- #1
Hypo86
- 1
- 0
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} =<br /> \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} =<br /> \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!
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} =<br /> \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} =<br /> \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!