Symbol representations of observables over the similitude group

Abstract

Generalized coherent states on Lie groups are constructed and these are used to represent quantum states and quantum observables as phase space functions in two different, but related, ways. We present concrete computations on the Similitude group whose elements are parametrized by translations, rotations and dilations in the plane. The basic idea is to obtain generalized coherent states from two basic ingredients of the Weyl-Wigner-Groenewold-Moyal formalism, the Weyl operator and Wigner function, which are in turn defined in terms of unitary representations of the group.