Path-integral over non-linearly realized groups and Hierarchy solutions

TL;DR

This paper develops a method to compute Green functions in the nonlinear sigma model by integrating a local functional equation, providing a perturbative definition of the path-integral over non-linearly realized SU(2).

Contribution

It introduces a novel perturbative approach to define the path-integral over non-linearly realized groups using the local functional equation.

Findings

Perturbative solution of the local functional equation for SU(2).

Explicit construction of Green functions in the nonlinear sigma model.

A new framework for handling non-linear group realizations in quantum field theory.

Abstract

The technical problem of deriving the full Green functions of the elementary pion fields of the nonlinear sigma model in terms of ancestor amplitudes involving only the flat connection and the nonlinear sigma model constraint is a very complex task. In this paper we solve this problem by integrating, order by order in the perturbative loop expansion, the local functional equation derived from the invariance of the SU(2) Haar measure under local left multiplication. This yields the perturbative definition of the path-integral over the non-linearly realized SU(2) group.