Deformation quantization on the cotangent bundle of a Lie group

TL;DR

This paper develops a comprehensive non-formal deformation quantization framework for the cotangent bundle of weakly exponential Lie groups, introducing integral formulas, algebraic structures, and operator representations.

Contribution

It provides the first complete theory of deformation quantization on these groups, including star-product formulas, algebraic extensions, and explicit examples.

Findings

Established a star-product integral formula.

Constructed a C*-algebra of observables.

Presented an operator representation in position space.

Abstract

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.