Algebraic Preliminaries for Field Theories in Space-Time Algebra

摘要

The algebra, generated by the gamma-matrices as introduced by Dirac in order to describe the behavior of electrons, turned out to be a faithful matrix representation of the real Clifford algebra of 4-dimensional Minkowski space-time M. As remarked by D. Hestenes it is possible indeed to give a matrix free treatment of the electromagnetic and the electro-weak fields. The paper gives a survey of algebraic structures of the Clifford algebra of M. Current field theories are based on such structures. Section 1 discusses the product structures and the involutions. In Section 2 the complex structures are discussed. They are based on the choice of a fixed unit pseudoscalar iota ('orientation'). Section 3 investigates algebraic structures depending on the choice of a fixed timelike vector epsilon ('time-axis'). It is found that the Lie algebras su(2) and su(2) x u(1) are isomorphic to substructures of the Clifford algebra. The Lie algebra su(3) can be rendered by the skew Hermitean traceless part of the tensor product of bivectors. In Section 4 the minimal ideals are discussed; they depend on the choice of a fixed spacelike vector upsilon ('z-axis'). Finally the interplay of these ideals with the Lie group SU(2) is studied.