from what I know: cross section in particles physics can be seen as throwing random darts on ballons on a wall, ie the wall is stationary. but if I want to work out the cross section of two laser beams at certain angles, the analogy above wouldn't hold would it? Since photons can never to stationary. So in the case of photon-photon scattering, what sort of approach would be required to find the cross section?
I am not sure if I fully understand your question but it seems that you simply want to go from potential scattering to particle-particle scattering (including quantum field theory). Is that correct? In both cases the fundamental entity is the so-called S-matrix S_{fi} which relates initial and final states of fixed momenta p_{i} and p_{f}. The S-matrix is related to the Hamiltonian H governing the time evolution of the system via the time evolution operator U. A well-known way to calculate S in perturbation theory is the so-called Dyson series which again can be used in both cases potential scattering and particle-particle scattering. In quantum field theory the Feynman diagrams are nothing else but graphical representations of Dyson series. http://en.wikipedia.org/wiki/S_matrix http://en.wikipedia.org/wiki/Dyson_series (in terms of an interaction potential V) http://en.wikipedia.org/wiki/Feynman_diagram (in terms of an interaction term in the Lagrangian) Btw.: Your example of light-light or photon-photon scattering is highly nontrivial. The reason is that in the QED Larangian (as in classical Maxwell theory) there is no interaction term for photons with photons, only for photons with matter currents (there is no term which involves a term with more than two photon fields A in the Lagrangian). That's why photon-photon scattering occures only in higher order Feynman diagrams mediated via photon-electron interaction and is highly suppressed (afaik not measurable experimentally) http://www.colinfahey.com/eclectic_images_2002/gg-scat.jpg
to be honest I'm not really familiar with these terms. What I was asking was relevant but not as profound as the information you have given (which I greatly appreciated :) ). it was just some misconception I had which I managed to clear a little earlier today. Thanks again for info, looks like I got a bit more studying to do (probably out of the course syllabus, but these do look like things that are nice to know)