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arXiv:quant-ph/0301172

Topics in Koopman-von Neumann Theory

D. Mauro

Published Jan 30, 2003• 70 citations
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Abstract

In this thesis we study several features of the operatorial approach to classical mechanics pionereed by Koopman and von Neumann (KvN) in the Thirties. In particular in the first part we study the role of the phases of the KvN states. We analyze, within the KvN theory, the two-slit experiment and the Aharonov-Bohm effect and we make a comparison between the classical and the quantum case. In the second part of the thesis we study the extension of the KvN formalism to the space of forms and Jacobi fields. We first show that all the standard Cartan calculus on symplectic spaces can be performed via Grassmann variables or via suitable combinations of Pauli matrices. Second we study the extended Hilbert space of KvN which now includes forms and prove that it is impossible to have at the same time a positive definite scalar product and a unitary evolution. Clear physical reasons for this phenomenon are exhibited. We conclude the thesis with some work in progress on the issue of quantization.

DOI
10.48550/arxiv.quant-ph/0301172

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In this thesis we study several features of the operatorial approach to classical mechanics pionereed by Koopman and von Neumann (KvN) in the Thirties. In particular in the first part we study the role of the phases of the KvN states. We analyze, within the KvN theory, the two-slit experiment and the Aharonov-Bohm effect and we make a comparison between the classical and the quantum case. In the second part of the thesis we study the extension of the KvN formalism to the space of forms and Jacobi fields. We first show that all the standard Cartan calculus on symplectic spaces can be performed via Grassmann variables or via suitable combinations of Pauli matrices. Second we study the extended Hilbert space of KvN which now includes forms and prove that it is impossible to have at the same time a positive definite scalar product and a unitary evolution. Clear physical reasons for this phenomenon are exhibited. We conclude the thesis with some work in progress on the issue of quantization.
In this paper we study the Hilbert space structure underlying the Koopman–von Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zero-forms that are the square integrable functions on phase space. They proved that in this Hilbert space the evolution is unitary for every system. In this paper we extend the KvN Hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space. We prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems. Vice versa, if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite. Identifying the one-forms with the Jacobi fields we provide a physical explanation of these phenomena. We also prove that the unitary/nonunitary character of the evolution is invariant under canonical transformations.
In this paper we continue the study started in [1] on the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the Thirties. In particular we show that the introduction of the KvN Hilbert space of complex and square integrable "wave functions" requires an enlargement of the set of the observables of ordinary classical mechanics. The possible role and the meaning of these extra observables is briefly indicated in this work. We also analyze the similarities and differences between non selective measurements and two-slit experiments in classical and quantum mechanics.
In this paper we study the Hilbert space structure underlying the Koopman-von Neumann operatorial formulation of classical mechanics. While the Hilbert space of zero-forms can be endowed with a positive definite scalar product and the evolution turns out to be unitary, we prove that this is not the case when we include higher forms. In this last case we explore all possible scalar products and prove that for those which are positive definite the evolution is not unitary and vice versa. This feature is due to the Grassmannian nature of the forms and it appears only in classical mechanics. It is known in fact that in a similar structure, which is supersymmetric quantum mechanics, this does not happen.
In this paper, we continue the study started in Ref. 1, of the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the 1930s. In particular, we show that the introduction of the KvN Hilbert space of complex and square integrable "wave functions" requires an enlargement of the set of the observables of ordinary classical mechanics. The possible role and the meaning of these extra observables is briefly indicated in this work. We also analyze the similarities and differences between non-selective measurements and two-slit experiments in classical and quantum mechanics.
The Koopman–von Neumann (KvN) formulation recasts deterministic classical mechanics in a Hilbert-space operator framework analogous to quantum mechanics. In this approach, the Liouville equation is expressed as a Schrödinger-type evolution equation generated by a Hermitian operator, yielding unitary propagation for phase-space-volume-preserving dynamics. We first clarify the relationship among the Koopman, Perron–Frobenius, and Liouville operators through an explicit example. Wilczek's construction shows that the KvN formalism is not only restricted to canonical Hamiltonian mechanics but this can be generalized any first-order classical dynamical system. Wilczek shows how a general autonomous first-order system can be generated by a Hermitian temporal-evolution operator after the introduction of auxiliary conjugate variables. We develop the KvN description of systems driven by Hamiltonian curl forces–position-dependent, non-conservative forces that are non-dissipative and preserve phase-space volume. Finally, we extend the KvN formalism to non-Hamiltonian systems using the Jacobi Last Multiplier and examine its connection to the Boltzmann H-theorem.