Unitary Representations of Hyperfinite Heisenberg Groups and Their Applications to Quantum Physics

这是一个跨学科的研究，包括数学基础，应用分析，数学物理和量子力学。超有限海森堡群的表示理论是由Ojima和Ozawa在1992年建立的，目的是为量子物理中非标准分析的系统应用提供一个统一的框架。研究结果包括与非标准方法相关的数学基础的结果，也包括与各种应用有关的应用分析的结果。以下是相对于量子物理学的应用特别重要的。Kelemen和罗宾逊用非标准分析方法重建了Glimm和Jaffe的phi^4_2模型。为了将非标准分析系统地应用于其他场模型的构造，我们在非标准酉表示理论的框架下推广了他们表示正则对易关系的非标准分析方法.作为应用，在此框架下重建了以下表示：Segal表示，相对论时间零点场和Araki-Woods表示。在超有限海森堡群表示的下一个应用中，我们构造了量子力学中单模电磁场的自伴相位算符。该算子自然地被认为是Pegg和巴内特提出的有限维空间上的近似相位算子的极限。该算子的谱测度是Helstrom发现的最优概率算子值测度的Naimark扩展。最近的一本量子光学教科书提到我们的结果如下：用这个算符计算相位在形式上类似于用Pegg-Barnett算符计算相位，并且在无穷小误差范围内给出正确的结果，因此相位的计算比Pegg-Barnett方法容易得多。

This is an interdisciplinary research including foundations of mathematics, applied analysis, mathematical physics, and quantum mechanics. The representation theory of hyperfinite Heisenberg groups was instituted by Ojima and Ozawa in 1992 in order to give a unified framework for systematic applications of nonstandard analysis to quantum physics. The research results include results in foundations of mathematics relative to the nonstandard method and also includes results in applied analysis concerning various applications. The following are of particular importance relative to applications to quantum physics. Kelemen and Robinson reconstructed the phi^4_2 model of Glimm and Jaffe with methods of nonstandard analysis. In order to apply nonstandard analysis to other constructions of field models systematically, we generalize their nonstandard analytical methods of representing the canonical commutation relations in the framework of the theory of nonstandard unitary representations. As applications, the following representations are reconstructed in this framework : the Segal representation, relativistic time zero fields, and the Araki-Woods representation. In the next application of a representation of a hyperfinite Heisenberg group, we constructed a self-adjoint phase operator of a single-mode electromagnetic field in quantum mechanics. This operator is naturally considered as the limit of the approximate phase operators on finite dimensional spaces proposed by Pegg and Barnett. The spectral measure of this operator is shown to be a Naimark extension of the optimal probability operator-valued measure found by Helstrom. A recent text book on quantum optics mentions our result as follows : The calculation of phase using this operator is formally similar to that using the Pegg-Barnett operator and gives the correct result within infinitesimal error, so that the calculation of phase becomes rather easier than the Pegg-Barnett method.