Quantization of a Scalar Field in Two Poincar\'e Patches of Anti-de Sitter Space and AdS/CFT

Two sets of modes of a massive free scalar field are quantized in a pair of Poincar\'e patches of Lorentzian anti-de Sitter (AdS) space, AdS$_{d+1}$ ($d \geq 2$). It is shown that in Poincar\'e coordinates $(r,t,\vec{x})$, the two boundaries at $r=\pm \infty$ are connected. When the scalar mass $m$ satisfies a condition $0 < \nu=\sqrt{(d^2/4)+(m\ell)^2} <1$, there exist two sets of mode solutions to Klein-Gordon equation, with distinct fall-off behaviors at the boundary. By using the fact that the boundaries at $r=\pm \infty$ are connected, a conserved Klein-Gordon norm can be defined for these two sets of scalar modes, and these modes are canonically quantized. Energy is also conserved. A prescription within the approximation of semi-classical gravity is presented for computing two- and three-point functions of the operators in the boundary CFT, which correspond to the two fall-off behaviours of scalar field solutions.