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 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 bounded by at most 25 10 − times the Planck length.

Hestenes' Tetrad and Spin Connections

“…in the limit of an infinitely large coupling constant. Both the constraint and the limit are explicated in the Kaluza-Klein model [10] − [12].…”

“…The Kaluza-Klein tetrad model is based on a constrained Yang-Mills formulation of the Dirac Theory [10] - [12]. In this formulation Hestenes' tensor fields F given by…”

“…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 [10] - [12]. In this model, the self-adjoint (symmetric) modes of the tetrad describe gravity, whereas, as in Hestenes' work, the isometric (rotational) modes of the tetrad together with a scalar field describe fermions.…”

Dirac equation: representation independence and tensor transformation

“…Those may be defined independently of the transformation of the wave function and the gamma matrices, and it was actually the spinor transformation scheme that was used previously when discussing the Hestenes tensor fields [30,[32][33][34]. Indeed all bispinor observables, such as the electric and chiral currents, energy-momentum and spin-polarization tensors, as well as the bispinor Lagrangian, can be expressed in terms of Hestenes' scalar and tetrad fields [35].…”

On the indeterminacy of the meter

“…For example, they often rely on Lorentz invariance considerations (i.e., a minimal length would be 16 On the former, see Bekenstein (1973) and Reifler and Morris (2003); on the latter, see Bernadotte and Klinkhamer (2007), Klinkhamer (2007), Stecker (2011), Christiansen et al (2011 and Laurent et al (2011). See Cunliff (2012) for criticism of some theoretical arguments for a minimal length.…”