[Paper Review] Independence of the existence of Pitowsky spin models

Ilijas Farah, Menachem Magidor|arXiv (Cornell University)|Dec 1, 2012

Quantum Mechanics and Applications10 references3 citations

TL;DR

This paper establishes the independence of the existence of Pitowsky spin models—functions assigning values to quantum spins that evade Bell-type no-go theorems—within Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). It proves that such models do not exist if the continuum is a real-valued measurable cardinal or in the random real model, showing that their existence depends on foundational set-theoretic assumptions, thus linking foundational physics to set theory.

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.

Motivation & Objective

• To investigate whether Pitowsky spin functions—non-measurable functions assigning values to spin-1/2 particles—can exist within standard set theory.
• To determine whether the existence of such functions is independent of ZFC, the standard axiomatic system for set theory.
• To explore the foundational implications of this independence for quantum mechanics, particularly regarding hidden variable theories and contextuality.
• To demonstrate that under certain set-theoretic assumptions (e.g., existence of a σ-additive extension of Lebesgue measure), Pitowsky functions cannot exist.
• To confirm that the non-existence of these functions is relatively consistent with ZFC, using models like the random real model.

Proposed method

• Uses the concept of a Pitowsky spin function F: S² → [-1,1] satisfying specific symmetry and integral conditions involving great circles and cosine-weighted averages.
• Applies measure-theoretic techniques to show that if a σ-additive extension of Lebesgue measure exists on P(ℝ), then any such Pitowsky function must be Borel-measurable.
• Employs results from Friedman and Fremlin on measure extensions and the random real model to show that in these models, all Pitowsky functions are Borel, hence cannot be non-measurable as required.
• Utilizes the notion of P-D-approximation for functions, showing that if a function is approximated by a Borel function in a filter of full outer measure sets, then it must itself be Borel.
• Analyzes KSP functions (Kochen-Specker type functions) with values in {0,1}, proving that their existence is also ruled out under the same assumptions.
• Applies the Portmanteau Theorem and Borel measurability of pushforward measures under continuous maps to establish the Borel nature of integrals involving rotation groups and Haar measure.

Experimental results

Research questions

• RQ1 Is the existence of Pitowsky spin functions independent of ZFC?
• RQ2 Can the existence of such functions be ruled out under stronger set-theoretic assumptions, such as the existence of a σ-additive extension of Lebesgue measure to P(ℝ)?
• RQ3 Does the non-existence of Pitowsky functions hold in the random real model of ZFC?
• RQ4 To what extent do foundational set-theoretic choices affect the possibility of constructing non-measurable hidden variable models in quantum mechanics?
• RQ5 Can the existence of KSP functions (violating Kochen-Specker contextuality) be similarly shown to be independent of ZFC?

Key findings

• If there exists a σ-additive extension of Lebesgue measure to the power set of ℝ, then no Pitowsky spin function can exist.
• In the random real model of ZFC, which satisfies ZFC but not the Continuum Hypothesis, no Pitowsky spin function exists.
• Under the assumption that the continuum is a real-valued measurable cardinal (a large cardinal axiom), no Pitowsky function exists.
• Any function that is P-D-approximated by a Borel function for a filter D of full outer measure sets must itself be Borel, which rules out non-measurable Pitowsky functions.
• KSP functions (with values in {0,1}) that almost violate Kochen-Specker contextuality are also ruled out under the same assumptions, as they too must be Borel.
• The integral I_{a,b,c}(F) for a KSP function F is equal to (5 - F(a))/8, and this expression is used to derive a contradiction if F is not Borel, thus proving its Borel nature.