Spaces of positive and negative frequency solutions of field equations in curved space–times. II. The massive vector field equations in static space–times

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Abstract
The space–times considered in this article are static, Vn×R, with compact space‐section manifolds without boundary, Vn, and such that the trajectories of the Killing vector field are geodesics. For the physical field of spin 1 and mass m≳0 in these space–times, field equations are solved in any adapted atlas, by the one‐parameter groups of unitary operators generated by scalar and vector Hamiltonians, i−1Tj−1, j=0,1, in Sobolev spaces Hjl−1(Vn) ×Hjl 1(Vn), lεR. Hilbert spaces of positive energy solutions of field equations, as well as those of reduced solutions and their canonical symplectic and complex structures, are determined. The existence and the uniqueness of Lichnerowicz’s (1−1) current on space–time are established, and the corresponding frequency‐solution Hilbert spaces are constructed. Within the framework of Segal, a definition of quantum field operators is given, leading to the postulated commutator for the physical field concerned.

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