Discreteness of Space from GUP in a Weak Gravitational Field

TL;DR

This paper explores how quantum gravity effects, modeled through GUP, lead to the quantization of space in various quantum equations, and investigates the influence of weak gravitational fields on this discreteness, suggesting universality of quantum gravity effects.

Contribution

It extends previous work on GUP-induced space quantization by analyzing the impact of weak gravitational fields, showing that quantization persists under gravity and indicating the universality of quantum gravity effects.

Findings

Quantization of length, area, and volume persists with weak gravity.

The nature of space quantization becomes more complex under gravity.

Quantum gravity effects are likely universal across different conditions.

Abstract

Quantum gravity effects modify the Heisenberg's uncertainty principle to a generalized uncertainty principle (GUP). Earlier work showed that the GUP-induced corrections to the Schr\"odinger equation, when applied to a non-relativistic particle in a one-dimensional box, led to the quantization of length. Similarly, corrections to the Klein-Gordon and the Dirac equations, gave rise to length, area and volume quantizations. These results suggest a fundamental granular structure of space. In this work, it is investigated how spacetime curvature and gravity might influence this discreteness of space. In particular, by adding a weak gravitational background field to the above three quantum equations, it is shown that quantization of lengths, areas and volumes continue to hold. However, it should be noted that the nature of this new quantization is quite complex and under proper limits, it reduces to cases without gravity. These results suggest that quantum gravity effects are universal.