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1、AlgebraicQuantumFieldTheory∗HansHalvorson†(withanappendixbyMichaelM¨uger‡)February14,2006AbstractAlgebraicquantumfieldtheoryprovidesageneral,mathematicallyprecisedescriptionofthestructureofquantumfieldtheories,andthendrawsoutconse-quencesofthisstructureb
2、ymeansofvariousmathematicaltools—thetheoryofoperatoralgebras,categorytheory,etc..GiventherigorandgeneralityofAQFT,itisaparticularlyapttoolforstudyingthefoundationsofQFT.ThispaperisasurveyofAQFT,withanorientationtowardsfoundationaltopics.Inadditiontocov
3、eringthebasicsofthetheory,wediscussissuesrelatedtononlocality,theparticleconcept,thefieldconcept,andinequivalentrepresen-tations.WealsoprovideadetailedaccountoftheanalysisofsuperselectionrulesbyDoplicher,Haag,andRoberts(DHR);andwegiveanalternativeproofo
4、fDoplicherandRobert’sreconstructionoffieldsandgaugegroupfromthecategoryofphysicalrepresentationsoftheobservablealgebra.ThelatterisbasedonunpublishedideasduetoJ.E.Robertsandtheabstractdualitytheo-remforsymmetrictensor∗-categories,aself-containedproofofwh
5、ichisgivenintheappendix.Contents1AlgebraicProlegomena41.1vonNeumannalgebras..............................41.2C∗-algebrasandtheirrepresentations......................61.3TypeclassificationofvonNeumannalgebras.................81.4Modulartheory............
6、......................10∗ForthcominginHandbookofthePhilosophyofPhysics,editedbyJeremyButterfieldandJohnEarman.HHwishestothank:MichaelM¨ugerforteachinghimabouttheDoplicher-RobertsTheorem;theeditorsfortheirhelpfulfeedbackandpatience;andDavidBaker,TracyLup
7、her,andDavidMalamentforcorrections.MMwishestothankJulienBichonforacriticalreadingoftheappendixandusefulcomments.†DepartmentofPhilosophy,PrincetonUniversity,PrincetonNJ,US.hhalvors@princeton.edu‡DepartmentofMathematics,RadboudUniversity,Nijmegen,NL.mueg
8、er@math.ru.nl12StructureoftheNetofObservableAlgebras132.1Netsofalgebras,basicproperties........................132.2Existence/uniquenessofvacuumstates/representations............142.3TheReeh-SchliederTheorem..........................172
