A New Look at the Path Integral of Quantum Mechanics

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1、hep-th/yymm.nnnnANewLookAtThePathIntegralOfQuantumMechanicsEdwardWittenSchoolofNaturalSciences,InstituteforAdvancedStudyEinsteinDrive,Princeton,NJ08540USAandDepartmentofPhysics,StanfordUniversityPaloAlto,CA94305AbstractarXiv:1009.6032v1[hep-th]30Sep2010TheFeynmanpathi

2、ntegralofordinaryquantummechanicsiscomplexifiedanditisshownthatpossibleintegrationcyclesforthiscomplexifiedintegralareassociatedwithbranesinatwo-dimensionalA-model.ThisprovidesafairlydirectexplanationoftherelationshipoftheA-modeltoquantummechanics;sucharelationshiphasbe

3、enexploredfromseveralpointsofviewinthelastfewyears.ThesephenomenahaveananalogforChern-Simonsgaugetheoryinthreedimensions:integrationcyclesinthepathintegralofthistheorycanbederivedfromN=4superYang-Millstheoryinfourdimensions.Hence,undercertainconditions,aChern-Simonspa

4、thintegralinthreedimensionsisequivalenttoanN=4pathintegralinfourdimensions.Contents1Introduction32IntegrationCyclesForQuantumMechanics62.1Preliminaries...................................62.2TheBasicFeynmanIntegral...........................82.3AnalyticContinuation....

5、..........................92.4TheSimplestIntegrationCycles.........................112.5ReviewOfMorseTheory.............................142.6ANewIntegrationCycleForTheFeynmanIntegral..............162.7IAndTheA-Model...............................192.8InterpretationInS

6、igma-ModelLanguage....................212.9ThePhysicalModel................................242.9.1TheBoundaryConditionAndLocalization...............262.10RecoveringTheHilbertSpace..........................283Hamiltonians303.1RerunningTheStoryWithAHamiltonian..........

7、.........303.2HamiltoniansAndSuperpotentials.......................333.2.1Hyper-KahlerSymmetries........................353.2.2AnExample................................363.2.3GeneralAnalysis.............................3714RunningTheStoryInReverse394.1Sigma-ModelsInOn

8、eDimension........................394.2Sigma-ModelsInTwoDimensions........................455AnalogsWithGaugeFields475.1Supe