Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Eugene Lytvynov

TL;DR

The article surveys how point processes on a locally compact Polish space arise as joint spectral measures of particle densities associated with CAR/CCR representations in Fock spaces endowed with quasi-free vacuum states. It synthesizes the projection spectral theorem with the $K$-transform framework to connect operator families and their correlation measures to concrete point processes, including Poisson, determinantal, permanental, and hafnian types. By detailing both the general spectral machinery and explicit kernel-based constructions (Hermitian and $J$-Hermitian), the work clarifies how quasi-free states encode correlation structures and yield configuration-space measures. The findings provide a rigorous bridge between infinite-dimensional harmonic analysis, operator algebras, and stochastic point processes, enabling systematic derivations of point-process models from quantum-like particle-density formalisms.

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.

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

TL;DR

The article surveys how point processes on a locally compact Polish space arise as joint spectral measures of particle densities associated with CAR/CCR representations in Fock spaces endowed with quasi-free vacuum states. It synthesizes the projection spectral theorem with the -transform framework to connect operator families and their correlation measures to concrete point processes, including Poisson, determinantal, permanental, and hafnian types. By detailing both the general spectral machinery and explicit kernel-based constructions (Hermitian and -Hermitian), the work clarifies how quasi-free states encode correlation structures and yield configuration-space measures. The findings provide a rigorous bridge between infinite-dimensional harmonic analysis, operator algebras, and stochastic point processes, enabling systematic derivations of point-process models from quantum-like particle-density formalisms.

Abstract

In this review paper, we demonstrate that several classes of point processes in a locally compact Polish space 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.

Paper Structure

This paper contains 12 sections, 3 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Point process and its correlation measure
  3. Spectral representation of the correlation measure
  4. Correlation measure of a family of Hermitian operators
  5. Quasi-free states on the CAR and CCR algebras
  6. Fock spaces
  7. Examples of application of Theorem \ref{['cdyre6u3']}
  8. Determinantal point process with a Hermitian kernel
  9. Determinantal point process with a $J$-Hermitian kernel
  10. Poisson point process
  11. Permanental point process
  12. Hafnian point process

Key Result

Theorem 1

Let $\Phi \subset H \subset \Phi'$ be a real standard triple, and let $\Psi \subset \mathcal{F} \subset \Psi'$ be a complex standard triple. Assume we are given a family $(A(\varphi))_{\varphi \in \Phi}$ of Hermitian operators in $\mathcal{F}$ such that Then, each operator $A(\varphi )$, $\varphi \in\Phi$, is essentially self-adjoint in $\mathcal{F}$, and we denote its closure by $(\tilde{A}(\var

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3: LM
  • Remark 4