Twisted Pseudo-differential Operators on Type I Locally Compact Groups

TL;DR

This paper develops a twisted global quantization framework for symbols on certain locally compact groups, linking representation theory, operator algebras, and quantum mechanics, especially for nilpotent Lie groups with magnetic fields.

Contribution

It introduces a new twisted pseudo-differential calculus on type I groups using crossed product $C^*$-algebras, connecting it to quantum mechanics and scalar quantization.

Findings

Established a twisted quantization for symbols on $ ext{G} imes ext{G}^ atural$

Connected the theory to scalar quantization of cotangent bundles for nilpotent Lie groups

Linked the framework to quantum observables in magnetic fields

Abstract

Let $\G$ be a locally compact group satisfying some technical requirements and $\wG$ its unitary dual. Using the theory of twisted crossed product $C^*$-algebras, we develop a twisted global quantization for symbols defined on $\G\times\wG$ and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle $T^*(\G)$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.