Eigenfunction Expansions of Functions Describing Systems with Symmetries

Abstract

Physical systems with symmetries are described by functions containing kine- matical and dynamical parts. We consider the case when kinematical symmetries are de- scribed by a noncompact semisimple real Lie group G. Then separation of kinematical parts in the functions is fulfilled by means of harmonic analysis related to the group G. This separation depends on choice of a coordinate system on the space where a physical system exists. In the paper we review how coordinate systems can be chosen and how the corre- sponding harmonic analysis can be done. In the first part we consider in detail the case when G is the de Sitter group SO0(1,4). In the second part we show how the corresponding theory can be developed for any noncompact semisimple real Lie group.