  We investigate the idea of a "general boundary" formulation of quantum field
theory in the context of the Euclidean free scalar field. We propose a precise
definition for an evolution kernel that propagates the field through arbitrary
spacetime regions. We show that this kernel satisfies an evolution equation
which governs its dependence on deformations of the boundary surface and
generalizes the ordinary (Euclidean) Schroedinger equation. We also derive the
classical counterpart of this equation, which is a Hamilton-Jacobi equation for
general boundary surfaces.
