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

Let $X$ be a locally compact Polish space and $σ$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $σ$ 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 $μ$ in $X$ is hafnian with correlation kernel $\mathbb K(x,y)$ if, for each $n\in\mathbb N$, the $n$th correlation function of $μ$ (with respect to $σ^{\otimes n}$) 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 $Π_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying $\int_Δ\mathbb E(|G(x)|^2)σ(dx)<\infty$ for each compact $Δ\subset X$. Then the Cox process $Π_R$ with $R(x)=|G(x)|^2$ is a hafnian point process. The main result of the paper is that each such process $Π_R$ 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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