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
| dc.creator | Lytvynov, Eugene | |
| dc.creator | Mei, Lin | |
| dc.date | 2006-08-14 | |
| dc.date.accessioned | 2026-07-07T07:21:44Z | |
| dc.date.available | 2026-07-07T07:21:44Z | |
| dc.description | 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 $\star$-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. | |
| dc.identifier | https://arxiv.org/abs/math/0608334 | |
| dc.identifier | http://arxiv.org/abs/math/0608334 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115407 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 47B15, 60G55, 81S05, 81S25 | |
| dc.title | 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 | |
| dc.type | text |