Fractional and noncommutative spacetimes

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 nonrotation-invariant but cyclicity-preserving measure of κ-Minkowski spacetime. 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 κ-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. © 2011 American Physical Society.