Now I am wondering which ingredients of Quantum Field Theory are actually essential for these computations:
Do we necessarily need to consider a QUANTUM theory, i.e. do we always have to quantize the fermion fields (I know that it's possible to compute e.g. Mott scattering with a classical (not quantized) Coulomb potential)?
Do we necessarily need to consider a FIELD theory or could we also go back from QFT to QM and consider waves instead of fields, using the (nonrelativistic) Schroedinger equation?
All in all, could you please tell me which are the reasons why we have to use which specific framework for all these kinds of different scattering processes (Moller, Mott, Bhabha, Bremsstrahlung, Rutherford, Synchrotron, Compton, etc.)? When is it, for example, important to treat the electron as a relativistic particle in a relativistic framework?