In this talk the model of a massive uncharged particle interacting with a radiation field, widely referred to as the Nelson model, is treated.
We start with an introduction to the mathematical background necessary to introduce a Hilbert space operator describing the energy of this system. We discuss some basic properties of operators on the so-called Fock space, the Hilbert space quantum field theory is built upon. Especially, the necessity of an ultraviolet cutoff for a well-defined model is discussed. We also look into some properties of Nelsons model, namely, translation invariance and the resulting direct integral decomposition.
The concept of renormalization, i.e., removing the cutoff, is introduced. We refer to results originally going back to Nelson (1964) and compare them to more recent mathematical treatments of the model.
Another important question raised in mathematical physics is the existence of ground states. It is known, that infrared divergence suggests absence of ground states. When assuming the radiation field to be massless, the Nelson model is infrared divergent. We treat the idea of a rigorous proof for non-existence of ground states, in the end of the talk.