CLASSICAL AND QUANTUM EVOLUTIONS OF THE DESITTER AND THE ANTI-DESITTER UNIVERSES IN 2+1 DIMENSIONS

Two canonical formulations of Einstein gravity in 2 + 1 dimensions, na mely, the ADM formalism and Chern-Simons gravity (CSG), are investigat ed in the case of nonvanishing cosmological constant. General argument s for reducing phase spaces of the two formalisms are given when the s patial 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 equivalenc e between the ADM formalism and CSG is shown. Moreover, in the torus c ase the relations between their phase spaces, and therefore between th eir quantizations, are given in almost the same form as that in the ca se when the cosmological constant vanishes. There are, however, some m odifications, 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.