Novel approaches to numerical solutions of quantum field theories

Two new approaches to numerically solving Quantum Field Theories are presented.; The Source Galerkin technique is a direct approach to determining the generating functional of a theory by solving the Schwinger-Dyson equations. The properties of the Source Galerkin technique are tested by using it to determine the phase structure of the Ultralocal &phis;4 theory. A framework for applying this approach to solving O( N) Nonlinear Sigma model is constructed.; The Sinc Function approximation is a highly efficient method of numerically evaluating Feynman diagrams. In the present dissertation the Sinc Function approximation is applied to fermionic fields. The Sinc expanded versions of fermion and photon propagators are derived. The accuracy of this approximation is tested by a direct comparison of the Sinc expanded propagators with exact propagators and by performing several sample calculations of one loop QED diagrams. An analysis of computational properties of the Sinc Function approach is presented.