Nonstandard representations of the canonical commutation relations

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, this paper generalizes their nonstandard analytical methods of representing the canonical commutation relations in the framework of the theory of nonstandard unitary representations. For this purpose, such notions are introduced and examined in detail as nonstandard hulls of representations, approximate representations, and their standardizations. As applications, the following representations are reconstructed in this framework: the Segal representation, relativistic time zero fields, and the Araki-Woods representation.