Non-perturbative methods in quantum field theory. Beyond the mean-field, the random phase approximation

Hadronic physics in field theory is a difficult subject and a field of active research. The problem is highly non-perturbative because one cannot use a perturbative development of QCD in the low energy sector. The equivalence between field theory and many-body problem leads us to apply well-known non-perturbative many-body techniques like the mean-field approximation (or Gaussian approximation) and the random phase approximation (RPA). Beyond the mean-field where the only correlations incorporated in the calculations are those between one particle and a 'mean-field' potential, the RPA take into account correlations between particles. To set up the formalism, we apply different types of RPA (namely standard, renormalized and with the Dyson equations approach) to one of the more simple quantum field theory with interaction, the λφ⁴ scalar theory. We show that a phase transition occurs, due to a dynamical symmetry breaking. The critical parameter is close to other results obtained by lattice calculations or cluster techniques. We compute also finite temperature results in mean field approximation. We study also a realistic effective model for the chiral phase transition, the linear-σ model. The Goldstone theorem is restored, to the contrary to the Gaussian approximation. Finally we go beyond renormalized RPA correlations in the anharmonic oscillator case and show that the first RPA correlations are sufficient to improve greatly the mean-field calculations. (author)