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PV: Plenarvorträge

PV VII

PV VII: Plenarvortrag

Mittwoch, 20. März 2002, 09:00–09:45, Oper

Noncommutative Geometry and Fundamental Interactions: The First Ten Years — •Jose Mariano Gracia-Bondia — Departamento de Fisica, Universidad de Costa Rica, 2060 San Pedro, Costa Rica

As a branch of mathematics noncommutative geometry (NCG), i.e. Alain Connes’ program of unification of mathematics under the aegis of quantum operator theory, is some twenty years old. As such, it is well established, exhibiting an impressive list of contributors and a process of vigorous development. The application to fundamental physics is barely ten years old and was triggered by well-known peculiarities and unanswered questions in the standard model of particle interactions. The approaches, both by Connes and by the humility: far from trying to dictate from first principles what the entrails of the subatomic world should be, they undertook to translate in geometric terms (necessarily noncommutative!) what Nature is trying to tell us. In the process a geometric understanding of spontaneous symmetry breaking and the interpretation of the Higgs field as yet another gauge field emerged naturally.

More recently, the “top-down” approach (going back to Snyder) has again gained dominance. String theory and NCG discovered their points of contact. Moyal-type algebras were found to be central in the dynamics of open strings rigidified by magnetic fields. It had been remarked earlier that it was possible to do quantum field theory on noncommutative spaces. Both facts combined resulted in a powerful fad. The mathematics of this variant of NCG is identical with the mathematics of (nonrelativistic, spinless) quantum mechanics, in the unfamiliar Groenewold-Moyal way, and so the latter was rediscovered by string theorists. While noncommutative spaces constitute a fantastic “theoretical laboratory” for testing what is fundamental and what is not in quantum field theory, and for exploring possible approaches to quantum gravity, the abundant literature on the subject is somewhat disappointing.

We review the main issues and developments so far: the noncommutative reconstructions of the standard model, the relation to string theory, noncommutative quantum field and gauge theory, their renormalizability or lack thereof, anomalies, “experimental” constraints, and the Connes-Kreimer reinterpretation of renormalization theory. In particular, we discuss the likelihood that NCG will live up to its promise for the physics of fundamental interactions.