Fractional and noncommutative spacetimes

TL;DR

This paper explores the relationship between fractional and noncommutative spacetimes, showing how measures in these frameworks relate at different scales and establishing a mapping based on algebraic structures.

Contribution

It introduces a scale-dependent mapping between fractional and noncommutative spacetimes, connecting log-oscillatory and power-law measures through algebraic and cyclicity properties.

Findings

Near the fundamental scale, fractional measures match -k Minkowski measures.

At larger scales, fractional measures become power-law, interpolating between -k Minkowski and canonical spacetime.

The results are based on a braiding formula applicable to nonlinear algebra mappings.

Abstract

We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of \kappa-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncommutative spacetime defined by a certain nonlinear algebra of the coordinates, which interpolates between \kappa-Minkowski and canonical spacetime. These results are based upon a braiding formula valid for any nonlinear algebra which can be mapped onto the Heisenberg algebra.