{
  "id": "2204.10808",
  "version": "v1",
  "published": "2022-04-21T01:28:37.000Z",
  "updated": "2022-04-21T01:28:37.000Z",
  "title": "Spinors in $\\mathbb{K}$-Hilbert Spaces",
  "authors": [
    "V. V. Varlamov"
  ],
  "comment": "29 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a structure of the $\\mathbb{K}$-Hilbert space, where $\\mathbb{K}\\simeq\\mathbb{R}$ is a field of real numbers, $\\mathbb{K}\\simeq\\mathbb{C}$ is a field of complex numbers, $\\mathbb{K}\\simeq\\mathbb{H}$ is a quaternion algebra, within the framework of division rings of Clifford algebras. The $\\mathbb{K}$-Hilbert space is generated by the Gelfand-Naimark-Segal construction, while the generating $C^\\ast$-algebra consists of the energy operator $H$ and the generators of the group $SU(2,2)$ attached to $H$. The cyclic vectors of the $\\mathbb{K}$-Hilbert space corresponding to the tensor products of quaternionic algebras define the pure separable states of the operator algebra. Depending on the division ring $\\mathbb{K}$, all states of the operator algebra are divided into three classes: 1) charged states with $\\mathbb{K}\\simeq\\mathbb{C}$; 2) neutral states with $\\mathbb{K}\\simeq\\mathbb{H}$; 3) truly neutral states with $\\mathbb{K}\\simeq\\mathbb{R}$. For pure separable states that define the fermionic and bosonic states of the energy spectrum, the fusion, doubling (complexification) and annihilation operations are determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-21T01:28:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert space",
      "pure separable states",
      "operator algebra",
      "neutral states",
      "quaternionic algebras define"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}