Path Integral Invariance under Point Transformations

TL;DR

This paper proposes a covariant, geometry-based definition of the path integral formalism for the Lagrangian, emphasizing the invariance under point transformations and analyzing the geometric reasons behind potential changes in the integral.

Contribution

It introduces a covariant framework for path integrals that preserves the classical limit without altering potentials or measures, focusing on the geometric aspects of path sets.

Findings

Path integral invariance depends on the preservation of path sets under transformations.

Explicit kernel calculations in different coordinate systems show consistent results.

Geometric explanations clarify when and why the integral may change under point transformations.

Abstract

We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms nor searching for a strange measure, but using as a framework the geometry of the spaces considered. We focus our attention on the set of paths used to join succesive points in the discretization if the time-slicing definition is used to calculate the integral.If this set of paths is not preserved when performing a point transformation, the integral may change. The reasons for this are geometrically explained. Explicit calculation of the Kernel in polar coordinates is made, yielding the same system as in Cartesian coordinates.