Noncommutative differential geometry with higher order derivatives

TL;DR

This paper develops a noncommutative geometric framework on the real line that incorporates second-order derivatives, enabling new models for scalar and gauge theories within this mathematical setting.

Contribution

It introduces a novel noncommutative differential geometry model including higher-order derivatives, expanding the tools for physical theories.

Findings

Constructed a toy model of noncommutative differential geometry with second derivatives

Formulated simple scalar and gauge field theories in this new geometric setting

Demonstrated the potential for noncommutative geometry to model physical phenomena

Abstract

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry.