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 $L^1$ 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 $\mathfrak H = \ell^2(\mathbb N) \otimes \mathcal H$, 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 $*$-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 $\mathcal A_n$.