Foundations of Quantum Field Theory

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1、FoundationsofQuantumFieldTheoryMichaelR.Douglas1SimonsCenter/StonyBrookUniversityString-Math2011AbstractAsurveyofnonperturbativeandpotentiallyrigorousdefinitionsofquantumfieldtheory,andthequestionswewouldliketostudywiththem.MichaelR.Douglas(SimonsCenter)

2、FoundationsofQFTString-Math20111/38Outline1Introduction2MathematicsandphysicsHistoryMathandphysicspointsofviewFoundationaltopics3QuestionsrequiringbetterfoundationsRigoroususeandjustificationofphysicsmethodsConnectingapproachestononperturbativeQFTProvin

3、gthatQFTswithspecificpropertiesdonotexistRoughcharacterizationofthespaceofQFTs4ApproachestobetterfoundationsPresentationofaQFTAxiomaticQFTAlgebraicapproachesNewapproachesfromstatisticalmechanicsConstructiveQFTMichaelR.Douglas(SimonsCenter)FoundationsofQ

4、FTString-Math20112/38IntroductionQuantumfieldtheoryandstringtheoryhavehadasignificantimpactonmathematics,andmathematicsisessentialinstudyingquantumfieldtheoryandstringtheory,aswehaveseenineverytalkatthisconference.Thisleadstoanaturalquestion,whichmanyhave

5、raised:Howcanwemakemathematicallyprecisedefinitionsofquantumfieldtheoryandstringtheory?MichaelR.Douglas(SimonsCenter)FoundationsofQFTString-Math20113/38IntroductionThereisalonghistoryofmathematicallypreciseapproachestoquantumfieldtheory,includingaxiomatic

6、quantumfieldtheoryconstructivequantumfieldtheoryalgebraicquantumfieldtheoryfunctionalintegrationandapproximatingexpansionsvertexalgebrasandconformalfieldtheoryprobabilisticapproaches,suchasSchramm-Loewnerevolutionchiralalgebrasandfactorizationalgebrastopol

7、ogicalfieldtheoryandhighercategorytheoryMichaelR.Douglas(SimonsCenter)FoundationsofQFTString-Math20114/38IntroductionInthistalk,wewillDiscusswhat‘mathematicalprecision’wouldmeaninthiscontext.Discussquestionsforwhichthisisclearlyvaluable,independentofmet

8、hodologicalorsociologicalconsiderations.Surveysomeoftheapproaches,anddiscussnextstepswecouldtaketowardsansweringthesequestions.MichaelR.Douglas(SimonsCenter)FoundationsofQFTString-Math20115/38MathematicsandphysicsHistoryOneshouldprobabl