Non-perturbative methods in Hamiltonian formulation of field theories

We discuss two non-perturbative approaches, namely, the WKB approximation and collective field method in the framework of Hamiltonian formulation in field theories. The first part of the thesis consists of the application of a combined WKB and variational method to compact quantum electrodynamics in three space-time dimensions. The ground state wave-functional and energy are derived as also the shift in the energy due to the presence of external static charges. In the second part of the thesis, we formulate the collective field method in canonical Hamiltonian formalism and apply it to 0(N) Heisenberg spin-system in two dimensions. The mass-gap and the beta-function are derived for large N. Introduction of the collective field in Lagrangian path integral formalism is also discussed and 1/N corrections computed