Electromagnetic propagators in hyperbolic Robertson–Walker cosmologies

Abstract

Green functions (retarded, advanced, Feynman and Dyson propagators) are calculated for the electromagnetic field in Robertson–Walker cosmologies with hyperbolic 3-manifolds as spacelike slices. The starting point is the Proca equation, i.e., the Maxwell field with a finite photon mass for infrared regularization, in a static cosmology with simply connected hyperbolic 3-sections. The time and space components of the resolvent kernel are scalar and vectorial point-pair invariants, respectively, and this symmetry allows for an explicit evaluation in the spectral representation. It is found that the quantum propagators have a logarithmic infrared singularity, which drops out in the zero curvature limit. Retarded and advanced Green functions remain well defined in the limit of zero photon mass, and they admit a simple generalization, by conformal scaling, to expanding 3-spaces. In cosmologies with multiply connected hyperbolic 3-manifolds as spacelike sections, the four enumerated propagators are constructed by means of Poincaré series. The spectral decomposition of the Green functions is given in terms of Eisenstein series for a certain class of open hyperbolic 3-spaces, including those with Schottky covering groups corresponding to solid handle-bodies as spacelike slices.