Geometric Approaches to Quantum Field Theory PDF
Preview Geometric Approaches to Quantum Field Theory
Springer Theses Recognizing Outstanding Ph.D. Research Kieran Finn Geometric Approaches to Quantum Field Theory Springer Theses Recognizing Outstanding Ph.D. Research AimsandScope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected foritsscientificexcellenceandthehighimpactofitscontentsforthepertinentfield of research. For greater accessibility to non-specialists, the published versions includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor explainingthespecialrelevanceoftheworkforthefield.Asawhole,theserieswill provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributionsmadebytoday’syoungergenerationofscientists. Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes andshouldfulfillallofthefollowingcriteria (cid:129) TheymustbewritteningoodEnglish. (cid:129) ThetopicshouldfallwithintheconfinesofChemistry,Physics,EarthSciences, EngineeringandrelatedinterdisciplinaryfieldssuchasMaterials,Nanoscience, ChemicalEngineering,ComplexSystemsandBiophysics. (cid:129) Theworkreportedinthethesismustrepresentasignificantscientificadvance. (cid:129) Ifthethesisincludespreviouslypublishedmaterial,permissiontoreproducethis mustbegainedfromtherespectivecopyrightholder(amaximum30%ofthethesis shouldbeaverbatimreproductionfromtheauthor’spreviouspublications). (cid:129) They must have been examined and passed during the 12 months prior to nomination. (cid:129) Each thesis should include a foreword by the supervisor outlining the signifi- canceofitscontent. (cid:129) The theses should have a clearly defined structure including an introduction accessibletonewPhDstudentsandscientistsnotexpertintherelevantfield. IndexedbyzbMATH. Moreinformationaboutthisseriesat http://www.springer.com/series/8790 Kieran Finn Geometric Approaches to Quantum Field Theory Doctoral Thesis accepted by University of Manchester, Manchester, United Kingdom Author Supervisor Dr.KieranFinn Prof.ApostolosPilaftsis DepartmentofPhysics UniversityofManchester andAstronomy Manchester,UK UniversityofManchester Manchester,UK ISSN2190-5053 ISSN2190-5061 (electronic) SpringerTheses ISBN978-3-030-85268-9 ISBN978-3-030-85269-6 (eBook) https://doi.org/10.1007/978-3-030-85269-6 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicatedtomygrandmother,PeggyAnne Green(24thMay1939–23rd March2020) Supervisor’s Foreword Sinceantiquity,humanshavebeenfascinatedtounderstandbettertheoriginofthe Cosmos and, in particular, the way this is imprinted in the constellations of stars, the motion of planets, and other celestial objects. Back in the sixth century BC, Pythagoreans believed that the divine origin of the Cosmos could be understood geometricallybydescribingthedistancefromtheEarthofthefiveknownmoving stars (planets) as radii of regular solids (more often known as Platonic solids) all inscribable in three-dimensional spheres. Refinement of such approaches led later tothedevelopmentofaPtolemaicgeocentricsystemofdeferentsandepicycles,as wellastoalternative,butlessinfluential,schoolsofthoughtinitiatedbyAristarchus andSeleucuswhoproposedaheliocentricmodelaroundthethirdcenturyBC. InthesixteenthcenturyAD,TychoBrahe’smeasurementsofplanetarymotion, Galileo’s observations of the phases of Venus, as well as the works of Copernicus andKepler,stronglydisfavouredgeocentricmodels,andtheheliocentricsystemsaw wideracceptance.However,thediscrepanciesbetweenthetwomodelswentunre- solved until the seventeenth century AD with the formulation of Newton’s law of universalattraction.ThefullconsequencesofNewton’sframe-dependentdynamics wereeliminatedonlythroughtheintroductionoffictitiousforcesbyd’Alembertand CoriolisintheeighteenthcenturyAD.OurmodernviewisthatbothTycho’sgeocen- tricandheliocentricmodelsarefullyequivalentprovidedtheappropriatefictitious forcesareconsideredandarerelatedbymeansofacoordinatetransformation. While inertial frames might appear more physically intuitive than non-inertial frames,thereshouldbeapriorinoreasontopreferonesystemovertheother,apart from convenience or ease in calculations. In his nominal theory of General Rela- tivityinthetwentiethcenturyAD,EinsteinaddressedsuccessfullyNewton’sframe- dependentdynamicsbyintroducingafullydifferential-geometricdescriptionofthe Cosmos. Any form of energy or matter will give rise to a curved manifold upon which geodesics become the only allowed trajectory of motion for any object or elementaryparticle,includinglightitself.Einstein’spioneeringideasrevolutionised our understanding of fundamental physics, as they imply that true laws of nature shouldbeframecovariant,andsomanifestlyindependentofourdescription. vii viii Supervisor’sForeword With the advent of Quantum Mechanics and Quantum Field Theory in the last century,theissueofquantisingEinstein’stheoryofgravitationhasresultedinperhaps the most famous long-standing problem of the twenty-first century. Specifically, findingaUVcompletetheoryofQuantumGravityconstitutesahugemilestonein theadvancementoffundamentalphysics.Thusfar,severalapproacheshavebeenput forwardintheliterature,suchasloopquantumgravity,superstring,andmembrane theories,butalltheseprovedtohavelimitedsuccess. MotivatedbyEinstein’stheoryofGravitation,VilkoviskyandDeWitt(VDW)in the early 1980s proposed a covariant differential-geometric approach to Quantum Gravity,whichincludesnotonlythecoordinatespacebutanentirefieldspace.This wasanattractiveideaascovariantfield-spaceactionsmayresolvetheproblemsof gaugedependenceingaugequantumfieldtheories. The VDWeffective action maysolve anumber of problems thatwould require knowledgeoftheoff-shelldynamicsofaquantumsystem,whicharehighlyframe- dependent in the configuration space. In particular, problems of great interest and importance are as follows: (i) the gauge-independent definition of effective charges in non-Abelian gauge theories; (ii) the gauge-invariant description of unstable particle dynamics within the context of S-matrix theory; (iii) the proper field-reparameterization invariant definition of electroweak precision observables, includingVeltman’selectroweakparameter;(iv)theuniquefield-reparameterization invariantexpansionsofeffectivefieldtheoriessuchasStandardModelEffectiveField Theory(SMEFT);and(v)aframecovariantdescriptionofcosmologicalinflationin modelswithmultiplescalarfieldsthatmayactasinflatons. One pressing question regarding the VDW effective action was its uniqueness. Thisissueofuniquenesshasbeenthetopicofnumerousstudies.Inthesamecontext,it wasrecentlyrealisedthatinadditiontotheclassicalaction,thepath-integralmeasure responsible for quantizing the theory using Feynman’s approach to QFT can also introduceframedependencethroughthegravitysector,therebyruiningthecentral missionoftheVDWformalism.Fortunately,asitwillbeshowninthisthesis,this ambiguitycanbeeliminatedbyproperlydefiningthemodelfunctionsoftheaction andbyidentifyinganovelonethatgovernsthepath-integralmeasure. Another previously unsurmountable difficulty of the VDW formalism was the consistentinclusionoffermions.AswasalreadynotedbyDeWitt,thisconstituteda formidabletask,sinceafermionickinetictermislinearintimederivatives,andsoa field-spacemetricthatincludesfermionsdidnotseemtoexist.AlthoughDeWittwas firsttonotethatbosonsandfermionscouldgiverisetoafield-spacesupermanifold that includes Grassmannian coordinates, the identification of a proper metric for suchasupermanifoldremainedelusive.Infact,thisproblemwassoseriousthatit has prevented the VDW formalism from becoming as widespread in the existing literatureasitdeservestobe. The present Ph.D. dissertation provides a rigorous and self-consistent solution to this problem. It explicitly shows how the correct supersymmetric metric can be derivedfromtheactionofthetheory.Theuniquenessofthemetricisguaranteedfor allultralocalreparametrisationsofthefields,withtheconditionthatthefield-space supermanifold(whichislocallyflat)shouldmatchthekinetictermofafreefermionic Supervisor’sForeword ix fieldtheory.Thesuggestedapproachdifferscruciallyfromthemorewell-knownone ofSupersymmetry,asonedoesnotneedtopostulatetheexistenceofnewparticles ornewGrassmanniandimensions. Dr.KieranFinncompletedhisPh.D.attheDepartmentofPhysicsandAstronomy attheUniversityofManchesterinOctober2020,undermysupervision.HisPh.D. thesisgivesanumberofillustrativeexamplesinordertoshowcasehowtheVDW formalismworksnotonlyforscalartheories,butalsofortheorieswithfermionsand gravity. It is a beautiful scientific treatise that can pave the way for future devel- opments in the field when applied to more realistic theories such as the Standard Model. I believe that the following thesis makes a significant leap in advancing our knowledgetowardsacompletegeometrizationofourMicro-Cosmos. Manchester,UK Prof.ApostolosPilaftsis March2021 Abstract The ancient Greeks thought that all of creation should be describable in terms of geometry. In this thesis, we take a step towards realising this dream by applying the methods of differential geometry to modern ideas about particle physics and cosmologyintheformofquantumfieldtheory. We shall achieve this using the formalism of field-space covariance, in which the degrees of freedom in a quantum field theory are treated as coordinates on a Riemannianmanifold,knownasthefield-spacemanifold.Thisformalismallowsus todescribesuchtheoriesgeometrically,inawaythatismanifestlyinvariantunder arbitrarychoicessuchastheunits,spacetimecoordinates,orfieldvariablesused. In this thesis, we extend the applicability of field-space covariance to quantum field theories with gravitational and fermionic degrees of freedom. We show how to construct the field-space manifold for such theories and how to equip it with a naturalmetric.Thus,wearenowabletoapplythisformalismtoallrealistictheories ofparticlephysics,includingtheStandardModel. In addition, we show that the potential term in a quantum field theory can also bedescribedgeometricallythroughaprocessknownastheEisenhartlift.Weshow how,byintroducingnewdegreesoffreedomintothetheory,apotentialtermcanbe recastasthecurvatureoffieldspace. Finally,weapplyourgeometricmethodstothetheoryofinflation.Weconstruct a manifold that describes the evolution of inflation geometrically as a geodesic. We show that the tangent bundle of this manifold, equipped with a natural metric, providesafinitemeasureontheinitialconditionsofinflation,whichwecanuseto studyfinetuninginthesemodels. xi
The list of books you might like
