Interpretations of Quantum Mechanics in Terms of Beable Algebras

雄一郎 北島

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.

Jochen Pade

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Jeroen C. Vink

An interpretation of quantum mechanics in terms of classical concepts, ``beables,'' due to de Broglie, Bohm, and Bell (BBB) is generalized and further developed. By assuming that all physical quantities take discrete values on sufficiently small scales, we can use this interpretation to give trajectories for all possible quantities, including the position of a particle, its spin, etc. When applied to position, it is shown that, in the continuum limit, this interpretation reduces to the causal one of Bohm. As an illustration, the BBB trajectories are computed explicitly in two simple models.