Path integral in position-deformed Heisenberg algebra with strong quantum gravitational measurement

TL;DR

This paper constructs path integral propagators within a position-deformed Heisenberg algebra that models strong quantum gravitational effects, showing that gravity influences particle trajectories by effectively bending paths and reducing classical action values.

Contribution

It develops a method to compute path integral propagators in a position-deformed algebra, illustrating quantum gravity's impact on particle dynamics in Euclidean space.

Findings

Quantum gravity effects bound classical actions, allowing faster particle trajectories.

Actions decrease as quantum gravitational effects increase, indicating path bending.

Classical trajectories are modified by quantum gravitational deformation.

Abstract

Position-deformed Heisenberg algebra with maximal length uncertainty has recently been proven to induce strong quantum gravitational fields at the Planck scale (2022 J. Phys. A: Math. Theor.55 105303). In the present study, we use the position space representation on the one hand and the Fourier transform and its inverse representations on the other to construct propagators of path integrals within this deformed algebra. The propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. Since the effects of quantum gravity are strong in this Euclidean space, we show that the actions which describe the classical trajectories of both systems are bounded by the ordinary ones of classical mechanics. This indicates that quantum gravity bends the paths of particles, allowing them to travel quickly from one point to another. It is numerically observed by the decrease in values of classical actions as one increases the quantum gravitational effects.