Background independence and asymptotic safety in conformally reduced gravity

We analyze the conceptual role of background independence in the application of the effective average action to quantum gravity. Insisting on a background independent renormalization group (RG) flow the coarse graining operation must be defined in terms of an unspecified variable metric since no rigid metric of a fixed background spacetime is available. This leads to an extra field dependence in the functional RG equation and a significantly different RG flow in comparison to the standard flow equation with a rigid metric in the mode cutoff. The background independent RG flow can possess a non-Gaussian fixed point, for instance, even though the corresponding standard one does not. We demonstrate the importance of this universal, essentially kinematical effect by computing the RG flow of quantum Einstein gravity in the “conformally reduced” Einstein-Hilbert approximation which discards all degrees of freedom contained in the metric except the conformal one. Without the extra field dependence the resulting RG flow is that of a simple ${\varphi }^4$ theory. By including it one obtains a flow with exactly the same qualitative properties as in the full Einstein-Hilbert truncation. In particular it possesses the non-Gaussian fixed point which is necessary for asymptotic safety.

Figure 1

(a) The figure shows the RG flow on the $(g,\lambda )$ plane which is obtained from the CREH truncation with ${\eta }_N^{(\mathrm{kin})}$. The arrows point in the direction of decreasing $k$. (b) As in (a), but for a wider range of $g$ and $\lambda$ values.

Figure 2

(a) As in Fig. 1a, but with ${\eta }_N^{(\mathrm{pot})}$. The thick line is the boundary of the physical parameter space on which ${\eta }_N^{(\mathrm{pot})}$ diverges. (b) As in (a), but here also the “unphysical” flow beyond the boundary is shown.