Existence of ground states for infrared-critical models of quantum field theory

When considering models of nonrelativistic quantum mechanical particles interacting with a field of massless relativistic bosons, one encounters an infrared problem. Heuristically, this is due to the fact that small energy fluctuations can create an infinite number of low-energy bosons, which causes infrared-divergences. From a mathematical perspective, this leads to the expectation that such systems cannot have a ground state, i.e., a stationary state at the lowest possible energy. In this thesis, we study two types of infrared-critical models. In case of the translation-invariant Nelson model, which describes the interaction of a nonrelativistic spinless quantum mechanical particle linearly coupled to the boson field, we extend previous proofs for the absence of ground states at fixed total momentum to the ultraviolet-renormalized model. Along the way, we study the dependence of the renormalized operators on the total momentum. Especially, we prove that in the physical case the domains of the renormalized operators with different total momentum have only the trivial vector space as intersection. The second model we study is the spin boson model. It describes a two-state quantum mechanical system, called spin, again linearly coupled to a boson field. It was previously proven by perturbative methods that the model has a ground state even in the physical infrared-critical case. This is attributed to a cancellation of divergences due to an underlying symmetry of the model. We provide a new non-perturbative proof for this fact, which allows us to extend the result to more singular models. Further, we can give an explicit coupling constant, below which the ground state exists. It is conjectured that there exists a critical coupling constant, above which the ground state ceases to exist.