= Four-dimensional Majorana bilinear vanishing criterion
{title2=$(C\gamma^{(n)})^T=-(-1)^{n(n+1)/2}C\gamma^{(n)}$}
For a <Majorana spinor> $\lambda$ with odd <Grassmann variables> as components, $\lambda^TM\lambda$ sees only the antisymmetric part of $M$. With $C^T=-C$ and $(\gamma^a)^T=-C\gamma^aC^{-1}$, reversing an antisymmetrized gamma product gives the displayed transpose rule. Thus the same-spinor vector and bivector bilinears vanish, whereas scalar, pseudoscalar and three-form bilinears can be nonzero. Four-dimensional <Hodge duality> gives the same vanishing result for a bivector multiplied by $\gamma_5$. Commuting spinors instead see the symmetric part, so their selection rule is different.
Back to article page
