Star Product Reduction for Coisotropic Submanifolds of Codimension 1

TL;DR

This paper introduces a new reduction method for star products on phase spaces constrained by codimension 1 coisotropic submanifolds, explicitly computing the reduced star product and analyzing the impact of different constraint functions.

Contribution

It presents a constructive approach to compute reduced star products for codimension 1 coisotropic submanifolds without relying on symmetries or group actions.

Findings

The reduction method depends on the choice of the constraint function.

Different constraint functions can lead to inequivalent reduced star products.

Explicit examples demonstrate the method's applicability and limitations.

Abstract

We propose a reduction procedure that leads to a reduced star product on the reduced phase space of a `First Class'-constrained system, where no symmetries, group actions or the like are present. For the case that the coisotropic submanifold has codimension 1, we establish a constructive method to compute the reduced star product explicitly. Concluding examples show that this method depends crucially on the constraint function singled out to describe the constrained submanifold and not only on this submanifold itself, and that two different constraint functions for the same constraint submanifold will generally result in not only different but inequivalent reduced star products.