- #1
TroyElliott
- 59
- 3
For a Majorana neutrino in matter we have the equation $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})\nu_{L} = m\overline{\nu_{L}}.$$ A is to be considered constant.
Squaring, in the ultra-relativistic limit one obtains the dispersion relation
$$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ i.e.
$$p \simeq E -(\frac{mm^{\dagger}}{2E}+A).$$
What I have is $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})(-i(\gamma^{\mu}\partial_{\mu})^{\dagger}-A(\gamma_{0})^{\dagger})$$ and I know $$\gamma_{0}^{\dagger} = \gamma_{0}$$ and $$(\gamma^{\mu})^{\dagger} = \gamma^{0}\gamma^{\mu}\gamma^{0}.$$
But I am not seeing how to get $$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ from this. Any hints would be greatly appreciated!
Squaring, in the ultra-relativistic limit one obtains the dispersion relation
$$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ i.e.
$$p \simeq E -(\frac{mm^{\dagger}}{2E}+A).$$
What I have is $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})(-i(\gamma^{\mu}\partial_{\mu})^{\dagger}-A(\gamma_{0})^{\dagger})$$ and I know $$\gamma_{0}^{\dagger} = \gamma_{0}$$ and $$(\gamma^{\mu})^{\dagger} = \gamma^{0}\gamma^{\mu}\gamma^{0}.$$
But I am not seeing how to get $$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ from this. Any hints would be greatly appreciated!