The electric components of the <electromagnetic field tensor> are
$$
F_{0i}=\dot A_i^V-\partial_i\Phi_A,
$$
while $F_{ij}$ depends only on the transverse vector. The scalar-vector cross term integrates to zero because $\partial_iA_i^V=0$, so the scalar action is
$$
S_{\rm scalar}=\frac12\int dt\,d^3x\,a(\partial_i\Phi_A)^2.
$$
Varying the nondynamical scalar gives the <constraint equation in field theory>
$$
\boxed{\nabla^2\Phi_A=0}.
$$
For every nonzero Fourier momentum this fixes $\Phi_A=0$, confirming that a source-free massless vector has no propagating scalar polarization.

Solution

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

A constraint equation follows by varying a field component that enters an action without its own time derivative. It restricts initial data or determines a nondynamical field instead of supplying an independent propagating degree of freedom.


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