 Solution

Use signature $(+,-,-,-)$ and the free Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$. For positive $p^0=E_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$, substitution of the two frequency signs givesMultiplying by the opposite factor shows $p^2=m^2$. Each equation has two independent solutions for a massive field.

In the Dirac basis, ${\gamma }^0=\mathrm{diag}(I,-I)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }_i\\ -{\sigma }_i& 0\end{array}\right)$. For orthonormal two-component spin labels ${\xi }_s,{\eta }_s$, convenient solutions areThe two-block equations verify these expressions using $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2={\mathbf{p}}^{2}I$. At rest the positive-frequency solutions have only upper components and the negative-frequency solutions only lower components. Their normalizations are $u_s^{†}{u}_r=v_s^{†}{v}_r=2E_{\mathbf{p}}{\delta }_{sr}$ and ${\bar{u}}_s{u}_r=2m{\delta }_{sr}$, ${\bar{v}}_s{v}_r=-2m{\delta }_{sr}$. Their completeness relations areThe field expansion combines fermionic annihilation operators multiplying the positive-frequency waves with antiparticle fermionic creation operators multiplying the negative-frequency waves. The latter frequency sign is not the energy of a physical negative-energy antiparticle state.