{
  "id": "2405.02239",
  "version": "v1",
  "published": "2024-05-03T16:50:06.000Z",
  "updated": "2024-05-03T16:50:06.000Z",
  "title": "Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization",
  "authors": [
    "Md. Rafsanjany Jim",
    "S. Hasibul Hassan Chowdhury"
  ],
  "comment": "33 pages, no figure",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP"
  ],
  "abstract": "A gauge invariant mathematical formalism based on deformation quantization is outlined to model an $\\mathcal{N}=2$ supersymmetric system of a spin $1/2$ charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra $(C^{\\infty}(\\mathbb{R}^{2})[[\\vartheta]],*^r)$ of formal power series in $\\vartheta$ with coefficients in the commutative ring $C^{\\infty}(\\mathbb{R}^{2})$ was employed to construct the relevant observables, viz., SUSY Hamiltonian $H$, supercharge operator $Q$ and its adjoint $Q^{\\dag}$ all belonging to the $2\\times 2$ matrix algebra $\\mathcal{M}_{2}(C^{\\infty}(\\mathbb{R}^{2})[[\\vartheta]],*^r)$ with the help of a family of gauge-equivalent star products $*^{r}$. The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter $r$ but also the noncommutativity parameter $\\vartheta$. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of $\\vartheta$. The Witten index for the noncommutative SUSY Landau problem turns out to be $-1$ corroborating the fact that there is no broken supersymmetry for the model we are considering.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T16:50:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D55",
      "46L65",
      "46L87",
      "81Q60",
      "81R60",
      "20C25"
    ],
    "keywords": [
      "supersymmetric quantum mechanics",
      "deformation quantization",
      "noncommutative plane",
      "noncommutative susy landau problem turns",
      "nontrivial fermionic ground state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}