It is shown that the hallmark quantum phenomenon of contextuality is present in classical statistical mechanics (CSM). It is first shown that the occurrence of contextuality is equivalent to there being observables that can differentiate between pure and mixed states. CSM is formulated in the formalism of quantum mechanics (FQM), a formulation commonly known as the Koopman-von Neumann formulation (KvN). In KvN, one can then show that such a differentiation between mixed and pure states is possible. As contextuality is a probabilistic phenomenon and as it is exhibited in both classical physics and ordinary quantum mechanics (OQM), it is concluded that the foundational issues regarding quantum mechanics are really issues regarding the foundations of probability.
It is shown that the hallmark quantum phenomenon of contextuality is present in classical statistical mechanics (CSM). It is first shown that the occurrence of contextuality is equivalent to there being observables that can differentiate between pure and mixed states. CSM is formulated in the formalism of quantum mechanics (FQM), a formulation commonly known as the Koopman-von Neumann formulation (KvN). In KvN, one can then show that such a differentiation between mixed and pure states is possible. As contextuality is a probabilistic phenomenon and as it is exhibited in both classical physics and ordinary quantum mechanics (OQM), it is concluded that the foundational issues regarding quantum mechanics are really issues regarding the foundations of probability.
A new perspective on quantization is presented in this article, where the Koopman-von Neumann formulation (KvN) and ordinary non-relativistic quantum mechanics (OQM) are viewed as unitarily inequivalent representations of the same underlying structure. This approach to quantization is based on the idea that contextuality, rather than ℏ, is a fundamental aspect of quantum mechanics. The novel view is motivated by the limitations of traditional approaches to quantization together with previous research demonstrating contextuality in KvN. In this representation theoretic approach, both KvN and OQM are considered valid quantum theories separated by a superselection rule.
This article proposes a novel formulation of quantization, which considers the Koopman-von Neumann formulation (KvN) and ordinary non-relativistic quantum mechanics (OQM) as unitarily inequivalent representations of the same underlying structure. The approach is grounded in the notion that contextuality, rather than occurrence if Plank's constant, is the fundamental aspect of quantum mechanics. The motivation for this novel view arises from the limitations of traditional approaches to quantization, combined with previous research demonstrating contextuality in KvN. Using a representation-theoretic approach, both KvN and OQM are regarded as valid quantum theories, separated by a superselection rule.
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 a previous work by the author it was shown that the Koopman-von Neumann formulation of classical mechanics (KvN) exhibits quantum interference. As such it was there claimed that KvN ought to be considered as a proper quantum mechanical theory in a similar sense as ordinary non-relativistic quantum mechanics (OQM) is. In this article this claim is made manifest by showing that KvN and OQM can be taken as merely inequivalent representations resulting from one and the same quantization scheme. Of course, as the typical notion of quantization necessitates the canonical commutation relations (CCR) to hold, which they do not in KvN, the concept needs to be properly revised. To justify such a revision the typical reasons for necessitation the CCR, reductionism and quantization as 'turning brackets to commutators', are examined and found insufficient.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
The circumstances of measurement have more direct significance in quantum than in classical physics, where they can be neglected for well-performed measurements. In quantum mechanics, the dispositions of the measuring apparatus-plus-environment of the system measured for a property are a non-trivial part of its formalization as the quantum observable. A straightforward formalization of context, via equivalence classes of measurements corresponding to sets of sharp target observables, was recently given for sharp quantum observables. Here, we show that quantum contextuality, the dependence of measurement outcomes on circumstances external to the measured quantum system, can be manifested not only as the strict exclusivity of different measurements of sharp observables or valuations but via quantitative differences in the property statistics across simultaneous measurements of generalized quantum observables, by formalizing quantum context via coexistent generalized observables rather than only its subset of compatible sharp observables. Here, the question of whether such quantum contextuality follows from basic quantum principles is then addressed, and it is shown that the Principle of Indeterminacy is sufficient for at least one form of non-trivial contextuality. Contextuality is thus seen to be a natural feature of quantum mechanics rather than something arising only from the consideration of impossible measurements, abstract philosophical issues, hidden-variables theories, or other alternative, classical models of quantum behavior.