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Photon - photon scattering in QED at low energies can be accurately described by the Euler-Heisenberg effective Lagrangian. This only involves the photons (naturally) because the electron has been integrated out, $\omega \ll m$ where $\omega$ is the photon energy and $m$ is the electron mass.

What type of effective photon - photon interaction does this correspond to? I mean is the effective interaction between photons attractive or repulsive? Since the photon is massless I'm not even sure the interaction between two photons can be interpreted as attractive/repulsive to be honest.

At the same time one usually derives the attractive or repulsive nature of an interaction via the 2-2 scattering and its matrix element (and its sign) so this procedure seems well-defined for photon-photon as well. So it seems a definite sign can be obtained which would tell us if the effective interaction is attractive or repulsive.

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  • $\begingroup$ I agree with your suggestion that, absent any fields, there's no way for 2 photons to have either an attractive or repulsive force. Sadly, I've forgotten everything I learned for my Quallies in this area. $\endgroup$ Oct 11, 2021 at 14:46
  • $\begingroup$ @CarlWitthoft, the electron produces a non-zero interaction between photons. The only question is whether this interaction is attractive, repulsive, or can't be described by either type, but this latter possibility is unlikely, since usually the sign of the scattering amplitude will dictate the repulsive or attractive nature of the interaction. $\endgroup$ Oct 11, 2021 at 14:57
  • $\begingroup$ The way one derives e.g. the Coulomb potential from electron-electron scattering relies (among other things) on two approximations - a non-relativistic assumption of low velocities and an assumption that one of the particles scatters "off the other", i.e. one of them can be stationary. I don't see how one would apply either of these approximations to photons. $\endgroup$
    – ACuriousMind
    Oct 11, 2021 at 16:21
  • $\begingroup$ @ACuriousMind Clearly a non-relativistic assumption can not be taken for photons. That is why this is a non-trivial question :) But I don't see why one would need your second assumption, scattering "off the other". As soon as one looks at the non-relativistic limit one gets the Coulomb potential (or Yukawa potential for scalar fields). I'm guessing it makes sense to say "photons, subject to the Euler-Heisenberg Lagrangians, tend to stick together" or "photons tend to repell each other", in some approximation, like low energy $\omega$. $\endgroup$ Oct 11, 2021 at 19:56
  • $\begingroup$ Closely related: How does photon-photon interaction manifest in QED?. The last sentence in that question is "Will it act as an attractive or a repulsive force between photons, or even something else altogether?" $\endgroup$ Oct 12, 2021 at 2:36

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