Measuring a Kaluza-Klein radius smaller than the Planck length

Abstract

Hestenes has shown that a bispinor field on a Minkowski space-time is equivalent to an orthonormal tetrad of one-forms together with a complex scalar field. More recently, the Dirac and Einstein equations were unified in a tetrad formulation of a Kaluza-Klein model which gives precisely the usual Dirac-Einstein Lagrangian. In this model, Dirac's bispinor equation is obtained in the limit for which the radius of higher compact dimensions of the Kaluza-Klein manifold becomes vanishingly small compared with the Planck length. For a small but finite radius, the Kaluza-Klein model predicts the velocity splitting of single fermion wave packets. That is, the model predicts that a single fermion wave packet will split into two wave packets with slightly different group velocities. The observation of such wave packet splits would determine the size of the Kaluza-Klein radius. If wave packet splits were not observed in experiments with currently achievable accuracies, the Kaluza-Klein radius would be bounded by at most ${10}^{\ensuremath{-}25}$ times the Planck length.