Quantization of a Scalar Field in Two Poincaré Patches of Anti-de Sitter Space and AdS/CFT

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${}_{d+1}$ ($d\ge 2$). It is shown that in Poincaré 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.

35 pages, 8 figures; ver.2: Fig.5, fig. 6 and subsection 2.4 modified; ver.3: Abstract and subsection 2.4 changed. Two figures removed and one figure added. 33 pages

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