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

超有限海森堡群的酉表示及其在量子物理中的应用

基本信息

批准号：
08454039
负责人：
OZAWA Masanao
金额：
$ 4.03万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1998
项目状态：
已结题

项目摘要

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.

这是一项跨学科研究，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论由 Ojima 和 Ozawa 于 1992 年建立，旨在为非标准分析在量子物理中的系统应用提供统一的框架。研究成果包括与非标准方法相关的数学基础的成果，也包括涉及各种应用的应用分析的成果。以下内容对于量子物理学的应用特别重要。 Kelemen和Robinson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到场模型的其他构造中，我们在非标准酉表示理论的框架下推广了它们表示正则交换关系的非标准分析方法。作为应用，在该框架中重建了以下表示：Segal 表示、相对论时间零场和 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算子。该算子自然被认为是 Pegg 和 Barnett 提出的有限维空间上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。最近一本关于量子光学的教科书提到了我们的结果：使用该算子的相位计算在形式上与使用Pegg-Barnett算子的计算相似，并且在无穷小的误差内给出了正确的结果，因此相位的计算变得比Pegg-Barnett方法容易得多。