GAUGE THEORIES AND NON-COMMUTATIVE GEOMETRY Jean Iliopoulos To cite this version: Jean Iliopoulos. GAUGE THEORIES AND NON-COMMUTATIVE GEOMETRY. 2005. hal- 00005072v2 HAL Id: hal-00005072 https://hal.archives-ouvertes.fr/hal-00005072v2 Preprint submitted on 1 Jun 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LPTENS-05/17 GAUGE THEORIES AND NON-COMMUTATIVE GEOMETRY J. Iliopoulos Laboratoire de Physique Th´eorique CNRS-ENS 24 rue Lhomond, F-75231 Paris Cedex 05, France
[email protected] ABSTRACT It is shown that a d-dimensional classical SU(N) Yang-Mills theory can be formulated in a d + 2-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. Talk given at the Symposium in Honor of the 70th birthday of Julius Wess, 10-11 January 2005. Max-Planck Institut f¨ur Physik (Werner Heisenberg-Institut), M¨unchen. ccsd-00005072, version 2 - 1 Jun 2005 1 Introduction With Quantum Mechanics, W. Heisenberg introduced non-commutativity in phase space. This property was found to be crucial to solve the problem of the instability of the classical Rutherford atom due to the short-distance singularity of the Coulomb potential.