Interpretations of algebraic quantum theory in terms of beable algebras

Yuichiro Kitajima

Rob Clifton

Rob Clifton

John Bell once argued that one ought to select, out of the 'observables' of quantum theory, some subset of 'beables' that can be consistently ascribed determinate values. Moreover, this subset should be selected so as to guarantee (among other things) that we can dispense with the orthodox interpretation's loose talk about 'measurement values': "...the probability of a beable being a particular value would be calculated just as was formerly calculated the probability of observing that value". Working in the framework of C*-algebras (in particular, Segal algebras), I propose an algebraic characterization of those subsets of bounded observables of a quantum system that can have beable status with respect to any (fixed) state of the system. It turns out that observables with beable status in a state need not all commute (a possibility Bell himself does not consider), but they must at least form a certain kind of 'quasicommutative' subalgebra determined by the state. A virtue of the analysis is that it applies to beables with continuous spectra, usually neglected in discussions of the no-hidden-variables theorems. In the (very) special case where the algebra of observables for a system is representable on a finite-dimensional Hilbert space, I give a complete characterization of the maximal beable subalgebras determined by any state of the system; the infinite-dimensional case remains open. These results are discussed in relation to previous results of a similar nature, to 'no-collapse' interpretations of quantum mechanics, and to algebraic relativistic quantum field theory.

J. E. Roberts, Gert Roepstorff

Masanao Ozawa

While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.

Laura Ruetsche

This chapter argues that, due to the circumstances described in Chapter 7, techniques developed for interpreting ordinary QM lack straightforward extension to the interpretation of QM∞. These techniques are variations on the maximal beable approach inspired by John Bell, which reduces the task of interpreting a quantum theory to the task of isolating from its family of physical magnitudes a largest set of observables which can be attributed values simultaneously. The chapter links the satisfactory execution of this task—an execution which grounds explanations of quantum state preparation–to the presence of atoms in the von Neumann algebras typical of ordinary QM. What stymies the maximal beable approach to QM∞ is the ubiquity of non-atomic von Neumann algebras.