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

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 $k^{(n)}(x_1,\dots ,x_n)=\mathrm{haf}[\mathbb{K}(x_i,x_j){]}_{i,j=1,\dots ,n}$. Here $\mathrm{haf}(C)$ 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.

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