, Volume 107, Issue 1, pp 83-143

Mathematical quantum theory I: Random ultrafilters as hidden variables

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic ‘truth’ for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the ‘truth-values’ for such assertions are elements of iterated boolean measure-algebras \(\mathbb{A}\) (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).

The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a metamathematical interpretation of ideas sometimes considered disparate, ‘heuristic’, or simply ill-defined: the ‘collapse of the wave function’, for example; Everett's many worlds'-construal of quantum measurement; and a ‘natural’ product space of contextual (nonlocal) ‘hidden variables’.

More precisely, these constructions permit us to write down a category-theoretically natural correlation between ‘ideal outcomes’ of quantum measurements u of a ‘universal wave function’, and possible ‘worlds’ of an Everett-Wheeler-like many-worlds-theory.

The ‘universal wave function’, first, is simply a pure state of the Hilbert space (L 2([0, 1]) M in a model M an appropriate mathematical-physical theory T, where T includes enough set-theory to derive all the analysis needed for von Neumann-algebraic formulations of quantum theory.

The ‘worlds’ of this framework can then be given a genuine model-theoretic construal: they are ‘random’ models M(u) determined by M-random elements u of the unit interval [0, 1], where M is again a fixed model of T.

Each choice of a fixed basis for a Hilbert space H in a model of M of T then assigns ‘ideal’ spectral values for observables A on H (random ultrafilters on the range \(\mathbb{A}\) of A regarded as a projection-valued measure) to such M-random reals u. If \(\mathbb{L}\) is the ‘universal’ Lebesgue measure-algebra on [0, 1], these assignments are interrelated by the spectral functional calculus with value 1 in the boolean extension (V( \(\mathbb{L}\) )) M , and therefore in each M(u).

Finally, each such M-random u also generates a corresponding extension M(u) of M, in which ‘ideal outcomes’ of measurements of all observables A in states are determined by the assignments just mentioned from the random spectral values u for the ‘universal’ ‘position’-observable on L 2([0, 1]) in M.

At the suggestion of the essay's referee, I plan to draw on its ideas in the projected sequel to examine more recent ‘modal’ and ‘decoherence’-interpretations of quantum theory, as well as Schrödinger's traditional construal of time-evolution. A preliminary account of the latter — an obvious prerequisite for any serious ‘many-worlds’-theory, given that Everett's original intention was to integrate time-evolution and wave-function collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measure-algebras to reinterpret time-ordered processes of measurements (determined, for example, by a given Hamiltonian observable H in M) as individual measurements in somewhat more complexly defined extensions M(u) of M.

In plainer English: if one takes a little care to distinguish boolean- from measure-algebraic tensor-products of the ‘universal’ measure-algebra L, one can reinterpret formal time-evolution so that it becomes ‘internal’ to the ‘universal’ random models M(u).