{
  "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 $\\mathcal A_n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11591",
  "author_total": 2,
  "authors": [
    {
      "name": "Dimitrios Giannakis"
    },
    {
      "name": "Michael Montgomery"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70eqzwkbvc0g3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11591",
  "fulltext_url": null,
  "id": "70eqzwkbvc0g3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11591",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7202369662",
  "pdf_url": "https://arxiv.org/pdf/2608.11591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70eqzwkbvc0g3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.11591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Measure-free Koopman-von Neumann dynamics and noncommutative Geometry",
  "title_text": "Measure-free Koopman-von Neumann dynamics and noncommutative Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/70eqzwkbvc0g3",
  "venue": "arXiv",
  "versions": []
}