dorsal/arxiv
View SchemaIsotopic, Genotopic and Hyperstructural Liftings of Lie's Theory and their Isoduals
| Authors | Ruggero Maria Santilli |
|---|---|
| Categories | |
| ArXiv ID | physics/9902036 |
| URL | https://arxiv.org/abs/physics/9902036 |
| Journal | Alg.Groups Geom.15:473-498,1999 |
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 local-differential, Hamiltonian and canonical-unitary conditions of the original conception. We therefore outline three sequential generalized mathematics introduced by the author under the name of iso-, geno- and hyper-mathematics which are based on generalized, Hermitean, non-Hermitean and multi-valued units, respectively. The resulting iso-, geno- and hyper-Lie theories, for which the new mathematics were submitted, have been extensively used for the description of nonlocal-integral systems with action-at-a-distance Hamiltonian and short-range-contact non-Hamiltonian interactions in reversible, irreversible and multi-valued conditions, respectively. We then point out that conventional, iso-, geno- and hyper-Lie 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 hyper-mathematics, which constitute anti-isomorphic images of the original mathematics characterized by negative-definite 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.
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"abstract": "After reviewing the basic role of Lie\u0027s theory for the mathematics and\nphysics of this century, we identify its limitations for the treatment of\nsystems beyond the local-differential, Hamiltonian and canonical-unitary\nconditions of the original conception. We therefore outline three sequential\ngeneralized mathematics introduced by the author under the name of iso-, geno-\nand hyper-mathematics which are based on generalized, Hermitean, non-Hermitean\nand multi-valued units, respectively. The resulting iso-, geno- and hyper-Lie\ntheories, for which the new mathematics were submitted, have been extensively\nused for the description of nonlocal-integral systems with action-at-a-distance\nHamiltonian and short-range-contact non-Hamiltonian interactions in reversible,\nirreversible and multi-valued conditions, respectively. We then point out that\nconventional, iso-, geno- and hyper-Lie theories are unable to provide a\nconsistent classical representation of antimatter yielding the correct charge\nconjugate states at the operator counterpart. We therefore outline yet novel\nmathematics proposed by the author under the names of isodual conventional,\niso-, geno- and hyper-mathematics, which constitute anti-isomorphic images of\nthe original mathematics characterized by negative-definite units and norms.\nThe emerging isodual generalizations of Lie\u0027s theory have permitted a novel\nconsistent characterization of antimatter at all levels of study, from Newton\nto second quantization. The main message emerging after three decades of\ninvestigations is that the sole generalized theories as invariant as the\noriginal formulation, the sole usable in physics, are those preserving the\noriginal abstract Lie axioms, and merely realizing them in broader forms.",
"arxiv_id": "physics/9902036",
"authors": [
"Ruggero Maria Santilli"
],
"categories": [
"physics.gen-ph"
],
"journal_ref": "Alg.Groups Geom.15:473-498,1999",
"title": "Isotopic, Genotopic and Hyperstructural Liftings of Lie\u0027s Theory and their Isoduals",
"url": "https://arxiv.org/abs/physics/9902036"
},
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