From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

In this work a method is presented to derive the generating functional in path integral form for a system with an arbitrary number of degrees of freedom and constrained by general conditions. The method is applied to the case of the dynamics of an inextensible chain subjected to external forces. Next, the generating functional of the inextensible chain is computed assuming that the interactions are switched off. Finally, the generating functional of a two-dimensional nonlinear sigma model with O(3) symmetry is derived exploiting its similarities with the model describing the dynamics of the inextensible chain.