# About Majorana spinor $1$ form

+ 1 like - 0 dislike
115 views

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.

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}

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
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.