182 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
Geodesic Motion on Closed Spaces: Two Numerical Examples
The geodesic structure is very closely related to the trace of the Laplace
operator, involved in the calculation of the expectation value of the energy
momentum tensor in Universes with non trivial topology. The purpose of this
work is to provide concrete numerical examples of geodesic flows. Two manifolds
with genus are given. In one the chaotic regions, form sets of negligible
or zero measure. In the second example the geodesic flow, shows the presence of
measurable chaotic regions. The approach is "experimental", numerical, and
there is no attempt to an analytical calculation.Comment: version accepted for publicatio
N-dimensional sl(2)-coalgebra spaces with non-constant curvature
An infinite family of ND spaces endowed with sl(2)-coalgebra symmetry is
introduced. For all these spaces the geodesic flow is superintegrable, and the
explicit form of their common set of integrals is obtained from the underlying
sl(2)-coalgebra structure. In particular, ND spherically symmetric spaces with
Euclidean signature are shown to be sl(2)-coalgebra spaces. As a byproduct of
this construction we present ND generalizations of the classical Darboux
surfaces, thus obtaining remarkable superintegrable ND spaces with non-constant
curvature.Comment: 11 pages. Comments and new references have been added; expressions
for scalar curvatures have been corrected and simplifie
Weyl-Gauge Symmetry of Graphene
The conformal invariance of the low energy limit theory governing the
electronic properties of graphene is explored. In particular, it is noted that
the massless Dirac theory in point enjoys local Weyl symmetry, a very large
symmetry. Exploiting this symmetry in the two spatial dimensions and in the
associated three dimensional spacetime, we find the geometric constraints that
correspond to specific shapes of the graphene sheet for which the electronic
density of states is the same as that for planar graphene, provided the
measurements are made in accordance to the inner reference frame of the
electronic system. These results rely on the (surprising) general
relativistic-like behavior of the graphene system arising from the combination
of its well known special relativistic-like behavior with the less explored
Weyl symmetry. Mathematical structures, such as the Virasoro algebra and the
Liouville equation, naturally arise in this three-dimensional context and can
be related to specific profiles of the graphene sheet. Speculations on possible
applications of three-dimensional gravity are also proposed.Comment: 22 pages, 3 figures - two new references and few typos fixed, matches
published version by Annals of Physic
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
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
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.
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
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