# Pseudoscalar current of Majorana fields

Consider a Majorana spinor $$\Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right)$$ and an pseudoscalar current $\bar\Phi\gamma^5\Phi$. This term is invariant under hermitian conjugation: $$\bar\Phi\gamma^5\Phi\to\bar\Phi\gamma^5\Phi$$ but if I exploit the two component structure $$\bar\Phi\gamma^5\Phi=-\phi\phi+\phi^\dagger\phi^\dagger$$ the invariance under hermitian conjugation seems lost $$-\phi\phi+\phi^\dagger\phi^\dagger\to\phi\phi-\phi^\dagger\phi^\dagger.$$ Where is the catch?

• Conjugate the individual components then multiply together to cancel out the minus? Commented Nov 23, 2015 at 2:02
• I don't understand what are you trying to say, can you be more explicit? Commented Nov 23, 2015 at 8:12

Ok, i found the (silly) error: $$\bar\Phi\gamma^5\Phi=\Phi^\dagger\gamma^0\gamma^5\Phi$$ so under hermitian conjugation this becomes $$\Phi^\dagger\gamma^5\gamma^0\Phi=-\Phi^\dagger\gamma^0\gamma^5\Phi=-\bar\Phi\gamma^5\Phi$$ that imply $$\bar\Phi\gamma^5\Phi+h.c.=0$$ the same result that we found exploiting the two component structure.

First, you should sort out if the components of the spinor are c-numbers or Grassmann numbers in your problem. If they are c-numbers, then the pseudoscalar built on the Majorana spinor vanishes, if I am not mistaken.

• Exploiting the indexes the product $\phi\phi$ is $$\phi\phi=\phi^\alpha\phi_\alpha=\phi^\alpha\epsilon_{\alpha\beta}\phi^\beta$$ where $\epsilon_{\alpha\beta}$ is the usual antisymmetric symbol and $\phi^\alpha$ and $\phi^\beta$ are Grassman numbers. So the product $\phi\phi$ commute. Commented Nov 22, 2015 at 22:52
• @Cervantes: If your components are Grassmann numbers, my answer does not seem relevant, but I don't quite see how your comment is relevant:-) Commented Nov 23, 2015 at 1:58
• What I was trying to say is that even if the component of $\phi$ are Grassmann numbers the product $\phi\phi$ commute as if they are c-numbers. Commented Nov 23, 2015 at 7:53