Gravity and Electromagnetism in Noncommutative Geometry

TL;DR

This paper develops a unified geometric framework combining gravity and electromagnetism using noncommutative geometry, replacing extra dimensions with discrete points, and deriving an action similar to Kaluza-Klein theory.

Contribution

It introduces a noncommutative differential calculus approach to unify gravity and electromagnetism, serving as a discrete analogue of Kaluza-Klein theory.

Findings

Derived an action matching dimensionally reduced Kaluza-Klein theory.

Presented a noncommutative geometric formulation of gravity and electromagnetism.

Replaced continuous extra dimension with two discrete points.

Abstract

We present a unified description of gravity and electromagnetism in the framework of a $Z_2$ noncommutative differential calculus. It can be considered as a ``discrete version" of Kaluza-Klein theory, where the fifth continuous dimension is replaced by two discrete points. We derive an action which coincides with the dimensionally reduced one of the ordinary Kaluza-Klein theory.