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I know a little bit about electrodynamics but I don't understand the validity of Gauge Transformations.

In particular I am confused on how the theory can be consistent among different gauges, in addition I don't understand what is "real" versus a byproduct of mathematics.

Are the electric field and magnetic fields "real"? While potential is a mathematical construct?

Why are these field "real" versus taking the scalar and vector potentials to be real?

In addition when one uses the Coulomb gauge versus the Lorenz gauge in deriving radiation - how do we know they are consistent? (I think I read some slides saying the Coulomb gauge actually gave the wrong results).

What's going on?

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What is "real" in electrodynamics is what you measure. The fields can be measured without ambiguity due to their influence on charges and currents, and so these are the things that are real. Maxwell's equations directly reference these fields and contain only physical information.

Potentials, on the other hand, are defined through the relations $\vec{E} = -\nabla\Phi - \partial\vec{A}/\partial t$ and $\vec{B} = \nabla\times\vec{A}$, subject to the constraints imposed by Maxwell's equations. This is entirely a trick that we use to hopefully make life easier.

Now given these relationships, there is an obvious ambiguity in $\vec{A}$, where we can take $\vec{A}' = \vec{A} + \nabla\Lambda$ for any differentiable function $\Lambda$ and still get the same magnetic field. With this, we can also see that if we take $\Phi' = \Phi - \partial\Lambda/\partial t$, then our electric and magnetic fields in terms of the potentials are the same as they were before. These are gauge transformations. Due to the fact that the fields don't change, they are independent of our gauge, and so any physical consequences--as they depend on the fields--must be consistent in all gauges.

In general, knowing that deriving formulas in different gauges is consistent can be done by actually going in and verifying that they give the same fields by hand, or trusting that, because the fields are by construction invariant under gauge transformation, unless an error was made somewhere the answers will be consistent.

EDIT

With respect to the Aharonov-Bohm effect, yes in quantum mechanics things get a little stickier and the potentials have a more important role, however it still remains that anything measurable must be gauge invariant. And the Aharonov-Bohm effect is gauge invariant, since the phase difference between paths can be written in terms of the magnetic flux, which is something physical.

To be as explicit as possible: imposing a gauge condition cannot have any physical effect. These gauge degrees of freedom represent redundancies in describing the system using potentials, and so "fixing the gauge" by imposing some condition doesn't change the system, it just gets rid of these redundancies. If you change a gauge and see a measurable difference, you either did something wrong, or your theory isn't actually gauge invariant.

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  • $\begingroup$ Classically, that is true. But the Aharonov-Bohm effect shows that the potentials are not just mathecmatical tricks to make life easier, but they have a "reality" of their own. $\endgroup$ – NickD Jul 11 '18 at 0:43
  • $\begingroup$ Does imposing the gauge constraint have any physical effect on the system at all? $\endgroup$ – FourierFlux Jul 11 '18 at 1:13
  • $\begingroup$ @NickD The popular interpretation that this effect makes the potentials real is incorrect. The effect only disproves the principle of locality. $\endgroup$ – safesphere Jul 11 '18 at 1:35
  • $\begingroup$ @safesphere fields are local by construction. Giving up "locality" is giving up the concept of field. So I side with the "popular" interpretation. $\endgroup$ – my2cts Jul 11 '18 at 6:09
  • $\begingroup$ neither kinetic nor potential energy (electromagnetic or otherwise) can be measured by any instrument we still consider them real physical quantities. The same goes for ever more abstract quantities such as entropy, force, etc. $\endgroup$ – hyportnex Jul 11 '18 at 11:00
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The theory of electromagnetism, and physics in general, is a model of reality. It is not reality itself so we cannot claim reality for the E and B fields.

Having dealt with the philosophy, one may ask if E and B are enough to account for electromagnetism. My answer to this is no. The Aharonov-Bohm effect cannot be described in terms of E and B. More generally, we are unable to get around the four potential in quantum mechanics. It is impossible to describe electromagnetic or photon spin as a separate property using E and B. It is impossible to implement the Lorenz gauge in quantum mechanics without "breaking the gauge", as the trick is called. The Noether conservation laws that follow from the standard Lagrangian, which is expressed in E and B, cannot themselves be expressed in terms of E and B without an unwarranted modification. The conservation laws of gauge invariant electromagnetism exhibit numerous paradoxes.

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  • $\begingroup$ Well said. However, quantum electrodynamics follows from the U(1) symmetry, which says that the potentials are not real. Thus the interpretation of the AB effect as "real potentials" cannot possibly be correct. $\endgroup$ – safesphere Jul 12 '18 at 14:36
  • $\begingroup$ Symmetry is mathematics. It requires physical assumptions befit you can draw conclusions. All I am saying, sofar, is that classical electrodynamics is better formulated without gauge invariance and that QED needs quite a few tricks to maintain gauge invariance at no advantage. $\endgroup$ – my2cts Jul 12 '18 at 18:24
  • $\begingroup$ If E and B fields are not real, why are imposing restrictions on the potentials instead of the E and B fields? I guess my issue is, taking your view point it's not clear what you tack down as truth to make the gauge transformations consistent. $\endgroup$ – FourierFlux Jul 12 '18 at 19:20
  • $\begingroup$ @FourierFlux Please avoid the words "real" and especially "truth" for the reasons discussed above by hyportnex and myself. Which restrictions on the potentials do you mean? I don't want to guess. $\endgroup$ – my2cts Jul 12 '18 at 20:10
  • $\begingroup$ Imposing gauges is imposing restrictions, if you have no baseline truth than you're going in circles. May as well impose some restriction on E and B fields directly? $\endgroup$ – FourierFlux Jul 12 '18 at 20:20

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