Mathematical quantum theory I: Random ultrafilters as hidden variables
 William Boos
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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 mathematicalphysical) assertions — a framework, that is, in which the ‘truthvalues’ for such assertions are elements of iterated boolean measurealgebras \(\mathbb{A}\) (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).
The essay itself employs constructions of Takeuti's booleanvalued 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 illdefined: 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 categorytheoretically natural correlation between ‘ideal outcomes’ of quantum measurements u of a ‘universal wave function’, and possible ‘worlds’ of an EverettWheelerlike manyworldstheory.
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 mathematicalphysical theory T, where T includes enough settheory to derive all the analysis needed for von Neumannalgebraic formulations of quantum theory.
The ‘worlds’ of this framework can then be given a genuine modeltheoretic construal: they are ‘random’ models M(u) determined by Mrandom 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 projectionvalued measure) to such Mrandom reals u. If \(\mathbb{L}\) is the ‘universal’ Lebesgue measurealgebra 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 Mrandom 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 timeevolution. A preliminary account of the latter — an obvious prerequisite for any serious ‘manyworlds’theory, given that Everett's original intention was to integrate timeevolution and wavefunction collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measurealgebras to reinterpret timeordered 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 measurealgebraic tensorproducts of the ‘universal’ measurealgebra L, one can reinterpret formal timeevolution so that it becomes ‘internal’ to the ‘universal’ random models M(u).
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 Title
 Mathematical quantum theory I: Random ultrafilters as hidden variables
 Journal

Synthese
Volume 107, Issue 1 , pp 83143
 Cover Date
 19960401
 DOI
 10.1007/BF00413903
 Print ISSN
 00397857
 Online ISSN
 15730964
 Publisher
 Kluwer Academic Publishers
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 William Boos ^{(1)}
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 1. 1427 E. Davenport, 52245, Iowa City, IA, USA