Kaluza-Klein theories in the framework of Polymer Quantum Mechanics

Abstract

We provide a re-analysis of the $5D$ Kaluza-Klein theory by implementing the polymer representation of the dynamics, both on a classical and a quantum level, in order to introduce in the model information about the existence of a cut-off scale. We start by showing that, in the framework of semi-classical quantum mechanics, the $5D$ Bianchi I model admits a solution in which three space directions expand isotropically, while the remaining one is static, offering in this way a very valuable scenario to implement a Kaluza-Klein paradigm, identifying in such a static dimension the compactified one. We then analyse the behaviour of geodesic motion in the context of the polymer representation, as referred to a $5D$ space-time with a static dimension. We demonstrate that such a revised formulation allows overcoming one of the puzzling questions of the standard Kaluza-Klein model corresponding to the limit of the charge to mass ratio for a particle, inapplicable to any fundamental one. Indeed, here, such a ratio can be naturally attributed to particles predicted by the Standard Model and no internal contradiction of the theory arises on this level. Finally, we study the morphology of the field equation associated with a charged scalar particle, i.e. we analyse a Klein-Gordon equation, which fifth coordinate is viewed in the polymer representation. Here we obtain the surprising result that, although the Kaluza-Klein tower has a deformed structure characterized by irregular steps, the value predicted for the particle mass can be, in principle, set within the Standard Model mass distribution and hence, the problem of the Planckian value of such mass, typical of the standard formulation, is now overcome. However, a problem with the charge to mass ratio still survives in this quantum field formulation.