42 research outputs found
Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry
Gauge fields have a natural metric interpretation in terms of horizontal
distance. The latest, also called Carnot-Caratheodory or subriemannian
distance, is by definition the length of the shortest horizontal path between
points, that is to say the shortest path whose tangent vector is everywhere
horizontal with respect to the gauge connection. In noncommutative geometry all
the metric information is encoded within the Dirac operator D. In the classical
case, i.e. commutative, Connes's distance formula allows to extract from D the
geodesic distance on a riemannian spin manifold. In the case of a gauge theory
with a gauge field A, the geometry of the associated U(n)-vector bundle is
described by the covariant Dirac operator D+A. What is the distance encoded
within this operator ? It was expected that the noncommutative geometry
distance d defined by a covariant Dirac operator was intimately linked to the
Carnot-Caratheodory distance dh defined by A. In this paper we precise this
link, showing that the equality of d and dh strongly depends on the holonomy of
the connection. Quite interestingly we exhibit an elementary example, based on
a 2 torus, in which the noncommutative distance has a very simple expression
and simultaneously avoids the main drawbacks of the riemannian metric (no
discontinuity of the derivative of the distance function at the cut-locus) and
of the subriemannian one (memory of the structure of the fiber).Comment: published version with additional figures to make the proof more
readable. Typos corrected in this ultimate versio
Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
An infinite family of classical superintegrable Hamiltonians defined on the
N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a
common set of (2N-3) functionally independent constants of the motion. Among
them, two different subsets of N integrals in involution (including the
Hamiltonian) can always be explicitly identified. As particular cases, we
recover in a straightforward way most of the superintegrability properties of
the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of
constant curvature and we introduce as well new classes of (quasi-maximally)
superintegrable potentials on these spaces. Results here presented are a
consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians,
together with an appropriate use of the phase spaces associated to Poincare and
Beltrami coordinates.Comment: 12 page
Casimir energy in a small volume multiply connected static hyperbolic pre-inflationary Universe
A few years ago, Cornish, Spergel and Starkman (CSS), suggested that a
multiply connected ``small'' Universe could allow for classical chaotic mixing
as a pre-inflationary homogenization process. The smaller the volume, the more
important the process. Also, a smaller Universe has a greater probability of
being spontaneously created. Previously DeWitt, Hart and Isham (DHI) calculated
the Casimir energy for static multiply connected flat space-times. Due to the
interest in small volume hyperbolic Universes (e.g. CSS), we generalize the DHI
calculation by making a a numerical investigation of the Casimir energy for a
conformally coupled, massive scalar field in a static Universe, whose spatial
sections are the Weeks manifold, the smallest Universe of negative curvature
known. In spite of being a numerical calculation, our result is in fact exact.
It is shown that there is spontaneous vacuum excitation of low multipolar
components.Comment: accepted for publication in phys. rev.
Energy distribution of maxima and minima in a one-dimensional random system
We study the energy distribution of maxima and minima of a simple
one-dimensional disordered Hamiltonian. We find that in systems with short
range correlated disorder there is energy separation between maxima and minima,
such that at fixed energy only one kind of stationary points is dominant in
number over the other. On the other hand, in the case of systems with long
range correlated disorder maxima and minima are completely mixed.Comment: 4 pages RevTeX, 1 eps figure. To appear in Phys. Rev.
Conformal compactification and cycle-preserving symmetries of spacetimes
The cycle-preserving symmetries for the nine two-dimensional real spaces of
constant curvature are collectively obtained within a Cayley-Klein framework.
This approach affords a unified and global study of the conformal structure of
the three classical Riemannian spaces as well as of the six relativistic and
non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both
Newton-Hooke and Galilean), and gives rise to general expressions holding
simultaneously for all of them. Their metric structure and cycles (lines with
constant geodesic curvature that include geodesics and circles) are explicitly
characterized. The corresponding cyclic (Mobius-like) Lie groups together with
the differential realizations of their algebras are then deduced; this
derivation is new and much simpler than the usual ones and applies to any
homogeneous space in the Cayley-Klein family, whether flat or curved and with
any signature. Laplace and wave-type differential equations with conformal
algebra symmetry are constructed. Furthermore, the conformal groups are
realized as matrix groups acting as globally defined linear transformations in
a four-dimensional "conformal ambient space", which in turn leads to an
explicit description of the "conformal completion" or compactification of the
nine spaces.Comment: 43 pages, LaTe
Integrable potentials on spaces with curvature from quantum groups
A family of classical integrable systems defined on a deformation of the
two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed
through Hamiltonians defined on the non-standard quantum deformation of a sl(2)
Poisson coalgebra. All these spaces have a non-constant curvature that depends
on the deformation parameter z. As particular cases, the analogues of the
harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed.
Another deformed Hamiltonian is also shown to provide superintegrable systems
on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant
curvature that exactly coincides with z. According to each specific space, the
resulting potential is interpreted as the superposition of a central harmonic
oscillator with either two more oscillators or centrifugal barriers. The
non-deformed limit z=0 of all these Hamiltonians can then be regarded as the
zero-curvature limit (contraction) which leads to the corresponding
(super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde
Dirac equation from the Hamiltonian and the case with a gravitational field
Starting from an interpretation of the classical-quantum correspondence, we
derive the Dirac equation by factorizing the algebraic relation satisfied by
the classical Hamiltonian, before applying the correspondence. This derivation
applies in the same form to a free particle, to one in an electromagnetic
field, and to one subjected to geodesic motion in a static metric, and leads to
the same, usual form of the Dirac equation--in special coordinates. To use the
equation in the static-gravitational case, we need to rewrite it in more
general coordinates. This can be done only if the usual, spinor transformation
of the wave function is replaced by the 4-vector transformation. We show that
the latter also makes the flat-space-time Dirac equation Lorentz-covariant,
although the Dirac matrices are not invariant. Because the equation itself is
left unchanged in the flat case, the 4-vector transformation does not alter the
main physical consequences of that equation in that case. However, the equation
derived in the static-gravitational case is not equivalent to the standard
(Fock-Weyl) gravitational extension of the Dirac equation.Comment: 27 pages, standard LaTeX. v2: minor style changes, accepted for
publication in Found. Phys. Letter
Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
The superposition of the Kepler-Coulomb potential on the 3D Euclidean space
with three centrifugal terms has recently been shown to be maximally
superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by
finding an additional (hidden) integral of motion which is quartic in the
momenta. In this paper we present the generalization of this result to the ND
spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry
approach that makes use of the curvature parameter. The resulting Hamiltonian,
formed by the (curved) Kepler-Coulomb potential together with N centrifugal
terms, is shown to be endowed with (2N-1) functionally independent integrals of
the motion: one of them is quartic and the remaining ones are quadratic. The
transition from the proper Kepler-Coulomb potential, with its associated
quadratic Laplace-Runge-Lenz N-vector, to the generalized system is fully
described. The role of spherical, nonlinear (cubic), and coalgebra symmetries
in all these systems is highlighted.Comment: 14 pages; PACS: 02.30.Ik 02.40.K
(Super)integrability from coalgebra symmetry: formalism and applications
The coalgebra approach to the construction of classical integrable systems
from Poisson coalgebras is reviewed, and the essential role played by
symplectic realizations in this framework is emphasized. Many examples of
Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given,
and their Liouville superintegrability is discussed. Among them,
(quasi-maximally) superintegrable systems on N-dimensional curved spaces of
nonconstant curvature are analysed in detail. Further generalizations of the
coalgebra approach that make use of comodule and loop algebras are presented.
The generalization of such a coalgebra symmetry framework to quantum mechanical
systems is straightforward.Comment: 33 pages. Review-contribution to the "Workshop on higher symmetries
in Physics", 6-8 November 2008, Madrid, Spai
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