The projection spectral theorem, quasi-free states and point processes

Eugene Lytvynov

The projection spectral theorem, quasi-free states and point processes

Mathematical Physics 2025-12-15 v2 math.MP

Authors: Eugene Lytvynov

Abstract

In this review paper, we demonstrate that several classes of point processes in a locally compact Polish space $X$ appear as the joint spectral measure of a rigorously defined particle density of a representation of the canonical anticommutation relations (CAR) or the canonical commutation relations (CCR) in a Fock space. For these representations of the CAR/CCR, the vacuum state on the corresponding $*$ -algebra is quasi-free. The classes of point process that arise in such a way include determinantal and permanental point processes.

Keywords

point processes projection spectral theorem quasi-free states canonical anticommutation relations canonical commutation relations Fock space determinantal point processes

Cite

@article{arxiv.2508.19147,
  title  = {The projection spectral theorem, quasi-free states and point processes},
  author = {Eugene Lytvynov},
  journal= {arXiv preprint arXiv:2508.19147},
  year   = {2025}
}

Comments

This is a review paper for the special issue of Methods of Functional Analysis and Topology devoted to the the 100th anniversary of Yuri Makarovych Berezansky's birthday

The projection spectral theorem, quasi-free states and point processes

Abstract

Keywords

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