Path Integrals on sl(2,R) Orbits

TL;DR

This paper develops a path integral quantization method for orbits of the sl(2,R) Lie algebra, extending techniques from compact groups to non-compact cases, and computes traces in various representations with applications in mathematical physics.

Contribution

It introduces a novel path integral approach for quantizing sl(2,R) orbits, including hyperbolic slices, and analyzes the representation theory of SL(2,R) and its covers.

Findings

Quantization of elliptic and hyperbolic orbits achieved

Explicit trace formulas for group elements computed

Connection established between hyperbolic basis and Mellin transform

Abstract

We quantise orbits of the adjoint group action on elements of the sl(2,R) Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL(2,R) on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl(2,R) representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL(2,R) of PSL(2,R) as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantisation problem by indicating applications.