. Bub, Interpreting the Quantum World, Cambridge Univ. Press, Cambridge, 1997. w6x R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998. w7x D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998. w8x J. Zimba, R. Clifton in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 69. w9x R. Clifton, in: J. Butterfield, C. Pagonis ŽEds.., From Physics to Philosophy, Cambridge Univ. Press, Cambridge, 1999 ŽChap. 2.. w10x H. Halvorson, R. Clifton, Int. J. Theor. Phys. 38 Ž1999. 2441. w11x S. Kochen, in: P.J. Lahti, P. Mittelstaedt ŽEds.., Symposium on the Foundations of Modern Physics, World Scientific, Singapore, p. 151. w12x R. Clifton, Brit. J. Phil. Sci. 46 Ž1995. 33. w13x R. Clifton, Ann. ŽNY. Acad. Sci. 755 Ž1995. 570. w14x G. Bacciagaluppi, M. Hemmo, Stud. Hist. Phil. Mod. Phys. 27B Ž1996. 239. w15x G. Bacciagaluppi, M. Dickson, Found. Phys. 29 Ž1999. 1165. w16x D. Dieks in: R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998, p. 144. w17x R. Spekkens, J. Sipe, Contingent Determinate Variables in Modal Interpretations of Quantum Mechanics Ž2000., preprint. w18x R. Spekkens, J. Sipe, quant-phr0003092. w19x P. Vermaas in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 103. w20x G. Bacciagaluppi, M. Donald, P. Vermaas, Helv. Phys. Act. 68 Ž1995. 679. w21x P. Vermaas, Stud. Hist. Phil. Mod. Phys. 27 Ž1996. 133. w22x M. Dickson, in: R. Healey, G. Hellman ŽEds.., Quantum Measurement: Beyond Paradox, Univ. of Minnesota Press, Minneapolis, 1998, p. 160. w23x J. Ax, S. Kochen, quant-phr9905077.
R. Cliftonr Physics Letters A 271 (2000) 167–177 w24x M. Dickson, R. Clifton in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 9. w25x M. Hemmo, in: D. Dieks, P. Vermaas ŽEds.., The Modal Interpretation of Quantum Mechanics, Kluwer, Dordrecht, 1998, p. 253. w26x R. Haag, Local Quantum Physics, 2nd ed., Springer, Berlin, 1992. w27x S. Horuzhy, Introduction to Algebraic Quantum Field Theory, Kluwer, Dordrecht, 1988. w28x H. Baumgartel, M. Wollenberg, Causal Nets of Operator ¨ Algebras, Springer, Berlin, 1992. w29x A. Wightman, Ann. Inst. Henri Poincare´ 1 Ž1964. 403.
177
w30x M. Requardt, J. Phys. A: Math. Gen. 15 Ž1982. 3715. w31x R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, American Mathematical Society, R.I. Providence, 1997. w32x R. Werner, quant-phr9910077 w33x D. Dieks, Phys. Rev. A 49 Ž1994. 2290. w34x R. Herman, M. Takesaki, Comm. Math. Phys. 19 Ž1970. 142. w35x L. Barnett, Proc. Am. Math. Soc. 123 Ž1995. 543. w36x S. Sakai, C ) -Algebras and W ) -algebras, Springer, Berlin, 1971. w37x A. Connes, E. Størmer, J. Func. Anal. 26 Ž1978. 187. w38x S. Summers, R. Werner, Ann. Inst. Henri Poincare´ 49 Ž1988. 215.