Hooke’s symmetries and nonrelativistic cosmological kinematics. II: Irreducible projective representations

Abstract

The irreducible unitary projective representations of the Hooke groups are constructed by Mackey’s method of induced representations. In this manner, we obtain a classification of the true irreducible representations of the extended and ordinary Hooke group universal covering, and then a classification of the elementary quantum systems which have Hooke symmetries. The eigenvalues of their invariant operators label these irreducible representations, and the corresponding infinitesimal representations define their Lie algebras. When this makes sense, some physical interpretations are given.