Quantum Gravity in Everyday Life General Relativit

arXiv:gr-qc/0311082v124Nov2003PreprinttypesetinJHEPstyle-HYPERVERSIONQuantumGravityinEverydayLife:GeneralRelativityasanEffectiveFieldTheory∗C.P.BurgessPhysicsDepartment,McGillUniversity3600UniversityStreet,Montr´ealQu´ebec,Canada,H3A2T8.e-mail:cliff@physics.mcgill.ca˜cliff(lastmodified:9October2003)Abstract:Thisarticleismeantasasummaryandintroductiontotheideasofeffectivefieldtheoryasappliedtogravitationalsystems,ideaswhichprovidethetheoreticalfoundationsforthemodernuseofGeneralRelativityasatheoryfromwhichprecisepredictionsarepossible.Keywords:generaltheoryofrelativity,quantumgravity,effectivefieldtheories.∗ToappearinLivingReviewsofRelativity.Contents1.Introduction21.1AgainsttheSplitBrain21.2IdentifyingWheretheProblemsLie31.3ARoadMap42.EffectiveFieldTheories52.1TheUtilityofLow-EnergyApproximations52.2AToyExample72.2.1Massless-ParticleScattering72.3TheToyModelRevisited92.3.1ExhibitingtheSymmetry92.3.2TimelyPerformancetheLow-EnergyApproximation92.3.3ImplicationsfortheLow-EnergyLimit102.3.4RedundantInteractions112.4LessonsLearned122.4.1WhyareEffectiveLagrangiansnotMoreComplicated?132.5PredictivenessandPowerCounting142.5.1Power-CountingLow-EnergyFeynmanGraphs142.5.2ApplicationtotheToyModel182.6TheEffectiveLagrangianLogic192.6.1TheChoiceofVariables202.6.2RegularizationDependence202.7TheMeaningofRenormalizability223.Low-EnergyQuantumGravity233.1GeneralRelativityasanEffectiveTheory233.1.1RedundantInteractions243.2PowerCounting253.2.1IncludingMatter263.3EffectiveFieldTheoryinCurvedSpace313.3.1WhenShouldEffectiveLagrangiansWork?323.3.2GeneralPowerCounting353.3.3HorizonsandLargeRedshifts35–1–4.ExplicitQuantumCalculations404.1Non-relativisticPointMassesin3SpatialDimensions404.1.1DefinitionofthePotential404.1.2CalculationoftheInteractionPotential424.1.3Implications464.2Co-DimensionTwoandCosmicStrings475.Conclusions486.Acknowledgements501.IntroductionQuantummechanicsandGeneralRelativitywerediscoveredwithinadecadeofoneanotherclosetoacenturyago,andalmostimmediatelythesearchforaquantumtheoryofgravityhadbegun.Eversinceithasbeenapuzzleastowhattheoreti-calframeworkmightultimatelyreconcilethesetwotheorieswithoneanother.Thisreconciliationhasproventobedifficulttoachieve,andalthoughseveralpromis-ingproposalsnowexistnonecanyetclaimcompletetheoreticalandexperimentalvindication.1.1AgainsttheSplitBrainThelong-standingnatureofthisdifficultyhasdrivensomephysiciststoastateofintellectualdespair,whereintheyconcludethatacrisisexistsinphysicswhichmightbecalledthecrisisofthesplitbrain.Ononehand,quantummechanics(anditsoffspringquantumfieldtheory)providesanincrediblysuccessfuldescriptionofallknownnon-gravitationalphenomena,withagreementbetweenpredictionsandex-perimentsometimestakingplaceatthepart-per-billionlevel[1,2].Ontheotherhand,classicalgeneralrelativityisalsoextremelysuccessful,withitspredictionsbeingwelltestedwithinthesolarsystemandforsomebinarypulsarsystems[3].(Thecosmologicalevidencefordarkmatteranddarkenergyissometimesproposedasindicatingthefailureofgravityoverlongdistances[4],butatpresenttheevi-dencefornewgravitationalphysicsatlargedistancesdoesnotseemcompelling[5].)Theperceivedcrisisistheabsenceofanover-archingtheoreticalframeworkwithinwhichbothsuccessescanbeaccommodated.Ourbrainsareeffectivelysplitintotwoincommunicativehemispheres,withquantumphysicslivinginoneandclassicalgeneralrelativityintheother.Theabsenceofsuchaframeworkwouldindeedbeacrisisfortheoreticalphysics,sincerealtheoreticalpredictionsarenecessarilyapproximate.Controllableresults–2–alwaysrequiresomeunderstandingofthesizeofthecontributionsbeingneglectedinanygivencalculation.IfquantumeffectsinGeneralRelativitycannotbequantified,thismustundermineoursatisfactionwiththeexperimentalsuccessofitsclassicalpredictions.Itisthepurposeofthisarticletopresentthemodernpointofviewontheseissues,whichhasemergedsincetheearly1980’s.Accordingtothispointofviewthereisnosuchcrisis,becausetheproblemsofquantizinggravitywithintheex-perimentallyaccessiblesituationsaresimilartothosewhichariseinahostofothernon-gravitationalapplicationsthroughoutphysics.Assuch,thesizeofquantumcor-rectionscanbesafelyestimatedandareextremelysmall.Thetheoreticalframeworkwhichallowsthisquantificationistheformalismofeffectivefieldtheories,whoseexplanationmakesupthebetterpartofthisarticle.Insodoingweshallseethatalthoughtherecanbelittledoubtofthefinaloutcome,theexplicitdeterminationofthesizeofsub-leadingquantumeffectsingravityhasinmanycasescomeonlyrelativelyrecently,andacompletequantitativeanalysisofthesizeofquantumcor-rectionsremainsaworkinprogress.1.2IdentifyingWheretheProblemsLieThisisnottosaythattherearenochallengingproblemsremaininginreconcilingquantummechanicswithgravity.Onthecontrary,manyofthemostinterestingissuesremaintobesolved,includingtheidentificationofwhattherightobserv-ablesshouldbe,andunderstandinghowspaceandtimemightemergefrommoremicroscopicconsiderations.Fortherestofthediscussionitisusefultoseparatethesedeep,unsolvedissuesofprinciplefromthemoreprosaic,technicalproblemofGeneralRelativity’snon-renormalizability.