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On page 11 of the 3rd edition of Electricity and Magnetism by Edward M. Purcell and David J. Morin it says:

"we know of quantum phenomena in the electromagnetic field that represents a failure of superposition, seen from the viewpoint of the classical theory."

What quantum phenomena are they talking about?

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The classical principle of superposition of electromagnetic field refers to the proposition that the electromagnetic field cannot be scattered by the electromagnetic field. In other words, if two electromagnetic waves pass through a point (say you point two LASERs in a way so that their beams intersect), they really pass through each other without affecting each other in their further journey, i.e., they don't scatter each other.

This is violated in quantum mechanics due to loop-corrections (for a discussion on how loop-corrections are quantum corrections to classical results, see this post and links therein). In QED, two photons can scatter each other off in a process that, for example, might look like this.

Notice that this is a loop-level phenomenon. There is no term in the Lagrangian of the QED that could be called a self-interaction term for the electromagnetic field (say, unlike in QCD where there are gluon-gluon self-interaction terms built into the Lagrangian itself). You can argue that a photon is not really scattering another photon--what it is doing is it is exchanging some electron/positron with another photon and that is what is responsible for the "apparent" scattering. But, in that sense, even two electrons don't scatter each other (there isn't any electronic self-interaction term in the QED Lagrangian either), they exchange a photon and that is what causes the electron-electron scattering (although, the electron-electron scattering of such kind can be realized at tree-level). So, you have to grant that the photons are really scattering each other off--through whatever mechanism.

This is what is meant when one says that the superposition principle for electromagnetic field no longer holds because the photons don't truly pass through each other at high enough energies--they really collide. For example, I found this 2014 press-release on the CERN website which has some discussion on using the LHC as a photon-photon collider (but I am very very ill-educated on the experimental prospects so someone can find more relevant links).

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