I Symmetrized Lagrangian (second quantization)

1. Feb 15, 2017

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 intrested 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 $(\partial_\mu \gamma^\mu \psi) \overline{\psi}$

Neutrino

2. Feb 16, 2017

haushofer

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

3. Feb 16, 2017

Neutrinos02

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.

4. Feb 18, 2017

haushofer

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