不定对合代数及其在Krein空间中的表示
Algebra with indefinite involution and its representation in Krein space
摘要：
例如，在量子场论中，常常不可避免地要引入一个不定度规空间，这是为了解释电磁场的明显协变量子化。我们给出了这种不定度规出现的两个更明显的数学原因。第一个概念是代数上对合的替换。对于一个代数${\cal A}$，其对合$\sdag$使得对合代数$({\cal A},\sdag)$的一个表示产生一个不定度量空间，我们用一个新的对合${\cal A}$ * 替换对合$\sdag$，使得$({\cal A},*)$是一个作用在一个具有正定度量的表示空间上的著名对合代数.这说明了两个不同构的对合代数通过对合的置换相互转化。第二个是对合动力系统$(({\cal A},*),{\bf Z}_{2},α)$的协变（希尔伯特空间）表示$({\cal H},π,U)$带来了代数${\cal A}$的Krein空间表示，其中替换了对合。例如，我们表示的异常CCR，CAR和伪Cuntz代数所产生的标准CCR，CAR和Cuntz代数。
英文摘要：
It is often inevitable to introduce an indefinite-metric space in quantum field theory, for example, which is explained for the sake of the manifestly covariant quantization of the electromagnetic field. We show two more evident mathematical reasons why such indefinite metric appears. The first idea is the replacement of involution on an algebra. For an algebra ${\cal A}$ with an involution $\sdag$ such that a representation of the involutive algebra $({\cal A},\sdag)$ brings an indefinite-metric space, we replace the involution $\sdag$ with a new one * on ${\cal A}$ such that $({\cal A},*)$ is a well-known involutive algebra acting on a representation space with positive definite metric. This explains that non-isomorphic two involutive algebras are transformed each other by the replacement of involution. The second is that a covariant (Hilbert space) representation $({\cal H},π,U)$ of an involutive dynamical system $(({\cal A},*),{\bf Z}_{2},α)$ brings a Krein space representation of the algebra ${\cal A}$ with the replaced involution. For example, we show representations of abnormal CCRs, CARs and pseudo-Cuntz algebras arising from those of standard CCRs, CARs and Cuntz algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
