2021/10/17 by В. М. Симулик, Simulik, V. M., Vyikon, I. I.
Mathematics · Physics and Astronomy · #81 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #F.4.3 #FOS: Physical sciences #General Physics (physics.gen-ph) #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2110.11406
openalex publication_date 2021/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Extended gamma matrix Clifford--Dirac and SO(1,9) algebras in the terms of 8 × 8 matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two isomorphic realizations CℓR(0,8) and CℓR(1,7) are considered. The corresponding gamma matrix representations of 45-dimensional SO(10) and SO(1,9) algebras, which contain standard and additional spin operators, are introduced as well. The SO(10), SO(1,9) and the corresponding CℓR(0,8), CℓR(1,7) representations are determined as algebras over the field of real numbers. The suggested gamma matrix representations of the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford algebras CℓR(0,8), CℓR(1,7) representations. Comparison with the corresponded algebras in the space of standard 4-component Dirac spinors is demonstrated. The proposed mathematical objects allow generalization of our results, obtained earlier for the standard Dirac equation, for equations of higher spin and, especially, for equations, describing particles with spin 3/2. The maximal 84-dimensional pure matrix algebra of invariance of the 8-component Dirac equation in the Foldy--Wouthuysen representation is found. The corresponding symmetry of the Dirac equation in ordinary representation is found as well. The possible generalizations of considered Lie algebras to the arbitrary dimensional SO(n) and SO(m,n) are discussed briefly.