Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918245611601920 |
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| author | Amin, Mustafa Walton, Mark A. |
| author_facet | Amin, Mustafa Walton, Mark A. |
| contents | The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of Hermitian operators and ascribe meaning to a coherent superposition of states, among other things. Here we consider the general KvN equation for a classical probability amplitude and show that its so-called gauge freedom allows the separation of variables. The amenability to Hilbert-space methods of the resulting KvN solutions is investigated. We construct superpositions from differently-gauged Liouvillian eigenstates, and find an orthonormal set among them. We find that some separable solutions describe the canonical ensemble with temperature related to the separation constant. Classical uncertainty relations arise naturally in the KvN formalism. We discuss one between the dynamical time and the Liouvillian in terms of the statistical description of classical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties Amin, Mustafa Walton, Mark A. Quantum Physics Mathematical Physics Classical Physics The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of Hermitian operators and ascribe meaning to a coherent superposition of states, among other things. Here we consider the general KvN equation for a classical probability amplitude and show that its so-called gauge freedom allows the separation of variables. The amenability to Hilbert-space methods of the resulting KvN solutions is investigated. We construct superpositions from differently-gauged Liouvillian eigenstates, and find an orthonormal set among them. We find that some separable solutions describe the canonical ensemble with temperature related to the separation constant. Classical uncertainty relations arise naturally in the KvN formalism. We discuss one between the dynamical time and the Liouvillian in terms of the statistical description of classical systems. |
| title | Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties |
| topic | Quantum Physics Mathematical Physics Classical Physics |
| url | https://arxiv.org/abs/2512.11148 |