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.
The functional renormalization group equations are derived for the conformally reduced gravity, in the framework of the Wegner–Houghton equation. It is argued, that the blocking introduces bilocal terms into the action, which can account for the evolution of the anomalous dimension. The phase structure exhibits the known structure including an ultraviolet attractive non-Gaussian fixed point.
The functional renormalization group equations are derived in Minkowski space–time for scalar models. It is shown that the couplings become complex, which can change the fixed point structure. New phases appear, and the models can build up complex fixed points.
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