About Majorana spinor $1$ form

Question

I am currently studying some basic materials about Majorana spinor 1 form.

I know usual (0-form) Majorana spinor very well(i guess). They satisfy the $\psi^c = \psi$ for majoran spinor $\psi$ and from them we can do some computation as follows \begin{align} &\overline{\lambda} \chi = \overline{\chi} \lambda \\ & \overline{\lambda} \gamma_\mu \chi = - \overline{\chi} \gamma_\mu \lambda \end{align} Deriving above formula one use $\psi^c = \psi$ and $\lambda_a \chi_b = -\chi_b \lambda_a$. and so on.

Here come back to the problem, for Majorana spinor 1 form, only surviving form of majorana spinor is known as $\overline{\psi} \gamma^a \psi \neq 0$ and $\overline{\psi} \gamma^{ab} \psi \neq 0$. From deriving above things i got stucked.

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I think my main problem is the lack of concept for Majorna spinor 1 form

First usual spinor 1 form that i know is \begin{align} A = A_\mu dx^{\mu} \end{align} In that sense i can do for Majorana spinor 1 form \begin{align} \psi = \psi_{\mu} dx^{\mu} \end{align}

What i want to know is for 1-form Majorna spinor case is still grassman property holds? $i.e$
\begin{align} \Psi \chi = -\chi \Psi \end{align} or it only holds for coefficient $\psi_\mu$, $\chi_\mu$ for one form? $i.e$ for $\Psi = \psi_\mu dx^\mu$, $\chi = \chi_\mu dx^\mu$ \begin{align} \psi_\mu \chi_\nu = - \chi_\nu \psi_\mu \end{align}

I know for the wedge product of $p$ and $q$ form gives \begin{align} w^{(p)} \wedge w^{(q)} = (-1)^{pq} w^{(q)} \wedge w^{(p)} \end{align}

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What i really want to do is following derivation for Majorna spinor 1 form \begin{align} \overline{\Psi} \gamma^5 \Psi = 0 \end{align}

\begin{align} \overline{\Psi} \gamma^5 \Psi = \Psi^{a} (C\gamma^5)_{ab} \Psi^b = \Psi^{a} \Psi^b(C\gamma^5)_{ab} = (-?)\Psi^b\Psi^{a} ( C \gamma^5)_{ab} \end{align}

Does the $-$ sign holds?

I can do the other leftover derivation except above parts.
If you know something please let me learn from you.

This post imported from StackExchange Physics at 2015-05-14 21:01 (UTC), posted by SE-user phy_math