### M Theory Lesson 10

Let's take a brief look at the history of solitons. In August 1834 Scott Russell observed a strange wave on a narrow channel of water near Edinburgh. After this he built wave tanks in his garden in order to study these solitary waves. Although he himself thought they were of fundamental importance, enthusiasm for solitons took quite a while to catch on.

In 1895 the KdV equation appeared. It describes non-linear shallow water waves. The modern Inverse Scattering method for solving non-linear PDEs was not developed until the 1960s, in the work of Kruskal et al. But earlier it was observed that the KdV equation is solved using the Weierstrass P function. That is, $u(x,t) = -P (x + ct) + \frac{c}{3}$.

Another application of the ISM is to the sine-Gordon equation. By the early 1980s physicists were studying quantum lattice analogues to such equations, and Kulish and Reshitikhin published an article (J. Sov. Math. 23 (1981) 2435) with some funny looking commutation relations. This was the birth of Quantum Groups. In no time at all the mathematicians were attacking these new algebras. In 1986 Drinfeld presented his seminal work at the ICM.

By the early 1990s there was already Kassell's fat yellow book, complete with an introduction to tensor categories! I remember having it on my desk at ANU in early '94. Apparently one day, when I wasn't around, Kulish walked past the desk and paused as he spotted Kassell's book. I'm told he was absolutely flabbergasted to discover that the QISM had become a huge industry.

In 1895 the KdV equation appeared. It describes non-linear shallow water waves. The modern Inverse Scattering method for solving non-linear PDEs was not developed until the 1960s, in the work of Kruskal et al. But earlier it was observed that the KdV equation is solved using the Weierstrass P function. That is, $u(x,t) = -P (x + ct) + \frac{c}{3}$.

Another application of the ISM is to the sine-Gordon equation. By the early 1980s physicists were studying quantum lattice analogues to such equations, and Kulish and Reshitikhin published an article (J. Sov. Math. 23 (1981) 2435) with some funny looking commutation relations. This was the birth of Quantum Groups. In no time at all the mathematicians were attacking these new algebras. In 1986 Drinfeld presented his seminal work at the ICM.

By the early 1990s there was already Kassell's fat yellow book, complete with an introduction to tensor categories! I remember having it on my desk at ANU in early '94. Apparently one day, when I wasn't around, Kulish walked past the desk and paused as he spotted Kassell's book. I'm told he was absolutely flabbergasted to discover that the QISM had become a huge industry.

## 2 Comments:

Half the problem with solitons is that they require non linear equations. The drunk who loses his wallet at closing time, searches for it underneath the street lamp, not necessarily where it was lost.

That's a general problem. People, from Thomas Young (1803) to Richard P. Feynman (1963), used the example of water waves as a transverse wave analogy to the transverse waves of particle physics.

But water waves are non linear - ie, their velocity depends on their amplitude.

So they totally different to light waves, whose velocity is independent of velocity, although there is a similarity with massive particles.

de Broglie states that

wavelength, lambda = h/p

which applies to all transverse waves. With slow massive particles, p = mv, while with photons p = E/c = E/(lambda * f).

Thus for slow massive particles,

lambda = h/(mv)

and for photons

lambda = h/[E/(lambda * f)]

= lambda * f * h/E

which tells us E = hf.

So for fermions, the velocity varies to accommodate changes in wavelength (or vice-versa), while for massless bosons the frequency is proportional to the energy.

There is a lot missing from the description of the photon. Maxwell's equations have the problem that the only way you can generate a curling magnetic field is to have an electric field which varies in time, which necessitates what Maxwell thought of as vacuum "displacement current".

Problem is, there is no displacement in the vacuum below the IR cutoff at about 1 fm from a charge. So according to this interpretation of QFT, there should not be any Maxwell radiation beyond the IR cutoff, when of course there is.

The problem is quite deep and is not mathematical as such. Maxwell's equations are a good model but the physical mechanism is more subtle.

Radiation does not require moving real charge: an electric field can do it, see http://electrogravity.blogspot.com/2006/04/maxwells-displacement-and-einsteins.html

The time-varying electric field in the photon acts just like accelerating charge, from the point of view of electric field acceleration causing the radiation which has the "displacement current" effects normally attributed to the displacement of charges.

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