How Does the Feynman Propagator Change When Multiple Photons Are Involved?

[Pants]

In QED, Feynman talks about P(A to B), the probability that a photon starting at a particular point in space will travel to a different point in space (I've since learned that P(A to B) is actually the Feynman propagator, unless I was midled).

What I'm wondering is - what happens if there are multiple photons? Taking the simplest case with two photons interacting, do things like photons turning into particle-antiparticle pairs need to be taken into account?

Thanks!
-Vince

 

[xepma]

Oh yes, you've certainly got that right. Interactions wil modify the propagator even in the case where you only have one photon. Remember that Feynman states that the propagator is basically a sum over all possible paths through which a photon can travel from A to B, with each path weighted by some weight factor (exp[i S] ).

For the interacting case there are many more ways in which a photon can travel from A to B. Now you also have to take into account -- as you correctly noted -- that a photon can temporarily turn into an particle/anti-particle pair, which afterwards annihilate again. But the photon can also do that twice, or three times, or in even more complex ways. All these different 'paths' that lead from A to B will contribute to the propagator. You will, however, quickly find that the more complex the intermediate process is, the smaller its contribution to the final propagator is.

For the case of two photons the same thing applies: photons can turn into particle/anti-particle pairs, and these pairs can then annihilate with each other (which is essentially how two photons can interact). You will find that the contribution of this intermediate process compared to the overal process is not that big at all -- which is not too suprising, because have you ever seen lightbeams collide?

Now, do keep in mind that you have to sum over all possible ways in which the photons can travel from A to B, so do not take these intermediate processes of particle creation too literal --- they are, in the end, nothing but a nice visualization way of interpreting the math. These intermediate states cannot be detected!

 

[Pants]

Thanks! I was under the impression that the Feynman propagator took this stuff into account for the case of a single photon, but it sounds like that has to be factored in on top of the probability of the simple path from place to place.

I remember a footnote in QED about taking multiple hops to get from one place to another, but I think that was about electrons rather than photons. I guess I'll have to check my copy when I get home.

 

[xepma]

Well, the Feynman propagator is actually a mathematical definition which applies to any quantum field theory. The precise mathematical expression depends on the theory (and type of particle) you are considering. The Feynman propagator for a photon in a theory with just photons and no interactions with electrons will be different from the Feynman propagator of a photon in QED. But you can also consider the Feynman propagator of an electron, a positron or a neutrino. They all have their own Feynman propagator.

By definition, the Feynman propagator is something like: given that a particle is located at x at time t, what is the amplitude that the particle will be at another position x' at a later time t' ? You can define these propagators for any type of particle, naturally, and they are always referred to as Feynman propagators.

So yes, the Feynman propagators takes all the interactions and intermediate states into account -- even though this is practically impossible to compute for interacting theories!