Covariant star product on symplectic and Poisson spacetime manifolds

TL;DR

This paper develops a covariant deformation quantization framework for tensor-valued differential forms on symplectic and Poisson manifolds, extending previous scalar-based structures and exploring potential applications in gravity and gauge theories.

Contribution

It introduces a covariant star product for tensor fields on symplectic and Poisson manifolds, generalizing existing scalar-based deformation quantization methods.

Findings

Covariant star product defined for tensor-valued differential forms.

Extension of star product to general Poisson manifolds with linear connection.

Potential applications discussed for gravity and gauge theories.

Abstract

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made.