Renormalization of Quantum Electrodynamics in a Classical Gravitational Field

The divergences of the Green's functions of electrons and photons in a classical gravitational field are investigated and are found to be removable by the introduction of suitable counter terms into the Lagrangian. These counter terms are obtained by rewriting the conventional renormalization technique in a generally covariant way. It is shown that infinite renormalization constants identical to those appearing in conventional quantum electrodynamics are sufficient for the removal of all divergences also when a gravitational field is present. No other renormalization is necessary. The segregation of the divergences is accomplished by making use of the transformation properties of the Green's functions under (i) general coordinate transformations, (ii) Vierbein rotations, and (iii) gauge transformations.