Renormalization group flows and entropy in curved spacetimes

I discuss the derivation and applications of a non-perturbative Renormalization Group (RG) equation for gauge theories and quantum gravity in Lorentzian spacetimes. A key ingredient to derive the RG equation in Lorentzian spacetimes is the use of a local regulator, acting as an artificial mass for correlation functions. A local regulator is compatible with the unitarity and Lorentz invariance of the theory, and it gives raise to a local RG equation. A Hadamard-type point-splitting regularisation guarantees that the RG equation is finite. The RG equation depends on the choice of a state, a distinctive feature that is not present in the Euclidean case. If the effective average action describing the theory does not contain derivatives higher than second order, and it is local in the fields, an application of the renown Nash-Moser theorem proves that the RG equation for scalar fields admits local, exact solutions. For gauge theories, the symmetries of the theory are controlled by an extended Slavnov-Taylor identity, that can be studied with the cohomology of the Batalin-Vilkovisky operator. Assuming the Local Potential Approximation in the case of an interacting scalar field, in 3 dimensions the RG flow exhibits the well-known Wilson-Fisher fixed point. In 4 dimensions, the flow has no non-trivial fixed point in the vacuum, but it exhibits a novel non-trivial fixed point in the high-temperature limit of a thermal state for the free theory. Moreover, the Bunch-Davies state in de Sitter spacetime also has a non-trivial fixed point in the inflationary regime. Finally, the flow is applied to the case of quantum gravity. Taking into account only state- and background-independent terms, the RG flow exhibits a non-trivial fixed point in the Einstein-Hilbert truncation, providing a mechanism for Asymptotic Safety in Lorentzian quantum gravity. In the second part, I study the relative entropy for a free scalar field propagating over a dynamical, spherically symmetric black hole. Considering the back-reaction of the relative entropy on the black hole horizon, it is possible to write a flux balance equation for the generalised entropy of quantum and gravitational degrees of freedom. Similar considerations lead to a conservation law for the generalised entropy in de Sitter spacetime.