A Discretized Version of Kaluza-Klein Theory with Torsion and Massive Fields

TL;DR

This paper develops a discretized Kaluza-Klein model with torsion using noncommutative geometry, resulting in a theory that includes massive modes and complex interactions among tensor, vector, and scalar fields.

Contribution

It introduces a novel discretized Kaluza-Klein framework with internal torsion, allowing for the emergence of massive modes and a detailed geometric structure.

Findings

Presence of massive modes in the model

Unique determination of connection forms with torsion

Rich and complex action structure

Abstract

We consider an internal space of two discrete points in the fifth dimension of the Kaluza-Klein theory by using the formalism of noncommutative geometry developed in a previous paper \cite{VIWA} of a spacetime supplemented by two discrete points. With the nonvanishing internal torsion 2-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor, a pair of vector and a pair of scalar fields. Using the generalized Cartan structure equation we are able not only to determine uniquely the hermitian and metric compatible connection 1-forms, but also the nonvanishing internal torsion 2-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza-Klein type model with zero and massive modes.