Spaces of positive and negative frequency solutions of field equations in curved space–times. I. The Klein–Gordon equation in stationary space–times

Abstract

In stationary space–times Vn×R with compact space-section manifold without boundary Vn, the Klein–Gordon equation is solved by the one-parameter group of unitary operators generated by the energy operator i−1T−1 in the Sobolev spaces Hl(Vn) ×Hl−1(Vn). The canonical symplectic and complex structures of the associated dynamical system are calculated. The existence and the uniqueness of the Lichnerowicz kernel are established. The Hilbert spaces of positive and negative frequency-part solutions defined by means of this kernel are constructed.