{
  "id": "2107.04900",
  "version": "v1",
  "published": "2021-07-10T20:06:28.000Z",
  "updated": "2021-07-10T20:06:28.000Z",
  "title": "Symmetry Reduction of States I",
  "authors": [
    "Philipp Schmitt",
    "Matthias Schötz"
  ],
  "comment": "39 pages, comments welcome!",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA"
  ],
  "abstract": "We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra $g$. The key idea advocated for in this article is that the \"correct\" notion of positivity on a *-algebra $A$ is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares $a^* a$ with $a \\in A$, but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on $A$ thus depends on this choice of positivity on $A$, and the notion of positivity on the reduced algebra $A_{red}$ should be such that states on $A_{red}$ are obtained as reductions of certain states on $A$. In the special case of the *-algebra of smooth functions on a Poisson manifold $M$, this reduction scheme reproduces the coisotropic reduction of $M$, where the reduced manifold $M_{red}$ is just a geometric manifestation of the reduction of the evaluation functionals associated to certain points of $M$. However, we are mainly interested in applications to non-formal deformation quantization and therefore also discuss the reduction of the Weyl algebra, and of the polynomial algebra whose non-commutative analogs we plan to examine in future projects.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-10T20:06:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L30",
      "53D20",
      "81P16"
    ],
    "keywords": [
      "symmetry reduction",
      "positivity",
      "normalized positive hermitian linear functionals",
      "reduction scheme reproduces",
      "non-formal deformation quantization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}