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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. Phase-space Noncommutative Quantum Mechanics (QM): Quantum Field Theory Connection with Quantum Gravity and String/Mtheory 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. 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 Quantum Mechanics – Deformation Quantization Deformation quantization method: leads to a phase space formulation of QM alternative to the more conventional path integral and operator formulations. Self-adjoint operators Density matrix Product of operators Commutator C∞ functions in flat phase-space; Wigner Function (quasi-distribution); *-product (Moyal product); Moyal Bracket Quantum Mechanics – Deformation Quantization Weyl-Wigner map: Generalized coordinates: *-product: Kernel representation: Generalized Weyl-Wigner map: T : coordinate transformation non-canonical New variables (no longer satisfy the standard Heisenberg algebra): Generalized Weyl-Wigner map: Noncommutative Quantum Mechanics I SW map: Generalized coordinates: S=Sαβ constant real matrix Weyl-Wigner map: Noncommutative Quantum Mechanics II *-product: Moyal Bracket: Wigner Function: Independence of Wξz from the particular choice of the SW map: (a ) Two sets of Heisenberg variables related by unitary transformation: Linear diff (b) Two generalized Weyl-Wigner maps: Is A1(z)=A2(z)? From (a) and (b): Unitary transformation (a) linear: Bastos et al., J. Math. Phys. 49 (2008) 072101. Applications: Noncommutative Gravitational Quantum Well Dependence of the energy level (1st order in perturbation theory) on η; Bounds for noncommuative parameters, θ and η: O.B. et al, Phys.Rev. D 72 (2005) 025010. Vanishing of the Berry Phase. C.B. and O.B., Phys.Lett. A 372 (2008) 5556. Noncommutative Quantum Cosmology: Kantowski Sachs cosmological model Bastos et al. , Phys.Rev. D 78 (2008) 023516. Momentum NC parameter η allows for a selection of states. θ≠0 η=0 θ=0 η≠0