An AdS/dS duality for a scalar particle

The motion of a scalar particle in (d+1)-dimensional AdS space may be described in terms of the Cartesian coordinates that span the (d+2)-dimensional space in which the AdS space is embedded. Upon quantization, the mass hyperboloid defined in terms of the conjugate momenta turns into the wave equation in AdS space. By interchanging the roles of coordinates and conjugate momenta in the (d+2)-dimensional space we arrive at a dual description. For massive modes, the dual description is equivalent to the conventional formulation, as required by holography. For tachyonic modes, this interchange of coordinates and momenta establishes a duality between Euclidean AdS and dS spaces. We discuss its implications on Green functions for the various vacua.