Topological Aspects of Classical and Quantum(2+1)-dimensional Gravity

古典及び量子(2+1)次元重力の位相的側面

Soda Jiro

In order to understand (3+1)-dimensional gravity, (2+1)-dimensional gravity is studied as a toy model. Our emphasis is on its topological aspects, because (2+1)-dimensional gravity without matter fields has no local dynamical degrees of freedom. Starting from a review of the canonical ADM formalism and York's formalism for the initial value problem, we will solve the evolution equations of (2+1)-dimensional gravity with a cosmological constant in the case of g = 0 and g = 1, where g is the genus of Riemann surface. The dynamics of it is understood as the geodesic motion in the moduli space. This remarkable fact is the same with the case of (2+1)-dimensional pure gravity and seen more apparently from the action level. Indeed we will show the phase space reduction of (2+1)-dimensional gravity in the case of g = 1. For g ≥ 2, unfortunately we are not able to explicitly perform the phase space reduction of (2+1)-dimensional gravity due to the complexity of the Hamiltonian constraint equation. Based on this result, we will attempt to incorporate matter fields into (2+1)-dimensional pure gravity. The linearization and mini-superspace methods are used for this purpose. By using the linearization method, we conclude that the transverse-traceless part of the energy-momentum tensor affects the geodesic motion. In the case of the Einstein-Maxwell theory, we observe that the Wilson lines interact with the geometry to bend the geodesic motion. We analyze the mini-superspace naoclel of (2+1)-dimensional gravity with the matter fields in the case of g = 0 and y = 1. For g = 0, a wormhole solution is found but for g = 1 we can not find an analogous solution. Quantum gravity is also considered and we succeed to perform the phase space reduction of (2+1)-dimensional gravity in the case of g = 1 at the quantum level. From this analysis we argue that the conformal rotation is not necessary in the sense that the Euclidean quantum gravity is inappropriate for the full gravity.

Physics [ 420 ]

eng

Copyright(c) by Author

・A. Hosoya and J. Soda, Mod. Phys. Lett. A4 (1989) 2539, references

・J. Soda, to be published in Prog. Theor. Phys. Vol.83 No.4 (April), references

・Y. Fujiwara and J. Soda, to be published in Frog. Theor. Phys. Vol.83 No.4 (April). references

[DOI] http://dx.doi.org/10.1142/S0217732389002847 references

[DOI] http://dx.doi.org/10.1143/PTP.83.805 references

[DOI] http://dx.doi.org/10.1143/PTP.83.733 references

広島大学