The Classical Limit of Minimal Length Uncertainty Relation:Revisit with the Hamilton-Jacobi Method

TL;DR

This paper investigates how a minimal length scale affects classical gravitational phenomena like planetary precession, light deflection, and radar time delay using the Hamilton-Jacobi method, setting observational limits on the deformation parameter.

Contribution

It introduces a Hamilton-Jacobi approach to analyze classical effects of minimal length, providing new insights and bounds on deformation parameters in classical and relativistic contexts.

Findings

Limits on deformation parameter from observational data

Minimal length affects classical gravitational phenomena

Comparison with previous methods and results

Abstract

The existence of a minimum measurable length could deform not only the standard quantum mechanics but also classical physics. The effects of the minimal length on classical orbits of particles in a gravitation field have been investigated before, using the deformed Poisson bracket or Schwarzschild metric. In this paper, we use the Hamilton-Jacobi method to study motions of particles in the context of deformed Newtonian mechanics and general relativity. Specifically, the precession of planetary orbits, deflection of light, and time delay in radar propagation are considered in this paper. We also set limits on the deformation parameter by comparing our results with the observational measurements. Finally, comparison with results from previous papers is given at the end of this paper.