227 research outputs found
Eisenhart lift for higher derivative systems
The Eisenhart lift provides an elegant geometric description of a dynamical
system of second order in terms of null geodesics of the Brinkmann-type metric.
In this work, we attempt to generalize the Eisenhart method so as to encompass
higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian.
A consistent geometric description seems feasible only for a particular class
of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.Comment: V2: 12 pages, minor improvements, references added; the version to
appear in PL
Conformal Triality of the Kepler problem
We show that the Kepler problem is projectively equivalent to null geodesic
motion on the conformal compactification of Minkowski-4 space. This space
realises the conformal triality of Minkwoski, dS and AdS spaces.Comment: 4 pages, no figures. Some modification
Hidden Symmetries of Dynamics in Classical and Quantum Physics
This article reviews the role of hidden symmetries of dynamics in the study
of physical systems, from the basic concepts of symmetries in phase space to
the forefront of current research. Such symmetries emerge naturally in the
description of physical systems as varied as non-relativistic, relativistic,
with or without gravity, classical or quantum, and are related to the existence
of conserved quantities of the dynamics and integrability. In recent years
their study has grown intensively, due to the discovery of non-trivial examples
that apply to different types of theories and different numbers of dimensions.
Applications encompass the study of integrable systems such as spinning tops,
the Calogero model, systems described by the Lax equation, the physics of
higher dimensional black holes, the Dirac equation, supergravity with and
without fluxes, providing a tool to probe the dynamics of non-linear systems.Comment: 54 pages, review article. To be published in Rev. Mod. Phy
Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
The Eisenhart-Duval lift allows embedding non-relativistic theories into a
Lorentzian geometrical setting. In this paper we study the lift from the point
of view of the Dirac equation and its hidden symmetries. We show that
dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in
general gives rise to the non-relativistic Levy-Leblond equation in lower
dimension. We study in detail in which specific cases the lower dimensional
limit is given by the Dirac equation, with scalar and vector flux, and the
relation between lift, reduction and the hidden symmetries of the Dirac
equation. While there is a precise correspondence in the case of the lower
dimensional massive Dirac equation with no flux, we find that for generic
fluxes it is not possible to lift or reduce all solutions and hidden
symmetries. As a by-product of this analysis we construct new Lorentzian
metrics with special tensors by lifting Killing-Yano and Closed Conformal
Killing-Yano tensors and describe the general Conformal Killing-Yano tensor of
the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Lastly,
we show how dimensionally reducing the higher dimensional operators of the
massless Dirac equation that are associated to shared hidden symmetries it is
possible to recover hidden symmetry operators for the Dirac equation with flux.Comment: 18 pages, no figures. Version 3: some typos corrected, some
discussions clarified, part of the abstract change
Ricci-flat spacetimes admitting higher rank Killing tensors
Ricci-flat spacetimes of signature (2,q) with q=2,3,4 are constructed which
admit irreducible Killing tensors of rank-3 or rank-4. The construction relies
upon the Eisenhart lift applied to Drach's two-dimensional integrable systems
which is followed by the oxidation with respect to free parameters. In four
dimensions, some of our solutions are anti-self-dual.Comment: 12 page
Curvatronics with bilayer graphene in an effective spacetime
We show that in AB stacked bilayer graphene low energy excitations around the
semimetallic points are described by massless, four dimensional Dirac fermions.
There is an effective reconstruction of the 4 dimensional spacetime, including
in particular the dimension perpendicular to the sheet, that arises dynamically
from the physical graphene sheet and the interactions experienced by the
carriers. The effective spacetime is the Eisenhart-Duval lift of the dynamics
experienced by Galilei invariant L\'evy-Leblond spin particles
near the Dirac points. We find that changing the intrinsic curvature of the
bilayer sheet induces a change in the energy level of the electronic bands,
switching from a conducting regime for negative curvature to an insulating one
when curvature is positive. In particular, curving graphene bilayers allows
opening or closing the energy gap between conduction and valence bands, a key
effect for electronic devices. Thus using curvature as a tunable parameter
opens the way for the beginning of curvatronics in bilayer graphene.Comment: 8 pages, 3 figures. Revised version with additional materia
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