Power counting renormalizability of quantum gravity in Lifshitz spacetime

TL;DR

This paper investigates the renormalizability of quantum gravity theories in Lifshitz spacetime, analyzing spectral dimensions and proposing modifications to achieve renormalizability while maintaining four-dimensional Lorentz symmetry.

Contribution

It provides a detailed analysis of power counting renormalizability in Einstein and Einstein-Gauss-Bonnet gravity within Lifshitz spacetime, including new methods involving radial discretization.

Findings

Spectral dimension exceeds two in UV, indicating non-renormalizability for Einstein gravity.

Pure Einstein-Gauss-Bonnet gravity can be renormalizable under specific parameter tuning.

Radial discretization can restore renormalizability while preserving low-energy Lorentz symmetry.

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.