Path Integrated Geodesics and Distances

TL;DR

The paper develops a path-integral framework to quantize geometry by treating geodesics and distances with quantum weights: $L_Q = L_C e^{-\beta L_C}$ for geodesics and a distance functional ${\cal S}_Q$ defined via $e^{-i s/l_p}$. It shows that geodesic trajectories remain unchanged under the quantum correction when using affine parametrization, while distances exhibit distinct behavior across causal types: spacelike separations acquire a minimum length $l_p$ (with $l_Q^2 = l_C^2 + l_p^2$), lightlike separations remain unaffected, and timelike separations approach the classical value for large separations. The work further uncovers a quantum/statistical physics duality, linking spacelike geodesic quantization to Feynman-type path integration and timelike averages to Boltzmann-like thermodynamics, with a proposed maximum energy tying to finite distance resolution. These results hint at potential mechanisms for singularity resolution and offer a bridge between quantum geometry and statistical physics, while suggesting that averaging over geodesics corresponds to averaging over connections/metrics.

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.