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A Local Hidden Variable Model Using Phenomenological Velocity and Its Implications for Bell’s Theorem

Parker Emmerson

Published Mar 18, 2025• 0 citations
Publisher

Abstract

Bell’s theorem presents a fundamental limit on the correlations achievable by any local hidden variable theory, seemingly precluding such models from reproducing quantum mechanical predictions. In this paper, we introduce a local hidden variable model based on phenomenological velocity and curvature operator techniques. By constructing a mathematical framework that maintains locality and realism, we demonstrate that it is possible to reproduce the quantum mechanical correlation functions that violate Bell’s inequalities. This challenges the conventional understanding imposed by Bell’s theorem and suggests the need for a re-examination of the foundational assumptions in quantum mechanics.

DOI
10.22541/au.174231250.03742080/v1

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Bell’s theorem presents a fundamental limit on the correlations achievable by any local hidden variable theory, seemingly precluding such models from reproducing quantum mechanical predictions. In this paper, we introduce a local hidden variable model based on phenomenological velocity and curvature operator techniques. By constructing a mathematical framework that maintains locality and realism, we demonstrate that it is possible to reproduce the quantum mechanical correlation functions that violate Bell’s inequalities. This challenges the conventional understanding imposed by Bell’s theorem and suggests the need for a re-examination of the foundational assumptions in quantum mechanics.
Bell's theorem revealed that a local hidden-variable model cannot completely reproduce the quantum mechanical predictions. Bell's inequality provides an upper bound under the locality and reality assumptions that can be violated by correlated measurement statistics of quantum mechanics. Greenberger, Horne, and Zeilinger (GHZ) gave a more compelling proof of Bell's theorem without inequalities by considering perfect correlations rather than statistical correlations. This work presents a temporal analog of the GHZ argument that establishes Bell's theorem in time without inequalities.
Abstract Bell's theorem revealed that a local hidden-variable model cannot completely reproduce the quantum mechanical predictions. Bell's inequality provides an upper bound under the locality and reality assumptions that can be violated by correlated measurement statistics of quantum mechanics. Greenberger, Horne, and Zeilinger (GHZ) gave a more compelling proof of Bell's theorem without inequalities by considering perfect correlations rather than statistical correlations. In this work, we present a temporal analogue of GHZ argument that establishes Bell's theorem in time without inequalities .
Abstract The experimentally verified violation of Bell’s inequalities apparently implies that at least one of two beliefs, which are accepted as true in almost all scientific practice, must be false. These beliefs are: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. For many years, many theories have been proposed to reconcile the violation of Bell’s inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that Bell’s inequalities are equivalent to the conditions to decide the completeness of a theory according to Boolean logic. Therefore, any theory aimed to violate Bell’s inequalities must start by giving up Boolean logic. Two problems are therefore defined: the “soft” one is to explain the violation of Bell’s inequalities without violating (non-Boolean) Local Realism; the “hard” one is to predict the time values when single particles are detected, in such a way that the resulting number of coincidences violates Bell’s inequalities. A simple hidden variables model is introduced, which solves the “soft” problem even in an ideal setup. This is possible thanks to the use of vectors in real space as the hidden variables and the corresponding operation (projection), which do not hold to Boolean logic. This model reconciles the violation of Bell’s inequalities with Local Realism and should end a controversy that has lasted for decades. Regarding the “hard” problem, the introduced model is as incomplete as Quantum Mechanics is. It is claimed that solving the “hard” problem requires devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variable values as input data and to replace the statistical Born’s rule of Quantum Mechanics with a deterministic threshold condition.
The precision with which we can measure operators that do not commute with conserved quantities is limited by the need to preserve the associated global symmetries. We show how to construct a local hidden-variable model that violates Bell inequalities by interpreting this measurement error as altering the measure space of the hidden variables. This provides a physically-motivated example of a supermeasured model, where the statistical independence assumption used to form Bell inequalities is violated without there being a causal dependence of the measurement settings on the hidden variables (superdeterminism) or a causal dependence of the hidden variables on the measurement settings (retrocausality). The model also gives subtly different predictions to quantum mechanics, that could be tested experimentally.
Bell inequality is violated in quantum mechanics and in experiments. Bell's theorem, on the other hand, concludes that quantum physics or nature is hence nonlocal. Bell's derivation of such a theorem comes in two versions, BT1 in which he started with determinism assumption in his 1964 original paper, and BT2 in his 1976 paper where he started with a "local causality" condition (LC). In this paper we show that (LC) does not follow from the physical concept of local causality except in the determinism limit. A local hidden variable model for the singlet pair is given that yields the correct quantum predictions for pair correlations. We also show determinism flagrantly violates quantum predictions. Thus Bell's theorem has no scope of applicability on possible hidden variable theory for quantum mechanics.
Bell nonlocality is an intriguing property of quantum mechanics with far reaching consequences for information processing, philosophy and our fundamental understanding of nature. However, nonlocality is a statement about static correlations only. It does not take into account dynamics, i.e. time evolution of those correlations. Consider a dynamic situation where the correlations remain local for all times. Then at each moment in time there exists a local hidden-variable (LHV) model reproducing the momentary correlations. Can the time evolution of the correlations then be captured by evolving the hidden variables? In this light, we define dynamical LHV models and motivate and discuss potential additional physical and mathematical assumptions. Based on a simple counter example we conjecture that such LHV dynamics does not always exist. This is further substantiated by a rigorous no-go theorem. Our results suggest a new type of nonlocality that can be deduced from the observed time evolution of measurement statistics and which generically occurs in interacting quantum systems.