Variational dynamics in open spacetimes

We study the effect of non-vanishing surface terms at spatial infinity on the dynamics of a scalar field in an open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. Starting from the path-integral formulation of quantum field theory, we argue that classical physics is described by field configurations which extremize the action functional in the space of field configurations for which the variation of the action is well defined. Since these field configurations are not required to vanish outside a bounded domain, there can be a non-vanishing contribution of a surface term to the variation of the action. We then investigate whether this surface term has an effect on the dynamics of the action-extremizing field configurations. This question appears to be surprisingly nontrivial in the case of the open FLRW geometry, since surface terms tend to grow as fast as volume terms in the infinite volume limit. We find that surface terms can be important for the dynamics of the field at a classical and quantum level, when there are supercurvature perturbations.