# Academic browsing of the night

Reading a paper on "Electromagnetic Interactions of Molecules with Metal Surfaces". It explains Weyl's method for computing fields of a dipole over half-space (apparently much simpler than Sommerfeld's solution).

"Consider an oscillating point dipole...." Hmm, what's a dipole?.. (Browse through Jackson chapter 4.)

Oh right, spherical harmonics... dipole moment (in general, lowest non-vanishing multipole moment) is independent of origin if net charge is zero... You technically have to include a delta-function into coordinate-free expression for dipole fields... Same derivation states the mean value theorem for the fields... Hmmm, I once learned all those theorems (Browse the web for a bit)

-- oh right, mean value theorem for the potentials... and Earnshaw's theorem... Ok, back to the paper...

Paper expresses the dipole as a current source. Hmm, what's a current source?.. (Browse through Jackson chapter 5.)

Oh right, Biot-Savart law (Jackson claims it doesn't have a standalone meaning expressed via differential elements but has to be integrated... and to connect it w/ B fields of a single moving charge is a difficult problem involving relativity) --ok, integrate to get Ampere's law (B=...), take curl, 3 tricks; integrate by parts, get the curl B Maxwell's equation (w/ displacement current).

Ok, now how do we connect this to a radiating dipole?... (Browse forward towards Jackson chapter 9; get distracted by chapter 6:) Hmm, why do we use a Lorenz gauge?..

Oh right, it makes our wave equations for all components of the 4-potential look the same... Hmm, what's a gauge? (Browse Wikipedia.)

"QED is the simplest example of one... Hmm, what's QED?..

Oh right, it starts out with Dirac equation. Hmm, what's Dirac equation?..

Oh right. If you take the expression for energy-momentum invariant (E^2-p^2c^2=m^2c^4) and put in operators, you get Klein-Gordon equation -- which can describe spinless scalar particles, but at the time physicists didn't realize such things existed, and the equation didn't give a satisfactory picture of the electrons (why exactly?...) so the search went on.
Dirac wanted something like a Klein-Gordon, but first order in space and time derivatives (why exactly?...) So what does he do?.. Well, he takes the square root of the operator-form E^2-p^2c^2 -- by prepending some anti-commuting 4x4 matrices to get rid of the cross-terms. It's so simple and clever. Just go to Wikipedia and look at the equation. And then it turns out that he can write his anti-commuting matrices using blocks of Pauli matrices -- which until then served as a phenomenological description of Stern-Gerlach. Then it turns out that you can write a very simple expression for the Dirac equation that involves Pauli matrices operating on spinors -- but here's the problem: when brought into the rest frame, one of those spinors is a negative-energy eigenstate. If such states are allowed, then why don't electrons decay into lower and lower energy states until they hit minus infinity?.. So Dirac postulated that the vacuum is actually a sea of electrons filling all the negative energy levels, and any observed negative energy state is just a hole. BUT -- just like in the electron/hole theory of charge carriers in solid state -- a hole would have to be positively charged!.. Dirac thought it might be a proton; Weyl was one of the first to suggest that maybe there exists a positively-charged electron. Weyl...
CRAP. I was supposed to figure out Weyl's dipole solution...

p.s. -- creation/annihilation formalism of QFT obviates the need for postulating Dirac's sea

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