The generating functional for scalar field theories

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.