Boundary conditions and renormalized stress-energy tensor on a Poincar\'e patch of $\textrm{AdS}_2$

Quantum field theory on anti-de Sitter spacetime requires the introduction of boundary conditions at its conformal boundary, due essentially to the absence of global hyperbolicity. Here we calculate the renormalized stress-energy tensor $T_{\mu\nu}$ for a scalar field $\phi$ on the Poincar\'e patch of $\text{AdS}_2$ and study how it depends on those boundary conditions. We show that, except for the Dirichlet and Neumann cases, the boundary conditions break the maximal $\textrm{AdS}$ invariance. As a result, $\langle\phi^2\rangle$ acquires a space dependence and $\langle T_{\mu\nu}\rangle$ is no longer proportional to the metric. When the physical quantities are expanded in a parameter $\beta$ which characterizes the boundary conditions (with $\beta=0$ corresponding to Dirichlet and $\beta=\infty$ corresponding to Neumann), the singularity of the Green's function is entirely subtracted at zeroth order in $\beta$. As a result, the contribution of nontrivial boundary conditions to the stress-energy tensor is free of singular terms.