Semigroups and Scattering Theory for a Scalar Field on Curved Space-time

A classical theory is developed for the time development and scattering of a minimally coupled scalar eld on closed spacetimes that evolve from initial, to nal static states. The time development is obtained by reformulating the eld equation as an abstract Cauchy problem on a Hilbert space. Constraints are imposed on the metric that enable the use of semigroup theory, and the eld solution is obtained from Cauchy data via application of a two parameter semi-group of evolution operators. The scattering theory is also formulated on a Hilbert space, and the wave operators and scattering operator are constructed from the evolution operators. It is shown that semigroup theory most readily applies to spacetimes that undergo contraction.