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QED describes the interaction of two operator fields. Classical electrodynamics describes the interaction of a classical field with a point charged particle.

My question is, what limits do you apply to get from the former to the latter?

I think that getting from the operator EM field to the classical EM field, requires taking the "high particle number limit". I'm not sure about how to go from the operator spinor field to the point particle approximation. I think this involves the "high momentum limit" somehow, so that the uncertainty principle becomes less relevant and the point particle nature emerges. But I don't know how one would take the "high momentum limit" on an operator field.

Is it true that we take both of these classical limits simultaneously? I'm not sure how to do this. Which limit do we take first? Please answer a step-by-step road-map guide bridging these two theories. You don't need to write the lengthy mathematics. Just the steps in order / a summary of the mathematical steps.

Brendan Darrer
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Ryder Rude
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    There is no general procedure for "the classical limit". What do you need this limit for? For instance, if you just want to see how the Coulomb potential emerges from QED, see this question and its answers. – ACuriousMind Jul 17 '22 at 17:24
  • @ACuriousMind I need the roadmap to the full Maxwell equations and the Lorentz Force law on a point charged particle, starting from the interacting QED operator equations. I think the exact math would be lengthy, so I just need a roadmap. Just write a description of the mathematical steps you'd carry out and the end result of each step. – Ryder Rude Jul 17 '22 at 17:31
  • Classical electrodynamics works most naturally with charge- and current-densities, it doesn't need particles at all. – Hoody Jul 17 '22 at 19:39
  • Maxwell’s equation (and charge continuity equation) can and do describe the interaction between the electromagnetic fields ($\boldsymbol{A}$) and a scalar charged field $\phi$. You need not think of maxwells equations as only describing the effect of EM fields on charged point particles.. – Jagerber48 Jul 17 '22 at 20:16
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    yuggib's answer to the duplicate I just found and linked describes exactly a "two-step procedure" that corresponds to what you seem to be asking about here; – ACuriousMind Jul 18 '22 at 08:28
  • @ACuriousMind I just read it. It didn't talk about starting with an interacting EM and Dirac field, and taking the limit in two steps. It instead starts with only a Dirac field. How would one take the two limits, starting from the interacting operator equations in the Heisenberg picture? There are two interacting Heisenberg equations. I expect one to become Lorentz force law and the other to become Maxwell's equations. – Ryder Rude Jul 18 '22 at 11:10

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