Berezin-Type Operators on the Cotangent Bundle of a Nilpotent Group

TL;DR

This paper develops Berezin-type operators and quantization methods on the cotangent bundle of a nilpotent Lie group, linking coherent states, covariant symbols, and pseudo-differential operators for advanced mathematical analysis.

Contribution

It introduces a novel Berezin-Toeplitz quantization framework on the cotangent bundle of nilpotent groups, extending existing quantization techniques with new formalism and potential applications.

Findings

Defined coherent states and covariant symbols for nilpotent groups

Established a Berezin-Toeplitz quantization scheme

Connected the formalism to pseudo-differential operators

Abstract

We define and study coherent states, a Berezin-Toeplitz quantization and covariant symbols on the product between a connected simply connected nilpotent Lie group and the dual of its Lie algebra. The starting point is a Weyl system codifying the natural Canonical Commutation Relations of the system. The formalism is meant to complement the quantization of the cotangent bundle by pseudo-differential operators, to which it is connected in an explicit way. Some extensions are indicated, concerning $\tau$-quantizations and variable magnetic fields.