# On the (non)linearity of electromagnetism

As a student you are typically told that Maxwell's equations (ME) in vacuum are linear. However, it seems that for extremely high electromagnetic fields the equations for electromagnetism turn out to be nonlinear. I think examples are super-strong laser pulses (even if the precise mechanism is unclear to me). How is this seen in Maxwell's equations? Is it a matter of treating the vacuum as a "material" with some e.g. polarisation, such that ME turn nonlinear? Or does one need QED?

• Nonlinear like this or like this?
– J.G.
Mar 26 at 21:21
• @J.G. I guess the second, because I'm just considering electromagnetism in vacuum here.
– m137
Mar 26 at 22:14

Maxwell equations are linear... but they are also incomplete - they involve sources (i.e., currents and charges) that cannot be determined from these equations. To really solve Maxwell equations these equations need to be complemented by material equations, which relate the currents and charges to the EM fields, possibly in non-linear fashion - this is where the non-linearities come from.

Physical vacuum is a kind of medium, which can produce such a non-linear response under some conditions. In this sense it is different from vacuum understood as "nothing".

Semiconductor analogies to Schwinger limit
Schwinger limit - the situation where vacuum starts behaving as a non-linear medium - can be seen in two ways. And idealized view is that similar to Zener effect in semiconductors/insulators: on a band diagram one can view application of a constant electric field as tilting the energy bands, so that some of the electron energy states in conduction band have the energies corresponding to states of the valence band and valence electrons can tunnel to the conduction band. (image source). The reason why this requires very strong electric field is that the region over which such tunneling happens should be shorter than the electron mean free path - i.e., the distance on which an electron is scattered by other electrons, phonons, impurities, etc. Otherwise the effect is washed out.

The same reasoning is applicable to vacuum, where tunneling of a valence electron to the conduction band is replace by creating of an electron-positron pair.

Another way to see Schwinger limit (and here we see why it is non-linear) is by comparing it to non-linear optical processes. This is less idealized view, since we may potentially observe it using strong laser fields. In quantum languages non-linear effects describe interaction between photons, i.e., processes involving more than one photons, such as two-photon absorption/emission, third-harmonic generation, or Raman scattering (image source.) All these processes require presence of a medium, such as atoms/molecules or a bulk material. E.g., one could modify optical properties of a semiconductor in respect to a certain light frequency by illuminating it with a strong laser pulse, creating electron-hole pairs. In an extreme limit, presence of multiple electron-hole pairs makes an insulating material in a conducting one, that is a transparent material into a reflecting one, etc.

The reason why we need irradiation with a strong field is that the generated population of electrons and holes has to be sufficiently long-living in comparison to the recombination time.

Again, the same logic applies to physical vacuum, where conduction electrons and holes are replace by real electrons and positrons.

References:

Very strong fields lead to vacuum-polarization, i.e. they "create" matter (at the very least electron-positron pairs). Even well before the creation of "real" particles there will be perturbations due to "virtual" particles (which is just another way of saying that the non-linear terms are kicking in). None of this is included in Maxwell's theory. So, yes, you would need to go to, at least, QED. In general you would have to start calculations using the standard model which also includes other fields beyond electrons and, at some point, you would need a theory of quantum gravity to make the correct predictions because the energy density becomes so high that it starts bending space-time at the microscopic level.

• That sounds reasonable. When you say that "very strong fields lead to vacuum-polarization", can't these be taken into account (at least to first order) by writing down Maxwell's equations with an "effective medium"? This is what I was thinking...
– m137
Mar 26 at 21:09
• @m137 Maxwell's equations don't know anything about the structure of matter. The charge density in there doesn't tell us if we are talking about electrons, positrons, muons or uranium ions. What vacuum polarization creates, however, is structured matter. Initially it will only create electrons and positrons but it wouldn't stop there as we ramp up the field. Would it be possible to create a mean field approximation for the vacuum plasma? Possibly, but since the resulting annihilation processes have discrete energies, there will most likely be non-equilibrium thermodynamics issues. Mar 26 at 23:41
• @m137 yes, I think it can be done, and if I remember correctly the first correction is written in Jackson (might be misremembering though...) Mar 28 at 15:08