1206.0022 (Marco Cariglia)

Hidden symmetries of Eisenhart lift metrics and the Dirac equation with
flux    [PDF]

Marco Cariglia

The Eisenhart lift is an example of geometrisation of interactions. In this paper we study the Eisenhart lift from the point of view of hidden symmetries of the Dirac equation. We find the non-trivial result that - under the hypothesis that symmetry operators linear in the momenta cannot lift to symmetry operators of higher order - the phase space dynamics of the original and the lift theory are different. It is possible both to lift the Dirac equation with flux to the massless Dirac equation in higher dimension and to lift isometries, however in general each of the two theories can have hidden symmetries that are not present in the other. As a by-product of this analysis we prove a number of further results. First, we show explicitly how to dimensionally reduce the massless Dirac equation in n+2 dimensions to get the lower dimensional Dirac equation with flux, and the inverse operation. Second, we construct new Lorentzian metrics with special tensors by lifting Killing-Yano and Closed Conformal Killing-Yano tensors. Third, we describe the general Conformal Killing Yano tensor of the Eisenhart 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.

View original: http://arxiv.org/abs/1206.0022