Deformation Quantization and Reduction by Stages

TL;DR

This paper explores the Quantum-Koszul method for constructing star products on reduced phase spaces, demonstrating its relation to existing reduction techniques and establishing conditions for staged reduction with equivalence of multi-step and single-step processes.

Contribution

It introduces a unified framework for reduction by stages using the Quantum-Koszul method and proves the equivalence of multi-step and single-step star products.

Findings

Quantum-Koszul method unifies reduction techniques

Reduction by stages yields identical star products as single-step

Established equivariant tubular neighborhood theorem

Abstract

We discuss the Quantum-Koszul method for constructing star products on reduced phase spaces in the symplectic, regular case. It is shown that the reduction method proposed by Kowalzig, Neumaier and Pflaum for cotangent bundles is a special case of the Quantum-Koszul method. We give sufficient conditions that reduction by stages is possible in the Quantum-Koszul framework and show that the star product obtained by two steps is identical to that obtained by one step. In order to do so, we prove an equivariant version of the compatible tubular neighborhood theorem.