BRST Cohomology and Phase Space Reduction in Deformation Quantization
Abstract
In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a ``quantum'' ChevalleyEilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is ``sufficiently nice'', e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200050774
 arXiv:
 arXiv:math/9901015
 Bibcode:
 2000CMaPh.210..107B
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Symplectic Geometry
 EPrint:
 LaTeX2e, 34 pages, revised version: minor changes and corrected typos