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

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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.