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arXiv:1212.0110

Independence of the existence of Pitowsky spin models

Ilijas Farah, Menachem Magidor

Published Dec 1, 2012• 3 citations
arXivHTMLPDF

Abstract

In 1982 I. Pitowsky used Continuum Hypothesis to construct hidden variable models for spin-1/2 and spin-1 particles in quantum mechanics. We show that the existence of Pitowsky models is independent from ZFC.
DOI
10.48550/arxiv.1212.0110

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An argument, due originally to J. S. Bell, is somewhat simplified and made more specific. It deals primarily with a quantum mechanical system consisting of the spins of two spin-12 particles. It shows that a description of the quantum mechanical measurement of the spin components of these two particles by means of hidden parameters is impossible if we assume that the parameters determining the outcome of the measurement of the spin of each particle are independent of the direction in which we decide to measure the spin of the other particle. The mathematical reason for the impossibility is analyzed.
In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed "spin-[Formula: see text] functions" and later "Kolmogorovian models", which employs a nonstandard notion of probability. We describe Pitowsky's analysis and argue (with the benefit of hindsight) that his notion of hidden variables is in fact just super-determinism (and accordingly physically not relevant). Pitowsky's first construction uses the Continuum Hypothesis. Farah and Magidor took this as an indication that at some stage physics might give arguments for or against adopting specific new axioms of set theory. We would rather argue that it supports the opposing view, i.e., the widespread intuition "if you need a non-measurable function, it is physically irrelevant".