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Recently it came to my mind, that I have some basic knowledge about QFT and know im principle how to calculate scattering amplitudes (at least for the $\phi^4$-theory), but have no idea how to understand most of the phenomena in nature.

Consider for example the reflection of light on a mirror. It is clear, that I have to use QED to calculate scattering amplitudes but how can this be done? There are millions of atoms that scatter light, so I will probably have to use a statistical approach?

It's not urgent but I would nevertheless be interested. At least I would like to know, if it is in principle possible, to obtain well known results like reflection or refraction directly and rigorously from (interaction) QFT. I know the arguments from Feynmans QED book and am not looking for plausibility arguments.

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You should ask Feynman, really! Feynman answers the questions (which indeed involves your particular problem in understanding things), after completing his lecture on the QED in Aukland. See: http://www.youtube.com/watch?v=eLQ2atfqk2c&list=PLi8UukytWC07rxW6Q9tLCMp1ofxcSR‌​JAb&index=2 at around 1:08:00. –  Godparticle Jun 9 at 13:47
    
I'm looking for mathematical methods to really calculate scattering amplitudes (so S-Matrix elements). It should be mathematically rigorous as much as possible. I'm not afraid of long and nasty calculations, but I don't see how it could be done. –  Wessi990 Jun 9 at 14:21
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More Phys.SE questions about optics and QED: physics.stackexchange.com/search?q=is%3Aq+optics+qed –  Qmechanic Jun 9 at 14:28

1 Answer 1

Lubos Motl , a member of this site, has a blog entry on how the classical fields emerge from QFT.

From the introduction :

I will discuss two somewhat different situations which however cover almost every example of a classical logic emerging from the quantum starting point:

  1. Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)
  2. Decoherence which makes us interpret absorbed particles as point-like objects and which makes generic superpositions of macroscopic objects unfit for well-defined questions about classical facts

.......

However, in the rest of this section, I want to focus on another way how to see classical physics of fields emerge out of large ensembles of photons, one that mimics the thermodynamic limit of statistical physics (even in the context of classical mechanics).

.....

for me the crux of the argument lies in the observation

The photons also have polarizations so the wave function has many components, too. I don't want to scare you by the indices but the wave function of a single photon mathematically looks like the (complexified) classical electromagnetic potential A⃗ (x,y,z), with some extra subtleties. (But its interpretation is different!)

which connects the classical electromagnetic field with the individual photons.

One does not need to carry the cumbersome ensemble of photons in discussing macroscopic observations as, for example, reflections, scattering of light and optical ray drawings. It is enough that it is possible to do so, but it is much more convenient to use the classical version, as we use thermodnamics, and not statistical mechanics, when discussing the behavior of matter in bulk, though there exists a one to one correlation of the microscopic mechanics to the macroscopic emergent variables and distributions.

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Thanks, I will read it but it seems not be, what I'm looking for. I know how to describe the EM-field in QFT and (at least) in principle the interaction with electrons. I'm looking for methods to calculate scattering processes on surfaces. To me it's not really clear, how I could use methods from statistical physics in this case. Probably I will have to learn QFT of many-body systems first. –  Wessi990 Jun 9 at 13:47
    
The point I am making is "you do not need to do that. The classical calculations are adequate. If you want to calculate the temperature of a mass you do not have to think of the individual molecules energies. The classical thermodynamic tools are what you need. Using QFT for images would be like using a laser beam to dig a train tunnel –  anna v Jun 9 at 17:01
    
I know that it's not necessary and you don't learn anything new, nevertheless, I'm interested, if it is possible. –  Wessi990 Jun 9 at 19:29

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