  To discuss interactions it is convenient to introduce a new representation, the interaction picture, which is an alternative to the Schrödinger and the Heisenberg picture. For the interaction picture one splits up the Hamiltonian, which is the generator of time-translations, into two parts \(H = H_0 + H_{\textit{int}}\), where \(H_0\) describes the free system, i.e., without interaction. It is solved exactly, and gets absorbed in a re-definition of the fields. Then \(H_{\textit{int}}\) is the interaction part of the Hamiltonian, or short the ‘interaction Hamiltonian.’ Using the interaction picture is advantageous because the equations of motion as well as, under certain conditions, the commutation relations are the same for interacting fields as for free fields. Therefore, various results that were established for free fields can still be used in the case of interacting fields. The central instrument for the description of scattering is again the S-matrix, which expresses the connection between in and out states by specifying the transition amplitudes. In QED, for instance, a state \(|\textit{in}\rangle\) describes one particular configuration of electrons, positrons and photons, i.e., it describes how many of these particles there are and which momenta, spins and polarizations they have before the interaction. The S-matrix supplies the transition amplitude that this state goes over to a particular \(|\textit{out}\rangle\) state, e.g., that a particular counter responds after the interaction. The transition amplitude is squared to form a probability, and such probabilities are checked in experiments.