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

TL;DR

This paper develops a method to derive the generating functional for constrained stochastic systems, applies it to an inextensible chain, and connects it to the nonlinear sigma model with O(3) symmetry.

Contribution

It introduces a general method for deriving generating functionals for constrained systems and links the dynamics of inextensible chains to the nonlinear sigma model.

Findings

Derived the generating functional for inextensible chain dynamics.

Connected the chain model to the nonlinear sigma model with O(3) symmetry.

Provided a framework for analyzing constrained stochastic processes.

Abstract

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.