  Quantum electrodynamics (QED) deals with the relativistic interaction of
bosonic gauge fields and fermionic charged particles. In QED, global
conservation laws of angular momentum for light-matter interactions are
well-known. However, local conservation laws, i.e. the conservation law of
angular momentum at every point in space, remain unexplored. Here, we use the
QED Lagrangian and Noether's theorem to derive a new local conservation law of
angular momentum for Dirac-Maxwell fields in the form of the continuity
relation for linear momentum. We separate this local conservation law into four
coupled motion equations for spin and orbital angular momentum (OAM) densities.
We introduce a helicity current tensor, OAM current tensor, and spin-orbit
torque in the motion equations to shed light on on the local dynamics of
spin-OAM interaction and angular momentum exchange between Maxwell-Dirac
fields. We elucidate how our results translate to classical electrodynamics
using the example of plane wave interference as well as a dual-mode optical
fiber. Our results shine light on phenomena related to the spin of gauge
bosons.
