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The Structure of Gauge Theories in Almost Commutative Geometries © January 2011 THESTRUCTUREOFGAUGETHEORIESIN ALMOSTCOMMUTATIVEGEOMETRIES koen van den dungen January 2011 Supervisors: Dr. W.D. van Suijlekom Prof. dr. R.H.P. Kleiss IMAPP Koen van den Dungen: The Structure of Gauge Theories in Almost Commutative Geometries © January 2011. A Master’s Thesis in Physics & Astronomy. Radboud University Nijmegen, Faculty of Science, Institute of Mathematics, Astrophysics and Particle Physics, Department of Theoretical High Energy Physics. On the cover: an impression of electrodynamics on the two-point space. This thesis was typeset with LATEX 2#. The design of the typographic style is avail- able via CTAN as “classicthesis”. Revised Version as of January 14, 2011. ABSTRACT Within the framework of noncommutative geometry, the concept of almost commu- tative geometries can be used to describe models in physics. We shall study the properties of real even almost commutative geometries in their most general form. In particular, we show that any such almost commutative geometry describes a gauge theory, minimally coupled to a Higgs field, and we show how to recover the gauge group from the spectral triple. In the 4-dimensional case, we derive the general form of the bosonic Lagrangian, and show that it is conformally invariant up to the appearance of a dilaton field. We then apply the general framework to three examples, which step by step take us towards the description of the Standard Model of high energy physics. The first example of electrodynamics shows that it is also possible to describe abelian gauge theories in noncommutative geometry. In the second example of the Glashow-Weinberg-Salam model, we in particular show that the Higgs mechanism provides a spontaneous breaking of both the gauge sym- metry and the conformal symmetry. In the third example, we can then easily extend the results to describe the full Standard Model. iii PREFACE Seven years and four months, that is how much time has passed between follow- ing my first lecture at the university and finalizing my thesis. Though a study in physics should nominally take only five years, I have tried my best to extend this period, and with success. I am glad that I am able to say that the entire delay was time well-spent. I have spent many hours of my time in useful work for several students’ organizations. Although this led to a delay of my graduation, I consider it to be a valuable contribution to my personal academic development. However, there comes a point when it is time to finish and move on, and the time arrived to finish my Master. About one year ago, I needed to choose a research area for my Master’s thesis. Although I am a student in physics, I have always had an interest in mathematics. I am very grateful for the opportunity to do my Master’s research right on the borders between physics and mathematics. I chose to do my Master’s research in the field of noncommutative geometry, an area of mathematics that at the time was completely unknown to me. Therefore, a large portion of my time was spent on reading books and papers and reproduc- ing known results. The original plan was to produce a cosmological model from noncommutative geometry. However, as time went by, other interesting topics in noncommutative geometry revealed themselves to me, and I frequently changed directions. Essentially, I had to finish my thesis before I could say what it would be about. Now my thesis is finished. During the past year I have not only familiarized myself with (a part of) noncommutative geometry, but I have also had the chance to start developing my own ideas on some topics. Perhaps, the latter can be considered the most important result of my thesis. acknowledgment Of course, writing this thesis is not something I could have done all by myself. I would like to thank my supervisor, Walter van Suijlekom, for all his help and advice, for allowing me the flexibility to move freely from one topic to the other, and for pointing me in the right direction when needed. I thank my parents for their ongoing encouragement. I also thank everyone in our office and the rest of the department of high energy physics for providing a pleasant working environment, where hard work and fun really seem to go hand in hand. v SAMENVATTING VOOR LEKEN summary for laymen (in dutch) Gedurende het afgelopen jaar werd mij regelmatig door bekenden de vraag gesteld: “Waar gaat je scriptie eigenlijk over?” In veel gevallen werd deze vraag vrij snel gevolgd door: “Of begrijp ik daar toch niks van?” Het eenvoudigste antwoord dat ik op deze vragen kan geven is: “Nee, dat begrijp je inderdaad niet. ” Vaak waagde ik vervolgens toch een halfslachtige poging om uit te leggen waar ik mee bezig was, en ik vertelde dan zoiets als: “Ik bestudeer een deelgebied van de wiskunde dat niet-commutatieve meetkunde heet. Ik ben natuurkundestudent, en zit hiermee op het grens- vlak van de wiskunde en de natuurkunde. Een belangrijke toepassing van de niet-commutatieve meetkunde is dat je het Standaard Model van de natuurkunde hieruit kunt afleiden. Dit Standaard Model beschrijft alle elementaire deeltjes en hun onderlinge interacties, behalve de zwaar- tekracht. Bijkomend voordeel van de niet-commutatieve meetkunde is dat je, naast het Standaard Model, ook de zwaartekracht erbij cadeau krijgt. Zelf bestudeer ik wat voor andere modellen men zou kunnen af- leiden uit de niet-commutatieve meetkunde, en wat voor eigenschappen deze modellen in het algemeen hebben.” Ik heb echter zelden het idee gehad dat mensen ook daadwerkelijk veel wijzer zijn geworden van deze korte uitleg. Bij deze wil ik daarom de uitdaging aangaan om te proberen in lekentaal samen te vatten waar mijn scriptie ongeveer over gaat. Voor ik echt begin, wil ik je complimenteren met het feit dat je het hebt aangedurfd om deze scriptie open te slaan, en dat je samen met mij de uitdaging wilt aangaan om er iets van te begrijpen. Ik kan je niet beloven dat je na het lezen van deze samenvatting alles perfect begrijpt, maar ik zal mijn best doen om ervoor te zorgen dat je er in ieder geval iets van zult opsteken. meetkunde en zwaartekracht Laten we ons eerst een idee vormen van wat meetkunde is. De meetkunde geeft een beschijving van de ruimte. Hiermee beschrijft men bijvoorbeeld wat de afstand tussen twee punten is, of wat het verschil tussen recht en krom is. Ruimte en tijd zijn nauw met elkaar verbonden, en het samengevoegde concept ruimte-tijd kunnen we geheel meetkundig beschrijven. De notie van ruimtelijke kromming verstoort ons intuïtieve idee van wat recht en krom is. Een steen die we in de lucht werpen zal een boogvormige baan volgen en verderop weer op de grond terecht komen. Deze boogvormige baan is in onze ogen duidelijk krom, maar kan niettemin ook beschreven worden als een ‘rechtlijnige’ beweging op een gekromde ruimte. Op deze wijze kan de zwaartekracht worden beschreven als meetkundige theorie. De aantrekkingskracht van de aarde kromt de ruimte, en zorgt er zo voor dat de omhoog geworpen steen vanzelf weer naar beneden komt. vii het standaard model en niet-commutatieve meetkunde De zwaartekracht is echter niet de enige kracht in de natuur die we kennen. Bekend uit het dagelijkse leven zijn elektriciteit en magnetisme. Daarnaast bestaan boven- dien de sterke en zwakke kernkrachten, die zorgen voor stabiele atoomkernen of voor radioactief verval. Een omschrijving van alle elementaire deeltjes (de funda- mentele bouwstenen van alle materie), inclusief het effect van de drie genoemde interacties op deze deeltjes, wordt gegeven door het Standaard Model van de na- tuurkunde. Dit Standaard Model bezit een bepaalde symmetrie, genaamd ijksymmetrie. Deze ijksymmetrie levert een wiskundig elegante omschrijving van de interacties tussen elementaire deeltjes. Het feit dat materie massa heeft, verstoort echter de symme- trie. Een theoretische oplossing die de symmetrie herstelt, is de introductie van een nieuw deeltje, het Higgs-deeltje, in het Standaard Model. Dit theoretisch voor- spelde deeltje is echter het enige onderdeel van het Standaard Model dat nog niet is waargenomen. Op dit moment wedijveren de deeltjesversnellers van Fermilab (Chicago) en CERN (Genève) om de ontdekking hiervan. De eenvoudige meetkunde is niet in staat om, naast de zwaartekracht, ook de in- teracties van het Standaard Model te beschrijven. Voor deze overige interacties is het nodig om, aan ieder punt in de ruimte-tijd, extra interne vrijheidsgraden toe te voegen. Deze interne vrijheidsgraden beschrijven welke deeltjes er bestaan, en welke interacties ze met elkaar kunnen hebben. Hier komt de niet-commutatieve meetkunde om de hoek kijken. Zij maakt de meetkunde algemener, door naast de ruimte-tijd ook deze interne vrijheidsgraden te beschrijven. Via deze combi- natie van ruimte-tijd met interne vrijheidsgraden, beschrijft de niet-commutatieve meetkunde een natuurkundig model. De precieze vorm van dit natuurkundige mo- del hangt af van de vorm van de interne vrijheidsgraden. Bij een geschikte keuze van deze interne vrijheidsgraden blijkt dat het bijbehorende natuurkundige model precies het Standaard Model oplevert (inclusief het Higgs-deeltje), ditmaal echter gekoppeld aan de meetkundige beschrijving van de zwaartekracht. mijn scriptie In mijn scriptie beschrijf ik eveneens natuurkundige modellen, die door de combina- tie van ruimte-tijd met bepaalde interne vrijheidsgraden uit de niet-commutatieve meetkunde verkregen worden. Ik doe dit echter zonder te kiezen welke interne vrijheidsgraden er zijn. Dit levert een algemene beschrijving op van natuurkundige modellen in niet-commutatieve meetkunde. Ook zonder te weten welke interne vrij- heidsgraden er zijn, is het nog steeds mogelijk om de algemene eigenschappen van zulke natuurkundige modellen goed te beschrijven. In het bijzonder toon ik aan dat deze modellen altijd een ijksymmetrie bezitten. Ook het Higgs-deeltje komt in alle gevallen vanzelf tevoorschijn.
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