Project/Area Number  08454039 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nagoya University 
Principal Investigator 
OZAWA Masanao School of Informatics and Sciences, Nagoya University, Prfessor, 情報文化学部, 教授 (40126313)

CoInvestigator(Kenkyūbuntansha) 
伊藤 正之 名古屋大学, 情報文化学部, 教授 (60022638)
IHARA Shunsuke School of Informatics and Sciences, Prfessor, 情報文化学部, 教授 (00023200)
MATSUBARA Yo School of Informatics and Sciences, Associate Prfessor, 情報文化学部, 助教授 (30242788)
YASUMOTO Masahiro Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10144114)
SHINODA Juichi Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (30022685)
MORIMOTO Hiroshi Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (20115645)

Project Fiscal Year 
1996 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥6,300,000 (Direct Cost : ¥6,300,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1997 : ¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1996 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  nonstandard analysis / quantum optics / Heisenberg groups / canonical commutation relatins / unitary representations / hyperfinite / phase operators / quantizations / 超準解析学 / 量子光学 / Heisenberg(ハイゼンベルク)群 / 正準交換関係 / ユニタリ表現 / 超有限 / 位相作用素 / 量子化 / Heisenberg群 / 量子力学 / Schrodinger表現 / Z_1飽和的超準宇宙 / 超準殻 / 作用素測度 / Naimark拡大 / Hilberk■間 
Research Abstract 
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 ArakiWoods representation. In the next application of a representation of a hyperfinite Heisenberg group, we constructed a selfadjoint phase operator of a singlemode 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 operatorvalued 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 PeggBarnett operator and gives the correct result within infinitesimal error, so that the calculation of phase becomes rather easier than the PeggBarnett method.
