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

Abstract

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 ${\text{AdS}}_2$ 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 $\text{AdS}$ 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.

arXiv comment7 pages. Minor Correction. Matches published version

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• Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime
• Quantum superposition of boundary condition in ${\mathrm{PAdS}}_2$