Existence of point processes through families of commuting Hermitian operators

• Main Author: Mei, Lin
• Published: Swansea University 2007
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.752050

This dissertation is devoted to problems of existence and physical interpretation of some point processes. In the first part of the dissertation, we introduce the notion of the correlation measure of a family of commuting Hermitian operators. 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 m X. For such a family, we introduce the notion of a correlation measure and prove that, if this correlation measure exists and satisfies some condition of growth, then there exists a point process over X having the same correlation measure (in the sense of the classical theory of point processes). Furthermore, the operators a(Delta) can be realised as multiplication operators in the L2-space with respect to this point process. In particular, our result extends the criterion of existence of a point process from [6, 15], to the case of the topological space X, which is a standard underlying space in the theory of point processes. In the second part of the dissertation, we consider some important applications of our general results. We discuss particle densities of the quasi-free representation of the CAR and CCR, which lead to fermion (determinantal), and boson (permanental) point processes. We also discuss convolutions of these particle densities, which lead to point processes whose correlation functions are given through the Vere-Jones a-determinants.