Unitary representations of the hyperfinite Heisenberg group and the logical extension methods

In order to provide a general framework for applications of nonstandard analysis to quantum physics, the hyperfinite Heisenbeig group, which is a finite Heisenberg group in nonstandard universe, is formulated and its uni-tary representations are examined. The ordinary Schr\"odinger representation of the Heisenberg group is obtained by a suitable standardization of its internal representation. As an application, a nonstandard-analytical proof of noncom-mutative Parseval’s identity based on the orthogonality relations for unitary representations of finite groups is shown. This attempt is placed in a general framework, called the logical extension methods in $pl\iota ysics $ , whicti aims at the systeinatic applications of me $tl\iota ods $ of foundations of malhemalics to extending physical tlieories. The program and the achievement of $tl\iota e $ logical extension $metl\iota ods $ are cxplaiiicd in soitie detail. 885 1994 60-91 60 1