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 256-dimensional representations of the Clifford algebras in the terms of 8x8 Dirac gamma matrices are introduced. The corresponding gamma matrix representations of 45-dimensional 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 8-component 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 4-component 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 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 is found as well.