Functional methods and perturbation theory

The framework of functional integrals in field theory is convenient for presenting a unified view of the perturbation expansion according to the number of loops. We review the calculation of the generating functional for irreducible Green functions and its renormalization properties. Calculations of the effective potential and Z-function are carried up to the order of two loops for the self-coupled scalar field. This is applied to compute the coefficients of the Callan-Symanzik equation which describes the short distance behavior of the theory. In conclusion, we present some speculations concerning the positivity of the coupling constant, and its relation to the on-shell two-particle scattering amplitude