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arXiv 1401.4423math-phmath.MP
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非交换几何中的实维空间

Real dimensional spaces in noncommutative geometry

Bas Jordans

备注：
29 pages, 2 figures
发表信息：
Journal of Functional Analysis 268 (2015) p. 2820-2850
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摘要：

本文将谱三元组的乘积推广到半有限谱三元组的乘积。我们将证明有限可和性和正则性在乘积下保持不变。Connes和Marcolli为每个$z\in(0,\infty)$构造了一个${\rm II}_\infty$ -半有限谱三元组，它可以被看作是一个维数为$z$的几何空间.一个小的适应他们的建设产生一个类型${\rm I}$ -半有限谱三元组。我们将研究这些半有限谱三元组的性质。同时，我们也将避免需要一个红外截止计算的维谱。使用这个集合的半有限谱三元组和产品的半有限谱三元组，可以构建一个数学工具的尺寸和zeta函数正则化的量子场论。

英文摘要：

In this paper we will extend the product of spectral triples to a product of semifinite spectral triples. We will prove that finite summability and regularity are preserved under taking products. Connes and Marcolli constructed for each $z\in(0,\infty)$ a type ${\rm II}_\infty$-semifinite spectral triple which can be considered as a geometric space of dimension $z$. A small adaption of their construction yields a type ${\rm I}$-semifinite spectral triple. We will investigate the properties of these semifinite spectral triples. At the same time we will also avoid the need for an infra-red cutoff to compute the dimension spectrum. Using this collection of semifinite spectral triples and the product of semifinite spectral triples one can construct a mathematical tool for dimensional and zeta-function regularisation in quantum field theory.

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