On the 256-dimensional gamma matrix representation of the Clifford algebra Cl(1,7) and its relation to the Lie algebra SO(1,9)

The representations of the 256-dimensional Clifford algebra in the terms of $8 \times 8$ Dirac $\gamma$ matrices are suggested. The application to the 8-component Dirac equation is studied. Two isomorphic realizations $\textit{C}\ell^{\texttt{R}}$(0,8) and $\textit{C}\ell^{\texttt{R}}$(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 $\textit{C}\ell^{\texttt{R}}$(0,8)$, \textit{C}\ell^{\texttt{R}}$(1,7) representations are determined as algebras over the field of real numbers. Relationships between the suggested representations of the SO(m,n) and Clifford algebras are investigated. The existence of the 512-dimensional Clifford algebra in the terms of $8 \times 8$ Dirac gamma matrices is mentioned. The role of matrix representations of such algebras in the quantum field theory is briefly considered. Comparison with the corresponded algebras for standard 4-component Dirac equation 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.Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:0908.3106; text overlap with arXiv:1906.0701