Quantization of a Scalar Field in Two Poincaré Patches of Anti-de Sitter Space and AdS/CFT release_rev_f7cf9613-8c31-43cb-ae83-8b3364be17b5

Abstract

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, AdS_d+1 (d ≥ 2). It is shown that in Poincaré coordinates (r,t,x⃗), the two boundaries at r=±∞ are connected. When the scalar mass m satisfies a condition 0 < ν=√((d^2/4)+(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=±∞ 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.
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