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

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 $\mathbb K^T(x,y)=\mathbb K(y,x)$\mathrm{.}Wesaythatapointprocess$\mu$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 $k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]{i,j=1,\dots,n},$\mathrm{.}Here$\operatorname{haf}©$ denotes the hafnian of a symmetric matrix $C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process ${\mathrm{\Pi }}_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying $\int{\Delta}\mathbb E(|G(x)|^2)\sigma(dx)<\infty$foreachcompact$\Delta\subset X$. Then the Cox process ${\mathrm{\Pi }}_R$ with $R(x)=\mathrm{\mid }G(x){\mathrm{\mid }}^2$ is a hafnian point process. The main result of the paper is that each such process ${\mathrm{\Pi }}_R$ 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.