An AdS/dS duality for a scalar particle

George Siopsis

doi:10.1016/s0370-2693(02)02515-7

Abstract

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.

Highlights

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
A significant step in this direction was the recent proposal by Strominger [1] of a dS/CFT correspondence where the CFT lies in the infinite past of dS space
If one analyzes the behavior of the respective Green functions carefully, one discovers that dS Green functions may not be obtained by a double analytic continuation of their AdS counterparts

Summary

This algebra is realized in terms of Poincare coordinates

AdS hypersurface in terms of Poincare coordinates:. The Casimir (5) may be expressed in terms of the Poincare coordinates (z, xμ) (12) and the conjugate momenta (pz, pμ). Upon quantization, it turns into the Schrodinger (wave) equation zd−1. In this regime, both solutions Φ±q (eq (19)) may be acceptable leading to distinct theories and different Green functions hinting at symmetry breaking. For m2l2 < −d2/4, ν becomes imaginary and the unitarity bound on the corresponding conformal field theory is violated Both modes Φ±q (19) are normalizable under the inner product (cf (16))

Another set of modes of interest are the Euclidean modes

We obtain

The Dirac brackets in this gauge are

The two orthonormal sets are related to each other by