Non-commutative quantum mechanics from Drinfeld-Twist

Full text. We derive the non-commutative quantum mechanics from an Abelian twist deformation of a Hopf algebra structure. The abelian twist induces a constant operator θij associated to the commutator of the position operators Xi ([Xi;Xj ] = θij). θij is not a plain c-number and lives in the universal enveloping algebra of the Heisenberg algebra associated to the quantum system. We formulate the consistent quantization procedure compatible with both the Hopf algebra structure and its abelian (Drinfeld) twist deformation. Two possibilities naturally arises. In the first case we have to renounce using the quantization procedure based on creation and annihilation operators acting on a Fock vacuum and replace it instead with the Wigner's approach. It is based on superalgebra-valued operators acting on a vacuum state which is given by a highest weight representation of the associated superalgebra. A second possibility requires considering composite operators made with Heisenberg algebra generators as primitive elements in a bigger Lie algebra endowed with bialgebra and Hopf algebra properties. We obtain the Noether symmetries of first-quantized quantum mechanical systems in this framework. We derive the deformations induced by the Non-abelian twist. We discuss in particular applications to such systems like the harmonic oscillators, the hydrogen atom with Coulomb potential, as well as the modifications induced by more general type of potentials which retain some symmetries. (author)