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As the title indicates, I'm not sure if Maxwell's equations or Gauss' law indicate that charged bodies are the source of the EM fields. This is question comes from the reading of this paper.

As Mario Bunge (one of the physicists quoted in the paper) stated in his Foundation of Physics (1967):

the source hypothesis is an extra assumption: as our axiom systems show, fields and currents are conjoined but not causally associated: Only field changes are causally associated with charged bodies in case there are any in the region considered. [...] Both in CEM and in GR the “source”-free field equations have infinitely many nontrivial solutions, in particular waves. In short, the source hypothesis is an extra assumption that may but need not be made – not any more than its converse, the sink hypothesis. Consequently there is no reason to discard the “advanced” F’s, particularly as they need not be interpreted as advanced fields. [...] For example, j(x’, t+) might be interpreted as the current at x’, t+ associated with the field F+; i.e., F+ might be regarded as the part of the field that reacts on the current and influences its value at a later instant. Consequently the total solution (4.18) could be interpreted as describing both the action of the currents on the field and the latter’s reaction on the former. Whether or not this interpretation holds water is here immaterial: the point is that some of the difficulties of CEM are of a semantic nature and some of them are caused by tacit and controvertible assumptions such as that the charges produce fields but not conversely – a hypothesis that is clearly absent from the axiom basis of CEM.

Or, as Gustavo Romero points out in this paper, in reference to Bunge's view:

Bunge’s analysis of the electromagnetic field is remarkably lucid. Among other Issues, he points out that theory does not properly contain the hypothesis that charges are the sources of the field. Strictly speaking, electromagnetic theory is a theory of the interactions between fields and charged particles. The hypothesis that charges are sources of the field must be added to the Maxwell equations in order to discard the advanced contributions (determined by future events) to the solutions of the inhomogeneous Dalambertian equation. This is done by means of the application of the principle of causality. This hypothesis is logical, ontological and epistemologically independent from the rest of the theory. Something similar happens with the general theory of relativity: the hypothesis that matter is the cause of the curvature of spacetime (gravitational field) is a posteriori.

In another book, Matter and Mind, Bunge wrote:

The electromagnetic field that remains in a region of space after all the electric charges have been neutralized, and all the electric currents have been switched off, is a concrete though tenuous thing.

What are your thoughts on this? Is the "source hypothesis" an a posteriori addition to interpret the equations? Do the EM field exist even when there are no charged bodies to "create it"? Am I or the authors (or maybe, I'm misinterpreting the authors) not seeing something?

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  • $\begingroup$ They do, there's nothing ambiguous about it. $\endgroup$
    – AfterShave
    Mar 19, 2023 at 7:55
  • $\begingroup$ That‘s why one ME on electric fields says in vector notation „there are charges“, while the corresponding one for magnetic field says „there are no magnetic monopoles“. Experiments formulated as math, and vice versa. $\endgroup$
    – MS-SPO
    Mar 19, 2023 at 9:00
  • $\begingroup$ $\partial_\mu F^{\mu\nu}=J^\nu$ is analogous to $\frac{d}{dt}p=F$. Charges no more cause the electromagnetic field than forces cause momentum; what's caused are changes thereof. $\endgroup$
    – J.G.
    Mar 19, 2023 at 10:04
  • $\begingroup$ @AfterShave I added a little bit of info. I hope it could bring in something relevant. $\endgroup$ Mar 19, 2023 at 18:58

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this is a really interesting question about the nature of electromagnetism and its interpretation, from my experience the two ideas are similar in the fact that they yield the same result. If we take a look at the definition of electric field given by: $$\vec{E}(r) = \frac{1}{4\pi \epsilon_0}\int_v \frac{\rho(r')}{(r' - r)^2}dv' \hat{r}' $$ This is interesting because this suggests that there is a charged body that has a charge distribution (may it be constant or not) and this is what creates the electric field, in other words, here the electric charge precedes the electric field. This is interesting enough but if we take a look at Gauss' law in differential form we have: $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} $$ Where we have the volume density of the outward flux of the electric field from an infinitesimal volume around a given point and this suggests that it is the existence of that flow that creates the notion of a charge distribution. In this interpretation Electric fields precedes electric charge. We know both statements to be true so the question "what interpretation is correct?" naturally arises and my answer is I don't know but it is interesting to see that this is not necessarily a bad thing for as we know from quantum mechanics we have both the first and second quantization formalisms which are very interesting because in the first we study a particle and its constraints to end up with a wave function that represents its properties where as in the second quantization the particle (or particles) leaves the stage and the fields enter it. In other words in the first quantization the particles create the fields whereas in the second quantization the fields create the particles.

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