Abstract
Three dimensional Lorentzian general relativity with zero cosmological constant and spinless point particles are studied. The topology of spacetime is M x R, where M is a Riemann surface and global hyperbolicity is assumed. The system has only finite degrees of freedom corresponding to non-trivial topology, since all solutions of
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the vacuum Einstein equations are locally flat spaces in three dimensions. After a generic introduction to phase space reduction, the polygon model of 't Hooft is introduced. This finite dimensional Hamiltonian system is encoded by a number of Euclidean polygons. It arises as the foliation of the three dimensional universe with spacelike Eucliden polygons, whose edges are decorated by two real numbers, their lengths L and the "so-called" boost parameters h of the corresponding transition functions of the geometric structure. Such an object is an element g of the Lorentz group SO(2,1).
Particles are easily added to this model by cutting out wedges from the polygons corresponding to a new pair of glued edges meeting at the "point particle". The condition for the angle at the tip of the wedge is Tr g=cos(m) in suitable units, where m is interpreted as the mass of the particle, which determine the angle. The deficit from 2pi in the rest frame of the particle is equal to its mass. These vertices carry three dimensional curvature singularities as opposed to the trivalent ones.
This thesis contains two main results. One concerns the case of higher genus spacelike slice without particles. It is shown that the h variables parametrize Teichmüller space, (which is the reduced configuration space of physically distinct universes) if they are interpreted as geodesic lengths in hyperbolic space. Then the dual graph of the skeleton yields an ideal triangulation of a uniformizing hyperbolic with the same topology as the physical equal time surface and with constant curvature -1. Using this one can show that the constraints corresponding to the closure of the Euclidean polygon (in case there is only one) can always be solved by a clever choice of the chart corresponding to basepoint of the geodesic loops on the uniformizing surface. It is very likely that each physically distinct universe admits a slicing in terms of only one polygon. The question of generalization for particles is more difficult, and even explicit configurations are found where the construction does not work.
The second result is the derivation of the postulated symplectic structure from the first order action of general relativity by means of symplectic reduction. It is done in two steps: the first is a reduction from the (infinite dimensional) dynamical fields of the action to a finite set of variables describing a non-planar polygonal contour in a background Minkowski space. This covariant model is then gauge-fixed to arrive at the 't Hooft polygon model. In the last subsection aspects of quantization procedures from these and related starting points are discussed.
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