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I only know the application to high energy physics: you have a Lagrangian with an interaction term between some fields, and you can calculate the amplitude of some process. Once you have this amplitude, you can usually calculate two observables: cross sections (scattering processes) and decay widths (decay processes).

How does this work in condensed matter physics? You have a Hamiltonian, you can calculate amplitudes using the perturbation theory ... and then? What kind of observables can you calculate and how do they relate to the amplitude?