On bound states and non-trivial fixed points in quantum field theories

The calculation of the binding energies in Quantum Field Theories (QFT's) is a hard and long standing problem. Even for weakly coupled field theories like QED the extraction of information on bound states from the perturbative expansion is considered an "art". In the first part of this thesis we define those bound states that can be recovered from the perturbative expansion (threshold bound states) and calculate their mass in various models. It is shown that the method of Pade approximation and the Bethe-Salpeter equation (supplemented by a proof of absence of on mass shell singularities) provide a systematic way of calculating threshold bound states masses from the perturbative expansion of the S-matrix. We check these methods on 1+1 and 2+1 dimensional models where there exist a good expansion for the S-Matrix i.e. either weak coupling or 1/N. In the second part of this thesis a less rigorous approach is taken. This part concentrates on the B=2 sector of the SU (3) Skyrme model. We show that one can generate classical configurations (skyrmions) corresponding to bound states of two particles of an effective field theory (the Skyrme model), starting from classical solutions of the euclidean SU(3) Yang-Mills theory. The parity doubling of the ground state in this sector is also investigated. The third and last part deals with non trivial fixed points in QFT's. It is shown that the infra-red fixed point of the chiral phase transition in d=3 is the critical Gross- Neveu model. This is yet another proof to the nonperturbative renormalizability of four- Fermi interactions in 2+1 dimensions. The critical exponents of this phase transition are calculated within the 1/N expansion. The renormalizability of the GN model is also demonstrated explicitly to next to leading order in l/N by calculating β function, effective potential and ultraviolet dimension of various operators. [more abstract]Science, Faculty ofPhysics and Astronomy, Department ofGraduat