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On Quantization, Contextuality & the Koopman-von Neumann Formulation

Fritiof Wallentin

Published Apr 14, 2023• 2 citations
Publisher

Abstract

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.

DOI
10.20944/preprints202302.0144.v2

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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.
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.
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.
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.
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.
Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of ``classical'' wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S(K)(q,p,t)S^{(K)}(q,p,t)S(K)(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing ∂∂p\frac{\partial }{\partial p}∂p∂​ by 0 and p by ℏı∂∂q\frac{\hbar }{\imath } \frac{\partial }{\partial q}ℏ​∂q∂​ , thus providing an explanation for the common quantization rules.