Noncommutative Geometry and Loop Quantum Gravity : Loops , Algebras and Spectral Triples

Spectral triples have recently turned out to be relevant for different approaches that aim at quantizing gravity and the other fundamental forces of nature in a mathematically rigorous way. The purpose of this workshop was to bring together researchers mainly from noncommutative geometry and loop quantum gravity –two major fields that have used spectral triples independently so far– in order to share their results and open issues. Mathematics Subject Classification (2000): 58B34, 46Lxx, 58J42, 81R15, 83C45. Introduction by the Organisers The workshop “Noncommutative Geometry and Loop Quantum Gravity: Loops, Algebras and Spectral Triples” has been organized by Christian Fleischhack (Paderborn), Matilde Marcolli (Pasadena), and Ryszard Nest (Copenhagen). This meeting was attended by 23 researchers from 8 countries, including several younger postdocs and two PhD students. We enjoyed 16 talks lasting about 50 to 75 minutes plus discussions. As there were no “official” talks after lunch until 4 pm and also no talks in the evening, there was a large amount of time left for informal discussions. The task of defining both a consistent and mathematically rigorous theory of quantum gravity is one of most challenging undertakings in modern theoretical physics. It is widely expected that at Planck scale the usual notions of smooth geometries have to be replaced by something different. Various arguments point 374 Oberwolfach Report 09/2010 towards geometric notions becoming noncommutative, so that geometric measurements should correspond to noncommuting operators. In fact, noncommutative geometry (NCG) provides a remarkably successful framework for unification of all known fundamental forces. Mathematically, it mainly grounds on the pioneering work of Connes, who related Riemannian spin geometries to a certain class of spectral triples over commutative C-algebras. Extending this formalism, Chamseddine and Connes demonstrated that the standard model coupled to gravitation naturally emerges from a spectral triple over an almost commutative C-algebra together with a spectral action. This way they even entailed experimentally falsifiable predictions in elementary particle physics. However, although fully implementing the idea of unification, this approach has remained essentially classical. Moreover, as the theory of spectral triples has only been developed for Riemannian manifolds, full general relativity needing Lorentzian geometries has not been tackled. Loop quantum gravity (LQG), on the other hand, is one of the most successful theories to quantize canonical gravity. Resting on a generalization of Dirac quantization by Ashtekar and Lewandowski, its decisive idea is to break down the quantization to finite-dimensional problems on graphs and then to reconstruct the continuum theory using projective/inductive limits over all graphs. Although the kinematical part of LQG is nicely understood, the dynamical part is vastly open territory – both mathematically and conceptually. This concerns mainly three, related issues: First of all, the spectral analysis of the quantum Hamiltonian constraint, responsible for time evolution, is very immature. Secondly, it is completely unknown how to reconstruct classical general relativity as a semiclassical limit of loop quantum gravity. And, instead of an emergent unification, matter has to be included by hand. Although NCG and LQG use very similar mathematical techniques – e. g. , operator algebras in general, or spectral encoding of geometry to be more specific –, their conceptual problems are rather complementary. Nevertheless, only recently, first steps to join the strengthes of both approaches have been made. In several papers since 2005, Aastrup and Grimstrup, later with one of the organizers (RN), have outlined how to construct a semifinite spectral triple for the full theory out of spectral triples based on a restricted system of nested graphs. One of the main tasks of the meeting was to bring together researchers from different fields – first of all, noncommutative geometry and loop quantum gravity, but also other fields like spectral triples on its own and axiomatic quantum field theory. For this, there were several introductory talks: • Hanno Sahlmann and Thomas Thiemann gave an overview on the origins and the current status of loop quantum gravity. Sahlmann focused on physical and kinematical issues, Thiemann on open issues concerning dynamics. • Giovanni Landi and Walter van Suijlekom presented introductions into noncommutative geometry. Whereas Landi spoke on general issues, Walter Noncommutative Geometry and Loop Quantum Gravity 375 van Suijlekom showed how one can encode the standard model of particle physics within the language of spectral triples. • Johannes Aastrup and Jesper Grimstrup demonstrated how spectral triples can fruitfully transfer ideas from noncommutative geometry into loop quantum gravity. • Klaus Fredenhagen and Rainer Verch introduced axiomatic quantum field theories as functors from the category of globally hyperbolic spacetimes into that of C-algebras. Fredenhagen concentrated on perturbation theory, i.e., such functors that are formal power series in ~. Verch used this framework to extend the notion of spectral triples to the Lorentzian case. Beyond these talks there have been more specialized ones: • Alan Carey described a generalization of spectral triples, so-called semifinite spectral triples. They arise naturally in the Aastrup-Grimstrup-Nest approach. • Matilde Marcolli and Jerzy Lewandowski studied further noncommutative structures arising in loop quantum gravity. Marcolli described how extended spin foams define noncommutative coordinate algebras; Lewandowski replaced the underlying structure group SU(2) of LQG by the quantum group SUq(2). • Victor Gayral and Thomas Krajewski spoke on quantum groups as well: Gayral from a more generalized perspective, Krajewski inspired by string theory. • Fedele Lizzi described noncommutative lattices that may lead to emerging spacetime. • Varghese Mathai and Raimar Wulkenhaar explained different types of deformation quantization. Mathai constructed noncommutative principal bundles and Wulkenhaar outlined why there should be non-perturbative quantum field theories over Moyal deformed R. The atmosphere within the workshop benefited very much from the liveliness of the discussions and questions, which occurred frequently before, during, and after the talks. From this point of view the meeting was very successful, on the one hand for enabling a significant exchange of ideas between researchers in the two major fields, and on the other side for presenting the results of the few scientists that work in the intersection of LQG and NCG. In particular the fact that for every talk usually at least half the audience was no specialist in the field covered in it, resulted in a very effective exchange of knowledge, from which both sides gained profit. Noncommutative Geometry and Loop Quantum Gravity 377 Workshop: Noncommutative Geometry and Loop Quantum Gravity: Loops, Algebras and Spectral Triples