# Mathematical quantum theory I: Random ultrafilters as hidden variables

DOI: 10.1007/BF00413903

- Cite this article as:
- Boos, W. Synthese (1996) 107: 83. doi:10.1007/BF00413903

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## 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)**.