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Hilbert Space Structure in Classical Mechanics: (I)

E. Deotto, Gozzi, E, Mauro, D

Published Aug 7, 2002• 16 citations
OpenAlex

Abstract

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.

Related papers

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/0208046•Journal of Mathematical Physics•Nov 22, 2003•23 citations

Hilbert space structure in classical mechanics. II

E. Deotto, E. Gozzi, D. Mauro

In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.

arXiv:quant-ph/0208047•Journal of Mathematical Physics•Nov 22, 2003•13 citations

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/0301172•Jan 30, 2003•70 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/0105112•International Journal of Modern Physics A•Apr 10, 2002•48 citations

Universal Local Symmetries and Nonsuperposition in Classical Mechanics

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In the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to include the evolution of differential forms. Because of their noninvariance, those extra observables have to be removed. This removal makes the superposition of states in the Koopman and von Neumann formulation, and as a consequence also in classical mechanics, impossible.

arXiv:1006.3029•Physical Review Letters•Oct 7, 2010•68 citations

Uncertainty and Wigner negativity in Hilbert-space classical mechanics

Mustafa Amin

Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.

quant-pharXiv:2602.10341•Feb 12, 2026•0 citations
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 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 analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.
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 the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to include the evolution of differential forms. Because of their noninvariance, those extra observables have to be removed. This removal makes the superposition of states in the Koopman and von Neumann formulation, and as a consequence also in classical mechanics, impossible.
Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.