Violation of the Robertson-Schr\"odinger uncertainty principle and   non-commutative quantum mechanics
  We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase-space (even if it violates the Robertson-Schr\"odinger uncertainty principle) is always a quantum state of an appropriate non-commutative extension of quantum mechanics. Conversely, all canonical non-commutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle. 
Catarina Bastos, Orfeu Bertolami, Nuno Costa Dias, Jo\~ao Nuno Prata
hep-th