Mathematical quantum theory. I: Random ultrafilters as hidden variables. (English) Zbl 0935.81002
This paper aims to provide a natural setting for the stochasticity of von Neumann’s classical formulation of non-relativistic quantum mechanics in a stochastic metatheory of random extensions \(V\)(A) of a set-theoretic universe \(V\) by Boolean algebras \(A\) of projections in a separable Hilbert space \(H\). These Boolean algebras are ranges of projection-valued selfadjoint operators on \(H\), i.e they are related to spectral values of the quantum observables. The ultimate aim of this approach is apparently to provide a framework of iterated measure-algebraic truth for mathematical assertions, a framework that is in which the truth-values for such assertions are elements of iterated Boolean measure-algebras.
As the quoted aims perhaps suggest, this long paper is an unusual mix of very technical and detailed mathematical analysis, which is sourced to Kolmogorov’s work, via Scott, Solovay, Krauss, and Takeuti, with excursions into the philosophical interpretive issues raised by this analysis. It is a strength of the author that his interpretive grand themes of this metatheoretic approach are not abandoned but revisited after the technical discussion, and it certainly makes for an unusual paper to combine very detailed mathematics with very broad brush flights of philosophy.
Much of this discussion is of a highly formal nature. However ample references to sources and related papers are provided, and two sections of the paper summarise first von Neumann’s analysis of quantum theory, and second provide a fairly thorough review of the basic theory of random extensions and Borel codes. Following these are a section on hidden states and the operator functional calculus, which reviews the nature of contextual and non-contextual hidden variable theories, with discussion of Gudder as well as Kochen and Specker’s results, relating these to the preceding Boolean-valued analysis. Section 4, entitled “Random hidden variables”, then uses the Boolean-valued tools developed earlier to distinguish hidden variable analyses of quantum theory that meet various conditions distinguished in the previous section, an analysis which he characterizes earlier as a metatheoretic sharpening of Gudder’s theorem.
In the final section the very technical discussions give way to philosophical considerations of the implications to the interpretation of quantum statistics. It is argued that the analysis in terms of random ultrafilters provides a secular interpretation of quantum theory, and also of Everett and Wheeler’s possible worlds, a kind of model-theoretic understanding which the author considers neutral as to an epistemic, metaphysical or virtual interpretation. However it is also claimed that most likely Einstein was wrong, that probabilities are intrinsic to the ontology of quantum systems, rather than epistemic. In a final philosophical flourish we are presented with blocks of quotations from Einstein and Hume on the one hand, Bohr and Keats (!) on the other, with the author himself opting for the latter two and leaving Bohr with the final word.
As the quoted aims perhaps suggest, this long paper is an unusual mix of very technical and detailed mathematical analysis, which is sourced to Kolmogorov’s work, via Scott, Solovay, Krauss, and Takeuti, with excursions into the philosophical interpretive issues raised by this analysis. It is a strength of the author that his interpretive grand themes of this metatheoretic approach are not abandoned but revisited after the technical discussion, and it certainly makes for an unusual paper to combine very detailed mathematics with very broad brush flights of philosophy.
Much of this discussion is of a highly formal nature. However ample references to sources and related papers are provided, and two sections of the paper summarise first von Neumann’s analysis of quantum theory, and second provide a fairly thorough review of the basic theory of random extensions and Borel codes. Following these are a section on hidden states and the operator functional calculus, which reviews the nature of contextual and non-contextual hidden variable theories, with discussion of Gudder as well as Kochen and Specker’s results, relating these to the preceding Boolean-valued analysis. Section 4, entitled “Random hidden variables”, then uses the Boolean-valued tools developed earlier to distinguish hidden variable analyses of quantum theory that meet various conditions distinguished in the previous section, an analysis which he characterizes earlier as a metatheoretic sharpening of Gudder’s theorem.
In the final section the very technical discussions give way to philosophical considerations of the implications to the interpretation of quantum statistics. It is argued that the analysis in terms of random ultrafilters provides a secular interpretation of quantum theory, and also of Everett and Wheeler’s possible worlds, a kind of model-theoretic understanding which the author considers neutral as to an epistemic, metaphysical or virtual interpretation. However it is also claimed that most likely Einstein was wrong, that probabilities are intrinsic to the ontology of quantum systems, rather than epistemic. In a final philosophical flourish we are presented with blocks of quotations from Einstein and Hume on the one hand, Bohr and Keats (!) on the other, with the author himself opting for the latter two and leaving Bohr with the final word.
Reviewer: R.Wallace Garden (Thames)
MathOverflow Questions:
Set theoretical realizations of the hidden variables program in quantum mechanicsMSC:
| 81P05 | General and philosophical questions in quantum theory |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
| 03G12 | Quantum logic |
Keywords:
Boolean-valued models; hidden variables; quantum theory; stochasticity; non-relativistic quantum mechanics; random extensions of a set-theoretic universe; selfadjoint operators on Hilbert space; iterated Boolean measure algebras; operator functional calculus; interpretation of quantum statistics; random ultrafiltersReferences:
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