Classical limit for Lie algebras

Abstract

We present a general method to construct the classical canonical coordinates for any Lie algebra. The symplectic 2-form, which defines the Poisson structure is constructed explicitly. In this way we are able to define the corresponding classical action for a Lie algebra, which can be subsequently used in recovering the quantum limit, e.g., through a functional integral approach. We pay special attention to the topological structure of the classical phase space, which plays a crucial role in recovering the correct form of the classical equations of motion and eventually in the requantization procedure. The classical phase space turns out to be “curved” and this induces the appearance of “gauge potentials”. The classical mechanics of such systems is analogous to the Dirac's constrained dynamics. We detail these constructions for the case of SU(2) and SU(3) Lie algebras. The main differences for other Lie algebras arise from a larger number of degrees of freedom.