Linear-in-potential charged-scalar Lagrangian

The densitycontains the term linear in the potential from minimal coupling, but omits $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\square +m^2)\varphi +ie(\partial \cdot A)\varphi +2ieA\cdot \partial \varphi =0$, together with its complex conjugate. The omitted quadratic interaction must not be added silently when solving this specified theory.

For $\delta \varphi =-iϵ\varphi$, the conserved current is $J^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2e{A}^{\mu }|\varphi {|}^2$. The conjugate field equations implywhich directly cancels the divergence of the explicit background term. Conservation requires no Lorenz gauge condition; constancy of total charge additionally requires vanishing boundary flux.

For the linear-in-potential charged-scalar Lagrangian, $\pi ={\dot{\varphi }}^{\ast }-ie{A}_0{\varphi }^{\ast }$ and ${\pi }^{\ast }=\dot{\varphi }+ie{A}_0\varphi$. With $A_{\mu }=(A_0,-\mathbf{A})$, the Hamiltonian density isThe first product includes $+e^2{A}_0^2|\varphi {|}^2$ after the Legendre transform; that term can arise in canonical variables even though the original Lagrangian has no interaction quadratic in $A$.