On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR

Abstract

Let X be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators a ( Δ ) indexed by all measurable, relatively compact sets Δ in X (a quantum stochastic process over X). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over X having the same correlation measure. Furthermore, the operators a ( Δ ) can be realized as multiplication operators in the L 2 -space with respect to this point process. In the proof, we utilize the notion of ⋆-positive definiteness, proposed in [Y.G. Kondratiev, T. Kuna, Harmonic analysis on the configuration space I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 201–233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space X, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.