  This two-paper series addresses and fixes the long-standing gauge invariance
problem of angular momentum in gauge theories. This QED part reveals: 1) The
spin and orbital angular momenta of electrons and photons can all be
consistently defined gauge invariantly. 2) These gauge-invariant quantities can
be conveniently computed via the canonical, gauge-dependent operators (e.g,
$\psi ^\dagger \vec x \times\frac 1i \vec \nabla \psi$) in the Coulomb gauge,
which is in fact what people (unconsciously) do in atomic physics. 3) The
renowned formula $\vec x\times(\vec E\times\vec B)$ is a wrong density for the
electromagnetic angular momentum. The angular distribution of angular-momentum
flow in polarized atomic radiation is properly described not by this formula,
but by the gauge invariant quantities defined here. The QCD paper
[arXiv:0907.1284] will give a non-trivial generalization to non-Abelian gauge
theories, and discuss the connection to nucleon spin structure.
