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 formula_1 minimally coupled to a gauge field formula_2. The dynamics is given by the Lagrangian density formula_3 where formula_4 is the electromagnetic field strength, formula_5 is the covariant derivative of the field formula_6 , formula_7 is the electric charge and formula_8 is the potential for the complex scalar field. This model is invariant under gauge transformations parameterized by formula_9 formula_10 If the potential is such that its minimum occurs at non-zero value of formula_11, this model exhibits the Higgs mechanism. This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to formula_7 times the minimum value of formula_11. As shown in 1973 by Nielsen and Olesen, this model, in formula_14 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 formula_15) and appears as a topological charge associated with the topological current formula_16 These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions.