Non-perturbative approach in model field theories

In the domain of strong interaction physics where perturbation series in quantum electrodynamics cannot be applied, non-perturbative approaches have been attempted. With this method of approach, the following types of problems have been studied. A one-dimensional model Hamiltonian is considered and a continued fraction approach is applied to find the energy expectation value of the system and to enumerate the possible physical connections with the various mathematical results of the analytical theory of continued fraction. Another Lagrangian model consisting of two interacting boson fields is considered and the connection of poles of the propagator function with energy-eigenvalue equation of the Hamiltonian system is shown. It is shown how in a full-fledged relativistic quantum field theory these ideas can be extended to obtain a continued fraction representation for the meson propagator. These results are discussed in the ladder model of the Bethesalpeter type. An interesting feature of the non-perturbation approach is to show the existence of a Lehman type representation for meson propagator function within the framework of the approximation schemes. Various other intricate mathematical points in non-polynomial Langrangian theories have also been discussed. (A.K.)