Removing the Faddeev-Popov zero modes from Yang-Mills theory in spacetimes with compact spatial sections

TL;DR

This paper demonstrates that zero modes of Faddeev-Popov ghosts in Yang-Mills theory on spacetimes with compact spatial sections can be removed in a way that preserves BRST invariance, leading to a de Sitter-invariant effective theory.

Contribution

It introduces a method to remove zero modes of Faddeev-Popov ghosts in Yang-Mills theory, maintaining gauge invariance and constructing a de Sitter-invariant quantum theory.

Findings

Zero modes can be removed while preserving BRST invariance.

The effective theory with zero modes removed is equivalent to a modified propagator approach.

The method applies to a wide class of spacetimes with compact spatial sections.

Abstract

It is well known that in de Sitter space the (free) minimally coupled massless scalar field theory does not admit any de Sitter-invariant Hadamard state. Related to this is the fact that the propagator for the massive scalar field corresponding to the de Sitter-invariant vacuum state diverges in the massless limit, with the infrared-divergent term being a constant. Since the Faddeev-Popov ghosts for the covariantly quantized Yang-Mills theory are minimally coupled massless scalar fields, it might appear that de Sitter symmetry would be broken in the ghost sector of Yang-Mills theory in de Sitter space. It is shown in this paper that the modes responsible for de Sitter symmetry breaking can be removed in a way consistent with BRST invariance and that a de Sitter-invariant theory can be constructed. More generally, it is shown that the spatially constant modes (the zero modes) of the Faddeev-Popov ghosts can be disposed of in a wide class of spacetimes with compact spatial sections. Then, the effective theory obtained by removing the zero modes, which contains a non-local interaction term, is shown to be equivalent to the theory corresponding to using a Faddeev-Popov-ghost propagator with the constant infrared-divergent term removed, provided that one can freely integrate by parts in the spacetime integral at the vertex for the ghost interaction term.