Non-Commutativity of Effective Space-Time Coordinates and the Minimal Length

TL;DR

This paper develops a framework for effective, non-commutative space-time coordinates that imply a minimal length scale, linking to Snyder's coordinates and Deformed Special Relativity, and offering physical insights into extra dimensions and deformed momenta.

Contribution

It introduces a novel set of effective space-time coordinates that are non-commutative and relate to existing theories like Snyder's coordinates and Deformed Special Relativity.

Findings

Effective coordinates imply a minimal length scale.

Connection established with Snyder's coordinates.

Relation to five-dimensional Deformed Special Relativity.

Abstract

Considering that a position measurement can effectively involve a momentum-dependent shift and rescaling of the "true" space-time coordinates, we construct a set of effective space-time coordinates which are naturally non-commutative. They lead to a minimum length and are shown to be related to Snyder's coordinates and the five-dimensional formulation of Deformed Special Relativity. This effective approach then provides a natural physical interpretation for both the extra fifth dimension and the deformed momenta appearing in this context.