Levent Kurt

Propagation of ultra-short solitons in stochastic Maxwell's equations

We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwells equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwells equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwells equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwells equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.

Higher-order corrections to the short-pulse equation

Using renormalization group techniques, we derive an extended short-pulse equation as an approximation to a nonlinear wave equation. We investigate the new equation numerically and show that the new equation captures efficiently higher-order effects on pulse propagation in cubic nonlinear media. We illustrate our findings using one- and two-soliton solutions of the first-order short-pulse equation as initial conditions in the nonlinear wave equation.

Grassmann Numbers and Clifford-Jordan-Wigner Representation of Supersymmetry

The elementary particles of Physics are classified according to the behavior of the multi-particle states under exchange of identical particles: bosonic states are symmetric while fermionic states are antisymmetric. This manifests itself also in the commutation properties of the respective creation operators: bosonic creation operators commute while fermionic ones anticommute. It is natural therefore to study bosons using commuting entities (e.g. complex variables), whereas to describe fermions, anticommuting variables are more naturally suited. In this paper we introduce these anticommuting- and at first sight unfamiliar- variables (Grassmann numbers) and investigate their properties. In particular, we briefly discuss differential and integral calculus on Grassmann numbers. Work supported in part by DOE contracts No. DE-AC-0276-ER 03074 and 03075; NSF Grant No. DMS-8917754.

Supergroups in Critical Dimensions and Division Algebras

We establish a link between classical heterotic strings and the groups of the magic square associated with Jordan algebras, allowing for a uniform treatment of the bosonic and superstring sectors of the heterotic string.

Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras

We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.

Algebraic formulation of hadronic supersymmetry based on octonions: new mass formulas and further applications

A special treatment based on the highest division algebra, that of octonions and their split algebraic formulation is developed for the description of diquark states made up of two quark pairs. We describe symmetry properties of mesons and baryons through such formulation and derive mass formulae relating π, ρ, N and Δ trajectories showing an incredible agreement with experiments. We also comment on formation of diquark-antidiquark as well as pentaquark states and point the way toward applications into multiquark formulations expected to be seen at upcoming CERN experiments. A discussion on relationship of our work to flux bag models, string pictures and to string-like configurations in hadrons based on spectrum generating algebras will be given.

Invariance Properties of the Exceptional Quantum Mechanics (F4) and Its Generalization to Complex Jordan Algebras (E6)

We consider a case in which the octonionic observables form a Jordan Algebra. Then the automorphism group turns out to be an exceptional group F 4 or E 6 and we are led to a gauge field theory of quarks and leptons based on exceptional groups. Some relations of octonion and split octonion algebras and their relation to algebra of quarks are explicitly shown.

Root Structures of Infinite Gauge Groups and Supersymmetric Field Theories

We show the relationship between critical dimensions of supersymmetric fundamental theories and dimensions of certain Jordan algebras. In our approach position vectors in spacetime or in superspace are endowed with algebraic properties that are present only in those critical dimensions. A uniform construction of super Poincar? groups in these dimensions will be shown. Some applications of these algebraic methods to hidden symmetries present in the covariant and interacting string Lagrangians and to superparticle will be discussed. Algebraic methods we develop will be shown to generate the root structure of some infinite groups that play the role of gauge groups in a second quantized theory of strings.