Non-Commutative Mechanics as a modification of space-time
We formulate non-relativistic classical and quantum mechanics in the non-commutative two dimensional plane. The approach we use is based on the Galilei group, where the non-commutativity is seen as a central extension upon identification of the boost generators with the position operator. We perform a systematic study of the free particle, defined by the symmetries of the space time, which include the no-commutativity. The symmetries at the classical level are analyzed in terms of Noether's theorem. Canonical quantization is presented and the representation of the corresponding Heisenberg algebra is obtained. The path integral representation and Wigner distribution function in phase space are also discussed. We work out, both at the classical and at the quantum level, the harmonic oscillator, avoiding the use of the conventional non-canonical transformation that leads to a momentum dependent potential. We use Einsteins' model for a solid to corroborate that, according with intuition, as a consequence of the space fuzziness the entropy is a growing function of $θ$ in the low temperature regime.