Classical and Quantum Solutions of 2+1 Dimensional Gravity

Abstract

Recently 2+1 dimensional gravity theory, especially AdS3 has been studied extensively [1, 2]. It was shown to be equivalent to the 2+1 Chern-Simon theory [3] and has been investigated to understand the black hole thermodynamics, i.e. Hawking temperature [4] and others. The purpose of this report is to investigate the canonical formalism of the original 2+1 dimensional Einstein gravity theory instead of the Chern-Simon theory. For the spherically symmetric space-time, local conserved quantities (mass and angular momentum) are introduced and using them canonical quantum theory is defined. Constraints are imposed on state vectors and solved analytically. In order to extract the physical meaning to the wave function, we impose the de Broglie-Bohm interpretation [5, 6, 7] and derive the differential equation for the metrics, which include the quantum effect. After fixing the gauge choice, special solutions of the metrics are obtained. Especially the quantum effect of the closed de Sitter universe is obtained qsuantitatively. It is interesting to note that the birth of universe appeared as the real time tunneling in stead of the imaginary time tunneling in the WKB approximation by Vilenkin [8] and in the path integral method by Hartle and Hawking [9]. The strategy to obtain the solution is followed by our previous work [10].