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

In order to provide a general framework for applications of nonstandard analysis to quantum physics, the hyperfinite Heisenberg group, which is a finite Heisenberg group in the nonstandard universe, is formulated and its unitary representations are examined. The ordinary Schrödinger representation of the Heisenberg group is obtained by a suitable standardization of its internal representation. As an application, a nonstandard-analytical proof of noncommutative 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 physics, which aims at the systematic applications of methods of foundations of mathematics to extending physical theories. The program and the achievement of the logical extension methods are explained in some detail.