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Noncommutative Geometry, Gauge Theory and Renormalization

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Noncommutative Geometry, Gauge Theory and Renormalization Mathematik Dissertationsthema Noncommutative Geometry, gauge theory and renormalization Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakult¨at der Westf¨alischen Wilhelms-Universit¨at M¨unster vorgelegt von Axel Marcillaud de Goursac aus Versailles (Frankreich) -2009- Dekan: Prof. Dr. Dr. hc Joachim Cuntz Erster Gutachter: Prof. Dr. Pierre Bieliavsky Zweiter Gutachter: Prof. Dr. Stefan Waldmann Weitere Mitglieder der Pr¨ufungskommission: Prof. Dr. Michel Dubois-Violette (Vorsitzender) Prof. Dr. Thiery Masson Prof. Dr. Pierre Vanhove Prof. Dr. Jean-Christophe Wallet Prof. Dr. Raimar Wulkenhaar Tag der m¨undlichen Pr¨ufung: 10.06.2009 Tag der Promotion: Contents 1 Noncommutative Geometry 15 1.1 Topology and C∗-algebras . 15 1.1.1 Definitions . 15 1.1.2 Spectral theory . 16 1.1.3 Duality in the commutative case . 17 1.1.4 GNS construction . 18 1.1.5 Vector bundles and projective modules . 19 1.2 Measure theory and von Neumann algebras . 20 1.2.1 Definition of von Neumann algebras . 20 1.2.2 Duality in the commutative case . 21 1.3 Noncommutative differential geometry . 22 1.3.1 Algebraic geometry . 22 1.3.2 Differential calculi . 23 1.3.3 Hochschild and cyclic homologies . 27 1.3.4 Spectral triples . 31 2 Epsilon-graded algebras 33 2.1 General theory of the ε-graded algebras . 33 2.1.1 Commutation factors and multipliers . 33 2.1.2 Definition of ε-graded algebras and properties . 38 2.1.3 Relationship with superalgebras . 42 2.2 Noncommutative geometry based on ε-derivations . 44 2.2.1 Differential calculus . 44 2.2.2 ε-connections and gauge transformations . 47 2.2.3 Involutions . 49 2.3 Application to some examples of ε-graded algebras . 52 2.3.1 ε-graded commutative algebras . 52 2.3.2 ε-graded matrix algebras with elementary grading . 52 2.3.3 ε-graded matrix algebras with fine grading . 59 3 Renormalization of QFT 65 3.1 Wilsonian renormalization of scalar QFT . 65 3.1.1 Scalar field theory . 65 3.1.2 Effective action and equation of the renormalization group . 69 3.1.3 Renormalization of the usual ϕ4 theory in four dimensions . 72 3.2 Relation with BPHZ renormalization . 76 3.2.1 Power-counting . 77 3.2.2 BPHZ subtraction scheme . 78 3.2.3 Beta functions . 80 3.3 Renormalization of gauge theories . 80 3.3.1 Classical theory and BRS formalism . 81 3.3.2 Algebraic renormalization . 84 4 QFT on Moyal space 89 4.1 Presentation of the Moyal space . 89 4.1.1 Deformation quantization . 89 4.1.2 The Moyal product on Schwartz functions . 91 4.1.3 The matrix basis . 92 4.1.4 The Moyal algebra . 93 4.1.5 The symplectic Fourier transformation . 96 4.2 UV/IR mixing on the Moyal space . 97 4.3 Renormalizable QFT on Moyal space . 100 4.3.1 Renormalization of the theory with harmonic term . 100 4.3.2 Principal properties . 104 4.3.3 Vacuum configurations . 107 4.3.4 Possible spontaneous symmetry breaking? . 112 4.3.5 Other renormalizable QFT on Moyal space . 113 5 Gauge theory on Moyal space 117 5.1 Definition of gauge theory . 117 5.1.1 Gauge theory associated to standard differential calculus . 117 5.1.2 U(N) versus U(1) gauge theory . 120 5.1.3 UV/IR mixing in gauge theory . 121 5.2 The effective action . 121 5.2.1 Minimal coupling . 122 5.2.2 Computation of the effective action . 123 5.2.3 Discussion on the effective action . 129 5.3 Properties of the effective action . 131 5.3.1 Symmetries of vacuum configurations . 132 5.3.2 Equation of motion . 132 5.3.3 Solutions of the equation of motion . 134 5.3.4 Minima of the action . 135 5.3.5 Extension in higher dimensions . 135 5.4 Interpretation of the effective action . 136 5.4.1 A superalgebra constructed from Moyal space . 137 5.4.2 Differential calculus and scalar theory . 138 5.4.3 Graded connections and gauge theory . 139 5.4.4 Discussion and interpretation . 142 Zusammenfassung Die nichtkommutative Geometrie bildet als wachsendes Gebiet der Mathematik einen vielversprechenden Rahmen f¨urdie moderne Physik. Quantenfeldtheorien ¨uber nichtkom- mutativen R¨aumenwerden zur Zeit intensiv studiert. Sie f¨uhrenzu einer neuen Art von Divergenzen, die ultraviolett-infrarot Mischung. Eine L¨osungdieses Problems wurde von H. Grosse und R. Wulkenhaar durch Hinzuf¨ugeneines harmonischen Terms zur Wirkung des φ4-Modells gefunden. Dadurch wird diese Quantenfeldtheorie ¨uber dem Moyal-Raum renormierbar. Ein Ziel dieser Doktorarbeit ist die Verallgemeinerung dieses harmonischen Terms auf Eichtheorien ¨uber dem Moyal-Raum. Basierend auf dem Grosse-Wulkenhaar-Modell wird eine neue nichtkommutative Eichtheorie eingef¨uhrt,die begr¨undeteChancen hat, renormierbar zu sein. Die wichtigsten Eigenschaften dieser Eichtheorie, insbesondere die Vakuumskonfigurationen, werden studiert. Schließlich wird mittels eines zu einer Super- algebra assoziierten Derivationskalk¨ulseine mathematische Interpretation dieser neuen Wirkung gegeben. Die Arbeit enth¨alt neben diesen Ergebnissen eine Einf¨uhrung zur nichtkommutativen Geometrie und zu ε-graduierten Algebren sowie eine Einf¨uhrung in die Renormierung von Quantenfeldtheorien f¨urSkalarfelder (nach Wilson und BPHZ) und Eichfelder. R´esum´e De nos jours, la g´eom´etrie non-commutative est un domaine grandissant des math´ema- tiques, qui peut apparaˆıtre comme un cadre prometteur pour la physique moderne. Les th´eoriesquantiques des champs sur des “espaces non-commutatifs” ont en effet ´et´e tr`es´etudi´ees, et sont sujettes `aun nouveau type de divergence, le m´elange ultraviolet- infrarouge. Cependant, une solution a r´ecemment ´et´eapport´ee`ace probl`eme par H. Grosse et R. Wulkenhaar en ajoutant `al’action d’un mod`elescalaire sur l’espace non- commutatif de Moyal, un terme harmonique qui la rend renormalisable. Un des buts de cette th`ese est l’extension de cette proc´edureaux th´eories de jauge sur l’espace de Moyal. En effet, nous avons introduit une nouvelle th´eorie de jauge non- commutative, fortement reli´eeau mod`elede Grosse-Wulkenhaar, et candidate `ala renor- malisabilit´e. Nous avons ensuite ´etudi´eses propri´et´es les plus importantes, notamment ses configurations du vide. Finalement, nous donnons une interpr´etation math´ematique de cette nouvelle action en terme de calcul diff´erentiel bas´esur les d´erivations, associ´e`a une superalg`ebre. Ce travail contient, outre les r´esultats mentionn´esci-dessus, une introduction `ala g´eom´etrienon-commutative, une introduction aux alg`ebres ε-gradu´ees, d´efinies dans cette th`ese,et une introduction `ala renormalisation des th´eories quantiques de champs scalaires (point de vue wilsonien et BPHZ) et de jauge. Abstract Nowadays, noncommutative geometry is a growing domain of mathematics, which can ap- pear as a promising framework for modern physics. Quantum field theories on “noncom- mutative spaces” are indeed much investigated, and suffer from a new type of divergence called the ultraviolet-infrared mixing. However, this problem has recently been solved by H. Grosse and R. Wulkenhaar by adding to the action of a noncommutative scalar model a harmonic term, which renders it renormalizable. One aim of this thesis is the extension of this procedure to gauge theories on the Moyal space. Indeed, we have introduced a new noncommutative gauge theory, strongly related to the Grosse-Wulkenhaar model, and candidate to renormalizability. We have then studied the most important properties of this action, and in particular its vacuum configurations. Finally, we give a mathematical interpretation of this new action in terms of a derivation-based differential calculus associated to a superalgebra. This work contains among the results of this PhD, an introduction to noncommutative geometry, an introduction to ε-graded algebras, defined in this thesis, and an introduction to renormalization of scalar (wilsonian and BPHZ point of view) and gauge quantum field theories. Danksagung An erster Stelle m¨ochte ich mich bei Prof. Dr. Jean-Christophe Wallet und Prof. Dr. Raimar Wulkenhaar f¨urdie Betreuung dieser Doktorarbeit bedanken. Ich bin ihnen sehr dankbar, mich in die nichtkommutative Geometrie und die Quantumfeldtheorie aus mathematischer und physikalischer Sicht eingewiesen zu haben. Ich habe sehr die wis- senschaftliche Begeisterung von Prof. Wallet und die P¨adagogik von Prof. Wulkenhaar gesch¨atzt, sowie ihre Geduld und Aufmerksamkeit, so dass ich besonders gute Arbeit- sumst¨andegenossen habe. Ich danke auch Prof. Dr. Dr. hc Joachim Cuntz, Dekan des Mathematischen Instituts der Westf¨alische Wilhelms-Universit¨atM¨unster,und Prof. Dr. Henk Hilhorst, Direktor des Laboratoire de Physique Th´eorique der Universit¨at Paris-Sud XI, f¨urihren Empfang in diesem doppelten deutschen-franz¨osischen und mathematischen-physikalischen Rahmen, der sehr bereichernd gewesen ist. Mein herzlicher Dank geht an Prof. Dr. Pierre Bieliavsky und Prof. Dr. Stefan Waldmann, die bereit waren, diese Doktorarbeit zu lesen und zu begutachten, sowie an Prof. Dr. Michel Dubois-Violette, Prof. Dr. Thierry Masson und Prof. Dr. Pierre Vanhove f¨urihre Bereitschaft, Mitglieder der Jury zu sein. Ich danke besonders Prof. Masson, dessen Vorlesungsskriptum mich zur nichtkommutativen Geometrie hingezogen hat, und der mich vieles auf diesem Gebiet gelehrt hat. Es hat mich gefreut, mit ihm zusammenzuarbeiten, nicht zu vergessen seine Hilfe bei LATEX-Problemen. Dann m¨ochte ich Dr. Adrian Tanasa f¨urseine Zusammenarbeit und seine wertvollen Ratschl¨age danken, und auch allen Kollegen, mit denen ich erfolgreich diskutiert habe: Dr. Johannes Aastrup, Dipl.-Phys. Eric Cagnache, Prof. Dr. Yvon Georgelin, Dr. Jesper Grimstrup, Dr. Razvan Gurau, Prof. Dr. John Madore, Prof. Dr. Jacques Magnen, Dr. Pierre Martinetti, Dr. Mario Paschke, Prof. Dr. Vincent Rivasseau, Dr. Fabien Vignes-Tourneret, Prof. Dr. Wend Werner, und allen, die ich hier nicht nennen kann.
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