Semiclassical States on Lie Algebras

TL;DR

This paper extends a semiclassical approximation technique from the canonical algebra to general finite-dimensional Lie algebras, enabling analysis of quantum systems with more complex symmetry structures.

Contribution

It develops a method to apply semiclassical states to Lie algebras with Casimir elements, broadening the scope of effective quantum analysis beyond canonical cases.

Findings

Extended semiclassical techniques to Lie algebras with Casimir elements.

Established conditions for consistent Casimir restriction during semiclassical truncation.

Provided explicit treatment of semiclassical states on universal enveloping 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.