  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.
