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

TL;DR

This paper develops a 256-dimensional gamma matrix representation of the Clifford algebra Cl(1,7) and explores its relation to the Lie algebra SO(1,9), extending previous work to higher spin equations and identifying invariance algebras.

Contribution

It introduces new gamma matrix representations for Clifford and Lie algebras in real 8x8 matrices, generalizing earlier results for Dirac equations to higher spin and arbitrary dimensions.

Findings

Constructed gamma matrix representations of SO(10) and SO(1,9) in real 8x8 matrices.

Identified the maximal 84-dimensional invariance algebra of the 8-component Dirac equation.

Compared new algebraic structures with standard Dirac spinor representations.

Abstract

Extended gamma matrix Clifford--Dirac and SO(1,9) algebras in the terms of $8 \times 8$ matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. 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. The suggested gamma matrix representations of the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford algebras $\textit{C}\ell^{\texttt{R}}$(0,8)$, \textit{C}\ell^{\texttt{R}}$(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.