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    $\begingroup$ That isn't really a full quantum theory -- this is like proposing to quantize the harmonic oscillator by writing $m \partial_t^2 x = - k x$ as $m \partial_t^2 \hat{x} = - k \hat{x}$. That doesn't actually fix $\hat{x}(t)$ unless you bring in the canonical momentum $\hat{p}$ as well. But when you try to do that with E&M you run into the same problems as in the standard QFT textbooks, which arise because $A_\mu$ is a gauge field. $\endgroup$
    – knzhou
    Commented Aug 29, 2023 at 1:36
  • $\begingroup$ @knzhou : That's interesting. Can you explain more? I.e. don't we have a natural conjugate that we could assume from prior theory, like just the usual wave conjugate, knowing that in free space we should have linear EM waves? $\endgroup$
    – The_Sympathizer
    Commented Aug 29, 2023 at 1:38
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    $\begingroup$ Sure, to elaborate, if you go ahead and compute the conjugate momentum in the usual way, you'll find that the one for $A^0$ just vanishes. You can get rid of this problem by gauge fixing $A^0 = 0$, but then you lose Lorentz invariance. $\endgroup$
    – knzhou
    Commented Aug 29, 2023 at 1:40
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    $\begingroup$ There are many misconceptions on this question. For example, Maxwell's equations are usually derived from assuming the Lagrangian for it, and we use the same Lagrangian for QED, just swapping the fields for operators. knzhou correctly points out that the derivatives will need reinterpretation. I want to point out that "problematic products of operator distributions" is a solved problem. Under causal perturbation theory, careful operator product splitting makes all the UV divergence go away, so all terms are well-defined. $\endgroup$
    – naturallyInconsistent
    Commented Aug 29, 2023 at 2:17
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    $\begingroup$ @naturallyInconsistent: Digging more though I note that your answer here - physics.stackexchange.com/q/768988 - seems to present a rather less rosy picture of the situation than this comment would seem to imply. What's going on? (Note also that being able to solve the equations, by the way, I'd differentiate from the idea of them being well-defined in principle. A 3-body problem is "not easy to solve", but it is well-defined in principle. And it's the idea of WDIP that these lines of questioning are intended to explore.) $\endgroup$
    – The_Sympathizer
    Commented Aug 29, 2023 at 3:01