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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 degenerateunder the global invariance $\phi \to \phi + \hbox{constant}$. We then quantizethe massless scalar field as a gauge theory by constructing a BRST-invariantquantum action. The corresponding gauge-breaking term is a non-local one of theform $S^{\rm GB}={1\over {2\alpha V}}\bigl(\int_{S^4} dV \phi \bigr)^2$ where$\alpha$ is a gauge parameter and $V$ is the volume of $S^4$. It allows us tocorrectly treat the zero mode problem. The quantum theory is invariant underSO(5), the symmetry group of $S^4$, and the associated two-point functions haveno infrared divergence. The well-known infrared divergence which appears bytaking the massless limit of the massive scalar field propagator is therefore agauge artifact. By contrast, the massless scalar field theory on de Sitterspace $dS^4$ - the lorentzian version of $S^4$ - is not invariant under thesymmetry group of that spacetime SO(1,4). Here, the infrared divergence isreal. 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. Similarconsiderations also apply to massive scalar field theories for exceptionalvalues of the mass parameter (corresponding to the discrete series of the deSitter group).

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