Abstract
In this paper, we consider the relation between the super-renormalizable theories of quantum gravity studied by Biswas, Gerwick, Koivisto, and Mazumdar [Phys. Rev. Lett. 108, 031101 (2012)] and Modesto [arXiv:1107.2403; arXiv:1202.0008] and an underlying noncommutativity of space-time. For one particular super-renormalizable theory, we show that at the linear level (quadratic in the Lagrangian) the propagator of the theory is the same one we obtain starting from a theory of gravity endowed with $\ensuremath{\theta}$-Poincar\'e quantum groups of symmetry. Such a theory is over the so-called $\ensuremath{\theta}$-Minkowski noncommutative space-time. We shed new light on this link and show that, among the theories considered in these references, there exists only one nonlocal 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 preexistent works in the literature and discuss preservation of the equivalence principle in our framework.
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