The path-integral is a formulation of quantum mechanics equivalent to the standard one, offering a new point oh view of the subject which is more intuitive than the usual approach. In this work we firstly present the Dirac orginal idea and then we describe how Feynman developed it. Therefore we apply the path-integral approach to the case of a scalar field theory, introducing the concept of generating functional. Finally we illustrate two alternative representations of the generating functional, dual to the Schwinger formalism, discussing how they lead to a realation with the Hermite polynomials. We also introduce the Schwinger-Dyson equation and express it through the representations introduced.
The physical system modelled is a periodic lattice that provides a background for interacting fermions. An effective model of the interacting fermions at zero temperature may be studied using standard techniques of many-body quantum mechanics. All the physical information of such a system is encoded in its Schwinger's functions that tell us how states evolve in time and what the matrix elements of all operators are. It is natural to do calculations on the interacting system by treating it as a perturbation of the corresponding noninteracting system, as the eigenstates and energy levels of the latter are known. All Schwinger functions may be conveniently cast in the path integral formalism, and combined into a generating functional. The Schwinger functions are shown to be sums of Feynman diagrams, where the electron propagators blow-up at the Fermi surface. Power counting is described and used to reveal subdiagram divergences. Renormalization via the renormalized Gallavotti-Nicolo Tree Expansion is implemented in an attempt to circumvent these subdiagram divergences and obtain finite values for the Schwinger functions. It is shown that if the Fermi surface has at worst quadratic singularities, the perturbation coefficients of the renormalizing counterterms are finite and remain so as the infrared cutoff is removed.