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A New Quantization Map

D. Mauro

Published Jan 1, 2003• 7 citations
OpenAlex

Abstract

In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irreducibly. 1

Related papers

From Koopman–von Neumann theory to quantum theory

U. Klein

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $$S^{(K)}(q,p,t)$$ 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 $$\frac{\partial }{\partial p}$$ by 0 and p by $$\frac{\hbar }{\imath } \frac{\partial }{\partial q}$$ , thus providing an explanation for the common quantization rules.

arXiv:1705.07427•Quantum Studies Mathematics and Foundations•May 31, 2017•25 citations

Quantization & the Koopman-von Neumann Formulation

Fritiof Wallentin

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.

•Nov 28, 2022•0 citations

A No-Go Theorem for Nonlinear Canonical Quantization

Miroslav Engliš

We want to point out the following strengthening of the classical theorem of Groenewold and van Hove: There exists no mapping Op from polynomial observables on the phase space into linear operators on which would map Poisson brackets into commutators, the position and momentum observables p and q into the usual (Schrödinger) position and momentum operators, and would obey the von Neumann rule for and . The point is that neither linearity, nor continuity etc. of Op are assumed.

Communications in Theoretical Physics•Mar 15, 2002•3 citations

On Quantization, Contextuality & the Koopman-von Neumann Formulation

Fritiof Wallentin

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.

•Feb 8, 2023•1 citation

From classical to quantum mechanics: “How to translate physical ideas into mathematical language”

Hervé Bergeron

Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].

arXiv:quant-ph/0102084•Journal of Mathematical Physics•Sep 1, 2001•12 citations

On the von Neumann rule in quantization

Olaf Müller

We show that any linear quantization map into the space of self-adjoint operators in a Hilbert space violates the von Neumann rule on postcomposition with real functions.

arXiv:1903.10494•Journal of Mathematical Physics•Aug 1, 2019•3 citations
In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irreducibly. 1
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.
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.
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.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].