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Small quibbles at electromagnetic field - seems to be some electric and magnetic being swapped.
Edit: try now - I accidentally copied the capitalisation from the discussion topic heading and now it is fixed
Small quibble: The article doesn't exist!!!
Edit: I see you fixed the link =).
Thanks for cross-checkinmg this, David.
But time is short, internet connections on trains are weak, and I am having trouble seeing at one glance which lines of this entry I should be paying attention to. Could you give me more details?
I put in little comments marked by 'DR', so just search for them. It's the section on the modernised version of Dirac's argument on quantisation of electric charge, but the article says this is quantisation of magnetic charge.
Oh, I see. Why don’t you just fix it? Right now some “Anonymous Coward” has locked the entry, so I can’t fix it.
I fixed one already, but then thought: the error is so obvious (iDirac's quantisation condition is often the first physical application of topology one learns) that perhaps it was done on purpose for good reason (say, switching the role of magnetic and electric for reasons of duality or something), so just flagged the rest and raised it here.
But wait, David, the Dirac argument does give the quantization of the magnetic charge.
Wikipedia backs me up (see the second paragraph) in saying that Dirac's study of the existence of a magnetic monopole lead him to the conclusion that the existence of a magnetic charge was only consistent if electric charge was quantised.
Dirac's argument might be applied the other way around, I don't know, but in the interest of accuracy... :)
It is both electric and magnetic, not either or :)
In other words, it is the product of electric and magnetic charge that is discrete.
hmm, of course. In the interest of historical motivation, I'd like to see the article follow Dirac. But we should probably put Eric's point in as well (I'm sorry, I'm at home and can't edit: thanks, spam blocker )-:)
I like the revival of interest in physics in nlab. There is the following thought I already alluded to few times: if Urs looks understands the point of view of Baez-Dolan and Freed-Hopkins-Lurie as identifying the QFT with sort of representation theory of higher groupoids, let me remind you of a curious fact that most of the so-called character formulas like Weyl, Demazure and other character formulas can be obtained alternativey as the fixed point formulas of the localization type (Duistermaar, Atiyah-Bott). In fact the elliptic genus of Witten is also introduced in semiclassical expansion of Feynman integral on the loop space, which reduces to certain Witten localization formula (see the book by Richard Szabo avilable on arxiv on semiclassical approximation to path integrals for background), and we know that Witten genus is one of the main motivations for the topological modular forms business. Now what I alluded few times is that character formulas are giving most valuable information on the representation theory, so why we would not try to get other way around: to try to find the appropriate character formulas for bordism n-categories, and then to define path integral in a way which would be tailored toward getting these formulas as localization fixed point formulas for the integral. I think if this were possible to do, this would be a major step in understanding the mathematical foundations of QFTs.
If I had the time I would add more clarifications to the entry, now, but I can’t. Just notice that the integral of the curvature 2-form of a line bundle on a Lorentzian space over a spacelike 2-sphere measures the magnetic charge enclosed in that 2-sphere. At the same time it is the non-torsion part of the Chern-class of the line bundle. So what Dirac really found was the quantization of the Chern-class, as later understood.
Of course this only works if the support of the magnetic charge is always removed from the manifold, because otherwise we would have no line bundle at all, because the 2-form is not closed where the magnetic current has support. It was then ven later realized by Freed in his remarkable article that the magnetic current is itself a differential refinement of a line gerbe . Then the bundle is generalized to a twisted bundle, twisted by that gerbe, and the magnetic charge quantizaiton is now the fact that the Dixmier-Douady class of the gerbe is “quantized”.
Have to rush off…
Then the bundle is generalized to a twisted bundle, twisted by that gerbe, and the magnetic charge quantizaiton is now the fact that the Dixmier-Douady class of the gerbe is “quantized”.
This is earlier, due Brylinski, I think.
This is earlier, due Brylinski, I think.
I don’t think Brylinski actually included the magentic charge density. In any case, I think the clear physical picture did not appear before Freed’s notes.
I had totally forgotten about this thread here on the entry electromagnetic field
to reply to the original question by David: no, there was no accidental swapping of “electric” and “magentic”. It is both electric and magnetic charge that are quantized by Dirac’s old argument.
I have tried to expand, polish and rework the entry to make this and other things more clear. But again I ran out of time and energy. But at least the charge quantization argument is briefly indicated now.
I have moved the section geometric origin of inhomogeneous media from electromagnetic field to its own entry and edited slightly. But this is still a bit cryptic to someone who does not already know what it’s about.
Hi, am i right that in this section $\lambda_{ij}$ $=\lambda_j|_{U_i\cap U_j}-\lambda_i|_{U_i\cap U_j}$, where $\lambda_i \in C^{\infty}(U_i,\mathbb{R})$, so, that $d\lambda_i=A_i$?
(I think, no, but how is it exactly?)
am i right that in this section $\lambda_{ij}$ $=\lambda_j|_{U_i\cap U_j}-\lambda_i|_{U_i\cap U_j}$, where $\lambda_i \in C^{\infty}(U_i,\mathbb{R})$, so, that $d\lambda_i=A_i$?
Not in general. In general there is no $\lambda_i$ such that $d \lambda_i = A_i$, because in general $d A_i = F\vert_{U_i} \neq 0$. But if it so happens that the curvature vanishes, then there are such $\lambda_i$ and you could choose $\lambda_{i j}$ as you indicate. But locally there are still other choices possible, since with $d \lambda_i = A_i$ also $d (\lambda_i + c_i) = A_i$, for $c_i$ a constant function. It is these local choices that globally make a non-trivial Cech cocycle, even if the curvature vanishes globally.
Yes, my question clearly has sense only in the $F=0$ case, I forgot to take this into account. And really, if I take $\lambda_{ij}=\lambda_j|_{U_i\cap U_j}-\lambda_i|_{U_i\cap U_j}+c$ instead of $\lambda_{ij}=\lambda_j|_{U_i\cap U_j}-\lambda_i|_{U_i\cap U_j}$ then $d\lambda_{ij}+d\lambda_{jk}=d\lambda_{ik}$ remains valid. Thanks for the clarification, Urs.
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