I'm trying to derive the adjoint of the Dirac equation in standard relativistic quantum mechanics. We have the Dirac equation as follows :
$$(i\gamma^{\mu}\partial_\mu -m)\psi=0$$
To find it's adjoint, I first take the hermitian conjugate, and note that $(\gamma^\mu)^\dagger = \gamma^0 \gamma^\mu \gamma^0 $
Using this, I get the adjoint equation:
$$\bar{\psi}(-i\gamma^{\mu}\partial_\mu -m)=0$$
However, now I want to do the same thing, but I shall use $p_\mu$ instead of $i\partial_\mu$.
So I write the Dirac equation again,
$$(\gamma^{\mu}p_\mu -m)\psi=0$$
I can follow the same steps as above, and end up with the following:
$$\bar{\psi}(\gamma^{\mu}p_\mu^\dagger -m)=0$$
Now, if I use my intuition, that tells me $p_\mu^\dagger = p_\mu$, then comparing the two final solutions point to the following :
$$p_\mu=i\overrightarrow{\partial_\mu}$$
and,
$$p_\mu^\dagger=-i\overleftarrow{\partial_\mu}$$
The arrows represent the fact that in the adjoint case, the differential operator acts to the left unlike the oridinary case where it acts to the right.
This creates a multitude of problems and confusions for me.
Firstly, I know from QM, that the differential operator is anti-hermitian. However, in the first derivation, I've not used that, as some other posts pointed toward the fact that $\partial_\mu$ should be considered as a vector in this case. And so, $\partial_\mu^\dagger =\partial_\mu$. This is a bit confusing, and I'd be glad if someone can explain this to me. However, if I considered $\partial_\mu^\dagger =-\partial_\mu$ instead, I'd end up with the equation
$$\bar{\psi}(i\gamma^{\mu}\partial_\mu -m)=0$$
This is clearly incorrect because of the relative signs. However, if someone can point out why this is so, I'll be glad.
Secondly however, if I accept that $\partial_\mu^\dagger =\partial_\mu$, I run into a different problem. Since I have already compared and found $p_\mu=i\overrightarrow{\partial_\mu}$ and $p_\mu^\dagger=-i\overleftarrow{\partial_\mu}$.
Since momentum is hermitian, I must have $p_\mu^\dagger=p_\mu$, and that would mean $$\overrightarrow{\partial_\mu}=-\overleftarrow{\partial_\mu}$$
In ordinary quantum mechanics, $p=i\partial$, and $p^\dagger = (-i)(-\partial)=p$. Clearly that is not what is going on in here.
This doesn't make sense to me. Can someone help me out with these two questions, and show me where exactly am I going wrong, and help me reconcile the two alternate descriptions of the same thing.
Regards.
