The Torus Universe in the Polygon Approach to 2+1-Dimensional Gravity

In this paper we describe the matter-free toroidal spacetime in 't Hooft's polygon approach to 2+1-dimensional gravity (i.e. we consider the case without any particles present). Contrary to earlier results in the literature we find that it is not possible to describe the torus by just one polygon but we need at least two polygons. We also show that the constraint algebra of the polygons closes.Comment: 18 pages Latex, 13 eps-figure

Winding Solutions for the two Particle System in 2+1 Gravity

Using a PASCAL program to follow the evolution of two gravitating particles in 2+1 dimensions we find solutions in which the particles wind around one another indefinitely. As their center of mass moves `tachyonic' they form a Gott-pair. To avoid unphysical boundary conditions we consider a large but closed universe. After the particles have evolved for some time their momenta have grown very large. In this limit we quantize the model and find that both the relevant configuration variable and its conjugate momentum become discrete.Comment: 15 pages Latex, 4 eps figure

Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity

We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation.Comment: 58 pages, 3 eps figures, presentation of the classical theory improve

Two particle Quantummechanics in 2+1 Gravity using Non Commuting Coordinates

We find that the momentum conjugate to the relative distance between two gravitating particles in their center of mass frame is a hyperbolic angle. This fact strongly suggests that momentum space should be taken to be a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non commuting, Hermitian operators on this hyperboloid. We also find that there is a smallest distance between the two particles of one half times the Planck length.Comment: 18 pages Latex, 2 eps figure

The 2+1 Kepler Problem and Its Quantization

We study a system of two pointlike particles coupled to three dimensional Einstein gravity. The reduced phase space can be considered as a deformed version of the phase space of two special-relativistic point particles in the centre of mass frame. When the system is quantized, we find some possibly general effects of quantum gravity, such as a minimal distances and a foaminess of the spacetime at the order of the Planck length. We also obtain a quantization of geometry, which restricts the possible asymptotic geometries of the universe.Comment: 59 pages, LaTeX2e, 9 eps figure

Cyberlaundering: The Risks, the Responses

This Article discusses the potential use of electronic cash for money laundering and possible government responses to the problem. Parts I and II provide an overview of electronic cash. Part III explores the effects that electronic cash can have on money laundering. Part IV explains through a series of hypotheticals how cyberlaundering can occur. Part V analyzes the federal government\u27s response to the threat of money laundering with electronic cash. Part VI concludes the Article with suggestions

(2+1)-Gravity Solutions with Spinning Particles

We derive, in 2+1 dimensions, classical solutions for metric and motion of two or more spinning particles, in the conformal Coulomb gauge introduced previously. The solutions are exact in the $N$-body static case, and are perturbative in the particles' velocities in the dynamic two-body case. A natural boundary for the existence of our gauge choice is provided by some ``CTC horizons'' encircling the particles, within which closed timelike curves occur.Comment: 30 pages, LaTeX, no figure