- #1
BobaJ
- 38
- 0
- Homework Statement
- I have to show, the last equality in the charge-conjugation property of the current $$C\bar{\Psi}_a\gamma^{\mu}\Psi_bC^{-1}=\bar{\Psi}_a^c\gamma^{\mu}\Psi_b^c=-(\bar{\Psi}_a\gamma^{\mu}\Psi_b)^\dagger$$
- Relevant Equations
- ##\bar{\Psi}^c=-\Psi^TC^{-1}##
##\Psi^c=C\bar{\Psi}^T##
##C^{-1}\gamma^{\mu}C=-\gamma^{\mu T}##
##C=-C^{-1}=-C^\dagger=-C^T##
And I have to use that the fermionic quantum fields ##\Psi_a## and ##\Psi_b## anticommute.
I'm probably just complicating things, but I'm a little bit stuck with this problem.
I started with just plugging in the definitions for ##\bar{\Psi}_a^c## and ##\Psi_b^c##. So I get
$$\bar{\Psi}_a^c\gamma^{\mu}\Psi_b^c=-\Psi_a^TC^{-1}\gamma^{\mu}C\bar{\Psi}_b^T$$.
After this I used ##C^{-1}\gamma^{\mu}C=-\gamma^{\mu T}## to get:
$$=\Psi-a^T\gamma^{\mu T}\bar{\Psi}_b^T$$.
Is this the right way? How do I go onto show the equality? Thank you for your help.
I started with just plugging in the definitions for ##\bar{\Psi}_a^c## and ##\Psi_b^c##. So I get
$$\bar{\Psi}_a^c\gamma^{\mu}\Psi_b^c=-\Psi_a^TC^{-1}\gamma^{\mu}C\bar{\Psi}_b^T$$.
After this I used ##C^{-1}\gamma^{\mu}C=-\gamma^{\mu T}## to get:
$$=\Psi-a^T\gamma^{\mu T}\bar{\Psi}_b^T$$.
Is this the right way? How do I go onto show the equality? Thank you for your help.