33 research outputs found
The exotic Galilei group and the "Peierls substitution"
Taking advantage of the two-parameter central extension of the planar Galilei
group, we construct a non relativistic particle model in the plane. Owing to
the extra structure, the coordinates do not commute. Our model can be viewed as
the non-relativistic counterpart of the relativistic anyon considered before by
Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the
plane, the two parameters combine with the magnetic field to provide an
effective mass. For vanishing effective mass the phase space admits a
two-dimensional reduction, which represents the condensation to collective
``Hall'' motions and justifies the rule called ``Peierls substitution''.
Quantization yields the wave functions proposed by Laughlin to describe the
Fractional Quantum Hall Effect.Comment: Revised version, to appear in Phys. Lett. B. Souriau's scheme and its
relation of with the Faddeev-Jackiw hamiltonian reduction is explained. 11
pages, LaTex, no figure
Moving vortices in noncommutative gauge theory
Exact time-dependent solutions of nonrelativistic noncommutative Chern -
Simons gauge theory are presented in closed analytic form. They are different
from (indeed orthogonal to) those discussed recently by Hadasz, Lindstrom,
Rocek and von Unge. Unlike theirs, our solutions can move with an arbitrary
constant velocity, and can be obtained from the previously known static
solutions by the recently found ``exotic'' boost symmetry.Comment: Latex, 6 pages, no figures. A result similar to ours was obtained,
independently, by Hadasz et al. in the revised version of their pape
Non-commutative mechanics in mathematical & in condensed matter physics
Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1). Souriau's construction applied to the two-parameter central extension of the planar Galilei group leads to the ''exotic'' particle, which has non-commuting position coordinates. A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non-commutative parameter is momentum-dependent in this case, and can take the form of a monopole in momentum space
Non-commutative oscillator with Kepler-type dynamical symmetry
A 3-dimensional non-commutative oscillator with no mass term but with a
certain momentum-dependent potential admits a conserved Runge-Lenz vector,
derived from the dual description in momentum space. The latter corresponds to
a Dirac monopole with a fine-tuned inverse-square plus Newtonian potential,
introduced by McIntosh, Cisneros, and by Zwanziger some time ago. The
trajectories are (arcs of) ellipses, which, in the commutative limit, reduce to
the circular hodographs of the Kepler problem. The dynamical symmetry allows
for an algebraic determination of the bound-state spectrum and actually extends
to the conformal algebra o(4,2).Comment: 10 pages, 3 figures. Published versio
Nonrelativistic anyons in external electromagnetic field
The first-order, infinite-component field equations we proposed before for
non-relativistic anyons (identified with particles in the plane with
noncommuting coordinates) are generalized to accommodate arbitrary background
electromagnetic fields. Consistent coupling of the underlying classical system
to arbitrary fields is introduced; at a critical value of the magnetic field,
the particle follows a Hall-like law of motion. The corresponding quantized
system reveals a hidden nonlocality if the magnetic field is inhomogeneous. In
the quantum Landau problem spectral as well as state structure (finite vs.
infinite) asymmetry is found. The bound and scattering states, separated by the
critical magnetic field phase, behave as further, distinct phases.Comment: 19 pages, typos corrected; to appear in Nucl. Phys.
Enlarged Galilean symmetry of anyons and the Hall effect
Enlarged planar Galilean symmetry, built of both space-time and field
variables and also incorporating the ``exotic'' central extension is
introduced. It is used to describe non-relativistic anyons coupled to an
electromagnetic field. Our theory exhibits an anomalous velocity relation of
the type used to explain the Anomalous Hall Effect. The Hall motions,
characterized by a Casimir of the enlarged algebra, become mandatory for some
critical value(s) of the magnetic field. The extension of our scheme yields the
semiclassical effective model of the Bloch electron.Comment: LaTeX, 7 pages. No figures. One more reference adde
The non-linear Schr\"odinger equation and the conformal properties of non-relativistic space-time
The cubic non-linear Schr\"odinger equation where the coefficient of the
nonlinear term is a function only passes the Painlev\'e test of Weiss,
Tabor, and Carnevale only for , where and are constants.
This is explained by transforming the time-dependent system into the
constant-coefficient NLS by means of a time-dependent non-linear
transformation, related to the conformal properties of non-relativistic
space-time. A similar argument explains the integrability of the NLS in a
uniform force field or in an oscillator background.Comment: Thoroughly revised version, in the light of new interest in
non-relativistic conformal tranformation, with a new reference list. 8 pages,
LaTex, no figures. To be published in Int. J. Theor. Phy
Galilean noncommutative gauge theory: symmetries & vortices
Noncommutative Chern-Simons gauge theory coupled to nonrelativistic scalars
or spinors is shown to admit the ``exotic'' two-parameter-centrally extended
Galilean symmetry, realized in a unique way consistent with the Seiberg-Witten
map. Nontopological spinor vortices and topological external-field vortices are
constructed by reducing the problem to previously solved self-dual equations.Comment: Updated version: some statements rephrased and further references
added. LaTex, 17 pages, no figure
Non-relativistic Maxwell-Chern-Simons Vortices
The non-relativistic Maxwell-Chern-Simons model recently introduced by Manton
is shown to admit self-dual vortex solutions with non-zero electric field. The
interrelated ``geometric'' and ``hidden'' symmetries are explained. The theory
is also extended to (non-relativistic) spinors. A relativistic, self-dual
model, whose non-relativistic limit is the Manton model is also presented. The
relation to previous work is discussed.Comment: 20 pages plain TeX. Revised: minor errors corrected and symmetries
explained in a clearer way. Version as will appear in Ann. Phys. (N.Y.