2+12+1 Covariant Lattice Theory and t'Hooft's Formulation

We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo $2 \pi$. A cyclic Hamiltonian implies that ``time'' is quantized. However, it turns out that this Hamiltonian is {\it constrained}. If one chooses an internal time and solves this constraint for the ``physical Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it a la Dirac}, the ``internal time'' observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4.Comment: 10 pages of text. One figure available from J.A. Zapata upon reques

Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity

We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representa- tions of the fundamental group in ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g = 1 and g = 2 (N = 0).Comment: 34 pages, Mexico preprint ICN-UNAM-93-1

Topological Lattice Gravity Using Self-Dual Variables

Topological gravity is the reduction of general relativity to flat space-times. A lattice model describing topological gravity is developed starting from a Hamiltonian lattice version of B\w F theory. The extra symmetries not present in gravity that kill the local degrees of freedom in $B\wedge F$ theory are removed. The remaining symmetries preserve the geometrical character of the lattice. Using self-dual variables, the conditions that guarantee the geometricity of the lattice become reality conditions. The local part of the remaining symmetry generators, that respect the geometricity-reality conditions, has the form of Ashtekar's constraints for GR. Only after constraining the initial data to flat lattices and considering the non-local (plus local) part of the constraints does the algebra of the symmetry generators close. A strategy to extend the model for non-flat connections and quantization are discussed.Comment: 22 pages, revtex, no figure

Taming the Heat Flux Problem: Advanced Divertors Towards Fusion Power

The next generation fusion machines are likely to face enormous heat exhaust problems. In addition to summarizing major issues and physical processes connected with these problems, we discuss how advanced divertors, obtained by modifying the local geometry, may yield workable solutions. We also point out that: (1) the initial interpretation of recent experiments show that the advantages, predicted, for instance, for the X-divertor (in particular, being able to run a detached operation at high pedestal pressure) correlate very well with observations, and (2) the X-D geometry could be implemented on ITER (and DEMOS) respecting all the relevant constraints. A roadmap for future research efforts is proposed

Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness

By investigating the canonical commutation rules for gravitating quantized particles in a 2+1 dimensional world it is found that these particles live on a space-time lattice. The space-time lattice points can be characterized by three integers. Various representations are possible, the details depending on the topology chosen for energy-momentum space. We find that an $S_2\times S_1$ topology yields a physically most interesting lattice within which first quantization of Dirac particles is possible. An $S_3$ topology also gives a lattice, but does not allow first quantized particles.Comment: 23 pages Plain TeX, 3 Figure

Hamiltonian formulation and analysis of a collisionless fluid reconnection model

The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie-Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad-Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation also is used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.Comment: 4 figure

Regge calculus and Ashtekar variables

Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way.Comment: 10 pages, uses harvmac. DFUPG 83/9