Classical and Fluctuating Paths in Spaces with Curvature and Torsion

TL;DR

This paper discusses a new mapping method that accurately transforms classical and quantum path integrals from flat to curved spaces, accounting for curvature and torsion effects crucial for quantum systems and bosonization.

Contribution

It introduces a novel transformation procedure inspired by plastic deformation theory, revealing a quantum equivalence principle and correcting path integral measures in curved spaces.

Findings

The Jacobian from nonholonomic transformations cancels curvature terms in path integrals.

Corrected measures improve the description of quantum systems like the hydrogen atom.

The method is essential for bosonization and quantum field theories in curved geometries.

Abstract

This conference talk elaborates on a recently discovered mapping procedure by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed correctly into those in curved space. This procedure evolved 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, there seems to exists a quantum equivalence principle which determines the measures of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations produce a nontrivial Jacobian in the path measure which in a curved space produces an additional term proportional to the curvature scalar canceling a similar term found earlier by DeWitt from a naive formulation of Feynman's time-sliced path integral. This cancelation is important in correctly describing semiclassically and quantum mechanically various systems such as the hydrogen atom, a particle on the surface of a sphere, and a spinning top. It is also indispensible for the process of bosonization, by which Fermi particles are redescribed in terms of Bose fields.