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

Let X be a locally compact Polish space and sigma a nonatomic reference measure on X (typically X = R-d and sigma is the Lebesgue measure). Let X-2 there exists (x, y) bar right arrow K(x, y) is an element of C-2x2 be a 2 x 2-matrix-valued kernel that satisfies K-T (x, y) = K(y, x). We say that a point process mu in X is hafnian with correlation kernel K(x, y) if, for each n is an element of N, the nth correlation function of mu (with respect to sigma(circle times n)) exists and is given by k((n)) (x(1), ..., X-n) = haf[K(x(i), x(j))]i,j=i, ..., 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 FIR is a Poisson point process in X with random intensity R(x). Let G(x) be a complex Gaussian field on X satisfying f(Delta) E(vertical bar G(x)vertical bar(2))sigma(dx) < infinity for each compact Delta subset of X. Then the Cox process Pi(R) with R(x) = vertical bar G(x)vertical bar(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 (CCRs), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.

Automated summary

The paper discusses hafnian point processes on locally compact Polish spaces and their relationship with Cox processes involving Gaussian fields, as well as their application in quantum mechanics.