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In the seemingly standard treatment of classical electromagnetism (cf. Qmechanic's answer here) we decide to work entirely within the framework of classical field theory. In particular, we define our dynamic variables as the electromagnetic potential $\mathcal{A}$ and associated matter fields $\psi$ and then define a Lagrangian $\mathcal{L}(\mathcal{A}, \psi)$. Concretely, we might study a theory of the electron in which we let $\psi$ be a Grassmann-valued Dirac field. But, this sort of treatment seems to stem from attempting to classicalize quantum field theory.

Is there a way to treat classical electromagnetism in which particles are treated using Lagrangian mechanics and the electromagnetic field is treated using Lagrangian field theory in a unified manner? This might entail, for instance, writing down some particle-field Lagrangian.

(Of course, textbook electromagnetism uses a particle-field formalism, but such formalism seems put together in an ad-hoc manner.)

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  • $\begingroup$ I object to the characterisation that that sort of treatment is "attempting to classicalise QFT" because we are treating the classical version of the QFT governing equations prior to quantising those fields. i.e. those are perfectly valid classical fields in their own right, just that, yes, ultimately we want the quantum versions. $\endgroup$ Commented Jul 29 at 5:18
  • $\begingroup$ Does my answer here answer you question? physics.stackexchange.com/a/795528/31895 $\endgroup$ Commented Jul 29 at 9:10

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Assuming the fields die off asymptotically at infinity, you can write the action for a charged point particle with coordinates $X^\mu(\tau)$ (where $\tau$ is the particle's worldline) as $$ S_q = q \int d^4 x \int d\tau \delta^{(4)}(x^\mu - X^\mu(\tau)) A_\mu(x) \frac{d X^\mu}{d\tau} = q \int d\tau A_\mu(X) \frac{d X^\mu}{d\tau} $$ You can add this to the action for a free point particle $$ S_{pp} = m \int d\tau = m \int d \tau \sqrt{-\eta_{\mu\nu} \frac{d X^\mu }{d \tau} \frac{d X^\nu}{d\tau}} $$ and the Maxwell action for the fields $$ S_{EM} = \int d^4 x \left(-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}\right) $$ to derive an action with an electromagnetic field coupled to a point particle $$ S\left[A_\mu(x), X^\mu(\tau)\right] = S_{EM}[A] + S_{pp}[X] + S_q[A, X] $$


Following some interesting discussion in the comments, I should mention that this action is meant to be understood in an effective field theory sense, valid when the relevant wavelengths/physical scales are larger than the true size of the charged particle. Classical physics tends to become singular and break down near true point particles, requiring some kind of regularization; this will happen if "self force" effects become important. In other words, the above action (like the Lorentz force law for a point particle) implicitly assumes you are going to be looking at the motion of a point particle in some external field, and not worry about the effect that the particle's field has on itself. One avenue toward dealing with finite size effects is to include higher order corrections in the effective theory, for example as described in Section 2.6 of arxiv.org/abs/hep-ph/0701129 in the context of gravitational waves. However, ultimately if you want to probe length scales that are the size of the particle, the point particle approximation is bound to break down. A resolution to this within classical mechanics would be to say the point particle actually has some finite radius. However, when dealing with, say, an electron, the actual resolution is that quantum field theory becomes relevant at some scale. The "self energy" of the electron can then understood in terms of Feynman diagrams with loops, that renormalize the mass of the electron. Anyway, the point is that the above action is to be understood as an approximation that is valid so long as you are willing to consider physics coarse grained over large enough length scales (just as with the Lorentz force law, which the above action reproduces).

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  • $\begingroup$ It is not possible to combine the field term with the interaction term in this straightforward manner, because the terms are undefined for net EM field in presence of a point particle. Either the particle has to have some size (and additional degrees of freedom) or the field multiplying the particle four-velocity has to be different from net EM field. $\endgroup$ Commented Jul 29 at 9:13
  • $\begingroup$ @JánLalinský It's a standard trick so long as you aren't interested in radiation reaction forces, eg physics.stackexchange.com/questions/701748/…. I'm not disagreeing there are subtleties but it's not completely useless. $\endgroup$
    – Andrew
    Commented Jul 29 at 11:27
  • $\begingroup$ You could also view it as the leading term in an EFT expansion that would incorporate effects of finite particle size at higher orders, similar to Section 2.6 of arxiv.org/abs/hep-ph/0701129. $\endgroup$
    – Andrew
    Commented Jul 29 at 11:34
  • $\begingroup$ The action as you wrote it is just a picture, it is not mathematically defined, due to delta multiplying a singularity. If you derive equations of motion for a particle from this action, you'll get a non-predictive equation $dp/dt = qF(x,t)$ with function $F$ singular at the particle. The value of $F$ there can be then additionally postulated to have any value, it is not constrained by the equations of motion. The action does not link the actual force to the field function, and this means we have used the wrong action. $\endgroup$ Commented Jul 29 at 18:10
  • $\begingroup$ @JánLalinský So long as you're willing to accept that your point of view also means that the Lorentz force law combined with Maxwell's equations for $E$ and $B$ is similarly ill defined, I'll accept that what I wrote does not cover all situations (such as when a self force becomes important). On the other hand, at some point you need to replace the idea of a point particle with QED, so maybe it's not so important to find a completely mathematically self consistent theory of a classical charged point particle. $\endgroup$
    – Andrew
    Commented Jul 29 at 18:12

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