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

Abstract

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)-(x,y) 2×2 be a 2 × 2-matrix-valued kernel that satisfies T(x,y) = (y,x). We say that a point process μ in X is hafnian with correlation kernel (x,y) if, for each n , the nth correlation function of μ (with respect to σ - n) exists and is given by k(n)(x 1,...,xn) =haf[(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 (|G(x)|2)σ(dx) < ∞ for each compact 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.