Bilinear Covariants of the Dirac Spinors

Abstract

There must exist 16 linearly independent 4 × 4 matrices, which we denote by EquationSource %MathType!MTEF!2!1!+-% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaada % qadaqaaiabfo5ahnaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMca % aaGaayPadaGaeqySdeMaeqOSdigaaa!3E0F!\widehat{\left( {{\Gamma ^n}} \right)}\alpha \beta $$. It turns out that one can construct 16 (a complete set) of these EquationSource%MathType!MTEF!2!1!+-% MathType!MTEF!2!1!+- % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaacq % qHtoWrdaWgaaWcbaGaamOBaaqabaaakiaawkWaaaaa!3945! $$\widehat{{\Gamma _n}}$$ (n =1, ..., 16) from the Dirac matrices and their products. We write

$$ \eqalign{ & \widehat{{\Gamma ^s}}\mathop = \limits_{(1)} 11,\widehat{\Gamma _{^\mu }^V}\mathop = \limits_{(4)} \gamma \mu ,\widehat{\Gamma _{^{\mu v}}^T}\mathop = \limits_{(6)} \widehat{{\sigma _{\mu v}}} = \widehat{ - {\sigma _{\scriptstyle v\mu \hfill \atop \scriptstyle \hfill} }} \cr & \widehat{{\Gamma ^P}} = i{\gamma ^0}{\gamma ^1}{\gamma ^2}{\gamma ^3} = {\gamma _5} \equiv {\gamma ^5},\widehat{\Gamma _{^\mu }^A}\mathop = \limits_{(4)} {\gamma _5}\gamma \mu \cr} $$

((5.1))

and verify step by step the postulated properties of the EquationSource%MathType!MTEF!2!1!+-% MathType!MTEF!2!1!+- % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaacq % qHtoWrdaWgaaWcbaGaamOBaaqabaaakiaawkWaaaaa!3945! $$\widehat{{\Gamma _n}}$$ as well as some extra ones (also cf. Example 3.1). First we shall prove that in (5.1) there are indeed 16 matrices. This is easily done by adding the values written in brackets below the symbols. The upper indices of the matrices (“S”, “V”, “T”, “P”, and “A”) have the meaning “scalar”, “vector”, “tensor”, “pseudovector”, and “axial vector”, and these specifications will become clear in the following.