Noncommutative Geometry and a Discretized Version of Kaluza-Klein Theory with a Finite Field Content

TL;DR

This paper develops a noncommutative geometric framework on a space-time with two discrete points, leading to a discretized Kaluza-Klein model with finite fields, massive modes, and potential cosmological implications.

Contribution

It introduces a novel noncommutative geometric approach to discretize Kaluza-Klein theory with a finite field content and explores its complex structure.

Findings

Finite field content derived from metric-compatible torsion-free connection.

Emergence of massive modes in the discretized model.

Possible implications for cosmological constant presence.

Abstract

We consider a four-dimensional space-time supplemented by two discrete points assigned to a $Z_2$ algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalised vielbein, we study the metric structure. Metric compatible torsion free connection defines a unique finite field content in the model and leads to a discretized version of Kaluza-Klein theory. We study some special cases of this model that illustrate the rich and complex structure with massive modes and the possible presence of a cosmological constant.