Zero modes, Euclideanization, and quantization

We consider the massless scalar field on the four-dimensional sphere $S^4$. Its classical action $S=\frac{1}{2}\int {S^4}^{}\mathrm{dV}{\left(\nabla \phi \right)}^2$ is degenerate under the global invariance $\phi \to \phi +\mathrm{const}$. We then quantize the massless scalar field as a gauge theory by constructing a Becchi-Rouet-Stora-Tyutin-invariant quantum action. The corresponding gauge-breaking term is a nonlocal one of the form $S^{\mathrm{GB}}=\left(\frac{1}{2\alpha V)}\right(\int {S^4}^{}\mathrm{dV}{\phi )}^2$ 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 O(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 ${\mathrm{dS}}^4$, the Lorentzian version of $S^4$, is not invariant under the symmetry group of that spacetime O(1,4). Here, the infrared divergence is real. Therefore, the massless scalar quantum field theories on $S^4$ and ${\mathrm{dS}}^4$ cannot be linked by analytic continuation. In this case, because of zero modes, the Euclidean approach to quantum field theory does not work. Similar consideration also apply to massive scalar field theories for exceptional values of the mass parameter.