Nonabelian Bosonization as a Nonholonomic Transformations from Flat to Curved Field Space

TL;DR

This paper demonstrates how nonholonomic transformations can map flat space path integrals to curved space, revealing a crucial Jacobian term that ensures quantum equivalence, especially in nonabelian bosonization.

Contribution

It introduces a nonabelian generalization of the Hubbard-Stratonovich transformation and clarifies the role of Jacobian terms in bosonization involving curved field spaces.

Findings

The Jacobian produces a curvature-dependent energy term canceling earlier naive results.

Nonholonomic transformations correctly relate flat and curved space path integrals.

The nonabelian Hubbard-Stratonovich formula is developed for bosonization applications.

Abstract

There exists a simple rule by which path integrals for the motion of a point particle in a flat space can be transformed correctly into those in curved space. This rule arose from well-established methods in the theory of plastic deformations, where crystals with defects are described mathematically by applying nonholonomic coordinate transformations to ideal crystals. In the context of time-sliced path integrals, this has given rise to a {\em quantum equivalence principle\/} which determines the measure of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations are accompanied by a nontrivial Jacobian which in curved spaces produces an additional energy proportional to the curvature scalar, thereby canceling an equal term found earlier by DeWitt from a naive formulation of Feynman's time-sliced path integral in curved space. The importance of this cancelation has been documented in various systems (H-atom, particle on the surface of a sphere, spinning top). Here we point out its relevance in the process of bosonizing a nonabelian one-dimensional quantum field theory, whose fields live in a flat field space. Its bosonized version is a quantum-mechanical path integral of a point particle moving in a space with constant curvature. The additional term introduced by the Jacobian is crucial for the identity between original and bosonized theory. A useful bozonization tool is the so-called Hubbard-Stratonovich formula for which we find a nonabelian version.