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

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
批准号:
08454039
负责人:
OZAWA Masanao
金额:
$4.03万
依托单位:
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
项目状态:
已结题
起止时间:
1996 至 1998

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中文摘要
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英文摘要
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.
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M.Ozawa: "Quantum state reduction and the quantum Bays principle" Quantum Communication, Computing, and Measurement (Plenum, New York, 1997). 233-241
M.Ozawa:“量子态约简和量子贝斯原理”量子通信、计算和测量(Plenum,纽约,1997 年)。
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Masanao Ozawa: "Phase Operator Problem and Macroscopic Extemsiom of Quatuam Mechanics" Annals of Physics.
Masanao Ozawa:“量子力学的相算子问题和宏观极端”物理学年鉴。
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M.Ozawa: "On the concept of wave packet reduction II: Operational approach" J.Japan Assoc.Phil.Sci.24. 9-15 (1996)
M.Ozawa:“关于波包缩减的概念 II:操作方法”J.Japan Assoc.Phil.Sci.24。
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M.Ozawa: "Quantum nondemolition monitoring of universal quantum computers" Phys.Rev.Lett.80. 631-634 (1998)
M.Ozawa:“通用量子计算机的量子非拆除监测”Phys.Rev.Lett.80。
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62
    Study of Quantum Foundations and Quantum Set Theory
    Study of the Probabilistic Interpretation of Quantum Set Theory
    • 批准号:
      15K13456
    • 项目类别:
      Grant-in-Aid for Challenging Exploratory Research
    • 资助金额:
      $2.5万
    • 财政年份:
      2015
    • 负责人:
      OZAWA Masanao
    • 依托单位:
    Mathematical Studies of Fundamental Principles of Quantum Theory
    • 批准号:
      26247016
    • 项目类别:
      Grant-in-Aid for Scientific Research (A)
    • 资助金额:
      $26.12万
    • 财政年份:
      2014
    • 负责人:
      OZAWA Masanao
    • 依托单位:
    Study of Quantum Set Theory Aiming at an Interpretation of Elements of Reality in Quantum Theory
    • 批准号:
      24654021
    • 项目类别:
      Grant-in-Aid for Challenging Exploratory Research
    • 资助金额:
      $2.5万
    • 财政年份:
      2012
    • 负责人:
      OZAWA Masanao
    • 依托单位:
    海外基金