BRST quantization of the massless minimally coupled scalar field in de Sitter space (zero modes, euclideanization and quantization)

TL;DR

This paper develops a BRST quantization method for the massless scalar field on a four-sphere, resolving zero mode issues and showing the gauge artifact nature of infrared divergences, contrasting with de Sitter space.

Contribution

It introduces a gauge-invariant BRST quantization approach for massless scalar fields on S^4, addressing zero modes and infrared divergences, and distinguishes Euclidean and Lorentzian cases.

Findings

Infrared divergence on S^4 is a gauge artifact.

The quantum theory on S^4 is SO(5) invariant with no IR divergence.

On dS^4, IR divergence is real and not removable by gauge fixing.

Abstract

We consider the massless scalar field on the four-dimensional sphere $S^4$. Its classical action $S={1\over 2}\int_{S^4} dV (\nabla \phi)^2$ is degenerate under the global invariance $\phi \to \phi + \hbox{constant}$. We then quantize the massless scalar field as a gauge theory by constructing a BRST-invariant quantum action. The corresponding gauge-breaking term is a non-local one of the form $S^{\rm GB}={1\over {2\alpha V}}\bigl(\int_{S^4} dV \phi \bigr)^2$ where $\alpha$ is a gauge parameter and $V$ is the volume of $S^4$. It allows us to correctly treat the zero mode problem. The quantum theory is invariant under SO(5), the symmetry group of $S^4$, and the associated two-point functions have no infrared divergence. The well-known infrared divergence which appears by taking the massless limit of the massive scalar field propagator is therefore a gauge artifact. By contrast, the massless scalar field theory on de Sitter space $dS^4$ - the lorentzian version of $S^4$ - is not invariant under the symmetry group of that spacetime SO(1,4). Here, the infrared divergence is real. Therefore, the massless scalar quantum field theories on $S^4$ and $dS^4$ cannot be linked by analytic continuation. In this case, because of zero modes, the euclidean approach to quantum field theory does not work. Similar considerations also apply to massive scalar field theories for exceptional values of the mass parameter (corresponding to the discrete series of the de Sitter group).