On the spaces of positive and negative frequency solutions of the Klein-Gordon equation in curved space-times

In a space-time $\text{(V}{}_{\mathrm{n}}\text{×}\text{R}\text{;g)}$ with V_n closed (n ≠ 2) satisfying certain global conditions, we can write the Klein-Gordon equation, relative to a suitable class of atlases, in the evolution form du/dt = T^-1(t)u, on Sobolev spaces K^l(V_n) = H^l(V_n) × H^l?1(V_n), where the spectrum of T^-1(t) is imaginary. Following papers by T. Kato and J. Kisyński we prove the existence of the evolution operator for this equation. The space $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(V}{}_{\mathrm{n}}\text{)}$ has a natural strongly-symplectic structure ω. We determine the explicit form of complex-structure-positive operators of this structure. We prove that any two such operators, say J₁, J₂, are symplectically equivalent, (i.e. there is a symplectic transformation S such that J₂ = SJ₁S^-1). Spaces of positive and negative frequency solutions are then unique modulo symplectic equivalence. Each operator J determines a regular kernel on space-time which satisfies the properties of the kernel postulated by A. Lichnérowich in his program of quantization of fields in curved space-times. We carry out explicit calculations in the case of Robertson-Walker space-times. If an additional condition is satisfied by the given space-time, a unique complex-structure-positive operator can be selected in a natural way. This condition is satisfied by globally stationary space-times.