Hermitian Operators in Dirac Equation

[Higgsono]

In the dirac equation we have a term which is proportional to $$\alpha p$$. In the book they say that
$$\alpha$$ must be an hermitian operator in order for the Hamiltonian to be hermitian.

As I understand, we require this because we want $$(\alpha p)^\dagger = \alpha p$$.

But $$(\alpha p)^\dagger = p^\dagger \alpha^\dagger = p \alpha$$, and so the order of the operators still change.

So if we just require both operators to be hermitian their product will still change if we take the hermitian conjugate.

 

[blue_leaf77]

$p$ acts on spatial part whereas $\alpha$ on spin part so the two operators commute.

 

[Higgsono]

$p$ acts on spatial part whereas $alpha$ on spin part so the two operators commute.

ok, thanks!