Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hilbert space L2(R3) and the Einstein's relativistic equation in the space L2(R3,dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hilbert space L2(R3) with the help of unique symmetric symplectic operator, the time evolution unitary operator U(t) has been fixed for the Klein Gordon eqution, incorporating either the positive or negative frequencies, in the infinite dimensional Hilbert space L2(R3). (author)