Noncommutative coherent states and related aspects of Berezin-Toeplitz quantization

TL;DR

This paper constructs and analyzes noncommutative coherent states derived from unitary irreducible representations of a specific nilpotent Lie group, exploring their properties and Berezin-Toeplitz quantization in noncommutative quantum mechanics.

Contribution

It introduces new noncommutative coherent states based on UIRs of a nilpotent Lie group and studies their quantization and semi-classical limits.

Findings

Reproducing kernels for noncommutative coherent states computed.

Berezin-Toeplitz quantization applied to noncommutative phase space.

Semi-classical asymptotics analyzed for these states.

Abstract

In~this paper, we construct noncommutative coherent states using various families of unitary irreducible representations (UIRs) of $\g$, a connected, simply connected nilpotent Lie group, that was identified as the kinematical symmetry group of noncommutative quantum mechanics for a system of 2-degrees of freedom in an earlier paper. Likewise described are the degenerate noncommutative coherent states arising from the degenerate UIRs of $\g$. We~then compute the reproducing kernels associated with both these families of coherent states and study Berezin-Toeplitz quantization of the observables on the underlying 4-dimensional phase space, analyzing in particular the semi-classical asymptotics for both these cases.