Abstract:
The main focus of this thesis is to elaborate rather unorthodox methods throughout a diverse range of mathematical techniques, which allow us to study quantum field theoretical models in a nonperturbative context. The thesis is divided into two parts: In the first part, we use semigroup integral to evaluate zeta–function regularized determinants. This is especially powerful for nonpositive operators such as the Dirac operator. In order to understand fully the quantum effective action, one should know not only the potential term but also the leading kinetic term. For this purpose, we use the Weyl type of symbol calculus to evaluate the determinant as a derivative expansion. The technique is applied both to a spin–0 bosonic operator and to the Dirac operator coupled to a scalar field. The latter is important for the Yukawa model. In the second part, we study the relativistic Lee model on static Riemannian manifolds. The relativistic Lee model is an overly simplified version of the Yukawa model, which is amenable to a nonperturbative treatment. Understanding of it could shed light on the Yukawa model. The model is constructed nonperturbatively through its resolvent, which is based on the so–called principal operator and the heat kernel techniques. It is shown that making the principal operator well defined dictates how to renormalize the parameters of the model. The renormalization of the parameters is the same in the light–front coordinates as in the instant form. Moreover, the renormalization of the model on Riemannian manifolds agrees with the flat case. The asymptotic behaviour of the renormalized principal operator in the large number of bosons limit implies that the ground state energy is positive. In 2 + 1 dimensions, the model requires only a mass renormalization. We obtain rigorous bounds on the ground state energy for the n–particle sector of the (2 + 1)–dimensional model.
