Going beyond perturbation theory: Parametric Perturbation Theory

We devise a {\sl non--perturbative} method, called {\sl Parametric Perturbation Theory} (PPT), which is alternative to the ordinary perturbation theory. The method relies on a principle of simplicity for the observable solutions, which are constrained to be linear in a certain (unphysical) parameter. The perturbative expansion is carried out in this parameter and not in the physical coupling (as in ordinary perturbation theory). We provide a number of nontrivial examples, where our method is capable to resum the divergent perturbative series, {\sl extract} the leading asymptotic (strong coupling) behavior and {\sl predict} with high accuracy the coefficients of the perturbative series. In the case of a zero dimensional field theory we prove that PPT can be used to provide the imaginary part of the solution, when the problem is analytically continued to negative couplings. In the case of a $\phi^4$ lattice model 1+1 and of elastic theory we have shown that the observables resummed with PPT display a branch point at a finite value of the coupling, signaling the transition from a stable to a metastable state. We have also applied the method to the prediction of the virial coefficients for a hard sphere gas in two and three dimensions; in this example we have also found that the solution resummed with PPT has a singularity at finite density. Predictions for the unknown virial coefficients are made.