In this talk, we consider physical models for the interaction of quantum mechanical matter particles with a quantized bosonic field. More precisely, we discuss the Pauli-Fierz and polaron models in 3D, where the matter particles are treated nonrelativistically and the influence of possibly very singular exterior (classical) magnetic fields is accounted for as well. Furthermore, we address the semirelativistic Nelson model in 2D where the matter particles have a relativistic dispersion relation. While the quantized field is minimally coupled and ultraviolet-regularized in the Pauli-Fierz model, it is linearly coupled and ultraviolet singular in the polaron and Nelson models. For all three models, we present Feynman-Kac formulas for the semigroups generated by the (semibounded) Hamiltonians, and we mention some of their applications in spectral theory.

Our results on the polaron and semirelativistic Nelson models are based on joint work with Benjamin Hinrichs (University of Paderborn).