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

TL;DR

This paper reviews how certain point processes, including determinantal and permanental types, can be represented as spectral measures of particle densities in Fock space representations of CAR and CCR, emphasizing the role of quasi-free states.

Contribution

It establishes a connection between point processes and spectral measures of particle densities in Fock space, highlighting the significance of quasi-free states for these processes.

Findings

Determinantal and permanental point processes arise as spectral measures.

Quasi-free states are fundamental in representing these point processes.

The framework links point processes with quantum algebra representations.

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.