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."
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Why does the linearity of Maxwell's equations prevents interaction of light by light?
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Why would interaction necessary need non-linearity?
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Is a vertex rule necessary non-linear?
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Is experimental observation of light-by-light scattering one proof that light is not "wave-only"?
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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?
Best Answer
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.