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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?

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    $\begingroup$ Two issues: (1) Dirac's equation is not enough, you need also the equations for electromagnetic fields (Maxwell), (2) Dirac equation is a single particle equation, it misses many-body dynamics. $\endgroup$
    – Mauricio
    Dec 31, 2022 at 19:22
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    $\begingroup$ Faulty premise that QED has unified anything, theoretically or otherwise $\endgroup$
    – fertilizerspike
    Dec 31, 2022 at 19:30
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    $\begingroup$ "...we also know that the configuration of stationary action under the QED Lagrangian is actually just a solution of the Dirac equation" <-- what?!?! where did this come from? $\endgroup$
    – Prahar
    Dec 31, 2022 at 19:54
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    $\begingroup$ QED is a QFT, quantum field theory. Quantum field theories follow the postulates of quantum mechanics, but have been formulated to study scatterings and decays, using Feynman diagrams. The Dirac equation is relevant to the extent that the field on which creation and annihilation operators work in quantum field theories, are represented for calculation by the plane wave solution of the electron field, muon field,... depending on the interaction studied. No, the Dirac equation is essential , but it cannot describe QED, $\endgroup$
    – anna v
    Dec 31, 2022 at 21:14
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    $\begingroup$ @AdamHerbst - The Dirac equation has a 0 on the right-hand-side of the equation in the link. The equation of motion for QED is different. $\endgroup$
    – Prahar
    Dec 31, 2022 at 21:15

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This is something often overlooked in discussions about Electromagnetism: technically, the correct way of solving an E&M problem is to solve the coupled system of equations given by the Maxwell equations and whatever equation rules the charged matter being considered. This is, of course, way more difficult than the two problems often considered in E&M courses:

  1. given a predetermined arrangement of matter, solve for the E&M fields;
  2. given a predetermine E&M field, solve for the evolution of charged matter. Notice we very often simplify the actual problem (solving both matter and fields at the same time) to the cases in which one of the parts is already given.

One of the reasons we do this is because most E&M courses will focus quite a lot on point-like charges, and it happens that point-like charges are inconsistent in E&M. The problem of solving Maxwell's equations for a point-like source and the equations of motion for this charge simultaneously is not well-posed. The radiation reaction effects affect the charge too much, and one can't really get a solution (think that the fields induced by the charge are infinite at the charge's location, and hence you have infinite forces acting). This is not a failure of E&M, but rather of the point-like charge approximation.

If one considers a charge distribution modeled by a field, one often can solve the complete system of equations consistently. The Dirac equation is such an example, and of course, it is a particularly interesting one. However, it isn't the only one. One could also consider a charged fluid, for example. Of a charged scalar field (for theoretical purposes).

So could you actually use the Dirac equation to determine the production of EM field as well as its effect on matter?

No. The Dirac equation is the equation of motion for the charged matter. The fields are still determined by the Maxwell equations. The correct way to approach this classically would be to solve the coupled system of equations given by the Maxwell equations sourced by a Dirac field and the Dirac equation.

Wald's Advanced Classical Electromagnetism has quite interesting discussions on this matter, in case it interests you.

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