Abstract
For any fundamental quantum field theory, unitarity, renormalizability, and relativistic invariance are considered to be essential properties. Unitarity is inevitably connected to the probabilistic interpretation of the quantum theory, while renormalizability guarantees its completeness. Relativistic invariance, in turn, is a symmetry which derives from the structure of spacetime. So far, the perturbative attempt to formulate a fundamental local quantum field theory of gravity based on the metric field seems to be in conflict with at least one of these properties. In quantum Ho\v{r}ava gravity, a quantum Lifshitz field theory of gravity characterized by an anisotropic scaling between space and time, unitarity and renormalizability can be retained while Lorentz invariance is sacrificed at high energies and must emerge only as approximate symmetry at low energies. I review various approaches to perturbative quantum gravity with a particular focus on recent progress in the quantization of Ho\v{r}ava gravity, supporting its theoretical status as a unitary, renormalizable and ultraviolet-complete quantum theory of gravity.
Highlights
The search for a consistent quantum theory of gravity can be dated back almost 90 years to the work of Rosenfeld [1]
It could be considered a particular strength of the bottomup approach that it is agnostic about the gravitational degrees of freedom in the UV: the low-energy limit of the effective field theory (EFT) defines the field variables, symmetries, and particle spectrum
In contrast to the local but unphysical UV divergences, true predictions of the quantum theory are connected with infrared (IR) effects that arise from longrange interactions dominated by massless particles
Summary
The search for a consistent quantum theory of gravity can be dated back almost 90 years to the work of Rosenfeld [1]. By construction the effective description breaks down at a finite energy scale and does not extend to the arbitrarily high energies required for a fundamental theory of quantum gravity In this respect, the non-perturbative asymptotic safety program towards quantum gravity might offer a solution in providing a consistent ultraviolet completion [41,42,43,44]. Recent proposals, which involve different quantization prescriptions for the ghost, preserve unitarity but instead lead to a violation of micro-causality [47, 48] In view of these problems, it has been suggested to explore the consequences of the assumption that Lorentz invariance is not a fundamental symmetry but only emerges as an approximate symmetry at low energies.
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