Constructing the classical limit for quantum systems on compact semisimple Lie algebras

TL;DR

This paper develops a general method to derive the classical limit of quantum systems based on compact semisimple Lie algebras, extending previous specific cases like su(2) and su(3).

Contribution

It provides a unified construction for the classical limit applicable to any compact semisimple Lie algebra, with explicit formulas and analysis of uniqueness.

Findings

Classical limit depends on the sequence of representations used.

Explicit formulas for classical limits are derived.

In most cases, the classical limit is not unique.

Abstract

We give a general construction for the classical limit of a quantum system defined in terms of generators of an arbitrary compact semisimple Lie algebra, generalizing known results for the $\mathfrak{su}_2$ and $\mathfrak{su}_3$ cases. The classical limit depends on the physical problem in question and is determined by the sequence of representations by which it is reached. Only in the simplest cases it is unique. We present explicit formulae useful in determining the classical limit in all important cases.