On Koopman-von Neumann Waves II

E. Gozzi, D. Mauro

Published Jan 1, 2003• 22 citations

Abstract

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.

Related papers

ON KOOPMAN–VON NEUMANN WAVES II

E. GOZZI, D. MAURO

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.

arXiv:quant-ph/0306029International Journal of Modern Physics AMar 29, 200417 citations

ON KOOPMAN–VON NEUMANN WAVES

D. Mauro

In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently from what happens in quantum mechanics. The role of the phases associated to these classical "wave functions" is analyzed in detail. In this framework we also perform the analog of the two-slit interference experiment and compare it with the quantum case.

arXiv:quant-ph/0105112International Journal of Modern Physics AApr 10, 200248 citations

The History of Hilbert-Space Formulations of Classical Physics

Jacob A. Barandes

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.

physics.hist-phCross-listedarXiv:2607.17408The European Physical Journal HJul 21, 20261 citation

Topics in Koopman-von Neumann Theory

D. Mauro

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.

arXiv:quant-ph/0301172Jan 30, 200370 citations

Hilbert space structure in classical mechanics. I

E. Deotto, E. Gozzi, D. Mauro

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.

arXiv:quant-ph/0208046Journal of Mathematical PhysicsNov 22, 200323 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, 20220 citations