Boundary conditions and renormalized stress-energy tensor on a Poincaré patch of ${\mathrm{AdS}}_2$

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 $\varphi$ on the Poincaré patch of ${\mathrm{AdS}}_2$ 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 AdS invariance. As a result, $⟨{\varphi }^2⟩$ acquires a space dependence and $⟨T_{\mu \nu }⟩$ 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.