Higher derivative quantum gravity

In [24] (together with A. O. Barvinsky) and in [25] I studied the R+R² higher-derivative quantum gravity with quadratic lagrangian of general type. From the previous investigations it has been known that this theory is asymptotically free in `physical' region of the coupling constants specified by the condition of stability of the flat spacetime, i.e. by the absence of tachyons. We reexamined the ultraviolet behavior of this theory and found the calculations of previous authors to be in error in the conformal sector. We showed that, in fact, the conformal coupling is not asymptotically free but has exactly the opposite `zero-charge' behavior in the `physical' region of coupling constants. Moreover, we showed that the stability condition of the flat spacetime is inconsistent with the asymptotic freedom in the conformal sector regardless of the presence of any low spin $(s\le 1)$ matter fields. Further, we showed that the theory can be asymptotically free only in `unphysical' region of coupling constants, specified by the positive definite Euclidean action.

This kind of behavior indicates that the conformal coupling of quantum excitations on the flat background becomes strong at high energies, the flat space becomes unstable, a phase transition occurs and a new stable curved vacuum appears which is nothing but a classical condensate of conformal excitations.

We calculated for the first time the off-shell one-loop counterterms of the reparametrization-invariant and gauge-independent Vilkovisky's effective action and established in this way the high-energy behavior of Einstein coupling constant (otherwise gauge-dependent) too.