Holomorphic Quantization in Constant Curvature Backgrounds

TL;DR

This paper develops a holomorphic quantization method for free particles in two-dimensional constant curvature spaces, providing explicit spectra and wave functions, and linking to representation theory of SL(2,R).

Contribution

It introduces a novel holomorphic quantization scheme based on Lagrangian embeddings for particles in curved backgrounds, with explicit examples and connections to representation theory.

Findings

Quantization scheme applied to various geometries including plane, torus, sphere, hyperbolic plane.

Recovered Hamiltonian spectra and wave functions for particles in these spaces.

Provided geometric interpretation of Repka's decomposition of SL(2,R) tensor products.

Abstract

We present a holomorphic quantization scheme for free point particles on two-dimensional constant curvature Riemannian backgrounds. The procedure is based on a Lagrangian embedding of the particle configuration space into a product of coadjoint orbits of the background isometry group. Examples are provided by particles on the plane, torus, sphere, and hyperbolic plane, with or without a monopole field. We elaborate the method by recovering the Hamiltonian spectrum and the wave functions on such spaces. As a by-product, we obtain a geometric and physical interpretation of Repka's result on the decomposition of tensor products of $\mathbf{SL}(2,\mathbb{R})$ discrete series representations.