52 research outputs found

### Newton-Hooke type symmetry of anisotropic oscillators

The rotation-less Newton--Hooke - type symmetry found recently in the Hill
problem and instrumental for explaining the center-of-mass decomposition is
generalized to an arbitrary anisotropic oscillator in the plane. Conversely,
the latter system is shown, by the orbit method, to be the most general one
with such a symmetry. Full Newton-Hooke symmetry is recovered in the isotropic
case. Star escape from a Galaxy is studied as application.Comment: Updated version with more figures added. 34 pages, 7 figures.
Dedicated to the memory of J.-M. Souriau, deceased on March 15 2012, at the
age of 9

### Maxwell - Chern - Simons topologically massive gauge fields in the first-order formalism

We find the canonical and Belinfante energy-momentum tensors and their
nonzero traces. We note that the dilatation symmetry is broken and the
divergence of the dilatation current is proportional to the topological mass of
the gauge field. It was demonstrated that the gauge field possesses the `scale
dimensionality' d=1/2. Maxwell - Chern - Simons topologically massive gauge
field theory in 2+1 dimensions is formulated in the first-order formalism. It
is shown that 6x6-matrices of the relativistic wave equation obey the Duffin -
Kemmer - Petiau algebra. The Hermitianizing matrix of the relativistic wave
equation is given. The projection operators extracting solutions of field
equations for states with definite energy-momentum and spin are obtained. The
5x5-matrix Schrodinger form of the equation is derived after the exclusion of
non-dynamical components, and the quantum-mechanical Hamiltonian is obtained.
Projection operators extracting physical states in the Schrodinger picture are
found.Comment: 18 pages, correction in Ref. [5

### Interpolating Action for Strings and Membranes - a Study of Symmetries in the Constrained Hamiltonian Approach

A master action for bosonic strings and membranes, interpolating between the
Nambu--Goto and Polyakov formalisms, is discussed. The role of the gauge
symmetries vis-\`{a}-vis reparametrization symmetries of the various actions is
analyzed by a constrained Hamiltonian approach. This analysis reveals the
difference between strings and higher branes, which is essentially tied to a
degree of freedom count. The cosmological term for membranes follows naturally
in this scheme. The conncetion of our aproach with the Arnowitt--Deser--Misner
representation in general relativity is illuminated.Comment: LaTex, 23 pages; discussion on ADM representation included and new
references adde

### Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem

$N$ "exotic" [alias non-commutative] particles with masses $m_a$, charges
$e_a$ and non-commutative parameters $\theta_a$, moving in a uniform magnetic
field $B$, separate into center-of-mass and internal motions if Kohn's
condition e_a/m_a=\const is supplemented with e_a\theta_a=\const. Then the
center-of-mass behaves as a single exotic particle carrying the total mass and
charge of the system, $M$ and $e$, and a suitably defined non-commutative
parameter $\Theta$. For vanishing electric field off the critical case $e\Theta
B\neq1$, the particles perform the usual cyclotronic motion with modified but
equal frequency. The system is symmetric under suitable time-dependent
translations which span a (4+2)- parameter centrally extended subgroup of the
"exotic" [i.e., two-parameter centrally extended] Newton-Hooke group. In the
critical case $B=B_c=(e\Theta)^{-1}$ the system is frozen into a static
"crystal" configuration. Adding a constant electric field, all particles
perform, collectively, a cyclotronic motion combined with a drift perpendicular
to the electric field when $e\Theta B\neq1$. For $B=B_c$ the cyclotronic motion
is eliminated and all particles move, collectively, following the Hall law. Our
time-dependent symmetries are reduced to the (2+1)-parameter Heisenberg group
of centrally-extended translations.Comment: 12 pages, no figures. A minor error and some typos corrected, one
reference adde

### Time dependent solitons of noncommutative Chern-Simons theory coupled to scalar fields

We study one- and two-soliton solutions of noncommutative Chern-Simons theory
coupled to a nonrelativistic or a relativistic scalar field. In the
nonrelativistic case, we find a tower of new stationary time-dependent
solutions, all with the same charge density, but with increasing energies. The
dynamics of these solitons cannot be studied using traditional moduli space
techniques, but we do find a nontrivial symplectic form on the phase space
indicating that the moduli space is not flat. In the relativistic case we find
the metric on the two soliton moduli space.Comment: 22 pages, 2 figures, JHEP3 style. v2: This paper is a thoroughly
revised version. We thank P.A. Horvathy, L. Martina and P.C. Stichel for
illuminating comments that led us to reconsider some of our previously
reported results; see note added at the end of the paper. v3:
Acknowledgements adde

### Acceleration-Enlarged Symmetries in Nonrelativistic Space-Time with a Cosmological Constant

By considering the nonrelativistic limit of de-Sitter geometry one obtains
the nonrelativistic space-time with a cosmological constant and Newton-Hooke
(NH) symmetries. We show that the NH symmetry algebra can be enlarged by the
addition of the constant acceleration generators and endowed with central
extensions (one in any dimension (D) and three in D=(2+1)). We present a
classical Lagrangian and Hamiltonian framework for constructing models
quasi-invariant under enlarged NH symmetries which depend on three parameters
described by three nonvanishing central charges. The Hamiltonian dynamics then
splits into external and internal sectors with new non-commutative structures
of external and internal phase spaces. We show that in the limit of vanishing
cosmological constant the system reduces to the one presented in [1] which
possesses accelaration-enlarged Galilean symmetries.Comment: 13 pages; small changes like a couple of footnotes et

### 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

### Super-extended noncommutative Landau problem and conformal symmetry

A supersymmetric spin-1/2 particle in the noncommutative plane, subject to an
arbitrary magnetic field, is considered, with particular attention paid to the
homogeneous case. The system has three different phases, depending on the
magnetic field. Due to supersymmetry, the boundary critical phase which
separates the sub- and super-critical cases can be viewed as a reduction to the
zero-energy eigensubspace. In the sub-critical phase the system is described by
the superextension of exotic Newton-Hooke symmetry, combined with the conformal
so(2,1) ~ su(1,1) symmetry; the latter is changed into so(3) ~ su(2) in the
super-critical phase. In the critical phase the spin degrees of freedom are
frozen and supersymmetry disappears.Comment: 12 pages, references added, published versio

### 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 $F(t,x)$ only passes the Painlev\'e test of Weiss,
Tabor, and Carnevale only for $F=(a+bt)^{-1}$, where $a$ and $b$ 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

### Landau Analog Levels for Dipoles in the Noncommutative Space and Phase Space

In the present contribution we investigate the Landau analog energy
quantization for neutral particles, that possesses a nonzero permanent magnetic
and electric dipole moments, in the presence of an homogeneous electric and
magnetic external fields in the context of the noncommutative quantum
mechanics. Also, we analyze the Landau--Aharonov--Casher and
Landau--He--McKellar--Wilkens quantization due to noncommutative quantum
dynamics of magnetic and electric dipoles in the presence of an external
electric and magnetic fields and the energy spectrum and the eigenfunctions are
obtained. Furthermore, we have analyzed Landau quantization analogs in the
noncommutative phase space, and we obtain also the energy spectrum and the
eigenfunctions in this context.Comment: 20 pages, references adde

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