Resummed perturbative series of scalar quantum field theories in two-particle-irreducible formalism

In this thesis the two-particle-irreducible (2PI) formalism is investigated with several applications, particular emphasis on renormalizability. In the O(N) symmetric scalar quantum field theory formulated with auxiliary fields it is pointed out that, statements recently appeared in the literature which raised doubts on renormalizability, are wrong. Counterterms are constructed in details in the NLO of the large-N expansion. The renormalizability is demonstrated at the same level of the approximation also with eliminating the auxiliary field. In the one component \phi^4 model (at T=0) a method using renormalization conditions is developed, which gives directly finite equations for the 1- and 2-point functions. This solves the problems of numerical precision of the counterterm formulated renormalization. In the two-loop 2PI approximation the equivalence of the new method and the procedure formulated with counterterms is shown, both analytically and numerically. For the counterterm method also a new algorithm is developed, which accelerates the numerical convergence. In the U(N)xU(N) model, an approximate solution of the field equations in the large-N limit is found in the broken phase, for a condensate proportional to the unit matrix. The approximation assumes the presence of a mass-hierarchy between the scalar and pseudoscalar fields. The validity of the latter assumption is investigated in the whole parameter space numerically. The stability of the solution against a more general condensate is checked. It is shown that, there exists a region of the parameter space, where the condensate corresponding to the "8" direction of the SU(3) classification can arise spontaneously. This solution is found to be metastable, but could transform into the true ground state, when a moderate external source is applied along the "8" direction.