Put $\widetilde c=\widetilde a+\widetilde b$. Substituting the <transverse-longitudinal decomposition> $V_\mu=A_\mu+\partial_\mu\pi$ and integrating cross terms by parts gives
$$
\mathcal L\simeq
\widetilde a A_\mu\Box A^\mu+m^2A_\mu A^\mu
-\widetilde c(\Box\pi)^2+m^2\partial_\mu\pi\partial^\mu\pi.
$$
The equations are
$$
(\widetilde a\Box+m^2)A_\mu=0,
\qquad
\Box(\widetilde c\Box+m^2)\pi=0,
$$
together with $\partial_\mu A^\mu=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 2 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-301/2/b/i/solution