{
  "id": "math/0608334",
  "version": "v1",
  "published": "2006-08-14T12:27:32.000Z",
  "updated": "2006-08-14T12:27:32.000Z",
  "title": "On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR",
  "authors": [
    "Eugene Lytvynov",
    "Lin Mei"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $X$ be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators $a(\\Delta)$ indexed by all measurable, relatively compact sets $\\Delta$ in $X$ (a quantum stochastic process over $X$). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over $X$ having the same correlation measure. Furthermore, the operators $a(\\Delta)$ can be realized as multiplication operators in the $L^2$-space with respect to this point process. In the proof, we utilize the notion of $\\star$-positive definiteness, proposed in [Y. G. Kondratiev and T.\\ Kuna, {\\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\\bf 5} (2002), 201--233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space $X$, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-14T12:27:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B15",
      "60G55",
      "81S05",
      "81S25"
    ],
    "keywords": [
      "correlation measure",
      "commuting hermitian operators",
      "quasi-free representations",
      "particle densities",
      "point process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8334L"
    }
  }
}