Geometric quantization of finite Toda systems and coherent States

TL;DR

This paper develops a geometric quantization framework for finite Toda systems, utilizing coadjoint orbits and group representations to construct coherent states and quantum Hilbert spaces.

Contribution

It introduces a novel geometric quantization approach for Toda systems based on coadjoint orbits and constructs explicit coherent states and quantum spaces.

Findings

Constructed a unitary representation of the group for the Toda system.

Derived Rawnsley coherent states for the quantized system.

Identified finite-dimensional quantum Hilbert spaces for the system.

Abstract

Adler had shown in 1979 that the Toda system can be given a coad- joint orbit description. We quantize the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with pos- itive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module . We find the Rawnsley coherent states after a completion of the above space of sections. We also find non-unitary finite dimensional quantum Hilbert spaces for the system.