Comment on the quantum modes of the scalar field on AdS d + 1 spacetime

The problem of the quantum modes of the scalar free field on anti-de Sitter backgrounds with an arbitrary number of space dimensions is considered. It is shown that this problem can be solved by using the same quantum numbers as those of the nonrelativistic oscillator and two parameters which give the energy quanta and respectively the ground state energy. This last one is known to be just the conformal dimension of the boundary field theory of the AdS/CFT conjecture. Pacs 04.62.+v The recent interest in propagation of quantum scalar fields on anti-de Sitter (AdS) spacetime is due to the discovery of the AdS/Conformal field theory-correspondence [1]. One of central points here is the relation between the field theory on the (d + 1)-dimensional AdS (AdS d+1) spacetime and the conformal field theory on its d-dimensional Minkowski-like boundary (M d). There are serious arguments that the local operators of the conformal field theory on M d correspond to the quantum modes of the scalar field on AdS d+1 [2]. Actually, for d = 3 [3, 4] as well as for any d [5] it is proved that the conformal dimension in boundary field theory is equal with the ground state energy on AdS d+1 [2]. Moreover, it is known that the energy spectrum is discrete and equidistant [5] its quanta wavelength being just the hyperboloid radius of AdS d+1 .