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I have been studying QFT from Timo Weigand’s lecture notes and in the chapter ‘Quantisation of spin-1 fields’, he describes the Feynman rules for QED and after some examples, there is subsection named ‘Non-linearities’, where it has been written that loop diagrams induce an interaction between the photons and as an example, at one loop photons scatter due to a process of the form

Photon Scattering

And then he adds that such effects are absent in the classical theory, where light don’t interact with each other due to the linear structure of classical theory and it is interesting that such QED effects break the linearity of classical optics. Now, this topic isn’t discussed further in the notes and I am wondering if anyone can explain what does the author mean by ‘Linear structure of the classical theory’ and how it prevents the interaction of light?

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Timo Weigand is referring to the fact that Maxwell's equations are linear (inhomogeneous) PDEs, or equivalently, the Maxwell action is quadratic in the photon field. For the photon field to be interacting at the classical level, the action should contain cubic or higher terms, which it doesn't.

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  • $\begingroup$ Further: Einstein's equation is an example of a classical model that doesn't have this property. When viewed as the field equations of a perturbation $h_{\mu \nu}$ on flat spacetime, it becomes a set of non-linear inhomogeneous PDEs. $\endgroup$ May 20 '20 at 20:32
  • $\begingroup$ $\uparrow$ Right. $\endgroup$
    – Qmechanic
    May 20 '20 at 20:34

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