Semiclassical States on Lie Algebras

Abstract

The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere), has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by "effectively" fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.