# In classical physics (classical electrodynamics), why linearity of Maxwell's equations prevent interaction of electromagnetic waves?

In classical physics (classical electrodynamics), electromagnetic waves don't interact. In quantum mechanics, they could. In this article on light-by-light scattering: https://arxiv.org/abs/1702.01625 , the introduction states that there is a connection between linearity of equations and possibility of interaction: "One of the key features of Maxwell’s equations is their linearity in both the sources and the fields, from which follows the superposition principle. This forbids effects such as light-by-light (LbyL) scattering, γγ→γγ, which is a purely quantum-mechanical process."

• Why does the linearity of Maxwell's equations prevents interaction of light by light?

• Why would interaction necessary need non-linearity?

• Is a vertex rule necessary non-linear?

• Is experimental observation of light-by-light scattering one proof that light is not "wave-only"?

• Bonus: if we use two torch lights that cross-each other, is there light scattered in some directions different to the axes of the two flashlights (assuming that we would have amazing experimental apparatus)? And if we use two laser beams instead of two torch lights?

## 4 Answers

Linearity implies the superposition principle. The superposition principle means that if $$\psi_1(x,t)$$ is a solution to the (vacuum) wave equation, and $$\psi_2(x,t)$$ is also another solution to the same equation, then $$\psi=\psi_1+\psi_2$$ will also be a solution. Interaction in the most general (non-QFT) sense would require that a wave configuration $$\psi_1(x)$$ "senses" the presence of another wave configuration $$\psi_2(x)$$, and gets modified by it. But the assumption that $$\psi_1(x)$$ is modified by $$\psi_2(x)$$ means that its time evolution is different compared to the situation that $$\psi_2$$ is not present. Hence, the time evolution of two wave configurations cannot be just a superposition which is ignorant of whether the respective other configuration is present or not.

If linearity precludes interaction in the above sense, then non-linearity is at least a necessary condition for interaction. It is not generally also a sufficient condition because any linear equation system can be made apparently nonlinear by a sufficiently complicated coordinate transform. And yet, the physics behind it cannot be changed by using different coordinates.

Light by light scattering is not a proof that light is not "wave-only". You can also include non-linear material laws into electrodynamics, which are able to describe numerous macroscopic phenomena you can look up in the Wikipedia article on nonlinear optics, which also cause light-by-light scattering (in matter). The point about being not "wave-only" is whether apparent macroscopic laws look more granular if you look closer. Just like the Navier-Stokes equations of hydrodynamics are only valid as long as you don't ever look with a microscope at the Brownian motion of dust particles suspended in the fluid, the Maxwell equations in matter are only valid macroscopically.

• It is not generally also a sufficient condition because any linear equation system can be made apparently nonlinear by a sufficiently complicated coordinate transform Is it valid for Electromagnetic fields? A coordinate (Lorentz) transform may linearly mix the $E$ and the $B$ fields, but that won't make it non linear. Also, the form of the Maxwell's equations won't change under a coordinate transform (e.g. one may have to write the curl in cylindrical coordinate, but that is still a linear theory). Apr 15, 2021 at 6:31
• @ArchismanPanigrahi I think oliver means the generalized coordinates describing the field itself, not spacetime coordinates. That is, defining new fields as nonlinear functions of $\psi$ or $\mathbf{E}$ or $\mathbf{B}$. Apr 15, 2021 at 10:51
• Do you agree that light by light scattering in vaccum conditions is a proof that light is not "wave-only" ? Apr 15, 2021 at 13:59
• @MathieuKrisztian: suppose we knew nothing about quantum field theory, what tells us that vacuum is not just another homogeneous, nonlinear material? So scattering in vacuum can't be a "proof". The real "proof" is the fact that quantum field theory allows to treat these effects more accurately. The scattering iself is just an indication that it could be somewhat different than a simple wave. Apr 15, 2021 at 14:08

What "interaction of electromagnetic waves" means, in essence, is that two electromagnetic waves pass through a region of space at the same time, and their forms after leaving that region are anything other than they would have been if each had passed through when the other wasn't there.

Linearity, on the other hand, means that the form of two electromagnetic waves after passing through the same place is the same as if they'd passed through at different times. So nonlinearity and interaction are essentially the same thing.

"One of the key features of Maxwell’s equations is their linearity in both the sources and the fields, from which follows the superposition principle. This forbids effects such as light-by-light (LbyL) scattering, γγ→γγ, which is a purely quantum-mechanical process."

"Maxwell’s equations are linear in both the sources and the fields" is technically correct but morally a lie, because it ignores the effect of the electromagnetic field on the sources. A complete consistent classical theory of EM with charges is nonlinear.

If you quantize just the linear part of the classical theory, you get a linear quantum theory. Only if you quantize the full classical theory do you get a nonlinear quantum theory, and that linearity was already present classically.

The real difference is that in the nonlinear classical theory, light beams will only interact in a region if charges are present, while in its quantum counterpart, they will interact even in vacuum. This is because charges are effectively present even in vacuum in the form of virtual-particle loops.

The language of interaction is based largely in quantum field theory. In that context, the quadratic part of the action produces the propagator of the field and any cubic or higher parts left over are lumped together and called the interactions. This is because these extra terms form the interaction vertices in Feynman diagrams.

From this point of view, the connection to linearity of the equations of motion is simply the observation that upon varying the action to produce the equations of motion, the quadratic terms will produce contributions to the equations of motion. On the other hand, cubic and higher terms in the action result in quadratic and higher contributions to the equations of motion...non-linearity.

From a classical point of view things are less clear because you would first need to define what you mean by "interact." One thing we can do however, is take the non-linear terms in the equations of motion and group them together so the resulting equations of motion look schematically like (linear)=(non-linear). If we want to lean on "intuition" from, say, Maxwell's equations, we can think about the right hand side as being like a source current (like charge distributions and currents in Maxwell's equations). It just happens to be one that these "sources" depend upon the fields themselves. If you like, you can then view this as the statement that the fields in a non-linear theory act as sources for themselves, and hence they will react to their own presence.

That's a bit of a stretch, but as I said, it really depends upon how you would like to define "interaction" in a purely classical context and how hard you're willing to lean on the ideas that come from quantum field theory.

I should note that the reason quantum field theory has light by light scattering, and this is a purely quantum effect, can be understood in several ways. If you like looking at diagrams, it should be a simple exercise to convince yourself that there are no tree-level diagrams with only external photon legs. To connect two external photon legs, the lowest loop way to do this would be to have an electron loop in the middle. Loop diagrams do not contribute to classical scattering processes, hence light by light scattering in QED is a genuine quantum effect.

Another way to see the same thing is to note that the quantum effective action for any theory is, at tree level, just the classical action. For QED, this means that higher order (cubic and higher) interaction terms start at the 1-loop calculation of the action. So if you were to try and write down the equations of motion for the vacuum expectation of the fields in QED, you would need to consider quantum corrections to the action before encountering a non-linear term in the equations of motion.

Interaction and linearity/non-linearity are NOT to be confused. Interaction is ability to exchange physical quantities (across space if you wish). Linearity is a description of a particular exchange (interaction). Two EM fields interact linearly, in the vacuum of course. They do not "affect" each other. That is not to say the do not interact.

On the other hand see, for example, two Solitons of a KdV. Both Solitons will emerge as they were before the collision, still during collision the do "interact". The shape of the two merged is very different than the individual shapes, i.e. not a simple sum. Hirota showed that it is described by the sum of the two plus a quadratic term.

Field Theory hint: This particular example can generate some degree of confusion, as it did in my view, because in a more general context the EM fields (rather photons) are the "interaction". So it seems as you are asking about the interaction of the interaction. Rather think of the effect on an "exploratory" particle. Said particle will feel both fields as the sum of the individual fields and no other contribution. My suggestion would be to use linearity or non-linearity as particular types of interactions and avoid being dragged into circular thinking such as non-linearity ==> interaction and linearity ==> non-interaction.