  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).
