Noncommutative Geometry and Quantum Field Theory

Abstract

The workshop Noncommutative Geometry and Quantum Field Theory, organised by Sergio Doplicher (Rome), Mario Paschke, Rainer Verch and Eberhard Zeidler (Leipzig) was held Oct. 23rd–Oct 29th, 2005.

There are several motivations to study the interplay of noncommutative geometry and quantum theory. First of all, it has frequently been argued from very different perspectives that physical spacetime might be noncommutative itself, i.e. it might best be described by a noncommutative $C^{\ast }$-algebra $\mathcal{A}$. The first day of the workshop was therefor mainly devoted to pedagogical reviews of these arguments, and in particular the appearance of a noncommutativity of spacetime in the most prominent approaches to a quantum theory of gravity. As a possible new research direction the problem was raised to consider the algebra $\mathcal{A}$ as part of the dynamics, that is to say to generalize (the sought for quantum version of) Einstein's equations for the metric of spacetime such that they also determine the (noncommuative) topology of spacetime dynamically via the coupling to matter fields. It has also been pointed out that such a program would stimulate new developments in mathematics.

The first and second day also saw reviews on the recent progress in pertubative constructions of quantum field theory over noncommutative spaces. It has, however, also been speculated by several speakers, that it might be possible to rigorously and nonpertubatively construct certain nontrivial interacting quantum theories over noncommuative spaces, even though this seems impossible for the corresponding theory on a commutative space.

Closely related important developments were reported, concerning Hopf algebras, progress in the Algebraic approach to Conformal Field Theory and its possible relations to a noncommutative index theory, and in the approach to Confinement and Renormalization Group in AQFT. Recent developments in String Theory, related to the subject of our meeting, were also reported.

Although quantum field theory can be formulated entirely in the language of operator algebras, geometrical entities play an essential, but not readily understood, role for the consistency and interpretation therof. Noncommutative Geometry, which studies the interrelation of operator algebras and geometry might therefor prove an indespensible tool on the way to a deeper understanding of quantum physics. A large number of talks in the workshop adressed possible applications of Noncommuative Geometry, and in particular the local index formula of Connes and Moscovici, to quantum field theory.

These applications of Noncommuative geometry require generalizations of Connes' notion of spectral triples, however. The fourth day of the meeting was therefor mainly devoted to the recent progress concerning the various generalisations required to study problems in quantum field theory, as well as to noncommutative formulations of the theory of Gravitation. The fact that the need of qantum physics will also stimulate new developments in mathematics (here the theory of operator algebras and Noncommutative Geometry), has been one of the main motivations to organise this meeting of mathematicians and physicists, which has brought about many concrete proposals for new research directions, but also accumulated many open questions. The workshop thus reflected very well the excellent prospects for progress in mathematics as well as in physics offered by this field of research.

On wednesday, after the excursion, N.P.Landsmann gave an evening talk on the classical debate between A.Einstein and N.Bohr, leading to very stimulating discussions on the foundations of quantum theory afterwards.

Discussions of this kind are rendered possible only by the unique atmosphere at the MFO. It is a pleasure to thank the director G.-M. Greuel, the admistration staff and the kitchen staff of the MFO for all their efforts to make this workshop so beneficial. We would also like to thank the state of Baden-Würtemberg, the main source of funding the MFO, and the European Union, whose support for the MFO enabled us to invite seven Ph.D. students in addition to the originally intended 26 senior researchers.