I Symmetrized Lagrangian (second quantization)

[Neutrinos02]

Hello,

I need some help to find the correct symmetrized Lagrangian for the field operators. After some work I guess that

$$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$

should be the correct Lagrangian but I'm not sure with this.

I'm also interested in the question of reordinger this Lagrangian in such a way that all \overline{\psi} are on the left and all \psi are on the right side. My problem: I don't know how to deal with products like (\partial_\mu \gamma^\mu \psi) \overline{\psi}

Thanks for your help.
Neutrino

 

[haushofer]

How do you define e.g. the second term?

 

[Neutrinos02]

How do you define e.g. the second term?

It should be [\overline{\psi}_a, \psi^a] := \sum_a \overline{\psi}_a \cdot \psi^a - \psi^a \cdot \overline{\psi}_a..
To ensure that the Lagrangian is hermitian we may add an aditional four divergence.

 

[haushofer]

I'm sorry, I only understand the first term where the barred psi comes first.