Power counting renormalizability of quantum gravity in Lifshitz spacetime

Abstract: We analyse the power counting renormalizability of the quantum field theory of Einstein or Einstein-Gauss-Bonnet gravity in D+2 dimensional Lifshitz spacetime. We show the spectral dimension becomes 2+(D/z) at the UV region where z is the critical exponent. Since it is larger than two, the quantum theory of Einstein gravity is not power counting renormalizable. For the pure Einstein-Gauss-Bonnet gravity, where Lifshitz spacetime is allowed only when the parameters are fine tuned, it happens that the graviton modes do not propagate and the quantum field theory is accidentally renormalizable when z>=D. Another method is discretizing the radial coordinate which changes the spectral dimension to 1+(D/z) at the UV region. Since our four dimensional spacetime is continuous, the four dimensional Lorentz symmetry is recovered at the low energy and the power counting renormalizability is still kept for z>=D, if the spacetime near the null singularity in Lifshitz spacetime is modified into AdS spacetime and the discrete radial direction is compactified like a brane world scenario. We also comment on the AdS/CFT correspondence.