Abstract
Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey this programme in terms of algebraic quantum theory.
Highlights
The mathematical formalism of quantum mechanics is open to interpretation
This article surveys a mathematical reformulation of quantum mechanics in terms of classical mechanics, intended to bring the interpretational issues with the former to a head
Known obstructions to hidden variable interpretations merely say that states cannot be located with exact precision in the state space, and are circumvented via open regions of states; the uncertainty principle cannot be expressed and poses no interpretational problem; the measurement problem is obviated because the new notion of state space incorporates all classical data resulting from possible measurements
Summary
The mathematical formalism of quantum mechanics is open to interpretation. For example, the possibility of deterministic hidden variables, the uncertainty principle, the measurement problem, and the reality of the wave function, are all up for debate. (The first and the last have rigorous restrictions: hidden variables by the Bell inequalities [1] and the Kochen–Specker theorem [2], discussed below, and reality of the wave function by the Pusey–Barrett–Rudolph theorem [3].) Classical mechanics shares none of those interpretational questions. If we take dynamics into account, the new notion of configuration space, called an active lattice: yields the same predictions as traditional quantum mechanics. This programme branches into a number of related themes, spread over the literature; see the extensive bibliography. We will use an n-level physical system as a running example to illustrate new notions (though many results have exceptions for n ≤ 2, and most interesting features occur in infinite dimension) The rest of this introduction summarizes the framework and discusses four salient features, before giving an overview of the rest of this article
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