A symmetry reduction scheme of the Dirac algebra without dimensional defects

Journal Article · Mon Feb 15 04:00:00 EST 2010 · Physics of Atomic Nuclei

DOI:https://doi.org/10.1134/S1063778810020122· OSTI ID:21443610

Dahm, R ^[1]

---

1. Universitaet Mainz, Zentrum fuer Datenverarbeitung (Germany)

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 Moebius 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)/U Sp(4) and U Sp(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 (Kaehlerian 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.

• OSTI ID: 21443610
• Journal Information: Physics of Atomic Nuclei, Journal Name: Physics of Atomic Nuclei Journal Issue: 2 Vol. 73; ISSN 1063-7788; ISSN PANUEO
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BOSON-EXCHANGE MODELS
CHIRAL SYMMETRY
DIRAC OPERATORS
ELEMENTARY PARTICLES
FIELD THEORIES
GEOMETRY
HADRONS
HERMITIAN OPERATORS
ISOSPIN
LIE GROUPS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MATHEMATICS
NONLINEAR PROBLEMS
PARTICLE MODELS
PARTICLE PROPERTIES
PARTICLES
PERIPHERAL MODELS
QUANTUM FIELD THEORY
QUANTUM OPERATORS
SIGMA MODEL
SYMMETRY
SYMMETRY GROUPS
THREE-DIMENSIONAL CALCULATIONS
U GROUPS
U-1 GROUPS