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### Hafnian point processes and quasi-free states on the CCR algebra

Eugene Lytvynov , Maryam Gharamah Ali Alshehri

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Swansea University Author:

• Accepted Manuscript under embargo until: 22nd November 2022

Abstract

Let $X$ be a locally compact Polish space and $\sigma$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb... Full description Published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics World Scientific Publishing https://cronfa.swan.ac.uk/Record/cronfa58782 No Tags, Be the first to tag this record! Abstract: Let$X$be a locally compact Polish space and$\sigma$a nonatomic reference measure on$X$(typically$X=\mathbb R^d$and$\sigma$is the Lebesgue measure). Let$X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$be a$2\times 2$-matrix-valued kernel that satisfies$\mathbb K^T(x,y)=\mathbb K(y,x)$. We say that a point process$\mu$in$X$is hafnian with correlation kernel$\mathbb K(x,y)$if, for each$n\in\mathbb N$, the$n$th correlation function of$\mu$(with respect to$\sigma^{\otimes n}$) exists and is given by$k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\,$. Here$\operatorname{haf}(C)$denotes the hafnian of a symmetric matrix$C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process$\Pi_R$is a Poisson point process in$X$with random intensity$R(x)$. Let$G(x)$be a complex Gaussian field on$X$satisfying$\int_{\Delta}\mathbb E(|G(x)|^2)\sigma(dx)<\infty$for each compact$\Delta\subset X$. Then the Cox process$\Pi_R$with$R(x)=|G(x)|^2$is a hafnian point process. The main result of the paper is that each such process$\Pi_R\$ is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCR), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free. College of Science