Quantum metrics with very low action in $R+R^2$ gravity

Abstract

We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as $r=h{L}_{Planck}$, with $h=0,1,...,N$ and $N$ up to $10^5$. The Lagrangian is of the form $\sqrt{g}(R+\hat{I}\pm R^2)$ (in units $c=\hslash =G=1$) and the action is positive-definite, allowing the use of a standard Metropolis algorithm with update probability $\exp (-{\hat{I}}^2\hat{I}”S)$. By minimizing the $R+R^2$ action with respect to conformal modes, Bonanno and Reuter have recently found analytical evidence of a non-trivial "rippled" ground state resembling a kinetic condensate of QCD. Our simulations at low but finite temperature ($T=β^{-1}$) also display strong localized oscillations of the metric, whose total action $S$ remains $\ll \hslash$ thanks to the indefinite sign of $R$. The average metric $⟨g_{rr}⟩$ is significantly different from flat space. The scaling properties of $S$ and $⟨g_{rr}⟩$ are investigated in dependence on $N$ and ${\hat{I}}^2$.

13 pages, 7 figures