First considerations on the generalized uncertainty principle for finite-dimensional discrete phase spaces

TL;DR

This paper develops a theoretical framework for finite-dimensional discrete phase spaces, extending the generalized uncertainty principle by incorporating topological effects and analyzing its applicability and limitations.

Contribution

It introduces a self-consistent framework for finite discrete phase spaces, expanding understanding of GUP with topological considerations.

Findings

Topological effects influence GUP in finite-dimensional phase spaces.

The framework clarifies conditions where GUP is valid.

Limitations of GUP in discrete systems are identified.

Abstract

Generalized uncertainty principle and breakdown of the spacetime continuum certainly represent two important results derived of various approaches related to quantum gravity and black hole physics near the well-known Planck scale. The discreteness of space suggests, in particular, that all measurable lengths are quantized in units of a fundamental scale (in this case, the Planck length). Here, we propose a self-consistent theoretical framework for an important class of physical systems characterized by a finite space of states, and show that such a framework enlarges previous knowledge about generalized uncertainty principles, as topological effects in finite-dimensional discrete phase spaces come into play. Besides, we also investigate under what circumstances the generalized uncertainty principle (GUP) works out well and its inherent limitations.