If I'm not mistaken, one great theoretical unification achieved by QED is that it eliminates the dualistic cause-and-effect picture of the electromagnetic interaction. That is, instead of having a field equation (Maxwell) and an equation of motion (Lorentz force), you in principle apply the Lagrangian to the entire configuration of matter + gauge field, and, I assume, the Lagrangian tends to select configurations (or perhaps more accurately, neighborhoods of configurations) that look more or less like the classical solutions, apart from e.g. the delocalization of the matter field.
But then, we also know that the configuration of stationary action under the QED Lagrangian is actually just a solution of the Dirac equation. So if QED describes both field production and motion of matter, then doesn't the Dirac equation do the same?
But I always thought of the Dirac equation as basically a relativistic Schrodinger equation, in which the external EM field $A_\mu$ is fixed and the equation only determines the resulting wavefunction -- that is, the counterpart of the classical equation of motion.
So could you actually use the Dirac equation to determine the production of EM field as well as its effect on matter? Or is there some subtlety I'm missing? Perhaps it has something to do with the $U(1)$ gauge symmetry of the Dirac equation not being local? But couldn't you just declare it to be local if you wanted?