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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