Jeffrey Bub

I define sublaltices of quantum propositions that can be taken as having determinate (but perhaps unknown) truth values for a given quantum state, in the sense that sufficiently many two-valued maps satisfying a Boolean homomorphism condition exist on each determinate sublattice to generate a Kolmogorov probability space for the probabilities defined by the slate. I show that these sublattices are maximal, subject to certain constraints, from which it follows easily that they are unique. I discuss the relevance of this result for the measurement problem, relating it to an early proposal by Jauch and Piron for defining a new notion of state for quantum systems, to a recent uniqueness proof by Clifton for the sublattice of propositions specified as determinate by modal interpretations of quantum mechanics that exploit the polar decompostion theorem, and to my own previous suggestions for interpreting quantum mechanics without the projection postulate

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system

The properties of classical and quantum systems are characterized by different algebraic structures. We know that the properties of a quantum mechanical system form a partial Boolean algebra not embeddable into a Boolean algebra, and so cannot all be co-determinate. We also know that maximal Boolean subalgebras of properties can be (separately) co-determinate. Are there larger subsets of properties that can be co-determinate without contradiction? Following an analysis of Bohrs response to the Einstein-Podolsky-Rosen objection to the complementarity interpretation of quantum mechanics, a principled argument is developed justifying the selection of particular subsets of properties as co-determinate for a quantum system in particular physical contexts. These subsets are generated by sets of maximal Boolean subalgebras, defined in each case by the relation between the quantum state and a measurement (possibly, but not necessarily, the measurement in terms of which we seek to establish whether or not a particular property of the system in question obtains). If we are required to interpret quantum mechanics in this way, then predication for quantum systems is quite unlike the corresponding notion for classical systems.

Quantum theory is a probabilistic theory that embodies notoriously striking correlations, stronger than any that classical theories allow but not as strong as those of hypothetical ‘super-quantum’ theories. This raises the question ‘Why the quantum?’—whether there is a handful of principles that account for the character of quantum probability. We ask what quantum-logical notions correspond to this investigation. This project isn’t meant to compete with the many beautiful results that information-theoretic approaches have yielded but rather aims to complement that work

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 determine (definite, sharp). We show how certain versions of the modal interpretation, Bohm's 'causal' interpretation, Bohr's complementarity interpretation, and the orthodox (Dirac-von Neumann) interpretation without the projection postulate can be recovered from the theorem. Bohr's complementarity and Einstein's 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 Bell's sense, as in Bohm's interpretation (where the preferred observable is position in configuration space).

In my reconstruction of Bohr's reply to the Einstein-Podolsky-Rosen argument, I pointed out that Bohr showed explicitly, within the framework of the complementarity interpretation, how a locally maximal measurement on a subsystem S2 of a composite system S1+S2, consisting of two spatially separated subsystems, can make determinate both a locally maximal Boolean subalgebra for S2 and a locally maximal Boolean subalgebra for S1. As it stands, this response is open to an objection. In this note, I show that meeting the objection requires a modification of the complementarity thesis concerning what propositions can be taken as determinate, or what observables can be raken to have values, in a given measurement context

It is argued that the measurement problem reduces to the problem of modeling quasi-classical systems in a modified quantum mechanics with superselection rules. A measurement theorem is proved, demonstrating, on the basis of a principle for selecting the quantities of a system that are determinate (i.e., have values) in a given state, that after a suitable interaction between a systemS and a quasi-classical systemM, essentially only the quantity measured in the interaction and the indicator quantity ofM are determinate. The theorem justifies interpreting the noncommutative algebra of observables of a quantum mechanical system as an algebra of “beables,” in Bell's sense

I formulate the interpretation problem of quantum mechanics as the problem of identifying all possible maximal sublattices of quantum propositions that can be taken as simultaneously determinate, subject to certain constraints that allow the representation of quantum probabilities as measures over truth possibilities in the standard sense, and the representation of measurements in terms of the linear dynamics of the theory. The solution to this problem yields a modal interpretation that I show to be a generalized version of Bohm's hidden variable theory. I argue that unless we alter the dynamics of quantum mechanics, or accept a for all practical purposes solution, this generalized Bohmian mechanics is the unique solution to the problem of interpretation.

Bell's proof purports to show that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. It is shown that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated. We consider an example of non- local hidden variable theory which reproduces the quantum statistics. A simple extension of the theory, which preserves the non- local character, alters the statistics in such a way that Bell's inequality is satisfied

Emilio Santos has argued (Santos, Studies in History and Philosophy of Physics http: //arxiv-org/abs/quant-ph/0410193) that to date, no experiment has provided a loophole-free refutation of Bell’s inequalities. He believes that this provides strong evidence for the principle of local realism, and argues that we should reject this principle only if we have extremely strong evidence. However, recent work by Malley and Fine (Non-commuting observables and local realism, http: //arxiv-org/abs/quant-ph/0505016) appears to suggest that experiments refuting Bell’s inequalities could at most confirm that quantum mechanical quantities do not commute. They also suggest that experiments performed on a single system could refute local realism. In this paper, we develop a connection between the work of Malley and Fine and an argument by Bub from some years ago [Bub, The Interpretation of Quantum Mechanics, Chapter VI(Reidel, Dodrecht,1974)]. We also argue that the appearance of conflict between Santos on the one hand and Malley and Fine on the other is a result of differences in the way they understand local realism

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed “strict,” and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell