We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that allows one to obtain an equation which is meaningful on generic Lorentzian backgrounds is the use of a local regulator, which keeps the theory covariant. As a proof of concept, the developed methods are used to show that non-trivial fixed points arise in quantum field theories in a thermal state and in the case of quantum fields in the Bunch–Davies state on the de Sitter spacetime.

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The goal of this paper is to prove an area law for the entropy of dynamical, spherically symmetric black holes from the relative entropy between coherent states of the quantum matter, generalising the results by Hollands and Ishibashi (2019 Class. Quantum Grav. 36 195001). We consider the relative entropy between a coherent state and a suitably chosen asymptotically vacuum state for a scalar quantum field theory propagating over a dynamical black hole. We use the conservation law associated to the Kodama vector field in spherically symmetric spacetimes, and the recent results on the relative entropy between coherent states found in Longo (2019 Lett. Math. Phys. 109 2587) and Casini et al (2019 Phys. Rev. D 99 125020). We consider the back-reaction of the quantum matter on the metric. From the conservation law associated with the Kodama vector field, we obtain an equation for the derivative along the outgoing lightrays of the generalised entropy, defined as the sum of the relative entropy between coherent states of the scalar field and one quarter of the horizon area.