Restoration of spontaneously broken continuous symmetries in de Sitter spacetime

We formulate a functional approach to scalar quantum field theory in (n+1) -dimensional de Sitter spacetime and solve the functional Schr\"odinger equation for the conformally and minimally coupled scalar fields in both the k=0 and k=1 gauges. We show that there is a natural initial condition, the requirement that the field energy remain finite as the scale factor a becomes small, which specifies a unique, time-dependent, de Sitter vacuum state. This initial condition is closely related to Hawking's prescription of including in the functional integral only those field configurations which are regular on the Euclidean section. The Green's functions constructed using this initial condition are explicitly shown to be the analytic continuation of those derived using the Euclidean path-integral formalism and the regularity (boundary) condition. These Green's functions are used to study the Hawking effect and the restoration of continuous symmetries. In particular we study the restoration of a broken O (2) symmetry of a ^4 theory. We argue that spontaneously broken continuous symmetries are always dynamically restored in de Sitter spacetime.