PQM and the GUP: implications of lattice dynamics and minimal uncertainties in Quantum Mechanics and Cosmology

We compare two alternative representations of quantum mechanics: Polymer Quantum Mechanics (PQM), which presents features similar to Loop Quantum Gravity and Loop Quantum Cosmology, and the Generalized Uncertainty Principle (GUP) representation, that through a modified Algebra yields an alternative uncertainty principle similar to those derived in String Theories and Brane Cosmology. These formalisms can be recast to apparently look similar, but while the GUP yields an absolute minimal uncertainty on position, PQM implements some kind of ultraviolet cut-off through a lattice and does not have a minimal uncertainty. Then we implement them on the anisotropic Bianchi I model in Misner-like variables on a semiclassical level: PQM always implies a removal of the singularities, while the GUP fails to do so, highlighting once again how the two representations are fundamentally incompatible.