Angular momentum in electrodynamics

When describing the motion of a charged particle in an electromagnetic field, the canonical momentum P (derived from the Lagrangian for this system) is not gauge invariant. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called kinetic momentum (used throughout this article), is (in SI units)

${\displaystyle \mathbf{p}=m\mathbf{v}=\mathbf{P}-<!--\; -\; -->e\mathbf{A}}$

where e is the electric charge of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is kinetic angular momentum, is given by

${\displaystyle \mathbf{K}=\mathbf{r}\times <!--\; \times \; -->(\mathbf{P}-<!--\; -\; -->e\mathbf{A})}$

The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.