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

Abstract

In a space-time (V n × R;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 K 1 2 (V n) 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 J 1, J 2, are symplectically equivalent, (i.e. there is a symplectic transformation S such that J 2 = SJ 1 S -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.