, Volume 107, Issue 1, pp 83143
Mathematical quantum theory I: Random ultrafilters as hidden variables
 William BoosAffiliated with
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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).
 Title
 Mathematical quantum theory I: Random ultrafilters as hidden variables
 Journal

Synthese
Volume 107, Issue 1 , pp 83143
 Cover Date
 199604
 DOI
 10.1007/BF00413903
 Print ISSN
 00397857
 Online ISSN
 15730964
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors

 William Boos ^{(1)}
 Author Affiliations

 1. 1427 E. Davenport, 52245, Iowa City, IA, USA