{
  "id": "2211.13284",
  "version": "v1",
  "published": "2022-11-23T20:06:47.000Z",
  "updated": "2022-11-23T20:06:47.000Z",
  "title": "Fuzzy hyperspheres via confining potentials and energy cutoffs",
  "authors": [
    "Gaetano Fiore"
  ],
  "comment": "Latex file, 40 pages, 3 figures",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP"
  ],
  "abstract": "We simplify and complete the construction of fully $O(D)$-equivariant fuzzy spheres $S^d_\\Lambda$, for all dimensions $d\\equiv D-1$, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. This is based on imposing a suitable energy cutoff on a quantum particle in $\\mathbb{R}^D$ subject to a confining potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff and the depth of the well diverge with $\\Lambda\\in\\mathbb{N}$. We show that the irreducible representation ${{\\bf\\pi}}_\\Lambda$ of $O(D\\!+\\!1)$ on the space of harmonic homogeneous polynomials of degree $\\Lambda$ in the Cartesian coordinates of $\\mathbb{R}^{D+1}$ (which we express via trace-free completely symmetric projections) is isomorphic to the Hilbert space $\\mathcal{H}_{\\Lambda}$ of the particle, as a reducible representation of $O(D)\\subset O(D\\!+\\!1)$. Moreover, we show that the algebra of observables is isomorphic to ${{\\bf\\pi}}_\\Lambda\\left(Uso(D\\!+\\!1)\\right)$. As $\\Lambda$ diverges (commutative limit) so does the dimension of $\\mathcal{H}_{\\Lambda}$, and we recover ordinary quantum mechanics on the sphere $S^d$; more formally, we have a fuzzy quantization of a coadjoint orbit of $O(D\\!+\\!1)$ that goes to the classical phase space $T^*S^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-23T20:06:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81R60",
      "22C05",
      "22D10",
      "22E46",
      "33C55"
    ],
    "keywords": [
      "confining potential",
      "fuzzy hyperspheres",
      "ordinary quantum mechanics",
      "equivariant fuzzy spheres",
      "hilbert space"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}