Write $\psi^P(x)=\eta_P\gamma^0\psi(x_P)$. The <chain rule> gives $\partial_0\psi(x_P)=(\partial_0\psi)(x_P)$ and $\partial_i\psi(x_P)=-(\partial_i\psi)(x_P)$. Using the <gamma matrix> relations $(\gamma^0)^2=1$ and $\gamma^i\gamma^0=-\gamma^0\gamma^i$,
$$
\begin{aligned}
(i\gamma^\mu\partial_\mu-m)\psi^P(x)
&=\eta_P\left(i\gamma^0\gamma^0\partial_0-i\gamma^i\gamma^0\partial_i-m\gamma^0\right)\psi(x_P)\\
&=\eta_P\gamma^0(i\gamma^\mu\partial_\mu-m)\psi(x_P)=0.
\end{aligned}
$$
Thus the <Dirac equation> is invariant under <parity symmetry in quantum field theory>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 305 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-305/1/c/solution