Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime

TL;DR

This paper develops a rigorous framework for the classical phase space and Hadamard states in the BRST formalism for gauge theories on curved spacetimes, establishing isomorphisms with existing approaches and criteria for non-degeneracy.

Contribution

It introduces a consistent definition of phase space in the BRST formalism and demonstrates its equivalence to the subsidiary condition approach for key gauge fields.

Findings

Phase space in BRST formalism is isomorphic to subsidiary condition approach.

Existence of Hadamard states is established for Maxwell and Yang-Mills fields.

Criteria for non-degeneracy of phase space are formulated in terms of BRST cohomology.

Abstract

We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the subsidiary condition approach of Hack and Schenkel in the case of Maxwell, Yang-Mills, and Rarita-Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang-Mills case is concluded from known results in the subsidiary condition (or Gupta-Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang-Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.