Notes on "quantum gravity" and non-commutative geometry

TL;DR

This paper provides a selective, phenomenological survey of non-commutative geometry's potential role in quantum gravity, emphasizing skepticism towards current approaches and advocating for a unified treatment of commutative and non-commutative manifolds.

Contribution

It offers a global overview of non-commutative geometry in quantum gravity, highlighting the importance of treating commutative and non-commutative spaces equally and questioning prevailing assumptions.

Findings

Non-commutative geometry offers a promising framework for quantum gravity.

Experimental results challenge some current quantum gravity theories.

Unified treatment of manifolds is crucial for progress in the field.

Abstract

I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide against just selling my mathematical wares, and for a survey, necessarily very selective, but taking a global phenomenological approach to its subject matter. After all, non-commutative geometry does not purport yet to solve the riddle of quantum gravity; it is more of an insurance policy against the probable failure of the other approaches. The plan is as follows: the introduction invites students to the fruitful doubts and conundrums besetting the application of even classical gravity. Next, the first experiments detecting quantum gravitational states inoculate us a healthy dose of skepticism on some of the current ideologies. In Section 3 we look at the action for general relativity as a consequence of gauge theory for quantum tensor fields. Section 4 briefly deals with the unimodular variants. Section 5 arrives at non-commutative geometry. I am convinced that, if this is to play a role in quantum gravity, commutative and non-commutative manifolds must be treated on the same footing; which justifies the place granted to the reconstruction theorem. Together with Section 3, this part constitutes the main body of the notes. Only very summarily at the end of this section we point to some approaches to gravity within the non-commutative realm. The last section delivers a last dose of skepticism. My efforts will have been rewarded if someone from the young generation learns to mistrust current mindsets.