Classical and quantum evolutions of the de Sitter and the anti-de Sitter universes in 2+1 dimensions

Two canonical formulations of Einstein gravity in 2+1 dimensions, namely, the ADM formalism and Chern-Simons gravity (CSG), are investigated in the case of nonvanishing cosmological constant. General arguments for reducing phase spaces of the two formalisms are given when the spatial hypersurface is compact. In particular, when the space has the topology of a sphere S² or a torus T², the spacetimes constructed from these two formulations can be identified and the classical equivalence between the ADM formalism and CSG is shown. Moreover, in the torus case the relations between their phase spaces, and therefore between their quantizations, are given in almost the same form as that in the case when the cosmological constant vanishes. There are, however, some modifications, the most remarkable one of which is that the phase space of the CSG is in 1 to 2 correspondence with that of the ADM formalism when the cosmological constant is negative.