Measuring a Kaluza-Klein radius smaller than the Planck length

TL;DR

This paper discusses a Kaluza-Klein model predicting that a fermion wave packet can split into two with different velocities if the extra dimension's radius is finite, implying the radius is extremely smaller than the Planck length if no splitting is observed.

Contribution

It introduces a model linking wave packet splitting to the size of extra dimensions, providing a method to measure or bound the Kaluza-Klein radius.

Findings

Wave packet splitting depends on the Kaluza-Klein radius.

No observed splitting implies the radius is at least 25 orders smaller than the Planck length.

The model unifies Dirac and Einstein equations in a Kaluza-Klein framework.

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 velocity splitting of single fermion wave packets. That is, the model predicts a single fermion wave packet will split into two wave packets with slightly different group velocities. 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 at least twenty five orders of magnitude smaller than the Planck length.