Normalisation of the $\gamma$-matrices

Question

I'm having a little difficulty with understanding the normalisation process of the $\gamma$-matrices.

In Thomson Modern Particle Physics 2013, the normalisation of the $\gamma$-matrices are quoted as:

$$ (\gamma^{\mu})^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0} $$ Where $\mu=0,1,2,3$ or sometimes just $\mu=0, k$ where, obviously $k=1,2,3$. I have attempted to start this given example, but I'm not sure on the next steps. Thus far, I have:

$$ (\gamma^{0})^{\dagger}=\gamma^{0} $$ $$ (\gamma^{k})^{\dagger}=-\gamma^{k} $$ I also know that $$ (\gamma^{0})^{2}=I\,\,\mathrm{and}\,\,(\gamma^{k})^{2}=-I $$ I'm just not sure how to put this together. If anyone could give a quick run through or some prods in the right direction that would be excellent.

Answer 1

While studying the $\gamma$-matrices, I also faced the same question. Here maybe an solution about this.

First, the convention is the same: $$(\gamma^{0})^{\dagger}=\gamma^{0}$$ $$(\gamma^{k})^{\dagger}=-\gamma^{k}$$ $$(\gamma^{0})^{2}=I\,\,\mathrm{and}\,\,(\gamma^{k})^{2}=-I$$ Additionally, here used the eqution, $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}I$$ Especially, for the $\gamma^{0}$ and $\gamma^{k}$, $$\{\gamma^{0},\gamma^{k}\}=0$$ Then we can see, $$(\gamma^{0} \gamma^{k})^{\dagger}=(\gamma^{k})^{\dagger} (\gamma^{0})^{\dagger}$$ and $$\gamma^{0} \gamma^{k}=-\gamma^{k} \gamma^{0}$$ so $$(-\gamma^{k} \gamma^{0})\dagger=(\gamma^{k})^{\dagger} (\gamma^{0})^{\dagger}$$ $$-\gamma^{0} (\gamma^{k})^{\dagger}=-\gamma^{k} \gamma^{0}$$ $$(\gamma^{k})^{\dagger}=\gamma^{0}\gamma^{k} \gamma^{0}$$ also we known that $$(\gamma^{0})^{\dagger}=\gamma^{0}\gamma^{0} \gamma^{0}$$ we can combined these together to see $$(\gamma^{\mu})^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}$$ Noticing that here we have already selected a specific representation.

Answer 2

• for $(\gamma^0)^2 = I_4$, so $(\gamma^0)^\dagger = \gamma^0$;
• for $(\gamma^i)^2 = -I_4$, so $(\gamma^i)^\dagger = -\gamma^i$

Than: ${\gamma_\mu}^\dagger = \gamma^0\gamma_\mu\gamma^0$