Maxwell's Equations & Gauge Invariance
Covariant electrodynamics and the gauge redundancy of the potentials.
Maxwell's equations in vacuum (Gaussian units):
$$\nabla \cdot \mathbf E = 4\pi\rho, \quad \nabla \cdot \mathbf B = 0,$$ $$\nabla \times \mathbf E = -\tfrac{1}{c}\partial_t \mathbf B, \quad \nabla \times \mathbf B = \tfrac{4\pi}{c}\mathbf J + \tfrac{1}{c}\partial_t \mathbf E.$$With $A^\mu = (\phi, \mathbf A)$ and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ (manifestly antisymmetric, $F_{\mu\nu} = -F_{\nu\mu}$):
$$\partial_\mu F^{\mu\nu} = \tfrac{4\pi}{c} J^\nu, \quad \partial_{[\lambda} F_{\mu\nu]} = 0.$$The gauge symmetry $A_\mu \to A_\mu + \partial_\mu \Lambda$ leaves $F_{\mu\nu}$ — and therefore all physics — invariant; this is the prototype for Yang–Mills theories. Two common gauge fixes:
- Lorenz gauge: $\partial_\mu A^\mu = 0$ (manifestly Lorentz covariant).
- Coulomb gauge: $\nabla \cdot \mathbf A = 0$ (convenient for radiation problems).
Interactive: travelling EM wave
$\mathbf E$ (red) and $\mathbf B$ (blue) oscillate perpendicular to each other and to the direction of propagation $\hat z$.