{
  "id": "1904.11211",
  "version": "v1",
  "published": "2019-04-25T08:45:34.000Z",
  "updated": "2019-04-25T08:45:34.000Z",
  "title": "Fock representations of multicomponent (particularly plekton) commutation relations",
  "authors": [
    "Alexei Daletskii",
    "Alexander Kalyuzhny",
    "Eugene Lytvynov",
    "Daniil Proskurin"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.OA"
  ],
  "abstract": "Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\\otimes 2}$ with norm $\\le1$, satisfying the Yang--Baxter equation. Bo\\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\\mathcal F(H)=\\bigoplus_{n=0}^\\infty\\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\\mathcal F(H)$. We then choose $H=L^2(X\\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\\mathbb R^d$ and $V:=\\mathbb C^m$ with $m\\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\\otimes 2}=L^2(X^2\\to V^{\\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\\ $(x,y)\\in X^2$, $C_{x,y}$ is a linear operator on $V^{\\otimes 2}$ with norm $\\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ is associated with plektons. For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-25T08:45:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L90",
      "81R10"
    ],
    "keywords": [
      "commutation relations",
      "fock representations",
      "particularly plekton",
      "annihilation operators",
      "multicomponent quantum system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}