Quantization of a scalar field in two Poincaré 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 Poincaré patches of Lorentzian anti-de Sitter (AdS) space, AdSd+1AdSd+1 (d≥2d≥2). It is shown that in Poincaré coordinates (r,t,x→), the two boundaries at r=±∞r=±∞ are connected. When the scalar mass m   satisfies a condition 0<ν=(d2/4)+(mℓ)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=±∞r=±∞ 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.