Noncommutative Spaces and the Theory of Geometrodynamics

TL;DR

This paper extends Connes' noncommutative geometry framework by generalizing spectral triples and the Dirac operator, leading to a new concept of generalized spacetime and a modified spectral action that recovers the original in a specific limit.

Contribution

It introduces a broader class of spectral triples with a generalized Dirac operator, enabling the formulation of a generalized spacetime within noncommutative geometry.

Findings

Development of a generalized covariant derivative

Construction of a generalized spacetime model

Recovery of Connes' spectral action in the appropriate limit

Abstract

This article is concerned with a generalisation of Connes' noncommutative framework. This is achieved by a general study of spectral triples, in particular through an analysis of the role played by the Dirac operator. The Dirac operator is used to construct a generalised covariant derivative, which is then employed to define a generalised spacetime. The spectral action is then modified to accommodate this generalisation and it is observed that in the appropriate limit, Connes' spectral action is recovered.