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 $g=0$ 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

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

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.