  Some theories, in particular those of current particle physics, are distinguished by being gauge invariant, which means that gauge transformations of certain terms do not change any observable quantities. A gauge transformation is a local symmetry transformation, with parameters that are smooth functions of the position \(\mathbf{x}\) and time \(t\). Requiring gauge invariance provides an elegant and systematic way of introducing models for interacting fields. Moreover, gauge invariance plays an important role in selecting theories. The prime example of an intrinsically gauge invariant theory is electrodynamics. In the potential formulation of Maxwell’s equations one introduces the vector potential \(\mathbf{A}\) and the scalar potential \(\phi\), which are linked to the magnetic field \(\mathbf{B}(\mathbf{x},t)\) and the electric field \(\mathbf{E}(\mathbf{x},t)\) by