Notes on "quantum gravity" and non-commutative geometry

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.

54 pages. Proceedings of the Summer School "New paths towards quantum gravity", Holbaek, Denmark; B. Booss-Bavnbek, G. Esposito and M. Lesch, eds. Springer, Berlin, 2010; pp. 3-58

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Cited by in corpus (10)

• An introduction to quantum gravity
• Dynamical Noncommutative Graphene
• Modular Theory, Non-Commutative Geometry and Quantum Gravity
• Two-dimensional pauli equation in noncommutative phase-space
• On the Three-Dimensional Pauli Equation in Noncommutative Phase-Space
• Windows on Quantum Gravity
• Non-commutative Einstein equations and Seiberg-Witten map
• Quantum general covariance
• Seiberg-Witten Map with Lorentz-Invariance and Gauge-Covariant Star Product
• Field Theory with Coordinate Dependent Noncommutativity