# Scalar electrodynamics

In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism.

The model consists of a complex scalar field $\phi(x)$ minimally coupled to a gauge field $A_\mu(x)$. The dynamics is given by the Lagrangian density

$\mathcal{L}= (D_\mu \phi)^* D^\mu \phi - U(\phi^*\phi) -\frac14 F_{\mu\nu}F^{\mu\nu}\ ,$

where $F_{\mu\nu}=(\partial_\mu A_\nu - \partial_\nu A_\mu)$ is the electromagnetic field strength,$D_\mu\phi=(\partial_\mu \phi - i e A_\mu \phi)$ is the covariant derivative of the field $\phi$, $e$ is the electric charge and $U(\phi^*\phi)$ is the potential for the complex scalar field. This model is invariant under gauge transformations parametrized by $\lambda(x)$

$\phi'(x) = e^{ie \lambda(x)}\phi(x)\quad\textrm{and}\quad A_\mu'(x)=A_\mu(x)+\partial_\mu \lambda(x)\ .$

If the potential is such that its minimum occurs at non-zero value of $|\phi|$, this model exhibits the Higgs mechanism. This can be seen by studying fluctuations about the lowest energy configuration, one sees that gauge field behaves as a massive field with its mass proportional to the $e$ times the minimum value of $|\phi|$. As shown in 1973 by Nielsen and Olesen, this model, in $2+1$ dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized (in units of $\tfrac{2\pi}{e}$) and appears as a topological charge associated with the topological current

$J_{top}^\mu =\epsilon^{\mu\nu\rho} F_{\nu\rho}\ .$

These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions.

## References

• Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2]