Dimensional Formalism Research Articles

TRANSFORMATIONS AND BRST-CHARGES IN (2+1)-DIMENSIONAL GRAVITATION

Canonical transformations relating the variables of the ADM-, Ashtekar’s and Witten’s formulations of gravity are computed in 2+1 dimensions. Three different forms of the BRST-charge are given in the (2+1)-dimensional Ashtekar formalism, two of them using Ashtekar’s form of the constraints and one of them using the forms suggested by Witten. The BRST-charges are of different rank.

Vector formalism for circularly symmetric laser beams

We present a finite dimensional vector formalism that describes the propagation of circularly symmetric beams in a simple and effective way. A propagation matrix operator is explicitly given that determines the amplitude as a function of the radius at any plane. Power transported to a circular or annular target is discussed in some detail. In particular, the problem of choosing a beam shape at the aperture that maximizes the power transported to a target has a simple solution in the new formalism.

Unified models from gauged supergroups. I. Dimensional reduction formalism. Unitary and orthosymplectic series

Principles are elaborated for constructing consistent grand unified models (GUT’s) plus Higgs sectors from dimensional reduction of gauged supergroups over graded coset spaces. A search of possible GUT groups is made for unitary and orthosymplectic supergroups over a space parametrized by a single pair of Grassmann coordinates.

Hamiltonian formulation of gravitating perfect fluids and the Newtonian limit

The canonical formalism is applied to self-gravitating perfect fluids with particular emphasis on recovering the correct nonrelativistic limit also in (quasi-) Hamiltonian form. We use essentially Lagrangian coordinates by considering the fluid defined by a map from space-time into a three-dimensional material manifold which is equipped with a volume element representing physically the matter (baryon number) density. By eliminating the coordinate freedom in this material space the usual matter conservation and (relativistic) Euler equations are recovered in a (3+1)-dimensional formalism which makes it very easy to compare them to their nonrelativistic limits. By splitting the 3-metric and its canonical momenta into a conformal part and the determinant we arrive at a system of evolution and constraint equations for the gravitational field that also has a well-defined Newtonian limit provided the geometric version of the Newtonian theory is also cast into an analogous (3+1)-dimensional form. Some of the evolution equations of the relativistic theory, however, become additional constraints in the limit which represents the freezing of the gravitational (or radiation) degrees of freedom. We then use this formalism to rederive the first-order post-Newtonian approximation and obtain the standard results in a flexible geometrical form since no gauge or coordinate conditions need be imposed in advance.

On the equivalence of the Einstein−Mayer and Einstein−Cartan theories for describing a spinning medium

It is shown that the five−dimensional formalism of the Einstein−Mayer theory of 1932 can be given a new interpretation and used to describe a continuous medium with intrinsic angular momentum. The equations of motion of a spinning particle are derived from the generalized field equations and are seen to be essentially the same as the Mathisson−Papapetrou equations. Finally the correspondence of the new approach with the theories based on Cartan’s torsion tensor

Improvements of the Properties of the Relativistic Parallel Plate Particle Analyzer

Using a three dimensional formalism previously established, the freedom to choose the coupling function g (θ, φ) between the polar angles defining the mean trajectory in the field-free region of the analyzer is exploited to improve the properties of the instrument. It is shown that a partial correction to the z variation of the image when θ and φ vary can be achieved yielding a maximum excursion Δz /z of 6% for a total θ excursion of Δθ = 18° at φ = const with a corresponding increase of resolving power by a factor of 2. Indications are also given on the possibility of cylindrization of the focal surface.