Path Integrated Geodesics and Distances

TL;DR

This paper investigates quantum corrections to geodesic distances using path integrals, revealing that while geodesics remain unaffected, distances exhibit quantum effects that depend on the geodesic type, with implications for singularity removal.

Contribution

It introduces a novel framework applying path integrals to quantum corrections of geodesic distances, distinguishing behaviors for different geodesic types and suggesting a role in singularity resolution.

Findings

Quantum corrections to distances differ for time-like, light-like, and space-like geodesics.

Light-like geodesics are unaffected by quantum corrections, preserving the causal structure.

Space-like geodesics have a minimum separation, potentially removing singularities.

Abstract

In this paper, the quantum corrections to the kinematics of geometry, specifically geodesics, are presented. This is done by employing the path integral over the geodesics. Interestingly, the geodesics do not see any modifications in this framework. However for the distances, it is demonstrated that these quantum corrections exhibit distinct behaviors for time-like, light-like, and space-like geodesics. For time-like geodesics, the maximum correction is the Planck length, which disappears when the classical separation vanishes. The light-like geodesics do not exhibit quantum corrections, meaning that the causal light cone remains the same in both classical and quantum frameworks under certain conditions. The quantum corrections for space-like geodesics impose a minimum on space-like separation, potentially playing a role in removing singularities by preventing null congruences from being closer than the Planck length. This framework also explores the correspondence between space-like/time-like geodesics and quantum/statistical physics.