How Feynman Diagrams Almost Saved Space

Feynman diagrams look to be pictures of processes that happen in space and time, and in a sense they are, but they should not be interpreted too ... Wecareaboutyourdata,andwe'dliketousecookiestogiveyouasmoothbrowsingexperience.Pleaseagreeandreadmoreaboutourprivacypolicy.AgreeHowFeynmanDiagramsAlmostSavedSpaceReadLaterShareCopied!CommentsReadLaterReadLaterFrankWilczekContributingColumnistJuly5,2016ViewPDF/PrintModedarkenergyFeynmandiagramshistoryofsciencephysicspodcastQuantizedColumnsquantumchromodynamicsquantumelectrodynamicsquantumphysicsspace-timetheoreticalphysicsAlltopicsQuantizedColumnsHowFeynmanDiagramsAlmostSavedSpaceByFrankWilczekJuly5,2016RichardFeynman'sfamousdiagramsembodyadeepshiftinthinkingabouthowtheuniverseisputtogether.ReadLaterSave Save Save Save Save Save RichardFeynmanlookedtiredwhenhewanderedintomyoffice.Itwastheendofalong,exhaustingdayinSantaBarbara,sometimearound1982.Eventshadincludedaseminarthatwasalsoaperformance,lunchtimegrillingbyeagerpostdocs,andlivelydiscussionswithseniorresearchers.Thelifeofacelebratedphysicistisalwaysintense.Butourvisitorstillwantedtotalkphysics.Wehadacoupleofhourstofillbeforedinner. IdescribedtoFeynmanwhatIthoughtwereexcitingifspeculativenewideassuchasfractionalspinandanyons.Feynmanwasunimpressed,saying:“Wilczek,youshouldworkonsomethingreal.”(Anyonsarereal,butthat’satopicforanotherpost.) Quantized Amonthlycolumninwhichtopresearchersexploretheprocessofdiscovery.Thismonth’scolumnist,FrankWilczek,isaNobelPrize-winningphysicistattheMassachusettsInstituteofTechnology. Lookingtobreaktheawkwardsilencethatfollowed,IaskedFeynmanthemostdisturbingquestioninphysics,thenasnow:“There’ssomethingelseI’vebeenthinkingalotabout:Whydoesn’temptyspaceweighanything?” Feynman,normallyasquickandlivelyastheycome,wentsilent.ItwastheonlytimeI’veeverseenhimlookwistful.Finallyhesaiddreamily,“IoncethoughtIhadthatonefiguredout.Itwasbeautiful.”Andthen,excited,hebegananexplanationthatcrescendoedinanearshout:“Thereasonspacedoesn’tweighanything,Ithought,isbecausethere’snothingthere!” Toappreciatethatsurrealmonologue,youneedtoknowsomebackstory.Itinvolvesthedistinctionbetweenvacuumandvoid. Vacuum,inmodernusage,iswhatyougetwhenyouremoveeverythingthatyoucan,whetherpracticallyorinprinciple.Wesayaregionofspace“realizesvacuum”ifitisfreeofallthedifferentkindsofparticlesandradiationweknowabout(including,forthispurpose,darkmatter—whichweknowaboutinageneralway,thoughnotindetail).Alternatively,vacuumisthestateofminimumenergy. Intergalacticspaceisagoodapproximationtoavacuum. Void,ontheotherhand,isatheoreticalidealization.Itmeansnothingness:spacewithoutindependentproperties,whoseonlyrole,wemightsay,istokeepeverythingfromhappeninginthesameplace.Voidgivesparticlesaddresses,nothingmore. Aristotlefamouslyclaimedthat“Natureabhorsavacuum,”butI’mprettysureamorecorrecttranslationwouldbe“Natureabhorsavoid.”IsaacNewtonappearedtoagreewhenhewrote: …thatoneBodymayactuponanotherataDistancethro’aVacuum,withouttheMediationofanythingelse,byandthroughwhichtheirActionandForcemaybeconveyedfromonetoanother,istomesogreatanAbsurdity,thatIbelievenoManwhohasinphilosophicalMattersacompetentFacultyofthinking,caneverfallintoit. ButinNewton’smasterpiece,thePrincipia,theplayersarebodiesthatexertforcesononeanother.Space,thestage,isanemptyreceptacle.Ithasnolifeofitsown.InNewtonianphysics,vacuumisavoid. ThatNewtonianframeworkworkedbrilliantlyfornearlytwocenturies,asNewton’sequationsforgravitywentfromtriumphtotriumph,and(atfirst)theanalogousonesforelectricandmagneticforcesseemedtodosoaswell.Butinthe19thcentury,aspeopleinvestigatedthephenomenaofelectricityandmagnetismmoreclosely,Newton-styleequationsprovedinadequate.InJamesClerkMaxwell’sequations,thefruitofthatwork,electromagneticfields—notseparatedbodies—aretheprimaryobjectsofreality. QuantumtheoryamplifiedMaxwell’srevolution.Accordingtoquantumtheory,particlesaremerelybubblesoffroth,kickedupbyunderlyingfields.Photons,forexample,aredisturbancesinelectromagneticfields. Asayoungscientist,Feynmanfoundthatviewtooartificial.HewantedtobringbackNewton’sapproachandworkdirectlywiththeparticlesweactuallyperceive.Indoingso,hehopedtochallengehiddenassumptionsandreachasimplerdescriptionofnature—andtoavoidabigproblemthattheswitchtoquantumfieldshadcreated. II. Inquantumtheory,fieldshavealotofspontaneousactivity.Theyfluctuateinintensityanddirection.Andwhiletheaveragevalueoftheelectricfieldinavacuumiszero,theaveragevalueofitssquareisnotzero.That’ssignificantbecausetheenergydensityinanelectricfieldisproportionaltothefield’ssquare.Theenergydensityvalue,infact,isinfinite. Thespontaneousactivityofquantumfieldsgoesbyseveraldifferentnames:quantumfluctuations,virtualparticles,orzero-pointmotion.Therearesubtledifferencesintheconnotationsoftheseexpressions,buttheyallrefertothesamephenomenon.Whateveryoucallit,theactivityinvolvesenergy.Lotsofenergy—infact,aninfiniteamount. Formostpurposeswecanleavethatdisturbinginfinityoutofconsideration.Onlychangesinenergyareobservable.Andbecausezero-pointmotionisanintrinsiccharacteristicofquantumfields,changesinenergy,inresponsetoexternalevents,aregenerallyfinite.Wecancalculatethem.Theygiverisetosomeveryinterestingeffects,suchastheLambshiftofatomicspectrallinesandtheCasimirforcebetweenneutralconductingplates,whichhavebeenobservedexperimentally.Farfrombeingproblematic,thoseeffectsaretriumphsforquantumfieldtheory. Theexceptionisgravity.Gravityrespondstoallkindsofenergy,whateverformthatenergymaytake.Sotheinfiniteenergydensityassociatedwiththeactivityofquantumfields,presenteveninavacuum,becomesabigproblemwhenweconsideritseffectongravity. Inprinciple,thosequantumfieldsshouldmakethevacuumheavy.Yetexperimentstellusthatthegravitationalpullofthevacuumisquitesmall.Untilrecently—seemoreonthisbelow—wethoughtitwaszero. PerhapsFeynman’sconceptualswitchfromfieldstoparticleswouldavoidtheproblem. III. Feynmanstartedfromscratch,drawingpictureswhosestick-figurelinesshowlinksofinfluencebetweenparticles.ThefirstpublishedFeynmandiagramappearedinPhysicalReviewin1949: Tounderstandhowoneelectroninfluencesanother,usingFeynmandiagrams,youhavetoimaginethattheelectrons,astheymovethroughspaceandevolveintime,exchangeaphoton,herelabeled“virtualquantum.”Thisisthesimplestpossibility.Itisalsopossibletoexchangetwoormorephotons,andFeynmanmadesimilardiagramsforthat.Thosediagramscontributeanotherpiecetotheanswer,modifyingtheclassicalCoulombforcelaw.Bysproutinganothersquiggle,andlettingitextendfreelyintothefuture,yourepresenthowanelectronradiatesaphoton.Andso,stepbystep,youcandescribecomplexphysicalprocesses,assembledlikeTinkertoysfromverysimpleingredients. Feynmandiagramslooktobepicturesofprocessesthathappeninspaceandtime,andinasensetheyare,buttheyshouldnotbeinterpretedtooliterally.Whattheyshowarenotrigidgeometrictrajectories,butmoreflexible,“topological”constructions,reflectingquantumuncertainty.Inotherwords,youcanbequitesloppyabouttheshapeandconfigurationofthelinesandsquiggles,aslongasyougettheconnectionsright. Feynmanfoundthathecouldattachasimplemathematicalformulatoeachdiagram.Theformulaexpressesthelikelihoodoftheprocessthediagramdepicts.Hefoundthatinsimplecaseshegotthesameanswersthatpeoplehadobtainedmuchmorelaboriouslyusingfieldswhentheyletfrothinteractwithfroth. That’swhatFeynmanmeantwhenhesaid,“There’snothingthere.”Byremovingthefields,he’dgottenridoftheircontributiontogravity,whichhadledtoabsurdities.Hethoughthe’dfoundanewapproachtofundamentalinteractionsthatwasnotonlysimplerthantheconventionalone,butalsosounder.Itwasabeautifulnewwaytothinkaboutfundamentalprocesses. IV. Sadly,firstappearancesproveddeceptive.Asheworkedthingsoutfurther,Feynmandiscoveredthathisapproachhadasimilarproblemtotheoneitwassupposedtosolve.Youcanseethisinthepicturesbelow.WecandrawFeynmandiagramsthatarecompletelyself-contained,withoutparticlestoinitiatetheevents(ortoflowoutfromthem).Theseso-calleddisconnectedgraphs,orvacuumbubbles,aretheFeynmandiagramanalogueofzero-pointmotion.Youcandrawdiagramsforhowvirtualquantaaffectgravitons,andtherebyrediscoverthemorbidobesityof“empty”space. Moregenerally,asheworkedthingsoutfurther,Feynmangraduallyrealized—andthenproved—thathisdiagrammethodisnotatruealternativetothefieldapproach,butratheranapproximationtoit.ToFeynman,thatcameasabitterdisappointment. YetFeynmandiagramsremainatreasuredassetinphysics,becausetheyoftenprovidegoodapproximationstoreality.Plus,they’reeasy(andfun)toworkwith.Theyhelpusbringourpowersofvisualimaginationtobearonworldswecan’tactuallysee. ThecalculationsthateventuallygotmeaNobelPrizein2004wouldhavebeenliterallyunthinkablewithoutFeynmandiagrams,aswouldmycalculationsthatestablishedaroutetoproductionandobservationoftheHiggsparticle. OnthatdayinSantaBarbara,citingthoseexamples,ItoldFeynmanhowimportanthisdiagramshadbeentomeinmywork.Heseemedpleased,thoughhecouldhardlyhavebeensurprisedathisdiagrams’importance.“Yeah,that’sthegoodpart,seeingpeopleusethem,seeingthemeverywhere,”herepliedwithawink. V. TheFeynmandiagramrepresentationofaprocessismostusefulwhenafewrelativelysimplediagramssupplymostoftheanswer.Thatistheregimephysicistscall“weakcoupling,”whereeachadditionalcomplicatinglineisrelativelyrare.Thatisalmostalwaysthecaseforphotonsinquantumelectrodynamics(QED),theapplicationFeynmanoriginallyhadinmind.QEDcoversmostofatomicphysics,chemistryandmaterialsscience,soit’sanamazingachievementtocaptureitsessenceinafewsquiggles. Asanapproachtothestrongnuclearforce,however,thisstrategyfails.Herethegoverningtheoryisquantumchromodynamics(QCD).TheQCDanaloguesofphotonsareparticlescalledcolorgluons,andtheircouplingisnotweak.Usually,whenwedoacalculationinQCD,ahostofcomplicatedFeynmandiagrams—festoonedwithmanygluonlines—makeimportantcontributionstotheanswer.It’simpractical(andprobablyimpossible)toaddthemallup. Ontheotherhand,withmoderncomputerswecangobacktothetrulyfundamentalfieldequationsandcalculatefluctuationsinthequarkandgluonfieldsdirectly.Thisapproachgivesbeautifulpicturesofanotherkind: Inrecentyearsthisdirectapproach,carriedoutonbanksofsupercomputers,hasledtosuccessfulcalculationsofthemassesofprotonsandneutrons.Inthecomingyearsitwillrevolutionizeourquantitativeunderstandingofnuclearphysicsoverabroadfront. VI. ThepuzzleFeynmanthoughthe’dsolvedisstillwithus,thoughithasevolvedinmanyways. Thebiggestchangeisthatpeoplehavenowmeasuredthedensityofvacuummoreprecisely,anddiscoveredthatitdoesnotvanish.Itistheso-called“darkenergy.”(Darkenergyisessentially—uptoanumericalfactor—thesamethingEinsteincalledthe“cosmologicalconstant.”)Ifyouaverageitovertheentireuniverse,youfindthatdarkenergycontributesabout70percentofthetotalmassintheuniverse. Thatsoundsimpressive,butforphysiciststhebigpuzzlethatremainsiswhyitsdensityisassmallasitis.Foronething,you’llremember,itwassupposedtobeinfinite,duetothecontributionoffluctuatingfields.Onebitofpossibleprogressisthatnowweknowawaytoescapethatinfinity.Itturnsoutthatforoneclassoffields—technically,thefieldsassociatedwithparticlescalledbosons—theenergydensityispositiveinfinity,whileforanotherclassoffields—thoseassociatedwithparticlescalledfermions—theenergydensityisnegativeinfinity.Soiftheuniversecontainsanartfullybalancedmixofbosonsandfermions,theinfinitiescancancel.Supersymmetrictheories,whichalsohaveseveralotherattractivefeatures,achievethatcancellation. Anotherthingwe’velearnedisthatinadditiontofluctuatingfields,thevacuumcontainsnon-fluctuatingfields,oftencalled“condensates.”Onesuchcondensateistheso-calledsigmacondensate;anotheristheHiggscondensate.Thosetwoarefirmlyestablished;theremaybemanyothersyettobediscovered.Ifyouwanttothinkofafamiliaranalogue,imagineEarth’smagneticorgravitationalfield,elevatedtocosmicproportions(andfreedofEarth).Thesecondensatesshouldalsoweighsomething.Indeed,simpleestimatesoftheirdensitygivevaluesfarlargerthanthatoftheobserveddarkenergy. We’releftwithanestimateofthedarkenergythatisfinite(maybe),butpoorlydeterminedtheoreticallyand,onthefaceofit,muchtoobig.Presumablythereareadditionalcancellationswedon’tknowabout.Themostpopularidea,atpresent,isthatthesmallnessofthedarkenergyisakindofrareaccident,whichhappenstooccurinourparticularcornerofthemultiverse.Thoughunlikelyapriori,itisnecessaryforourexistence,andthereforewhatwearefatedtoobserve. Thatstory,I’mafraid,isnotnearlysoelegantasFeynman’s“There’snothingthere!”Let’shopewecanfindabetterone.   ForanyoneinterestedinlearningmoreaboutFeynmandiagramsandquantumelectrodynamics,theauthorrecommendsFeynman’sbookQED:TheStrangeTheoryofLightandMatter. ThisarticlewasreprintedonWired.com. 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