# Electromagnetism theory and complex scalar field

I've got the following problem for classical field theory lecture:

Find equations of motion (equations of field?), canonical and symmetrical tensor of energy-momentum in electromagnetic field theory with complex scalar field with charge $q$ and potential $V$.

I've get lost - should we just get classical electrodynamics lagrangian and derive the energy-momentum tensor, or the potential V of scalar field (with gauge symmetry generating electromagnetism) actually changes the EM lagrangian?

I assume that by "potential $V$ of the scalar field" is meant "everything that stands on the right of $\partial \overline \phi \partial \phi$ in the scalar field's lagrangian", e.g. $V=-m^2 \overline \phi \phi$ for the complex KG field. Assuming that $V$ is invariant under $U(1)$ gauge transformations, you can introduce a covariant derivative in the usual way, by requiring that: $$D_{[A+\partial \alpha]}(e^{-i q\alpha (x)}\phi(x)) = e^{-iq\alpha (x)}D_{[A]}\phi (x),$$ which gives you the electromagnetic interaction of the scalar field. The full lagrangian reads:$$\mathscr L = D\overline \phi D \phi + V(\phi , \overline \phi)-\frac {1}{4}F^{\mu \nu }F_{\mu \nu}$$
The (canonical or symmetric) energy-momentum tensor can be calculated in the usual way, but note that, since the $A-\phi$ coupling depends on the derivatives $\partial \phi$, also the momentum components $T^{0i}$ will get contributions from the interaction lagrangian (compare with spinorial QED).