Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions

TL;DR

This paper compares the classical and quantum evolutions of de Sitter and anti-de Sitter universes in 2+1 dimensions using ADM and Chern-Simons formalisms, revealing their classical equivalence and phase space relations.

Contribution

It demonstrates the classical equivalence of ADM and Chern-Simons formalisms for 2+1D gravity with nonzero cosmological constant and explores their phase space relations.

Findings

Classical equivalence between ADM and Chern-Simons formalisms for certain topologies.

Phase space correspondence between the two formalisms varies with cosmological constant sign.

Modified phase space relations when the cosmological constant is negative.

Abstract

Two canonical formulations of the Einstein gravity in 2+1 dimensions, namely, the ADM formalism and the Chern-Simons gravity, are investigated in the case of nonvanishing cosmological constant. General arguments for reducing phase spaces of the two formalisms are given when spatial hypersurface is compact. In particular when the space has the topology of a sphere $S^{2}$ or a torus $T^{2}$, the spacetimes constructed from these two formulations can be identified and the classical equivalence between the ADM and the CSG is shown. Moreover in the $g=1$ 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 the one of the ADM when the cosmological constant is negative.