Renormalization of a Classical Gravitational Field Interacting with Quantized Matter Fields

The behavior of the mean value of the energy-momentum tensor of a set of quantized matter fields interacting with a classical gravitational field which is, in turn, produced by this mean value, is investigated. Singularities appear in the energy-momentum tensor corresponding to divergences of three different orders: ∞4, ∞2, and log ∞. These can be removed by the introduction of counter terms into Einstein's equation. The ∞4 singularity is removed by a ``cosmological term,'' the ∞2 singularity by a renormalization of the gravitation constant, and the log ∞ singularity by a counter term derivable from a Lagrangian which is quadratic in the Riemann tensor. The gravitational Green's function corresponding to this semiclassical approximation to the fully quantized theory is found to have the asymptotic behavior 1/p4 instead of 1/p2, and thus to have a much weaker singularity in the coordinate representation than the Green's function of the ``bare'' linearized theory.