Bulk versus boundary dynamics in anti–de Sitter spacetime

We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti–de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincaré coordinates. For ${\mathrm{AdS}}_3,$ the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\mathrm{SL}(2,R{)}_L\times \mathrm{SL}(2,R{)}_R$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincaré coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.