On the Lorentz symmetry in conformally reduced quantum gravity

The functional renormalization group treatment of the conform reduced Einstein-Hilbert gravity is extended by following the evolution of the time and space derivatives separately, in order to consider the Lorentz symmetry during the evolution. We found the Reuter fixed point in the ultraviolet region. It is shown that starting from the Gaussian fixed point the Lorentz symmetry breaks down in the vicinity of the Reuter fixed point. Similarly, in the broken phase it also breaks down in the infrared region close to a critical singularity scale. By calculating the anomalous dimension form the kinetic term of the action, we found a new relevant coupling belonging to the curvature.