Parity

It is useful to understand the effect of a parity inversion on a Dirac spinor. Again work with the Dirac equation and its parity inverted form in which $x_j\rightarrow -x_j$ and $x_4$ remains unchanged (the same for the vector potential).

$$\begin{eqnarray*} \gamma_\mu{\partial\over\partial x_\mu}\psi(x)+{mc\over\hbar}\... ...rtial\over\partial x_4}\right)S_P\psi+{mc\over\hbar}\psi&=&0 \\ \end{eqnarray*}$$

Since $\gamma_4$ commutes with itself but anticommutes with the $\gamma_i$, it works fine.

$$\bgroup\color{black} S_P=\gamma_4 \egroup$$

(We could multiply it by a phase factor if we want, but there is no point to it.)

Therefore, under a parity inversion operation

$$\psi'=S_P\psi=\gamma_4\psi$$

Since $\gamma_4=\pmatrix{1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & -1 & 0 \cr 0 & 0 & 0 & -1 \cr}$, the third and fourth components of the spinor change sign while the first two don't. Since we could have chosen $-\gamma_4$, all we know is that components 3 and 4 have the opposite parity of components 1 and 2.