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

Power counting renormalizability of quantum gravity in Lifshitz spacetime