2006/08/14 by Eugene Lytvynov, Lin Mei, Lytvynov, Eugene +1 · 2 citations
Mathematics · Physics and Astronomy · #47B15 #60G55 #81S05 #81S25 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:47B15 #msc:60G55 #msc:81S05 #msc:81S25
paper · pdf · doi:10.48550/arxiv.math/0608334
arxiv created 2006/08/14 · openalex publication_date 2006/08/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2-space with respect to this point process. In the proof, we utilize the notion of ⋆-positive definiteness, proposed in [Y. G. Kondratiev and T. Kuna, \it Infin. Dimens. Anal. Quantum Probab. Relat. Top. \bf 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.