Transcript Noncommutative Quantum Mechanics

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:
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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
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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
)
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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