Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
arXiv:1206.0022 · doi:10.1103/PhysRevD.86.084050
Abstract
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.
18 pages, no figures. Version 3: some typos corrected, some discussions clarified, part of the abstract changed
References in corpus (24)
- General Kerr-NUT-AdS Metrics in All Dimensions
- Gravitational Perturbations of Higher Dimensional Rotating Black Holes: Tensor Perturbations
- An instability of higher-dimensional rotating black holes
- Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
- `Hidden' Symmetries of Higher Dimensional Rotating Black Holes
- Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
- Hidden Symmetry of Higher Dimensional Kerr-NUT-AdS Spacetimes
- Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness
- Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime
- Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system
- Covariant hamiltonian dynamics
- Complete Set of Commuting Symmetry Operators for the Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes
- Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
- Symmetries of the Dirac operator with skew-symmetric torsion
- Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime
- A Note on Separability of Field Equations in Myers-Perry Spacetimes
- Klein tunneling in carbon nanostructures: a free particle dynamics in disguise
- Killing-Yano equations and G-structures
- Supersymmetry, holonomy and Kundt spacetimes
- Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group
- Hidden symmetry in the presence of fluxes
- Conserved quantities in non-abelian monopole fields
- Geometry of the energy landscape and folding transition in a simple model of a protein
- Uniqueness of Rotating Charged Black Holes in Five-Dimensional Minimal Gauged Supergravity