Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / b / Solution

The electric components of the electromagnetic field tensor arewhile $F_{ij}$ depends only on the transverse vector. The scalar-vector cross term integrates to zero because ${\partial }_i{A}_i^V=0$, so the scalar action isVarying the nondynamical scalar gives the constraint equation in field theoryFor every nonzero Fourier momentum this fixes ${\Phi }_A=0$, confirming that a source-free massless vector has no propagating scalar polarization.