Path integral in position-deformed Heisenberg algebra with maximal length uncertainty

Abstract

In this work, we study the path integral in a position-deformed Heisenberg algebra with quadratic deformation which implements both minimal momentum and maximal length uncertainties. We construct propagators of path integrals within this deformed algebra using the position space representation on the one hand and the Fourier transform and its inverse representations on the other. The result is remarkably similar to the one obtained by Pramanik (2022) from the Perivolaropoulos’s deformed algebra (Perivolaropoulos, 2017). Then, the propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. We also show that the actions which describe the classical trajectories of both systems are bounded by the ordinary ones of classical mechanics due to the existence of this maximal length. Consequently, particles of these systems travel faster from one point to another with low kinetic and mechanical energies.