{
  "annotation_id": "e209892c-0dc6-4e3b-9054-6e37a6ff52c6",
  "date_created": "2026-03-02T18:01:52.359000Z",
  "date_modified": "2026-03-02T18:01:52.359000Z",
  "file_hash": "1ff4172f2408f4b6e0b9dc25f14836065353c38d366d5be37c373484779d5dd8",
  "private": false,
  "record": {
    "abstract": "In this paper we study the Hilbert space structure underlying the Koopman-von\nNeumann (KvN) operatorial formulation of classical mechanics. KvN limited\nthemselves to study the Hilbert space of zero-forms that are the square\nintegrable functions on phase space. They proved that in this Hilbert space the\nevolution is unitary for every system. In this paper we extend the KvN Hilbert\nspace to higher forms which are basically functions of the phase space points\nand the differentials on phase space. We prove that if we equip this space with\na positive definite scalar product the evolution can turn out to be non-unitary\nfor some systems. Vice versa if we insist in having a unitary evolution for\nevery system then the scalar product cannot be positive definite. Identifying\nthe one-forms with the Jacobi fields we provide a physical explanation of these\nphenomena. We also prove that the unitary/non unitary character of the\nevolution is invariant under canonical transformations.",
    "arxiv_id": "quant-ph/0208046",
    "authors": [
      "E. Deotto",
      "E. Gozzi",
      "D. Mauro"
    ],
    "categories": [
      "quant-ph",
      "hep-th"
    ],
    "doi": "10.1063/1.1623333",
    "journal_ref": "J.Math.Phys. 44 (2003) 5902-5936",
    "title": "Hilbert Space Structure in Classical Mechanics: (I)",
    "url": "https://arxiv.org/abs/quant-ph/0208046"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b55dff80-06f7-4ae6-b881-c76bb49dbdb2",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}