{
  "annotation_id": "50a77d2d-afe7-4bc7-87f1-13bcd60b3d29",
  "date_created": "2026-03-02T18:01:53.165000Z",
  "date_modified": "2026-03-02T18:01:53.165000Z",
  "file_hash": "a3ad749d173d0e48817add9de12abef4feff92252bfc1975ec3935a4bfe882ee",
  "private": false,
  "record": {
    "abstract": "The function $\\gamma(x)=\\frac{1}{\\sqrt{1-x^2}}$ plays an important role in\nmathematical physics, e.g. as factor for relativistic time dilation in case of\n$x=\\beta$ with $\\beta=\\frac{v}{c}$ or $\\beta=\\frac{pc}{E}$. Due to former\nconsiderations it is reasonable to study the power series expansion of\n$\\gamma(x)$. Here its relationship to the binomial distribution is shown,\nespecially the fact, that the summands of the power series correspond to the\nreturn probabilities to the starting point (local coordinates, configuration or\nstate) of a Bernoulli random walk. So $\\gamma(x)$ and with that also proper\ntime is proportional to the sum of the return probabilities. In case of $x=1$\nor $v=c$ the random walk is symmetric. Random walks with absorbing barriers are\nintroduced in the appendix. Here essentially the basic mathematical facts are\nshown and references are given, most interpretation is left to the reader.",
    "arxiv_id": "quant-ph/0207045",
    "authors": [
      "Wolfgang Orthuber"
    ],
    "categories": [
      "quant-ph",
      "math-ph",
      "math.MP"
    ],
    "title": "A discrete and finite approach to past proper time",
    "url": "https://arxiv.org/abs/quant-ph/0207045"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b87c029a-bc9d-4a54-bd18-a0aab7b3c9f1",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}