{
  "annotation_id": "00b254c8-12e0-42a8-9814-0d569217dec3",
  "date_created": "2026-03-02T18:02:15.966000Z",
  "date_modified": "2026-03-02T18:02:15.966000Z",
  "file_hash": "34f5af2b615a51a2df52fd63f55137699f1b1fd5af037c58a53392bb36a9028d",
  "private": false,
  "record": {
    "abstract": "Complex numbers appear in the Hilbert space formulation of quantum mechanics,\nbut not in the formulation in phase space. Quantum symmetries are described by\ncomplex, unitary or antiunitary operators defining ray representations in\nHilbert space, whereas in phase space they are described by real, true\nrepresentations. Equivalence of the formulations requires that the former\nrepresentations can be obtained from the latter and vice versa. Examples are\ngiven. Equivalence of the two formulations also requires that complex\nsuperpositions of state vectors can be described in the phase space\nformulation, and it is shown that this leads to a nonlinear superposition\nprinciple for orthogonal, pure-state Wigner functions. It is concluded that the\nuse of complex numbers in quantum mechanics can be regarded as a computational\ndevice to simplify calculations, as in all other applications of mathematics to\nphysical phenomena.",
    "arxiv_id": "quant-ph/0504102",
    "authors": [
      "A. J. Bracken"
    ],
    "categories": [
      "quant-ph",
      "math-ph",
      "math.MP"
    ],
    "doi": "10.1016/S0034-4877(06)80005-4",
    "title": "Complex numbers and symmetries in quantum mechanics, and a nonlinear   superposition principle for Wigner functions",
    "url": "https://arxiv.org/abs/quant-ph/0504102"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "a883028d-a629-4f43-ba3a-5cbb8c48a8b8",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}