34 research outputs found
Review on possible gravitational anomalies
This is an updated introductory review of 2 possible gravitational anomalies
that has attracted part of the Scientific community: the Allais effect that
occur during solar eclipses, and the Pioneer 10 spacecraft anomaly,
experimented also by Pioneer 11 and Ulysses spacecrafts. It seems that, to
date, no satisfactory conventional explanation exist to these phenomena, and
this suggests that possible new physics will be needed to account for them. The
main purpose of this review is to announce 3 other new measurements that will
be carried on during the 2005 solar eclipses in Panama and Colombia (Apr. 8)
and in Portugal (Oct.15).Comment: Published in 'Journal of Physics: Conferences Series of the American
Institute of Physics'. Contribution for the VI Mexican School on Gravitation
and Mathematical Physics "Approaches to Quantum Gravity" (Playa del Carmen,
Quintana Roo, Mexico, Nov. 21-27, 2004). Updates to this information will be
posted in http://www.lsc-group.phys.uwm.edu/~xavier.amador/anomalies.htm
Hopf instantons in Chern-Simons theory
We study an Abelian Chern-Simons and Fermion system in three dimensions. In
the presence of a fixed prescribed background magnetic field we find an
infinite number of fully three-dimensional solutions. These solutions are
related to Hopf maps and are, therefore, labelled by the Hopf index. Further we
discuss the interpretation of the background field.Comment: one minor error corrected, discussion of gauge fixing added, some
references adde
Complete sets of invariants for dynamical systems that admit a separation of variables
Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated HamiltonâJacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = ij. The 2nâ1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the HamiltonâJacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion
Fermion Zero Modes in Odd Dimensions
We study the zero modes of the Abelian Dirac operator in any odd dimension.
We use the stereographic projection between a dimensional space and a
sphere embedded in a dimensional space. It is shown that the
Dirac operator with a gauge field of uniform field strengths in has
symmetries of SU()U(1) which is a subgroup of SO(). Using group
representation theory, we obtain the number of fermion zero modes, as well as
their explicit forms, in a simple way.Comment: 14 page
Multiple zero modes of the Dirac operator in three dimensions
One of the key properties of Dirac operators is the possibility of a
degeneracy of zero modes. For the Abelian Dirac operator in three dimensions
the construction of multiple zero modes has been sucessfully carried out only
very recently. Here we generalise these results by discussing a much wider
class of Dirac operators together with their zero modes. Further we show that
those Dirac operators that do admit zero modes may be related to Hopf maps,
where the Hopf index is related to the number of zero modes in a simple way.Comment: Latex file, 20 pages, no figure
Particle creation via relaxing hypermagnetic knots
We demonstrate that particle production for fermions coupled chirally to an
Abelian gauge field like the hypercharge field is provided by the microscopic
mechanism of level crossing. For this purpose we use recent results on zero
modes of Dirac operators for a class of localized hypermagnetic knots.Comment: Latex, 10 pages, no figure
Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature
A quantum sl(2,R) coalgebra is shown to underly the construction of a large
class of superintegrable potentials on 3D curved spaces, that include the
non-constant curvature analogues of the spherical, hyperbolic and (anti-)de
Sitter spaces. The connection and curvature tensors for these "deformed" spaces
are fully studied by working on two different phase spaces. The former directly
comes from a 3D symplectic realization of the deformed coalgebra, while the
latter is obtained through a map leading to a spherical-type phase space. In
this framework, the non-deformed limit is identified with the flat contraction
leading to the Euclidean and Minkowskian spaces/potentials. The resulting
Hamiltonians always admit, at least, three functionally independent constants
of motion coming from the coalgebra structure. Furthermore, the intrinsic
oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of
non-constant curvature are identified, and several examples of them are
explicitly presented.Comment: 14 pages. Based in the contribution presented at the Group 27
conference, Yerevan, Armenia, August 13-19, 200
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"
We review recent results on superintegrable quantum systems in a
two-dimensional Euclidean space with the following properties. They are
integrable because they allow the separation of variables in Cartesian
coordinates and hence allow a specific integral of motion that is a second
order polynomial in the momenta. Moreover, they are superintegrable because
they allow an additional integral of order . Two types of such
superintegrable potentials exist. The first type consists of "standard
potentials" that satisfy linear differential equations. The second type
consists of "exotic potentials" that satisfy nonlinear equations. For , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975
Massless geodesics in as a superintegrable system
A Carter like constant for the geodesic motion in the
Einstein-Sasaki geometries is presented. This constant is functionally
independent with respect to the five known constants for the geometry. Since
the geometry is five dimensional and the number of independent constants of
motion is at least six, the geodesic equations are superintegrable. We point
out that this result applies to the configuration of massless geodesic in
studied by Benvenuti and Kruczenski, which are matched to
long BPS operators in the dual N=1 supersymmetric gauge theory.Comment: 20 pages, no figures. Small misprint is corrected in the Killing-Yano
tensor. No change in any result or conclusion