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arXiv:1911.02481 · Entropy

Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories

Claudio Garola

Published Jan 18, 2021• 2 citations
arXivHTMLPDF

Abstract

Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.

DOI
10.3390/e23010121

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Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.
We have proposed in several recent papers a critical view of some parts of quantum mechanics (QM) that is methodologically unusual because it rests on analysing the language of QM by using some elementary but fundamental tools of mathematical logic. Our approach proves that some widespread beliefs about QM can be questioned and establishes new links with a classical view, which is significant in the debate on the interpretations of QM. We propose here a brief survey of our results, highlighting their common background. We firstly show how quantum logic (QL) can be embedded into classical logic (CL) if the embedding is required to preserve the logical order and not the algebraic structure, and also how QL can be interpreted as a pragmatic sublanguage within a pragmatic extension of CL. Both these results challenge the thesis that CL and QL formalize the properties of different and incompatible notions of truth. We then show that quantum probability admits an epistemic interpretation if contextuality is taken into account as a basic constituent of the language of QM, which overcomes the interpretation of quantum probability as ontic. Finally, we show that the proofs that QM is a contextual theory stand on a supplementary epistemological assumption that is usually unnoticed and left implicit. Dropping such assumption opens the way, at least in principle, to non-contextual interpretations of QM.
Interpretational paradoxes of quantum mechanics may be associated with the application of the structure of classical probability theory to systems whose probabilistic organization is fundamentally contextual. In this work, we propose an outline of an alternative probability theory in which the primary object is not an elementary outcome but a contextual whole—a class of alternatives defined by a common structural property. In this approach, probability is organized on two levels. The first level describes the probabilities of contextual wholes, while the second level describes the conditional probabilities of alternatives within those wholes. Individual alternatives do not possess autonomous probabilities outside their context. Their probabilities arise only as contextual projections of a more fundamental probabilistic structure. Within this model, quantum mechanics appears as a special case of a more general theory in which contextual wholes are described by superpositions of probability amplitudes. Classical probability theory emerges as a limiting case in which all contexts are mutually compatible and can be combined into a single global description. The proposed framework provides a possible conceptual foundation for understanding quantum probability, contextuality, interference, and hybrid probability distributions within a unified structure.
Abstract Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an “ontic extension” defines a nonseparable (global) random variable that generates physical correlations, and an “epistemic restriction” constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck’s constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.
This doctoral thesis in mathematics consists of three articles that explore the probabilistic structure of quantum mechanics and quantization from a novel perspective. The thesis adopts a probabilistic interpretation of quantum mechanics, which views the archetypical quantum experiments of Bell- and double-slit- type as violating the principle of non-contextuality, i.e., the assertion that all events and observables are always representable on one single Kolmogorovian probability space, rather than the principles of realism or locality. This probabilistic interpretation posits that quantum mechanics constitutes a probability theory that adheres to the principle of contextuality, and that quantum events explicitly occur at the level of measurement, rather than the level of that which is measured, as these are traditionally interpreted. The thesis establishes a natural connection between the probabilistic structure of quantum mechanics, specifically Born’s rule, and the frequentist interpretation of probability. The major conceptual step in establishing this connection is to re-identify quantum observables instead as unitary representations of groups, whose irreducible sub-representations correspond to the observable’s different possible outcomes, rather than primarily as self- adjoint operators. Furthermore, the thesis reformulates classical statistical mechanics in the formalism of quantum mechanics, known as the Koopman-von Neumann formulation, to demonstrate that classical statistical mechanics also adheres to the principle of contextuality. This finding is significant because it raises questions about the existence of a hidden-variable model of classical statistical mechanics of the kind as examined in Bell’s theorem, where this presumed hidden-variable model traditionally has been seen as that which distinguishes "classical" from "quantum" probability.A novel reformulation of quantization is proposed considering it rather in terms of the representation theory of Hamiltonian flows and their associated inherent symmetry group of symplectomorphisms. Contrary to the traditional view of quantization, this formulation can be regarded as compatible with the probabilistic interpretation of quantum mechanics and offers a new perspective on the quantization of gravity.
In this work, we focus on the philosophical aspects and technical challenges that underlie the axiomatization of the non-Kolmogorovian probability framework, in connection with the problem of quantum contextuality. This fundamental feature of quantum theory has received a lot of attention recently, given that it might be connected to the speed-up of quantum computers-a phenomenon that is not fully understood. Although this problem has been extensively studied in the physics community, there are still many philosophical questions that should be properly formulated. We analyzed different problems from a conceptual standpoint using the non-Kolmogorovian probability approach as a technical tool.
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an "ontic extension" defines a nonseparable (global) random variable that generates physical correlations, and an "epistemic restriction" constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck's constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.