1. Noncommutative Quantum Mechanics Catarina Bastos IBERICOS, Madrid 16th-17th April 2009 C. Bastos, O. Bertolami, N. Dias and J. Prata, J. Math. Phys. 49 (2008) 1. C. Bastos, O. Bertolami, N. Dias and J. Prata, Phys. Rev. D 78 (2008) 023516. C. Bastos and O. Bertolami, Phys. Lett. A 372 (2008) 5556.

2. Phase-space Noncommutative Quantum Mechanics (QM): Quantum Field Theory Connection with Quantum Gravity and String/M- theory Find deviations from the predictions of QM Presumed signature of Quantum Gravity.  Obtain a phase-space formulation of a noncommutative extension of QM in arbitrary number of dimensions;  Show that physical previsions are independent of the chosen SW map.

3. Noncommutative Quantum Mechanics   ij e  ij antisymmetric real constant (dxd) matrices  Seiberg-Witten map: class of non-canonical linear transformations  Relates standard Heisenberg-Weyl algebra with noncommutative algebra  Not unique  States of the system:  Wave functions of the ordinary Hilbert space  Schrödinger equation:  Modified ,  -dependent Hamiltonian  Dynamics of the system

4. Quantum Mechanics – Deformation Quantization  Self-adjoint operators C ∞ functions in flat phase-space;  Density matrix Wigner Function (quasi-distribution);  Product of operators *-product (Moyal product);  Commutator Moyal Bracket Deformation quantization method: leads to a phase space formulation of QM alternative to the more conventional path integral and operator formulations.

5. Weyl-Wigner map: *- product : Generalized coordinates: Quantum Mechanics – Deformation Quantization Kernel representation:

6. Generalized Weyl-Wigner map:  T : coordinate transformation non-canonical  New variables (no longer satisfy the standard Heisenberg algebra):  Generalized Weyl-Wigner map:

7. Noncommutative Quantum Mechanics I SW map: Generalized coordinates: S=S αβ constant real matrix Weyl-Wigner map :

8. Noncommutative Quantum Mechanics II Moyal Bracket: Wigner Function : *- product :

9. Independence of W ξ z from the particular choice of the SW map:  Two sets of Heisenberg variables related by unitary transformation:  Two generalized Weyl-Wigner maps:  Is A 1 (z)=A 2 (z)?  From (a) and (b):  Unitary transformation (a) linear: (a ) (b) Bastos et al., J. Math. Phys. 49 (2008) 072101. Linear diff

10. Applications: Noncommutative Gravitational Quantum Well Dependence of the energy level (1 st order in perturbation theory) on η; Bounds for noncommuative parameters, θ and η: Vanishing of the Berry Phase. Noncommutative Quantum Cosmology: Kantowski Sachs cosmological model Momentum NC parameter η allows for a selection of states. O.B. et al, Phys.Rev. D 72 (2005) 025010. C.B. and O.B., Phys.Lett. A 372 (2008) 5556. θ≠0 η=0 θ=0 η≠0 Bastos et al., Phys.Rev. D 78 (2008) 023516.