In the classical model of light as an EM wave, interference is a trivial consequence of the linearity of the wave equation. Now, if we model light as collections of photons, how is light interference explained? If I recall correctly there are explanations that arise from Feynman's path integral formalism. Is this correct?
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$\begingroup$ You could invoke Feynman's path integral formalism but it's really worth noting that in QED the electromagnetic field still exists and obeys Maxwell's equations, so of course there is interference. $\endgroup$– SturrumCommented Jun 26 at 14:48
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$\begingroup$ By "collection of photons" you really don't mean "collection of incoherent ones"... QED retains the coherence information of that "collection" which you are utilizing classically and swayed people in the 19th century! $\endgroup$– Cosmas ZachosCommented Jun 26 at 15:04
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1$\begingroup$ Not really. Interference only requires linearity. It doesn't require any particular equation of motion. $\endgroup$– FlatterMannCommented Jun 26 at 21:40
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$\begingroup$ @my2cts I think it is... with the right amount of experience. I am not sure the insight that interference is a non-effect caused by the absence of self-interaction is trivial on an educational level. We try to demonstrate it in high school with wave machines or wave pools, but how many students really "get it"? I certainly didn't get it until I was taking an undergrad class in the math department and a mathematician pointed it out. The endless occurrence of this question among laymen indicates to me that we are not doing a good job teaching physics. Or maybe it just can't be taught... $\endgroup$– FlatterMannCommented Jun 27 at 7:30
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$\begingroup$ @my2cts I am simply saying that superposition of two solutions forming a new solution only requires linearity. It doesn't matter whether that's a first or second order equation and what kind of equation it is. It doesn't have to be a wave equation. Like I said, the mathematicians are far ahead of us in the way they teach these things. We are still too 19th century centric in our educational habits IMHO. We need to start much earlier to teach abstract thinking skills in physics. $\endgroup$– FlatterMannCommented Jun 27 at 12:24
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2 Answers
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Now, if we model light as collections of photons, how is light interference explained?
It is explained the same as classically. Classically there is an electromagnetic field that obeys Maxwell’s equations. Quantum mechanically the photon field also approximately obeys Maxwell’s equations (although in the QFT case Maxwell’s equations are derived from the underlying gauge symmetry). So interference remains a consequence of the linearity of the field equations.
Note that I said “approximately” above. At very high intensities the photon field has a small amount of self-interaction. Two photons of sufficient energy have a small chance of producing a virtual electron and positron pair which annihilate to produce a new pair of photons. This is a very small non-linearity. Since interference is, as you mention, a result of the linearity of the equations, this small non-linearity will slightly disturb interference.
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$\begingroup$ ‘ in the QFT case Maxwell’s equations are derived from the underlying gauge symmetry’ Do you have references for this statement? $\endgroup$– my2ctsCommented Jun 26 at 21:58
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1$\begingroup$ I like this one: nicf.net/articles/classical-em $\endgroup$– DaleCommented Jun 27 at 2:11
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$\begingroup$ But classically, interferences (interference patterns) are a consequence of intensity being proportional to the modulus square of the electromagnetic fields. What is the equivalent of that, for the photon field? The probability density? $\endgroup$– agaminonCommented Jun 27 at 15:20
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Feynmans PI basically says if we consider many paths (like in a computer simulation) the sum/phase squared info shows paths that are full wavelength multiples are preferred (more probable). This mathematically coincides with the ancient Huygens/classical approach. One can say that the EM field is resonant.
You can talk about many photons but there is only one EM field in the apparatus, it reacts to all the electrons in the apparatus. The photons are dumb, yes they can superimpose, but the field guides all energy transmission. The excited electron (before photon emission) is strong contributor to the EM field, the field is where the interference/resonance is happening.
