Grassmann Electrodynamics and General Relativity

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, "inner" product), metric, inner product, Killing's vector fields, Hodge star operator, integral forms, co-differential and connection on odd Riemannian supermanifolds are introduced. The electrodynamics and Einstein relativity with anti-commuting variables only are formulated modifying the geometry beyond classical (even, bosonic) theories appropriately. Extension of these ideas to general supermanifolds is straightforward. All this "odd business" (in both meanings of the word "odd") is based on classical geometrical analogy.Comment: 16 pages. submitted to Journal of Geometry and Physic

Square-root quantization: application to quantum black holes

Two different ways of quantizing the relativistic Hamiltonian for radial motion in the field of Coulomb-like potential are compared. The results depend slightly on choice of time. In the case of Lorentzian time a Sommerfeld spectrum is recovered. Application to quantum black holes gives a sqrt{n} mass spectrum with about the same numerical factors.Comment: LaTeX file Talk given on the Second Conference on Constrained Dynamics and Quantum Gravity, Santa Margherita, Ligure, Italy, 17-21 September 199

A theory of thin shells with orbiting constituents

The self-gravitating, spherically symmetric thin shells built of orbiting particles are sstudied. Two new features are found. One is the minimal possible value for an angular momentum of particles, above which elleptic orbits become possible. The second is the coexistence of both the wormhole solutions and the elleptic or hyperbolic orbits for the same values of the parameters (but different initial conditions). Possible applications of these results to astrophysics and quantum black holes are briefly discussed.Comment: 22 pages, Latex, 10 eps figures. CERN preprint no. CERN-TH 2000-16

Felix Alexandrovich Berezin and his work

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page

Brane Universe: Global Geometry

The global geometries of bulk vacuum space-times in the brane-universe models are investigated and classified in terms of geometrical invariants. The corresponding Carter-Penrose diagrams and embedding diagrams are constructed. It is shown that for a given energy-momentum induced on the brane there can be different types of global geometries depending on the signs of a bulk cosmological term and surface energy density of the brane (the sign of the latter does not influence the internal cosmological evolution). It is shown that in the Randall-Sundrum scenario it is possible to have an asymmetric hierarchy splitting even with a $Z_2$-symmetric matching of "our" brane to the bulk.Comment: 23 pages, 34 figures, Talk given at the "Invisible Universe International Conference ", Paris, June 29 -July 3, 200