- #1
pellman
- 684
- 5
I find the Lagrangian associated with the Dirac equation given in texts as
[tex]\mathcal{L}=\bar{\psi}\left(i\gamma^\mu \partial_\mu - m\right)\psi[/tex]
or
[tex]\mathcal{L}=i\bar{\psi}\gamma^\mu \partial_\mu \psi- m\bar{\psi}\psi[/tex]
[tex]\mathcal{L}=i \psi^{\dagger}\gamma^0\gamma^\mu \partial_\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
Taking the hermitian conjugate of the expression on the right, we get
[tex](-i) \partial_\mu\psi^{\dagger}\left(-\gamma^\mu\right) \gamma^0 \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
[tex]=-i \partial_\mu\psi^{\dagger}\gamma^0\gamma^\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
[tex]=-i \partial_\mu\bar{\psi}\gamma^\mu \psi- m\bar{\psi}\psi[/tex]
which, as far as I can tell, is not equal to [tex]\mathcal{L}[/tex]
So if it is not hermitian, is that ok?
[tex]\mathcal{L}=\bar{\psi}\left(i\gamma^\mu \partial_\mu - m\right)\psi[/tex]
or
[tex]\mathcal{L}=i\bar{\psi}\gamma^\mu \partial_\mu \psi- m\bar{\psi}\psi[/tex]
[tex]\mathcal{L}=i \psi^{\dagger}\gamma^0\gamma^\mu \partial_\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
Taking the hermitian conjugate of the expression on the right, we get
[tex](-i) \partial_\mu\psi^{\dagger}\left(-\gamma^\mu\right) \gamma^0 \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
[tex]=-i \partial_\mu\psi^{\dagger}\gamma^0\gamma^\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]
[tex]=-i \partial_\mu\bar{\psi}\gamma^\mu \psi- m\bar{\psi}\psi[/tex]
which, as far as I can tell, is not equal to [tex]\mathcal{L}[/tex]
So if it is not hermitian, is that ok?