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Scattering of light by light does not occur in the solutions of Maxwell's equations (since they are linear and EM waves obey superposition), but it is a prediction of QED (the most significant Feynman diagrams have a closed loop of four electron propagators).

Has scattering of light by light in vacuum ever been observed, for photons of any energy? If not, how close are we to such an experiment?

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This was demonstrated by "Experiment 144" at SLAC in 1997. Here is a list of publications from that project, for instance "Positron Production in Multiphoton Light-by-Light Scattering", whose abstract reads:

A signal of 106±14 positrons above background has been observed in collisions of a low-emittance 46.6 GeV electron beam with terawatt pulses from a Nd:glass laser at 527 nm wavelength in an experiment at the Final Focus Test Beam at SLAC. The positrons are interpreted as arising from a two-step process in which laser photons are backscattered to GeV energies by the electron beam followed by a collision between the high-energy photon and several laser photons to produce an electron-positron pair. These results are the first laboratory evidence for inelastic light-by-light scattering involving only real photons.

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This is what I was looking for. One interesting aspect is that, since the invariant mass of one backscattered photon and one laser photon is insufficient for pair production, this is actually the reaction 5 (or more) gamma → e+ e- –  Keenan Pepper Nov 27 '10 at 23:14
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But this doesn't really answer the original question, since scattering of photons to produce an electron-positron pair makes perfect sense classically. The original question was about the purely quantum process in which two photons scatter into two photons. I'm not sure if it's been observed for "real" photons. –  Matt Reece Nov 28 '10 at 4:52
    
"Positron Production ..." paper in pdf forma is at e.g. wwwphy.princeton.edu/~kirkmcd/examples/accel/… . Some nice theory is at arxiv.org/pdf/hep-ph/9805507 –  sigoldberg1 Nov 28 '10 at 5:01
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I've a vague memory that light-on-light scattering puts an upper limit on the energy of photons that will propagate over cosmic distance scales. So one probe is very high energy gamma astronomy.

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Yes, but that should only put an upper bound on the cross section, right? You can't really use it to show that photon-photon scattering is definitely taking place, can you? –  Keenan Pepper Nov 27 '10 at 22:40
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@Keegan. If you see photons above some threshold from "local" phenomena (say our super cluster), and not from far ones and can rule out other processes... –  dmckee Nov 27 '10 at 22:56
    
Are you thinking of the limit on cosmic ray particle energies due to scattering with the microwave background? This is the "GZK Limit" –  DarenW Nov 29 '10 at 4:47
    
@DarenW no, but the mechanism is related. Above circa 400 TeV photons can pair produce on the CMB, and while the cross-section is small over cosmic distances it becomes significant. –  dmckee Feb 18 '13 at 20:23
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I guess you were asking about elastic photon-photon interaction. If that is the case, I remember reading some proposals (and argumentation of feasibility) of experiments using existing laser facilities in:

E. Lundström et al. Using High-Power Lasers for Detection of Elastic Photon-Photon Scattering. Phys. Rev. Lett. 96 no. 8, 083602 (2006). arXiv:hep-ph/0510076

Further references:

J. Lundin et al. Analysis of four-wave mixing of high-power lasers for the detection of elastic photon-photon scattering. Phys. Rev. A 74 no.4, 043821 (2006). arxiv:hep-ph/0606136.

Stephen D. H. Hsu and Brian M. Murray. Thermal gravity, black holes and cosmological entropy. Phys. Rev. D 73 no.4, 044017. arxiv:hep-th/0512033.

I have no idea if the experiments were really made, but those references are quite cool :)

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Long ago as a grad student (1960s) I saw a proposal to use intense fluxes of gamma rays from simultaneously detonated fission bombs (underground) to look for gamma-gamma scattering. I'm sure that the technical requirements would be daunting, not to mention the political problems!

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