Measure-free Koopman-von Neumann dynamics and noncommutative Geometry

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), $\mathcal{H}$, 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 $\mathfrak{H}={\mathrm{\ell }}^2(\mathbb{N})\otimes \mathcal{H}$, followed by a dilation of $L$ to an essentially skew-adjoint, free derivation, $\mathcal{D}$, acting on a weighted symmetric Fock space $\mathfrak{F}$ generated by $\mathfrak{H}$. We show that the unitary evolution generated by $\mathcal{D}$ consistently recovers the (generally, non-unitary) Koopman evolution of observables in a dense subalgebra of $\mathcal{H}$ 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 $({\mathcal{A}}_n,\mathfrak{F},-i\mathcal{D})$, $n\in \mathbb{N}$, wherein ${\mathcal{A}}_n$ are non-abelian $\ast$-subalgebras of $B(\mathfrak{F})$ generated by creation and annihilation operators and $-i\mathcal{D}$ plays the role of a Dirac operator inducing extended pseudometrics on the state spaces of ${\mathcal{A}}_n$.