A symmetry reduction scheme of the Dirac algebra without dimensional defects

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

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

Dahm, R ^[1]

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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, Vol. 73, Issue 2; Other Information: DOI: 10.1134/S1063778810020122; Copyright (c) 2010 Pleiades Publishing, Ltd.; ISSN 1063-7788
• Country of Publication: United States
• Language: English

Related Subjects

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