GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry

In this work, we formulate a generalized uncertainty principle with both position and momentum operators modified from their canonical forms. We study whether Lorentz symmetry is violated and whether it can be saved with these modifications. The requirement that Lorentz invariance is not violated places restrictions on the way the position and momentum operators can be modified. We also investigate the connection between general uncertainty principle and non-commutative geometry models, e.g.,laying out the connection between area/area operators and angular momentum in both models.