Dynamics of the semiclassical Einstein equations.

General relativity with higher-derivative terms could constitute a suitable route to quantize gravity. The gravitational Lagrangian with 8 terms preserves many of the good properties of simple general relativity2 and may lead to a framework in which quantum gravity would become renormalizable. However, it was soon realized that this kind of theory confronts two fundamental problems. The first difficulty was essentially that in general relativity with higherderivative terms one ultimately has either unitarity or renormalizability, but never both. 4' Renormalization-group techniques have been used by Julve and Tonin6 and Salam and Strathdee7 to move the ghost mass off to infinity, where it should become innocuous. Resummation based on the 1lN approximation was carried out by Tomboulis who showed that the real poles vanish in the 1/jiI expansion when gravity is coupled to N massless fermions, but one finds a complex pair of poles in the physical sheet thus violating the analyticity of the S matrix. The second difficulty was raised by Horowitz and %aid who studied~ the dynamics of Einstein equations modified by higher-order derivative terms and showed that flat spacetime then is either catastrophically unstable to conformally flat perturbations that grow exponentially on the Planck time scale, or oscillates at the Planck frequency resulting in radiation with higher energy than the most energetic cosmic-ray events by some seven orders of magnitude or so. It has been recently suggested, nevertheless, ' that in a theory with higher-order derivative terms modulated by a logarithmic dependence on 8, the graviton propagator ought to have no tachyon or unphysical-state poles, or complexcorjugate poles on the physical sheet, awhile the theory appears to be renormalizable and analytical. These kinds of theories could restore also a physical behavior for the semiclassical theory of back reaction. The purpose of this Brief Report is to investigate further this subject starting with a I theory of general relativity with higher-order derivative terms containing a logarithmic dependence on A. The stress-energy operator for a conformally invariant quantum field in a conformally flat spacetime has the general form9