Measure-free Koopman-von Neumann dynamics and noncommutative Geometry
Abstract
The Koopman-von Neumann formulation of classical statistical dynamics maps the isometric evolution of probability densities in L1 under the Liouville equation to a unitary evolution of quantum mechanical wavefunctions in L2 generated by the antisymmetric part of the Liouville operator. This approach enables the use of Hilbert space techniques to model the statistical evolution of observables. However, being an L2 method, it is not suitable for representing pointwise evolution along dynamical trajectories. We propose a Koopman-von Neumann framework that replaces the Lp spaces associated with a volume measure on the state space manifold, X, by a reproducing kernel Hilbert space (RKHS), ℋ, of continuous functions on X that satisfy joint analyticity conditions with respect to the generator (vector field), V, of the dynamics and its RKHS adjoint. Our scheme employs a dilation of the antisymmetric part of V to an essentially skew-adjoint operator, L, on the tensor product Hilbert space H = ℓ2(ℕ) ⊗ ℋ, followed by a dilation of L to an essentially skew-adjoint, free derivation, 𝒟, acting on a weighted symmetric Fock space F generated by H. We show that the unitary evolution generated by 𝒟 consistently recovers the (generally, non-unitary) Koopman evolution of observables in a dense subalgebra of ℋ generated by jointly analytic vectors, over a time interval that is uniformly bounded away from zero. We then build a family of weak spectral triples (𝒜n, F, −i 𝒟), n ∈ ℕ, wherein 𝒜n are non-abelian *-subalgebras of B(F) generated by creation and annihilation operators and −i 𝒟 plays the role of a Dirac operator inducing extended pseudometrics on the state spaces of 𝒜n.
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BibTeX
@misc{giannakis2026measurefree,
title = {{Measure-free Koopman-von Neumann dynamics and noncommutative Geometry}},
author = {Dimitrios Giannakis and Michael Montgomery},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11591},
eprint = {2608.11591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11591},
}
RIS
TY - UNPB
TI - Measure-free Koopman-von Neumann dynamics and noncommutative Geometry
AU - Dimitrios Giannakis
AU - Michael Montgomery
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.11591
UR - https://arxiv.org/abs/2608.11591
AB - The Koopman-von Neumann formulation of classical statistical dynamics maps the isometric evolution of probability densities in L^1 under the Liouville equation to a unitary evolution of quantum mechanical wavefunctions in L^2 generated by the antisymmetric part of the Liouville operator. This approach enables the use of Hilbert space techniques to model the statistical evolution of observables. However, being an L^2 method, it is not suitable for representing pointwise evolution along dynamical trajectories. We propose a Koopman-von Neumann framework that replaces the L^p spaces associated with a volume measure on the state space manifold, X, by a reproducing kernel Hilbert space (RKHS), ℋ, of continuous functions on X that satisfy joint analyticity conditions with respect to the generator (vector field), V, of the dynamics and its RKHS adjoint. Our scheme employs a dilation of the antisymmetric part of V to an essentially skew-adjoint operator, L, on the tensor product Hilbert space H = ℓ^2(ℕ) ⊗ ℋ, followed by a dilation of L to an essentially skew-adjoint, free derivation, 𝒟, acting on a weighted symmetric Fock space F generated by H. We show that the unitary evolution generated by 𝒟 consistently recovers the (generally, non-unitary) Koopman evolution of observables in a dense subalgebra of ℋ generated by jointly analytic vectors, over a time interval that is uniformly bounded away from zero. We then build a family of weak spectral triples (𝒜_n, F, −i 𝒟), n ∈ ℕ, wherein 𝒜_n are non-abelian *-subalgebras of B(F) generated by creation and annihilation operators and −i 𝒟 plays the role of a Dirac operator inducing extended pseudometrics on the state spaces of 𝒜_n.
ER -