Sultan Catto

On the time-dependent oscillator and the nonlinear realizations of the Virasoro group

Abstract Using the nonlinear realizations of the Virasoro group we construct the action of the conformal quantum mechanics (CQM) with additional harmonic potential. We show that SL (2, R ) invariance group of this action is nontrivially embedded in the reparametrization group of the time which is isomorphic to the centerless Virasoro group. We generalize the consideration to the Ermakov systems and construct the action for the time-dependent oscillator. Its symmetry group is also the SL (2, R )∼ SU (1,1) group embedded in the Virasoro group in a more complicated way.

Revisiting the role of octonions in hadronic physics

Octonions and their split versions are shown to be applicable to the solutions of a large number of problems in hadronic physics, from the foundations of exceptional groups that are used in grand unified theories, to heterotic strings, to the non-Desarguesian geometric property of space-time symmetries, twistors, harmonic superspace, conformal field theories, etc. Upon a brief review of these investigations we proceed to show how they are used in the unification of ancient and modern geometries, which in turn open new avenues for, and goes far beyond in providing, geometric foundations for the existence of internal symmetries such as color and flavor.

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.

Instanton solutions in supersymmetric chiral field theories

Abstract Instanton solutions in three and four dimensional supersymmetric chiral field theories are presented. Possible applications of the underlying theory is discussed.

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.

Scalar Mesons, Multiquark States and Supersymmetry

Quark models with potentials derived from QCD, including the quark‐diquark model for excited hadrons leading to a symmetry between mesons and baryons gives mass formulae in very good agreement with experiment and goes a long way in explaining the approximate symmetries and supersymmetries of the hadronic spectrum, including the symmetry breaking mechanism. We give a description of the only algebra describing color degrees of freedom based on split octonionic units and leading to a mathematical understanding of formation of diquarks as well as multiquark states.