Costruzione di stati di ground su spaziotempi statici dotati di bordo di tipo tempo: libertà nascoste e divergenze infrarosse.

In this work we review the construction of ground states focusing on a real scalar field whose dynamics is ruled by the Klein-Gordon equation on a large class of static spacetimes with a time-like boundary. As in the analysis of the classical equations of motion, when enough isometries are present, via a mode expansion the construction of two-point correlation functions boils down to solving a second order, ordinary differential equation on an interval of the real line. Using the language of Sturm-Liouville theory, most compelling is the scenario when one endpoint of such interval is classified as a limit circle, as it often happens when one is working on globally hyperbolic spacetimes with a timelike boundary. In this case, beyond initial data, one needs to specify a boundary condition both to have a well-defined classical dynamics and to select a corresponding ground state. Here, we take into account boundary conditions of Robin type by using well-known results from Sturm-Liouville theory, but we go beyond the existing literature by exploring an unnoticed freedom that emerges from the intrinsic arbitrariness of secondary solutions at a limit circle endpoint. Accordingly, we show that infinitely many one-parameter families of sensible dynamics are admissible. In other words, we emphasize that physical constraints guaranteeing the construction of full-fledged ground states do not, in general, fix one such state unambiguously. In addition, we provide, in full detail, an example on (1+1)-half Minkowski spacetime to spell out the rationale in a specific scenario where analytic formulae can be obtained.\\ Then, on $\text{PAdS}_2\times \mathbb{S}^2$, we construct the two-point correlation functions for the ground and thermal states admitting generalized $(\gamma,v)$-boundary conditions. We follow the prescription we have developed for two different choices of secondary solutions. For each of them, we obtain a family of admissible boundary conditions parametrized by $\gamma \in [0,\frac{\pi}{2}]$. We study how they affect the response of a static Unruh-DeWitt detector. The latter not only perceives variations of $\gamma$, but also distinguishes between the two families of secondary solutions in a qualitatively different, and rather bizarre, fashion. Our results highlight once more the existence of a freedom in choosing boundary conditions at a timelike boundary which is greater than expected and with a notable associated physical significance.\\ To conclude, we observe that, depending on the assigned boundary condition of Robin type, this procedure does not always lead to the existence of a suitable bi-distribution $\omega_2 \in \mathcal{D}'(\mathcal{M}\times\mathcal{M})$ due to the presence of infrared divergences. As a concrete example we consider a Bertotti-Robinson spacetime in two different coordinate patches. In one case we show that infrared divergences do not occur only for Dirichlet boundary conditions as one might expect a priori, while, in the other case, we prove that they occur only when Neumann boundary conditions are imposed at the time-like boundary.