- #1
Andrea M.
- 28
- 1
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?
$$
\Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right)
$$
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?