I think this paper has a very misleading title, as the result it presents is not as remarkable as to be actually proof of what is known as Maxwell's equations (they were formulated by Heaviside btw.) Basically what the paper does is that it defines operators for ρ, j, E based on the hindsight knowledge of Maxwell's equations and non-relativistic equation of motion for a particle; then it shows that it is possible to have both sets as Heisenberg equations. That's definitely not something a physicist would call deriving Maxwell's equations. Most of the thing claimed to be derived in the title is actually defined/assumed, the result is merely that the Heisenberg formalism, the commutation relations and non-relativistic equation of motion do not seem incompatible. The speed of light in the result is there purely because Dyson knows the result he needs to get.
I disagree. I would call this a proof but not a derivation. In other words, each step follows logically from the previous ones, even if there's no way that you would discover the final equations this way if you didn't already know or guess them.
In general, this is part of what makes reading mathematical papers so difficult; the steps of the proofs are almost never written down in the order they were discovered. All of the false starts and a lot of the intuition is discarded to yield a shorter but sometimes totally mysterious path from the assumptions to the conclusion.
Right. It's not possible to prove Maxwell's equations from first principles, we only "know" they are true from experience. In particular, there's no logical reason divergence of B has to be zero; logically there could be magnetic monopoles, but experience seems to indicate there are not.
(Not a physicist) So what's the result of applying Feynman's axioms to the analysis of whatever gedanken experiment that lead Einstein to special relativity? Does it further enlighten whatever paradox existed that required a resolution via special relativity?
I was lucky enough of having a good math teacher that understood and taught Lorentz and Einstein equations, he used to say "Once that you get it, you will laugh at them", here I am several years later still waiting for some of those laughs.
Aside: I'm not signed up for the site (not interested in tying that login to other accounts), but I wanted to comment that luisb's explanation of Eq. (13) seems to miss the point a little. That equation doesn't need to be derived from Eq. (4). Instead, Eq. (13) is just a statement that any antisymmetric 3x3 matrix can always be recharacterized as a (pseudo)vector "times" the totally antisymmetric Levi-Civita symbol epsilon_{ijk}. As the text says, Eq. (13) is the definition of H. Only after we've defined what H is can we then use Eq. (4) to provide a definition of E.
When I was an undergrad and graduate EE student, the "go to" book to help with electromagnetics was Div, Grad, Curl, and all that (http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161). Back in my era it was the 1st edition of the book, not the 4th.
Div, Grad, Curl helped a bit, but what really made it click for me was an excellent professor some other EE math-class-in-disguise that explained those vector calc operations in terms of divergence (source density) and flux (change in time/space).
As far as understanding the linked paper, I can't follow the proof either. Equations 1-4 I've never seen, 5-8 are Maxwell's Equations which are familiar but we wrote them with different notation, 9-18 are again equations I've never seen. The meat of the proof in 19-21 is built on 14 mystery equations and 4 that I recognize.
As a former EE I guess we didn't prove equations as much as take their existence as given and then figured out what that implied for the real world. ;) Other posters aroberge and wraithm have mentioned also needing QM from physics to follow the proof, which must be where the other equations are from!
Surely you have seen equation 1 (but perhaps with a slightly different notation). Each dot above an x represent a time derivative. So, the left-hand side is the second derivative of the position with respect to time aka the acceleration, times the mass. So, the first equation is simply F = ma (but, yeah... the notation can easily throw one off if one has not see it before.)
There are some interesting comments below the article. The article claims to derive Maxwell equations based on Newtonian mechanics (Galilean transformations), which sounded a little strange. I remember pretty well that one had to consider Lorentz transformations to obtain Maxwell equations. Sure enough, the comments indicate that this is required and the proof does not quite succeed.
OK I have a problem with this step: [x'_1,H_1]+[x'_2,H_2]+[x'_3,H_3] is equivalent to ∂_1H_1 + ∂_2H_2 + ∂_3H_3 = 0. In the whole paper until the eq. 18 Dyson talks about operators being functions of particle coordinate operators x_k and velocities x'_k. There is no mention of space coordinates. Suddenly he claims that some operator is equivalent to divergence of a field H that was never mentioned before. It is not clear how does he calculate what ∂_1H_1 is. I do not see how this makes any sense.
He is not claiming that the commutator is equivalent to the divergence, what he is saying is that [xl',Hl] = 0 is equivalent to div H = 0, which is different. If you click on the annotation on the left you will see that [xl',Hl] = 0 means that Hl is not a function of xl and so ∂_lH_l will be zero and consequently div H = Sum ∂_lH_l is also zero.
I know, I forgot to check the formula before clicking the submit button. I meant it as you explain. The problem is that this commutator equation implies nothing about partial derivatives of H with respect to spatial coordinates that EM fields are function of, because Dyson's x_l are not spatial coordinates (which are always real numbers), but they are non-relativistic Heisenberg operators of coordinates of considered particle (infinite matrices).
The difference is as important as the difference between coordinate of a space point and coordinate of a particle at some definite time. This confusion was probably caused by the unfortunate choice of notation, where particle coordinates were denoted as x_l, notation better spent on the general spatial coordinates. If we use, say, r_l to denote Heisenberg operators for the particle coordinates, it is clear that (not being a function of operators r_l) implies nothing about (not being a function of position x_l).
Let me simplify this argument. What Dyson is saying seems no different from saying that if you have a field v_l(x) giving velocity of water at general point x, this field is obviously not a function of coordinates r_l of a test particle and it follows that sum of derivatives ∂_lv_l is zero. Behold, we arrive at the conclusion that the flow must be incompressible, no need to assume anything from experience. All water must be incompressible! The absurdity of this conclusion is pretty obvious, and the reason it was obtained is clear: v_l is independent of r_l, but it depends on x_l, so nothing about partial derivatives ∂/∂x_l can be easily inferred.
It's great that you're concerned about civility on HN, and believe me I sympathize with the condition of not remembering things one has a degree in. But I'm afraid you misread aroberge—who was trying to reassure the reader that they needn't assume graduate physics—and unfortunately your reply crosses into incivility itself.
The Griffiths texts for E&M and Quantum were some of the best educational books I've ever read, and unfortunately both were stolen from me as an undergrad.
