A Remarkable New Identity Satisfied by the Dirac Matrices of a Bilocal Field Theory

Abstract

In 1925 Elie Cartan described `triality' \cite{CARTAN25}, \cite{CARTAN} as a symmetry between SO$(8; \mathbb{C})$ vectors and the two types of Spin$(8; \mathbb{C})$ spinor. It is known that the reduced generators of the Clifford algebra $\mathbb{C}_{8}$ defined on the real, eight-dimensional Euclidean space $\mathbb{E}_{8}$ satisfy an identity that guarantees the existence of matrix representations (acting on the vector and spinor bundles of $\mathbb{E}_{8}$) of triality. Analogously, let $\mathbb{E}_{4,4}$ denote a real eight-dimensional pseudo-Euclidean vector space that is endowed with an indefinite inner product with signature $(+,+,+,-\,;\,-,-,-,+)$. As a normed vector space, $\mathbb{E}_{4,4} \cong M_{3,1} \times {}^{*}\!M_{3,1}$, where $M_{3,1}$ and ${}^{*}\!M_{3,1}$ denote real four-dimensional Minkowski spacetimes, with opposite signatures. %Clearly, bilocal Minkowski field theories may be cast on the $\mathbb{E}_{4,4}$ spacetime. The reduced generators (i.e., the Dirac matrices) of the pseudo Clifford algebra $\mathbb{C}_{4,4}$ defined on $\mathbb{E}_{4,4}$ satisfy an identity $\,$ \cite{NASH86} $\,,\,$ \cite{NASH90} that guarantees the existence of invertible linear mappings between each of the two types of $\overline{S0(4,4; \mathbb{R})}$ spinor and the ${S0(4,4; \mathbb{R})}$ vector, thereby realizing matrix representations of triality that act on the vector and spinor bundles of the spacetime $\mathbb{E}_{4,4}$. In this note we generalize this identity (see Eq.[\ref{newIdentity}]).