The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity

TL;DR

This paper explores the connection between super-renormalizable quantum gravity theories and non-commutative spacetime symmetries, revealing that one such theory aligns with quantum group symmetry structures on -Minkowski spacetime.

Contribution

It demonstrates that a specific super-renormalizable quantum gravity theory can be described using  quantum group symmetries, linking quantum gravity with non-commutative geometry.

Findings

Identified a unique non-local, Lorentz-invariant super-renormalizable gravity theory with quantum group symmetry.

Established the equivalence of the propagator at linear level with a  non-commutative spacetime model.

Connected super-renormalizable gravity theories with non-commutative spacetime symmetries.

Abstract

In this paper we consider the relation between the super-renormalizable theories of quantum gravity (SRQG) studied in [arXiv:1110.5249v2, arXiv:1202.0008] and an underlying non-commutativity of spacetime. For one particular super-renormalizable theory we show that at linear level (quadratic in the Lagrangian) the propagator of the theory is the same we obtain starting from a theory of gravity endowed with {\theta}-Poincar\'e quantum groups of symmetry. Such a theory is over the so called {\theta}-Minkowski non-commuative spacetime. We shed new light on this link and show that among the theories considered in [arXiv:1110.5249v2, arXiv:1202.0008], there exist only one non-local and Lorentz invariant super-renormalizable theory of quantum gravity that can be described in terms of a quantum group symmetry structure. We also emphasize contact with pre-existent works in the literature and discuss preservation of the equivalence principle in our framework.