Noncommutative Noncommutative Quantum MechanicsQuantum Mechanics

Catarina BastosCatarina Bastos

IBERICOS, Madrid 16th-17th April IBERICOS, Madrid 16th-17th April 20092009

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 Phase-space Noncommutative Quantum Mechanics (QM):Quantum Mechanics (QM):

Quantum Field TheoryQuantum Field TheoryCConnection with Quantum Gravity and onnection with Quantum Gravity and String/M- theoryString/M- theory

Find dFind deviations from the predictions of eviations from the predictions of QM QM

PresumedPresumed signature of Quantum Gravity. 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 Noncommutative Quantum MechanicsMechanics

ijij e e ijij antisymmetric real constant antisymmetric real constant ((ddxxdd)) matrices matrices

Seiberg-Witten mapSeiberg-Witten map: class of non-canonical linear : class of non-canonical linear transformationstransformations Relates standard Heisenberg-Weyl algebra with Relates standard Heisenberg-Weyl algebra with

noncommutative algebranoncommutative algebra NNot uniqueot unique

States of the system:States of the system: Wave functions of the ordinary Hilbert spaceWave functions of the ordinary Hilbert space

Schrödinger equation:Schrödinger equation: Modified Modified ,,-dependent-dependent Hamiltonian Hamiltonian Dynamics of the systemDynamics of the system

Quantum Mechanics Quantum Mechanics –– Deformation Deformation QuantizationQuantization

Self-adjoint operators CSelf-adjoint operators C∞ ∞ functions in functions in flat phase-space;flat phase-space;

DDensity matrix Wigner Function ensity matrix Wigner Function (quasi-distribution);(quasi-distribution);

Product of operators *-product Product of operators *-product (Moyal product);(Moyal product);

Commutator Moyal BracketCommutator Moyal Bracket

Deformation quantization method: leads to a phase space formulation of QM alternative to the more conventional path integral and operator formulations.

Weyl-Wigner map:

*-product:

Generalized coordinates:

Quantum Mechanics Quantum Mechanics –– Deformation Deformation QuantizationQuantization

Kernel representation:

Generalized Weyl-Wigner map:Generalized Weyl-Wigner map:

T : coordinate transformation non-canonicalT : coordinate transformation non-canonical

NNew variables (no longer satisfy the standard ew variables (no longer satisfy the standard Heisenberg algebra):Heisenberg algebra):

Generalized Weyl-Wigner map: Generalized Weyl-Wigner map:

Noncommutative Quantum Noncommutative Quantum Mechanics IMechanics I

SW map:

Generalized coordinates:

S=Sαβ constant real matrixS=Sαβ constant real matrix

Weyl-Wigner map:

Noncommutative Quantum Noncommutative Quantum Mechanics IIMechanics II

Moyal Bracket:

Wigner Function:

*-product:

Independence of WIndependence of Wξξz z from the particular from the particular

choice of the SW map:choice of the SW map: TTwo sets of Heisenberg variables related by unitary wo sets of Heisenberg variables related by unitary

transformation:transformation:

TTwo generalized Weyl-Wigner maps:wo generalized Weyl-Wigner maps:

Is Is AA11(z)=A(z)=A22(z)(z)?? From (a) and (b): From (a) and (b):

Unitary transformation (a)Unitary transformation (a) linear: linear:

(a)

(b)

Bastos et al., J. Math. Phys. 49 (2008) 072101.

Linear diff

Applications:Applications:Noncommutative Gravitational Quantum WellNoncommutative Gravitational Quantum Well

Dependence of the energy level (1Dependence of the energy level (1stst order in order in perturbation theory) on η;perturbation theory) on η;

BBounds for noncommuative parameters, θ and η:ounds for noncommuative parameters, θ and η:

Vanishing of the Berry Phase. Vanishing of the Berry Phase.

Noncommutative Quantum Cosmology:Noncommutative Quantum Cosmology:Kantowski Sachs cosmological model Kantowski Sachs cosmological model

Momentum NC parameter η allows for a selection of Momentum NC parameter η allows for a selection of states.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.