{
  "annotation_id": "ac172352-18b6-4e4c-ae89-fefba1c07b13",
  "date_created": "2026-03-02T18:02:34.422000Z",
  "date_modified": "2026-03-02T18:02:34.422000Z",
  "file_hash": "da366c6479f38b9aea055d7ca625a2a9e75c670f82defbd92d42adefc60d7b1b",
  "private": false,
  "record": {
    "abstract": "In this article combinatorial aspects of normal ordering annihilation and\ncreation operators of a multi-mode boson system are discussed. The modes are\nassumed to be coupled since otherwise the problem of normal ordering is reduced\nto the corresponding problem of the single-mode case. To describe the normal\nordering in the multi-mode case for each mode a colour is introduced and\ncoloured contractions are considered. A depiction for coloured contractions via\ncoloured linear representations is given. In analogy to the single-mode case\nassociated coloured Stirling numbers are defined as coefficients appearing in\nthe process of normal ordering powers of the number operators. Several\nproperties of these coloured Stirling numbers are discussed.",
    "arxiv_id": "quant-ph/0701185",
    "authors": [
      "Toufik Mansour",
      "Matthias Schork"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1007/s11503-008-1009-6",
    "title": "On the normal ordering of multi-mode boson operators",
    "url": "https://arxiv.org/abs/quant-ph/0701185"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "26030458-32c8-49fb-8a2a-0027d251e4dd",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}