With matrix-valued fields, <charge conjugation> acts as
$$
C:\quad A_\mu(t,\mathbf x)\mapsto-A_\mu^T(t,\mathbf x),
\qquad
F_{\mu\nu}\mapsto-F_{\mu\nu}^T.
$$
Under <parity>, $A_0$ is even and $A_i$ is odd:
$$
P:\quad A_0(t,\mathbf x)\mapsto A_0(t,-\mathbf x),
\qquad
A_i(t,\mathbf x)\mapsto-A_i(t,-\mathbf x).
$$
Consequently $F_{0i}$ is odd and $F_{ij}$ is even. Under <time-reversal symmetry>, which is antiunitary, $A_0$ is even and $A_i$ is odd after $t\mapsto-t$; hence $F_{0i}$ is even and $F_{ij}$ is odd. Equivalently, the non-Abelian electric and magnetic fields transform as
$$
\begin{array}{c|ccc}
&C&P&T\\ \hline
\mathbf E&-\mathbf E^T&-\mathbf E&+\mathbf E\\
\mathbf B&-\mathbf B^T&+\mathbf B&-\mathbf B
\end{array}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-305/2/ii/solution