Yoon Seok Choun

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.

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.