Rob Clifton

A uniqueness theorem for ‘no collapse’ interpretations of quantum mechanics

Abstract We prove a uniqueness theorem showing that, subject to certain natural constraints, all ‘no collapse’ interpretations of quantum mechanics can be uniquely characterized and reduced to the choice of a particular preferred observable as determinate (definite, sharp). We show how certain versions of the modal interpretation, Bohms ‘causal’ interpretation, Bohrs complementarity interpretation, and the orthodox (Dirac-von Neumann) interpretation without the projection postulate can be recovered from the theorem. Bohrs complementarity and Einsteins realism appear as two quite different proposals for selecting the preferred determinate observable—either settled pragmatically by what we choose to observe, or fixed once and for all, as the Einsteinian realist would require, in which case the preferred observable is a ‘beable’ in Bells sense, as in Bohms interpretation (where the preferred observable is position in configuration space).

The definability of objective becoming in Minkowski spacetime

In his recent article ‘On Relativity Theory and Openness of the Future’ (1991), Howard Stein proves not only that one can define an objective becoming relation in Minkowski spacetime, but that there is only one possible definition available if one accepts certain natural assumptions about what it is for becoming to occur and for it to be objective. Stein uses the definition supplied by his proof to refute an argument due to Rietdijk (1966, 1976), Putnam (1967) and Maxwell (1985, 1988) that Minkowski spacetime leaves no room for objective becoming whatsoever. However, Steins proof does not seem to go far enough. By considering only what events have become from the standpoint of any given event, Steins uniqueness proof fails from the outset to allow for a more general kind of becoming whereby it is understood to occur from the standpoint of events on the particular worldlines followed by observers. This suggests that there may, after all, be more than one way to define objective becoming in Minkowski spacetime once each observers worldline is allowed to figure in the definition. This suspicion is further aroused by two recent proposals for objective, worldline-dependent becoming due to Peacock (1992) and Muller (1992) who advocate ways of defining becoming that are not equivalent to the definition Steins uniqueness proof delivers. Nevertheless, we show that Steins uniqueness proofcan be extended in a natural way to cover this more general kind of becoming, provided one does not enrich standard Minkowski spacetime by privileging certain sets of worldlines over others in an unwarranted manner. Thus we aim to reinforce Steins point that standard Minkowski spacetime does make room for objective becoming, but in essentially only one way, despite arguments and proposals to the contrary.

Independently Motivating the Kochen—Dieks Modal Interpretation of Quantum Mechanics

The distinguishing feature of ‘modal’ interpretations of quantum mechanics is their abandonment of the orthodox eigenstate–eigenvalue rule, which says that an observable possesses a definite value if and only if the system is in an eigenstate of that observable. Kochens and Dieks new biorthogonal decomposition rule for picking out which observables have definite values is designed specifically to overcome the chief problem generated by orthodoxys rule, the measurement problem, while avoiding the no-hidden-variable theorems. Otherwise, their new rule seems completely ad hoc. The ad hoc charge can only be laid to rest if there is some way to give Kochens and Dieks rule for picking out which observables have definite values some independent motivation. And there is, or so I will argue here. Specifically, I shall show that theirs is the only rule able to save Schrödingers cat from a fate worse than death, and sidestep the Bell–Kochen–Specker no-hidden-variables theorem, once we impose four independently natural conditions on such rules

The Properties of Modal Interpretations of Quantum Mechanics

Orthodox quantum mechanics includes the principle that an observable of a system possesses a well-defined value if and only if the presence of that value in the system is certain to be confirmed on measurement. Modal interpretations reject the controversial ‘only if’ half of this principle to secure definite outcomes for quantum measurements that leave the apparatus entangled with the object it has measured. However, using a result that turns on the construction of a Kochen–Specker contradiction, I argue that modal interpretations cannot deliver a metaphysically tenable conception of properties in quantum mechanics unless they also abandon the less controversial ‘if’ half of the orthodox principle.

Unremarkable contextualism: Dispositions in the Bohm theory

One way to characterize dispositions is to take them to be reducible to categorical properties plus experimental arrangements. We argue that this view applied to Bohm s ontological interpretation of quantum theory provides a good picture of the unremarkable nature of spin in that interpretation, and so explains how a simple realism of possessed values may be retained in the face of Kochen and Speckers theorem. With this in mind we discuss Redheads influential analysis of Kochen and Speckers theorem which does nor appear to allow for the above view.

QuasiBoolean algebras and simultaneously definite properties in quantum mechanics

We define and characterize a new abstract notion of “quasiBoolean algebra,” intermediate in nature between an (ortho)lattice and a Boolean algebra. It will turn out that such algebras are natural candidates for representing the simultaneously definite properties of a quantum system. We then prove a general theorem about maximal quasiBoolean subalgebras of an ortholattice which we use to derive a number of different proposals in the literature for what properties of a quantum system should be regarded as simultaneously definite.

Why modal interpretations of quantum mechanics must abandon classical reasoning about physical properties

Modal interpretations of quantum mechanics propose to solve the measurement problem by rejecting the orthodox view that in entangled states of a system which are nontrivial superpositions of an observables eigenstates, it is meaningless to speak of that observable as having a value or corresponding to a property of the system. Though denying this is reminiscent of how hidden-variable interpreters have challenged orthodox views about superposition, modal interpreters also argue that their proposals avoid any of the objectionable features of physical properties that beset hidden-variable interpretations, like contextualism and nonlocality. Even so, I shall prove that modal interpreters of quantum mechanics are still committed to giving up at least one of the following three conditions characteristic of classical reasoning about physical properties: (1) Properties certain to be found on measuring a system should be counted as intrinsic properties of the system. (2) If two propositions stating the possession of two intrinsic properties by the system are regarded as meaningful, then their conjunction should also correspond to a meaningful proposition about the system possessing a certain intrinsic property; and similarly for disjunction and negation. (3) The intrinsic properties of a composite system should at least include (though need not be exhausted by) the intrinsic properties of its parts. Conditions 1–3 are by no means undeniable. But the onus seems to be on modal interpreters to tell us why rejecting one of these is preferable to an ontology of properties incorporating contextualism and nonlocality.

Making Sense of the Kochen‐Dieks “No‐Collapse” Interpretation of Quantum Mechanics Independent of the Measurement Problem

The quantum measurement problem is really the by-product of adopting a particular, rather conservative, view about the conditions necessary for an observable to possess a definite value. Because this view seems to have become entrenched by von Neumann’s classic analysis of quantum measurement,’ I shall call it vNdefiniteness. It asserts that an observable possesses a definite value at any given time only if one of its eigenvalues can be predicted with certainty using the state vector that applies to the system at that time. One can do no better to see the awkward consequences that vN-definiteness has for quantum measurement than to recall Schrodinger’s simple, but striking, example.2 Think of a device that detects whether or not a radioactive atom has decayed. If the answer is vN-definitely Yes, the decay products trigger off a chain of events rapidly culminating in the death of a cat:

Uniqueness of prime factorizations of linear operators in quantum mechanics

Abstract We prove a recent conjecture of Healeys — that every non-null projection operator in a tensor-product Hilbert space has a unique factorization into prime projection operators — by deriving it from a more general uniqueness result that we show to hold for factorizations of arbitrary linear operators.