Renormalization and effective field theory slides Kevin Costello

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1、RenormalizationandeectiveeldtheoryConsiderscalareldtheoriesonaRiemanniancompactmanifoldM.SpaceofeldsisC1(M).ActionisoftheformZS()=L()MwhereL()isaLagrangian(forexample,L()=D+4whereDistheLaplacian).ConsiderscalareldtheoriesonaRiemanniancompactmanifoldM.Spaceofelds

2、isC1(M).ActionisoftheformZS()=L()MwhereL()isaLagrangian(forexample,L()=D+4whereDistheLaplacian).Fundamentalquantitiesofthetheory(correlationfunctions)aresupposedtobegivenbyfunctionalintegralsoftheformZeS()=~O():::O()1n2C1(M)whereOiaresomeotherfunctonsonC1(M)(obser

3、vables).Theseinnite-dimensionalintegralsdon'tmakesense.Idea:(Wilson,Kadano...)allphysicsuptoenergyisdescribedbyalowenergyeectivetheory.Idea:(Wilson,Kadano...)allphysicsuptoenergyisdescribedbyalowenergyeectivetheory.EectivetheoryisdescribedbyafunctionalSe[]:C1(M)!R[

4、0;)C1(M)[0;)isthespaceoflow-energyelds:sumsofeigenvectorsoftheLaplacianwitheigenvaluein[0;).Idea:(Wilson,Kadano...)allphysicsuptoenergyisdescribedbyalowenergyeectivetheory.EectivetheoryisdescribedbyafunctionalSe[]:C1(M)!R[0;)C1(M)[0;)isthespaceoflow-energyelds:s

5、umsofeigenvectorsoftheLaplacianwitheigenvaluein[0;).Fundamentalquantitesoftheeectivetheoryaregivenbyintegralsoverthenite-dimensionalspaceC1(M)[0;).If0<physicsatscalesbelow0isdeterminedbyphysicsatscalesbelow.If0<physicsatscalesbelow0isdeterminedbyphysicsatscalesbel

6、ow.Thus:Se[0]determinedbySe[].If0<physicsatscalesbelow0isdeterminedbyphysicsatscalesbelow.Thus:Se[0]determinedbySe[].Thisistherenormalizationgroupequation:(Z)Se[0]()=~logeSe[](L+H)=~LH2C1(M)[0;)1Herethelow-energyeldLisinC(M)[0;0).Allexperimentsoccur

7、atsomeniteenergy,andsomeasuresomeeectivetheorySe[].Allexperimentsoccuratsomeniteenergy,andsomeasuresomeeectivetheorySe[].Whateveraquantumeldtheoryis,itshouldgivelow-energyeectivetheoriesSe[]forall<1.Allexperimentsoccuratsomeniteenergy,andsomeasuresomeeectiveth

8、eorySe[].Whateveraquantumeldtheoryis,itshouldgivelow-energyeectivetheor