A Symmetry Reduction Scheme of the Dirac Algebra without Dimensional Defects

摘要

In relating the Dirac algebra to homogeneous coordinates of a projective geometry, we present a simple geometric scheme which allows to identify various Lie algebras and Lie groups well-known from classical physics as well as from quantum field theory. We introduce a 1-point-compactification and quaternionic Mobius transformations, and we use SU*(4) and a symmetry reduction scheme without dimensional defects to identify transformations and particle representations thoroughly. As such, two subsequent nonlinear sigma models SU*(4)/USp(4) and USp(4)/SU(2) x U(1) emerge as well as a possible double coset decomposition of SU*(4) with respect to SU(2) x U(1). Whereas the first model leads to equivalence classes of hyperbolic manifolds and naturally introduces coordinates and velocities, the second coset model leads to a Hermitian symmetric (vector) space (Kahlerian space) of real dimension 6, i.e., to a 3-dimensional complex space with a global symplectic and a local SU(2) x U(1) symmetry which allows to identify the (local) gauge group of electroweak interactions as well as under certain assumptions it admits compact SU(3) transformations as automorphisms of this 3-dimensional (hyper) complex vector space. In the limit of low energies, this geometric SU*(4) scheme naturally yields the (compact) group SU(4) to describe "chiral symmetry" and conserved isospin of hadrons as well as the low-dimensional hadron representations. Last not least, with respect to some of the SU*(4) generators we find a multiplication table which (up to signs) is identical with the octonions represented in the Fano plane.