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

Understand

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 $S^4$.