  We investigate the question of unitarity of evolution between hypersurfaces
in quantum field theory in curved spacetime from the perspective of the general
boundary formulation. Unitarity thus means unitarity of the quantum operator
that maps the state space associated with one hypersurface to the state space
associated with the other hypersurface. Working in Klein-Gordon theory, we find
that such an evolution is generically unitary given a one-to-one correspondence
between classical solutions in neighborhoods of the respective hypersurfaces.
This covers the case of pairs of Cauchy hypersurfaces, but also certain cases
where hypersurfaces are timelike. The tools we use are the Schroedinger
representation and the Feynman path integral.
