Abstract
In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed “spin- 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”.
Highlights
We briefly recall the notion on hidden variables and the two no-go theorems that will be relevant in this paper: Bell’s theorem [1], the groundbreaking first proof that local hidden variables are impossible; and the Greenberger-Horne-Zeilinger (GHZ) theorem [6]
A hidden variable model for a given quantum mechanical state | must predict the results that are guaranteed by quantum mechanics
It is natural to assume that a certain classical probabilistic mixture of different hidden variables represents |, and that we can represent a sequence of systems in state | by “randomly” picking hidden variables; and we require that the resulting frequencies are those predicted by quantum mechanics
Summary
We briefly recall the notion on hidden variables and the two no-go theorems that will be relevant in this paper: Bell’s theorem [1], the groundbreaking first proof that local hidden variables are impossible; and the Greenberger-Horne-Zeilinger (GHZ) theorem [6]. The GHZ theorem is simpler and stronger, as it shows that local hidden variables cannot be consistently assigned to a single system (of three particles); whereas Bell’s theorem shows that certain statistical frequencies cannot be reproduced by local hidden variables. More details can be found e.g., in Mermin’s paper [12]. We will only consider systems of one, two or three spin-1/2 particles. Σa denotes the spin observable in direction a (with possible values ±1); if we are dealing with more We will only consider systems of one, two or three spin-1/2 particles. σa denotes the spin observable in direction a (with possible values ±1); if we are dealing with more
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