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1、QUANTUMTHEORYOFGRAVITY.IIwherewenowusethepropagator8+inplaceofGinfrontoftheintegralsoastoobtaincorrectexternal-linewavefunctionsfortheSmatrix.Theonlyverticeswhichgetinsertedbythefactorinsquarebracketsin(20.36b)arethebareverticesS~andVt;&e.Therefore(O,co$'~0,—co)maybeexpressedint
2、hecompactform(12.20),butwithGreplacedbyS~,~providedthesymbol8/5pisnolongertakenliterallybutisunderstoodtoyieldGSsGwhenactingonGandS~+twhenactingonS,tohavenoeffectonV&„;&s,andtoinsert(inallpossibleways)intoany6ctitiousquantum"loopmerelyonemorevertexV&„&pItavinglhesameorientatt'ot
3、tasatttheotherverticesalreadyirttheloop.Withthisunderstandingitiseasytoseethat(20.35)thenyieldsalsoEqs.(12.21),(12.22),and(12.23),withthemodi6cationG~N~appliedtoallexternallines.Chronologicalproductamplitudesde6nedinthiswaymaybeuseddirectlyin(13.6)tocalculatetheSmatrix.Theconsis
4、tencyofthesesimpleruleswithpreviouslyobtainedresultsisreadilychecked.Forexample,ifnon-causalchainsarereinsertedintoFigs.2(b)and3(b)theresultingprimarydiagramsforthelowest-orderradiativecorrectionstotheone-andtwo-quantumamplitudesarepreciselythoseobtainedbythepresentprescription.
5、WenoteinparticularthesuKciencyoftheverticesS„andV~;~pandtheuniformorientationofthelatteraroundany6ctitiousquantumloop."ItwillbenotedthattheoperatorsS/by',whenredefinedinthisway,arestillcommutative.PHYSICALREVIEWVOLUMEj.62,NUMBER525OCTOBER&967QuantumTheoryofGravity.III.Applicatio
6、nsoftheCovariantTheory*BRvCES.DEWITTInstituteforAdvancedStudy,Princeton,itIetoJerseyDepartnientofPhysics,UniversityofNorthCarolina,ChapelHill,IiorthCarol&tat(Received25July1966;revisedmanuscriptreceived9January196'I)Thebasicmomentum-spacepropagatorsandvertices(includingthosefort
7、hefictitiousquanta)aregivenforboththeYang-Millsandgravitationalfields.Thesepropagatorsareusedtoobtainthecrosssectionsforgravitationalscatteringoftwoscalarparticles,scatteringofgravitonsbyscalarparticles,graviton-gravitonscattering,two-.gravitonannihilationofscalar-particlepairs,
8、andgravitonbremsstrah-lung.Specialfeaturesofthe
