Hafnian point processes and quasi-free states on the CCR algebra - Eugene Lytvynov (Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2022)
Let X be a locally compact Polish space and σ a nonatomic reference measure on X (typically X=Rd and σ is the Lebesgue measure). Let X2∋(x,y)↦K(x,y)∈C2×2 be a 2×2-matrix-valued kernel that satisfies KT(x,y)=K(y,x). We say that a point process μ in X is hafnian with correlation kernel K(x,y) if, for each n∈N, the nth correlation function of μ (with respect to σ⊗n) exists and is given by k(n)(x1,…,xn)=haf[K(xi,xj)]i,j=1,…,n. Here 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 ΠR is a Poisson point process in X with random intensity R(x). Let G(x) be a complex Gaussian field on X satisfying ∫ΔE(∣∣G(x)∣∣2)σ(dx)<∞ for each compact Δ⊂X. Then the Cox process ΠR with R(x)=|G(x)|2 is a hafnian point process. The main result of the paper is that each such process ΠR is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCRs), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.