Minimum length in 3D without modified commutators

In this work we investigate the Generalized Uncertainty Principle (GUP) in three dimensions. Using various constraints based on experiment, observations, or theory we find specific forms of the modified three dimensional position and momentum operators, leading to a modified commutator and a modified uncertainty principle for these operators (i.e. a GUP). We find that the three dimensional GUP exhibits some interesting phenomena that do not occur in one dimension, e.g. the minimal distance scales in the direction parallel to a particles velocity can be different from the perpendicular direction. We also discuss the application of this three dimensional GUP to possible experiments/observations such as photon dispersion from gamma ray bursts and the GZK cutoff.