I've been reading for pleasure about two-photon interaction experiments, and one thing that confuses me is why, for example, two photons in the visible spectrum cannot interact. Is this indeed the case, and if so, why?
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Photons are electrically neutral, so their interaction is by vacuum polarisation. The involvement of virtual charged particles makes the interaction difficult at energies $\ll m_e c^2$, $m_e$ the electron mass. (Difficult, but not impossible; pair production, of course, would be.) Eq. (1) here, first derived in 1935, shows the cross-section is proportional to the sixth power of the photons' CM energy. (It's also proportional to $\alpha^4$, with $\alpha\approx 1/137$ the fine-structure constant, so obviously $\alpha^4$ is very small.)
J.G.
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