11 research outputs found
Non-Commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts
Employing Maxwell's equations as the field theory of the photon, quantum
mechanical operators for spin, chirality, helicity, velocity, momentum, energy
and position are derived. The photon ``Zitterbewegung'' along helical paths is
explored. The resulting non-commutative geometry of photon position and the
quantum version of the Pythagorean theorem is discussed. The distance between
two photons in a polarized beam of given helicity is shown to have a discrete
spectrum. Such a spectrum should become manifest in measurements of two photon
coincidence counts. The proposed experiment is briefly described.Comment: Latex, 13 pages, 3 figure
On the Green's Function of the almost-Mathieu Operator
The square tight-binding model in a magnetic field leads to the
almost-Mathieu operator which, for rational fields, reduces to a
matrix depending on the components , of the wave vector in the
magnetic Brillouinzone. We calculate the corresponding Green's function without
explicit knowledge of eigenvalues and eigenfunctions and obtain analytical
expressions for the diagonal and the first off-diagonal elements; the results
which are consistent with the zero magnetic field case can be used to calculate
several quantities of physical interest (e. g. the density of states over the
entire spectrum, impurity levels in a magnetic field).Comment: 9 pages, 3 figures corrected some minor errors and typo
Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics
To build genuine generators of the rotations group in noncommutative quantum
mechanics, we show that it is necessary to extend the noncommutative parameter
to a field operator, which one proves to be only momentum dependent.
We find consequently that this field must be obligatorily a dual Dirac monopole
in momentum space. Recent experiments in the context of the anomalous Hall
effect provide for a monopole in the crystal momentum space. We suggest a
connection between the noncommutative field and the Berry curvature in momentum
space which is at the origine of the anomalous Hall effect.Comment: 4 page
Role of phason-defects on the conductance of a 1-d quasicrystal
We have studied the influence of a particular kind of phason-defect on the
Landauer resistance of a Fibonacci chain. Depending on parameters, we sometimes
find the resistance to decrease upon introduction of defect or temperature, a
behavior that also appears in real quasicrystalline materials. We demonstrate
essential differences between a standard tight-binding model and a full
continuous model. In the continuous case, we study the conductance in relation
to the underlying chaotic map and its invariant. Close to conducting points,
where the invariant vanishes, and in the majority of cases studied, the
resistance is found to decrease upon introduction of a defect. Subtle
interference effects between a sudden phason-change in the structure and the
phase of the wavefunction are also found, and these give rise to resistive
behaviors that produce exceedingly simple and regular patterns.Comment: 12 pages, special macros jnl.tex,reforder.tex, eqnorder.tex. arXiv
admin note: original tex thoroughly broken, figures missing. Modified so that
tex compiles, original renamed .tex.orig in source
From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics
In 1990, Dyson published a proof due to Feynman of the Maxwell equations
assuming only the commutation relations between position and velocity. With
this minimal assumption, Feynman never supposed the existence of Hamiltonian or
Lagrangian formalism. In the present communication, we review the study of a
relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's
magnetic angular momentum and Dirac magnetic monopole are the consequences of
the structure of the Lorentz Lie algebra defined by the Feynman's brackets.
Then, we extend these ideas to the dual momentum space by considering
noncommutative quantum mechanics. In this context, we show that the
noncommutativity of the coordinates is responsible for a new effect called the
spin Hall effect. We also show its relation with the Berry phase notion. As a
practical application, we found an unusual spin-orbit contribution of a
nonrelativistic particle that could be experimentally tested. Another practical
application is the Berry phase effect on the propagation of light in
inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland,
U.S.A., August 2006
The Noncommutative Harmonic Oscillator based in Simplectic Representation of Galilei Group
In this work we study symplectic unitary representations for the Galilei
group. As a consequence the Schr\"odinger equation is derived in phase space.
The formalism is based on the non-commutative structure of the star-product,
and using the group theory approach as a guide a physical consistent theory in
phase space is constructed. The state is described by a quasi-probability
amplitude that is in association with the Wigner function. The 3D harmonic
oscillator and the noncommutative oscillator are studied in phase space as an
application, and the Wigner function associated to both cases are determined.Comment: 7 pages,no figure