交换厄米特算子族的关联测度及其在CAR和CCR准自由表示粒子密度中的应用
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
摘要：
设$X$是局部紧的第二可数Hausdorff拓扑空间.我们考虑一族交换厄米算子$a(Δ)$，其指标是$X$（$X$上的量子随机过程）中的所有可测的、相对紧的集合$Δ$.对于这样一个家庭，我们介绍了相关措施的概念。证明了如果算子族具有满足某种增长条件的相关测度，则在$X$上存在具有相同相关测度的点过程.此外，运算符$a(Δ)$可以实现为关于该点过程的$L^2$ -空间中的乘法运算符。在证明中，我们利用了[Y. G. Kondratiev和T. Kuna，{ \it Infin.迪门斯Anal.量子概率相对顶部。} { \bf 5}（2002），201- 233]。特别地，我们的结果将该论文中点过程存在的判据推广到拓扑空间$X$的情形，该拓扑空间是点过程理论中的标准基础空间。作为应用，我们讨论了CAR和CCR的准自由表示的粒子密度，这导致费米子，玻色子，类费米子和类玻色子（例如，2阶的para-fermions和para-bosons）点过程。特别是，我们证明了任何费米子点过程对应于一个厄米内核可以推导出这种方式。
英文摘要：
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.
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