双局域场论Dirac矩阵满足的一个新恒等式
A Remarkable New Identity Satisfied by the Dirac Matrices of a Bilocal Field Theory
摘要：
1925年，埃利·嘉当将\cite{CARTAN25}，\cite{CARTAN}描述为SO $(8; \mathbb{C})$向量和两种自旋$(8; \mathbb{C})$旋量之间的对称性。众所周知，定义在实八维欧氏空间$\mathbb{E}_{8}$上的Clifford代数$\mathbb{C}_{8}$的约化生成元满足一个恒等式，该恒等式保证三元性的矩阵表示（作用于$\mathbb{E}_{8}$的向量和旋量束）的存在。类似地，令$\mathbb{E}_{4,4}$表示实八维伪欧几里德向量空间，其被赋予具有签名$(+,+,+,-\,;\,-,-,-,+)$的不定内积。作为赋范向量空间，$\mathbb{E}_{4,4} \cong M_{3,1} \times {}^{*}\!M_{3,1}$，其中$M_{3,1}$和${}^{*}\!M_{3,1}$表示具有相反签名的实四维闵可夫斯基时空。显然，双域闵可夫斯基场论可以适用于$\mathbb{E}_{4,4}$时空。减少的生成器（即，定义在$\mathbb{E}_{4,4}$上的伪Clifford代数$\mathbb{C}_{4,4}$的Dirac矩阵）满足恒等式$\,$ \cite{NASH86} $\,,\,$ \cite{NASH90}，该恒等式保证两种类型的$\overline{S0(4,4; \mathbb{R})}$旋量中的每一种与${S0(4,4; \mathbb{R})}$向量之间存在可逆线性映射，从而实现作用于时空$\mathbb{E}_{4,4}$的向量和旋量束的三重性的矩阵表示。在本说明中，我们推广了这个恒等式（见Eq. [ \ref{newIdentity} ]）。
英文摘要：
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}]).
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