Quantum effective potential in S1xR3

The functional integral formulation of quantum field theory is applied to the study of the vacuum state in spacetimes with S1xR3 topology. Such a global spacetime structure can be physically realized both in flat and in curved spacetime. In the first case one deals with finite temperature quantum field theories (if S1 is time-like) or with field theories in a spacetime with a compact space dimension (if S1 is spacelike). When curvature is present, a S1 time-like dimension is induced by the Wick rotation whenever the metric is endowed with an event horizon, and this leads to the thermal nature of the vacuum in these cases. We shall take into account here only conformally flat spacetimes. Finally we discuss in some details the topological restoration of a spontaneously broken symmetry and the strictly related problem of the mass dynamical generation. (author)