On the 256dimensional gamma matrix representation of the Clifford algebra Cl(1,7) and its relation to the Lie algebra SO(1,9)
Abstract
The 256dimensional representations of the Clifford algebras in the terms of 8x8 Dirac gamma matrices are introduced. The corresponding gamma matrix representations of 45dimensional SO(10) and SO(1,9) algebras, which contain the standard and additional spin operators, are introduced as well. The SO(10), SO(1,9) and corresponding Clifford algebras are determined as the algebras over the field of real numbers in the space of 8component spinors. The relationships between the suggested representations of the SO(m,n) and Clifford algebras are investigated. The role of matrix representations of such algebras in the quantum field theory is considered briefly. Our start from the corresponded algebras in the space of standard 4component Dirac spinors is mentioned. The proposed mathematical objects allow the generalization of our results, obtained earlier for the standard Dirac equation, for the equations of higher spin and, especially, for the equations, describing the particles with spin 3/2. The maximal 84dimensional pure matrix algebra of invariance of the 8component Dirac equation in the FoldyWouthuysen representation is found. The corresponding symmetry of the Dirac equation is found as well.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.11406
 Bibcode:
 2021arXiv211011406S
 Keywords:

 Physics  General Physics;
 High Energy Physics  Theory;
 81;
 F.4.3
 EPrint:
 18 pages. arXiv admin note: substantial text overlap with arXiv:0908.3106