Classical and quantum gravity with fractional operators

Following the same steps made for a scalar field in a parallel publication, we propose a class of perturbative theories of quantum gravity based on fractional operators, where the kinetic operator of the graviton is either made of fractional derivatives or a covariant fractional d'Alembertian. The classical action for each theory is constructed and the equations of motion are derived. Unitarity and renormalizability of theories with a fractional d'Alembertian are also considered. We argue that unitarity and power-counting renormalizability never coexist, although in some cases one-loop unitary and finiteness are possible. One of the theories is unitary and infrared-finite and can serve as a ghost-free model with large-scale modifications of general relativity.
Comment: 24 pages, 1 figure. This paper was originally part of arXiv:2102.03363v1. v1: discussion on symmetries expanded. v2: minor typos corrected. Supersedes the published version which was published without the author's consent and includes several uncorrected typos