Homework Help: Deriving equation of motion

1. May 20, 2015

Hypo86

Hi!

I have the following problem with some old lecture notes I recently had a look on.

I have two different fermions (1 and 2) with masses m1 and m2
and the following Lagrangian (where the mass term for fermion 2 is dropped, because we are only interested in the dynamics of fermion 1) of the form:

$$\mathcal{L} = \bar{\Psi}_{1} \left( i {\not}{\partial} - {\not}{A} \left( 1 - \gamma_{5} \right) - m_{1} \right) \Psi_{1},$$

where

$$A^{\mu} = \dfrac{G_{F}}{\sqrt{2}} \left[ \bar{\Psi}_{2} \gamma^{\mu} \left( 1 - \gamma_{5} \right) \Psi_{2} \right] =: (\varphi, \vec{A})$$

If we compute the equation of motion from that (Euler-Lagrange), one finds:

$$\left( i \dfrac{\partial}{\partial t} - \varphi \right) \Psi_{1} = \left[ \vec{\alpha} \cdot \left( \dfrac{1}{i} \nabla - \vec{A} \right) + \beta m_{1} \right] \Psi_{1},$$

where $\beta = \gamma^{0}$ and $\vec{\alpha} = \beta \vec{\gamma}$

I think this is not correct, because there are two $(1 - \gamma_{5})$ factors missing.
I find:

$$\left( i \dfrac{\partial}{\partial t} - \varphi ( 1 - \gamma_{5} ) \right) \Psi_{1} = \left[ \vec{\alpha} \cdot \left( \dfrac{1}{i} \nabla - \vec{A} ( 1 - \gamma_{5} \right) + \beta m_{1} \right] \Psi_{1}$$

Did I miss anything? I would be glad if someone could have a short look on it.
Thanks a lot!

2. May 25, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 25, 2015

nrqed

Hi, welcome to PF!

Can you double check what you wrote above? It does not make sense as stated. A is a gauge field, it should not be expressed in terms of the lagrangian of the second fermion. It looks like what you wrote for A^mu should be a term in the lagrangian, not A^mu