Physical significance of generalized boundary conditions: an Unruh-DeWitt detector viewpoint on ${\text{AdS}}_2\times {\mathbb{S}}^2$

On ${\text{AdS}}_2\times {\mathbb{S}}^2$, we construct the two-point correlation functions for the ground and thermal states of a real Klein-Gordon field admitting generalized $(\gamma ,v)$-boundary conditions. We follow the prescription recently outlined in [1] 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.