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Quasi- and Semi-Classical Quantum Field Theories

The study of the interaction between matter and radiation by means of Quantum Field Theories (QFT) represents a challenge due to the lack of a general, rigorous mathematical frame and the difficulty of the calculations involved. A successful strategy to overcome these obstacles is the derivation of effective models, which retain the important physical features of the microscopic ones, but are simpler to analyze. This is possible, for example, when the field alone is really intense or when both the number of particles and the excitations of the field is high, namely in the quasi-classical and semi-classical regimes, respectively. In the present talk we show how in these regimes, by introducing a semi-classical algebraic formulations for the QFT, the dynamics and the ground state energy of the QFT models converge in suitable topologies to the dynamics and ground state energy of effective Schrödinger models.