# The back way for deriving Maxwell's Equations: from charge conservation?

• timeant
In summary, the conversation discusses the derivation of Maxwell's equations from gauge symmetry and charge conservation. There is disagreement about the validity of certain publications and the role of gauge symmetry in deriving conservation laws. The thread is eventually closed due to high moderation and potential bias from one of the participants.

#### timeant

I found one article in 1993 talking about it.

[Unacceptable reference deleted by the Mentors]

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Delta2
This is my favourite one:
https://iopscience.iop.org/article/10.1088/0143-0807/36/6/065036/pdf

There is a bit more that can be said about the derivation regarding where that e comes from but will leave it there at the moment. It becomes clearer when you see the derivation from Guage symmetry:
https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

But it is well known the fundamental basis of EM is gauge symmetry. Noether's Theorem basically says symmetry leads to a conservation law and conversely. The conservation law from gauge symmetry is charge conservation, so it is hardly surprising that it also leads to Maxwell's equations. I have seen several 'derivations', and they all really boil down to gauge symmetry or charge conservation - plus relativity, of course.

Thanks
Bill

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Delta2 and Kolmo
From a local gauge symmetry you don't get a uniquely defined conserved quantity, because gauge symmetry is rather a redundance in the description of a physical situation, i.e., the vector potential is not uniquely defined from the dynamical equations but only up to a gauge transformation, but vector potentials that differ only by a gauge transformation describe the same physical situation, i.e., the indeterminacy of the potentials is irrelevant for the description of the physical situation.

The electric charge (or the electric-charge four-current) conservation follows from the corresponding global symmetry and is a necessary condition for the consistency of the gauge theory. For details see

https://www.osti.gov/servlets/purl/6129984/ (preprint)
https://doi.org/10.1119/1.16219 (paper)

Delta2, Kolmo and bhobba
[UPDATE -- Quote Box with link to unacceptable reference deleted]

Are those spam links at the end of the PDF paper that you linked to? What are those? They look misplaced in a scientific publication...

[UPDATE -- Image of spam in the unacceptable reference deleted]

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Delta2
berkeman said:
Are those spam links at the end of the PDF paper that you linked to? What are those? They look misplaced in a scientific publication...

https://www.physicsforums.com/attachments/283358

Yes, it is an issue that needs to be sorted out. We do not promote 'spam' here.

Thanks
Bill

EM gauge symmetry leads to conservervation laws having no physical meaning.
Electric charge conservation law can be derived from Dirac field's gauge symmetry by Noether theorem, not by EM field.

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Nowdays one must be very careful, whether you really are looking at a scientific publication or not. That's a great example. Having spam in a "scientific paper" let's look it at least much more suspicious than other publications. The claim in the abstract that that's something new is for sure wrong. For sure already Maxwell was aware of the continuity equation for electric charge and current following from his laws.

bhobba, Dale and Delta2
timeant said:
EM gauge symmetry does not lead to conservervation laws having no physical meaning.
Charge conservation law can be derived from Dirac field's gauge symmetry by Noether theorem, not by EM field.
It can be derived from the global symmetry not from the local one. That's a subtle point and one should think it through carefully. See the AJP paper quoted in #4.

bhobba
vanhees71 said:
It can be derived from the global symmetry not from the local one. That's a subtle point and one should think it through carefully. See the AJP paper quoted in #4.
Free EM fields are built up by ##A_{\mu}##. Electric charge density and current, which is irrelevant ##A_\mu##, are built by Dirac's ##\psi##.

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bhobba and weirdoguy
timeant said:

Update -- After a Mentor discussion the unacceptable reference in the OP has been deleted and the thread is reopened.

A few points:

(1) If it's a scientific paper, it doesn't have spam in it. Full stop.

(2) I don't believe you are here to ask questions. I believe you are here to push your own point of view. That's based on what you have written in this and other threads.

(3) That makes (1) even stronger. BTW, did you write it?

(4) To answer your original question, "The back way for deriving Maxwell's Equations: from charge conservation?", you can't. There are other theories with charge conservation that have different "Maxwell Equations" - Proca Electrodynamics and Calssical Yang-Miles to name but two.

bhobba, vanhees71 and berkeman
The moderation rate in this thread is significantly higher than the posting rate. A sure sign to close it.