La géométrie non commutative : un site de construction pour aller au delà de la vision ordinaire de l'espace-temps / Noncommutative geometry a building site to go beyond our ordinary vision of space-time

Cher lecteur, ce billet est avant tout une passerelle vers ce qui est à mes yeux la meilleure conférence d'Alain Connes disponible en ligne (cliquer sur le titre en gras de la conférence, sous la capture d'écran).  Il y défend avec conviction sa vision spectrale de l'espace-temps en analysant avec passion certains des résultats les plus profonds obtenus par les physiciens (renormalisation des théories quantiques des champs de jauge de Yang-Mills-Higgs et validation empirique du Modèle Standard de la physique des particules) à travers le prisme de son programme de géométrie non commutative. Il y est d'abord question de comprendre conceptuellement les avancées de la physique des hautes énergies pour aller au delà de notre vision naïve d'un espace-temps phénoménologique à quatre dimensions.

Dear reader, this post is mainly a gateway to the most inspiring conference of Alain Connes available on-line (personal opinion). He defends convincingly his vision of a spectral spacetime, analysing the deepest theoretical and experimental results obtained by physisicts (renormalisation of Yang-Mills-Higgs quantum gauge field theories and validation of the Standard Model of particle physics) through the prism of his noncommutative geometric program. The purpose is to understand conceptually the achievements of high energy physics to go beyond our naive vision of a four dimensional phenomenological space-time.

KITP Program: Mathematical Structures in String Theory (Aug 1 - Dec 16, 2005) 

Conference  Non-Commutative Geometry & Space-Time (Nov 17, 2005) 

Dr. Alain Connes, IHES, Bures-sur-Yvette

//last edit 26 April 2016

I provide below the introduction of a text written by Connes almost at the same time of the former lecture which is probably its most faithful transcription :

The transition Classical → Quantum is very simple to formulate in terms of the Feynman integral which affects each classical field configuration with the probability amplitude e^iS/ℏ. While this prescription works remarkably well for the quantization of the classical fields involved in the standard model provided one uses the technique of renormalization,this latter perturbative technique fails dramatically when one tries to deal with the gravitational field g_µν.

In many ways this result is not surprising. Indeed many of the basic notions of the traditional formalism of Quantum Field Theory (QFT), such as particles, scattering matrices, etc... heavily rely on the flat geometry of Minkowski space and the related Poincaré symmetry group. Treating the quantization of the g_µν in the same way would -if successful- produce a quantum field theory of the g_µν on Minkowski space: a strange result indeed when viewed from the geometric standpoint! The technical reason for the notorious difficulty of quantizing the g_µν in the traditional perturbative way is the clash with either renormalizability or unitarity. 

In some sense this clash contains a serious warning, namely that one should not try to rush but rather meditate the lessons of both general relativity and QFT before even starting to compute something. In this very short essay we shall describe a “spectral” point of view on geometry which allows to start taking into account the lessons from both sides. We shall first do that for renormalization and explain in rough outline the content of our recent collaborations with Dirk Kreimer and Matilde Marcolli. As far as general relativity is concerned, since the functional integral cannot be treated in the traditional perturbative manner, it relies heavily as a “sum over geometries” on the chosen paradigm of geometric space. This will give us the occasion to discuss, in the light of noncommutative geometry, the issue of “observables” in gravity and our joint work with Ali Chamseddine on the spectral action, with a first attempt to write down a functional integral on the space of noncommutative geometries.

Noncommutative geometry and physics

Alain Connes, 2006