For spin \( 1/2 \) baryons, the fields \( \Psi \) are expanded in terms of the Dirac spinors (positive energy solution shown here with \( \overline{u}u=1 \)) $$ u(k\sigma)=\sqrt{\frac{E(k)+m}{2m}} \left(\begin{array}{c} \chi\\ \\ \frac{\mathbf{\sigma}\mathbf{k}}{E(k)+m}\chi \end{array}\right), $$ with \( \chi \) the familiar Pauli spinor and \( E(k) =\sqrt{m^2 +|\mathbf{k}|^2} \). The positive energy part of the field \( \Psi \) reads $$ \Psi (x)={\displaystyle \frac{1}{(2\pi )^{3/2}} \sum_{\mathbf{k}\mathbf{\sigma}}u(k\mathbf{\sigma})\exp{-(ikx)}a_{\mathbf{k}\mathbf{\sigma}}}, $$ with \( a \) being a fermion annihilation operator.