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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Deterministic model of spin and statistics

Itamar Pitowsky

Published May 15, 1983• 91 citations
Publisher

Abstract

A deterministic model that accounts for the statistical behavior of random samples of identical particles is presented. The model is based on some nonmeasurable distribution of spin values in all directions. The mathematical existence of such distributions is proved by set-theoretical techniques, and the relation between these distributions and observed frequencies is explored within an appropriate extension of probability theory. The relation between quantum mechanics and the model is specified. The model is shown to be consistent with known polarization phenomena and the existence of macroscopic magnetism. Finally the possibility of a thought experiment which indicates a deviation from the predictions of quantum mechanics is described.

DOI
10.1103/physrevd.27.2316

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A deterministic model that accounts for the statistical behavior of random samples of identical particles is presented. The model is based on some nonmeasurable distribution of spin values in all directions. The mathematical existence of such distributions is proved by set-theoretical techniques, and the relation between these distributions and observed frequencies is explored within an appropriate extension of probability theory. The relation between quantum mechanics and the model is specified. The model is shown to be consistent with known polarization phenomena and the existence of macroscopic magnetism. Finally the possibility of a thought experiment which indicates a deviation from the predictions of quantum mechanics is described.
A theoretical investigation is made into the difficulties encountered in constructing a deterministic model for quantum mechanics and into the restrictions that can be placed on the form of such a model.The various implications of the known 11 impossi bil ity proofs II are examined.A possible explanation for the non-locality required by Bell's proof is suggested in terms of backwards-in-time causality.The efficacy of the Kochen and Specker proof is brought into doubt by showing that there is a possible way of avoiding its implications in the only known physically realizable situation to which it applies.Also, a new thought experiment is put forward to show that a particle's predetermined momentum and energy values cannot satisfy the laws of momentum and energy conservation without conflicting with the predictions of quantum mechanics.Special attention is paid to a particular class of deterministic models which appear not to have been explicitly studied before.The models considered are those for which the individual outcomes of measurements are not dependent on hidden variables associated with the measuring apparatus and for which the hidden variables of a particle do not need to be randomized after each measurement.Specific models of this type are constructed and examined for the cases of both discrete and continuous spectra of eigenvalues.In developing these models a need is found for a joint distribution for position and momentum.In relation to this point, it is demonstrated that the quantum mechanical wave equations actually point to a particular joint distribution expression, namely the Margenau and Hill distribution.The schemes considered are found to have certain shortcomings and possible steps towards more satisfactory models are suggested.
Quantum theory has played a significant role in modern philosophy both as a source of metaphysical ideas and as an important example of a 'scientific revolution'. In spite of the sixty or so years that have elapsed since its invention, a long lasting controversy concerning the interpretation and meaning of quantum theory prevails. Almost all authors, however, seem to agree on one major point, namely, that there could be no interpretation of this theory which is both realistic and local.;The purpose of this thesis is to demonstrate that this premiss is false and that a realistic, local and deterministic interpretation of quantum theory (at least of part of it) does exist, provided that we extend the classical concept of probability.;In order to establish this a 'quasi classical' probability theory is developed based on some non Lebesgue measurable 'events', which is then applied to account for spin-statistics. Finally I note how this model reflects on the problems of physical realism, locality, the status of probability theory and the philosophical foundations of mathematics.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
Abstract Quantum mechanics is the foundation of modern physics that is thought to be applicable to all physical phenomena, at least in principle. However, when applied to macroscopic bodies, the theory seems to be inconsistent. Wigner's friend and related thought experiments demonstrate that accounts by different observers described by the rules of quantum mechanics may be contradictory. Although still highly debated, such experiments seem to demonstrate an incompatibility of quantum mechanics with the usual rules of logic. Alternatively, one of the hidden assumptions in the thought experiments must be wrong. For instance, the argument is invalidated if macroscopic observers cannot be considered as physical systems described by the rules of quantum theory. Here we prove that there is a way to apply the rules of quantum mechanics to macroscopic observers while avoiding contradictory accounts of measurement by the observers. The key to this is the random noise that is ever present in nature and that represents the uncontrollable part of interaction between measured system and the surroundings in classical and quantum physics. By exploring the effect of the noise on microscopic and macroscopic bodies, we demonstrate that accounts of Wigner, the friend and other agents all become consistent. Our result suggests that the existing attempts to modify the Schrodinger equation to account for measurement results may be misguided. More broadly, the proposed mechanism for modeling measurements underlies the phenomenon of decoherence and is shown to be sufficient to explain the transition to Newtonian physics in quantum theory.
Present quantum theory, which is statistical in nature, does not predict joint probability distribution of position and momentum because they are noncommuting. We propose a deterministic quantum theory which predicts a joint probability distribution such that the separate probability distributions for position and momentum agree with usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive definite. The theory predicts a correlation between position and momentum in individual events.