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

TL;DR

This paper explores a new identity satisfied by Dirac matrices in a bilocal field theory context, extending triality concepts from Euclidean to pseudo-Euclidean spaces with applications to Minkowski spacetimes.

Contribution

It generalizes a known identity for Dirac matrices in pseudo-Euclidean space, enabling new matrix representations of triality between vectors and spinors in bilocal field theories.

Findings

Established a generalized identity for Dirac matrices in ,4 space.

Demonstrated the existence of invertible mappings between spinor and vector representations.

Extended triality symmetry concepts to bilocal Minkowski spacetimes.

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}]).