Isotopic, Genotopic and Hyperstructural Liftings of Lie's Theory and their Isoduals
Abstract
After reviewing the basic role of Lie's theory for the mathematics and physics of this century, we identify its limitations for the treatment of systems beyond the localdifferential, Hamiltonian and canonicalunitary conditions of the original conception. We therefore outline three sequential generalized mathematics introduced by the author under the name of iso, geno and hypermathematics which are based on generalized, Hermitean, nonHermitean and multivalued units, respectively. The resulting iso, geno and hyperLie theories, for which the new mathematics were submitted, have been extensively used for the description of nonlocalintegral systems with actionatadistance Hamiltonian and shortrangecontact nonHamiltonian interactions in reversible, irreversible and multivalued conditions, respectively. We then point out that conventional, iso, geno and hyperLie theories are unable to provide a consistent classical representation of antimatter yielding the correct charge conjugate states at the operator counterpart. We therefore outline yet novel mathematics proposed by the author under the names of isodual conventional, iso, geno and hypermathematics, which constitute antiisomorphic images of the original mathematics characterized by negativedefinite units and norms. The emerging isodual generalizations of Lie's theory have permitted a novel consistent characterization of antimatter at all levels of study, from Newton to second quantization. The main message emerging after three decades of investigations is that the sole generalized theories as invariant as the original formulation, the sole usable in physics, are those preserving the original abstract Lie axioms, and merely realizing them in broader forms.
 Publication:

arXiv eprints
 Pub Date:
 February 1999
 arXiv:
 arXiv:physics/9902036
 Bibcode:
 1999physics...2036S
 Keywords:

 Physics  General Physics
 EPrint:
 Latex document for the centennial of Lie's death