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Where can one find some concrete physical problems (with solutions) that illustrates the uselfullness and power of QFT? These must not be solvable by QM or SR alone.

It would be good if the problem utilizes the most central QFT concepts. I am trying to learn QFT but it looks like only some mathematical definitions.

Concrete QFT calculations which shows agreement with physical experiment to high accuracy.

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For an approach to scattering in a non-equilibrium environment, have a look at Quantum Kinetics in Transport and Optics of Semiconductors by Hartmut J. W. Haug & Antti-Pekka Jauho. The idea is mostly that you cannot really use standard QM if the number of particles in your system is not constant. – Claudius Dec 11 '12 at 9:52
Lamb shift calculation might be a good starting point. – twistor59 Dec 11 '12 at 10:24
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Right, twistor, that's a precision calculation in QFT, much like the most precise prediction in science, the anomalous moment of the electron. But even some tree-level processes such as positron-electron annihilation do totally need QFT. – Luboš Motl Dec 11 '12 at 10:41
This is exactly what the first chapter of Peskin and Schroder attempts to do, by outlining the calculation of the cross-section for $e^{+} + e^{-} \rightarrow \mu^{+} + \mu^{-}$ – Jerry Schirmer Dec 12 '12 at 17:24

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Textbooks frequently argue that QFT is necessary because it's the only way of combining the quantum mechanics of particles and special relativity. This is true, but there's a much simpler argument for QFT's necessity: Quantum field theory is necessary if you want a correct and logically coherent description of the physics of fields.

In particular: If an electron is in a superposition of position states, then its electric field must also be in a superposition. So we need operators to represent value of the electric field at various points. There are technical subtleties here, but the core point is very simple: As soon as you need to discuss the possibility that a field is in a superposition of states, you are doing quantum field theory. You can obscure this by choosing bad language, but the mathematics doesn't care.

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The classic textbook "Quantum Field Theory" by Mandl and Shaw (Revised Edition; Wiley and Sons) contains many applications. E.g. in chapter 8 you can find beautiful figures comparing theoretical predictions of differential cross sections to experimental data for processes such as Bhabba scattering. In chapter 9 you can find further applications, including comparison of the predicted muon magnetic moment with the experimental values of high precision measurements and Lamb shift for Hydrogen. In posterior chapters you can find applications of the quantum field theory of weak interactions to tauons or neutrinos and further comparison with experimental values.

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