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[en] This thesis consists of an introduction, four chapters, a discussion and an appendix. The introduction provides the background to the work covered in this thesis, as well as an outline of the structure of the thesis. Chapter 1 presents a study related to fermion zero modes. The aim is to prove the existence of a zero mode that Klinkhamer and Lee observed in their study of a fermion doublet coupled to a chiral SU(2) gauge field. The proof comprises analytical and numerical analysis on the stability of the solutions obtained in the study of the Dirac equation of the fermion. Chapter 2 sketches a new mechanism for deriving a discrete and bounded fermion mass spectrum, based on the work of Klinkhamer and the present author. The model theory used consists of two fermion fields interacting with a Higgs-like scalar field. An open extra dimension is introduced to this theory so that a set of explicit classical solutions to the equations of motion is obtained. When the wave functions are required to be normalizable in the extra dimension, the masses of the four-dimensional fermions naturally become bounded and discrete. Chapter 3 consists of an investigation of a theory on the gauged Lorentz group. In Minkowskian space-time, the pure Yang-Mills theory of this group, with spherical symmetry imposed on the gauge field, reduces to a new theory in a two-dimensional space-time. The reduced theory has a scalar field with four degrees of freedom and a quartic potential, and two abelian gauge fields. A problem that remains to be solved, however, is that the potential of the scalar field is not bounded from below. In Chapter 4, a method for deriving a set of identities of the correlation functions in quantum field theories is presented. It can be used to obtain a variational equation (resembling a differential equation) for the generating functional of a given theory. When the generating functional is expanded into a Taylor series in terms of the source field(s), the variational equation makes it possible to relate the correlation functions of different processes to one another. The calculation here is non-perturbative. One of the identities derived for the λ - φ"4 theory is tested and verified. The thesis ends with a discussion in order to address the questions left unanswered in the previous chapters. Some reflections and ideas are given here, in hope that they can stimulate interest in future research.