Show that the fermion operators (hat{P}^{mu}) and (hat{M}^{mu u}) are the generators of the Poincar group by

Show that the fermion operators \(\hat{P}^{\mu}\) and \(\hat{M}^{\mu u}\) are the generators of the Poincaré group by proving Eqs. (6.3.214), (6.3.212) and (6.3.213). Show that \(\hat{P}^{\mu}\) and \(\hat{M}^{\mu u}\) are Hermitian. (Hint: First prove before normal-ordering and then prove that normal-ordering only involves adding a real constant. While each step is relatively straightforward, this proof requires care and effort.)