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Entanglement entropies calculated in the framework of quantum field theory onclassical, flat or curved, spacetimes are known to show an intriguing area lawin four dimensions, but they are also notorious for their quadratic ultravioletdivergences. In this paper we demonstrate that the analogous entanglemententropies when computed within the Asymptotic Safety approach to backgroundindependent quantum gravity are perfectly free from such divergences. We arguethat the divergences are an artifact due to the over-idealization of a rigid,classical spacetime geometry which is insensitive to the quantum dynamics.

Finite Entanglement Entropy in Asymptotically Safe Quantum Gravity