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

Abstract

Let [Formula: see text] be a locally compact Polish space and [Formula: see text] a nonatomic reference measure on [Formula: see text] (typically [Formula: see text] and [Formula: see text] is the Lebesgue measure). Let [Formula: see text] be a [Formula: see text]-matrix-valued kernel that satisfies [Formula: see text]. We say that a point process [Formula: see text] in [Formula: see text] is hafnian with correlation kernel [Formula: see text] if, for each [Formula: see text], the [Formula: see text]th correlation function of [Formula: see text] (with respect to [Formula: see text]) exists and is given by [Formula: see text]. Here [Formula: see text] denotes the hafnian of a symmetric matrix [Formula: see text]. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process [Formula: see text] is a Poisson point process in [Formula: see text] with random intensity [Formula: see text]. Let [Formula: see text] be a complex Gaussian field on [Formula: see text] satisfying [Formula: see text] for each compact [Formula: see text]. Then the Cox process [Formula: see text] with [Formula: see text] is a hafnian point process. The main result of the paper is that each such process [Formula: see text] 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.