lectures on the geometry of quantization

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1、LecturesontheGeometryofQuantizationSeanBatesAlanWeinsteinDepartmentofMathematicsDepartmentofMathematicsColumbiaUniversityUniversityofCaliforniaNewYork,NY10027USABerkeley,CA94720USAsmb@math.columbia.edualanw@math.berkeley.edu1Contents1Introduction:TheHarmonicOscillator52Th

2、eWKBMethod82.1SomeHamilton-Jacobipreliminaries.......................82.2TheWKBapproximation............................113SymplecticManifolds173.1Symplecticstructures...............................173.2Cotangentbundles................................283.3Mechanicsonmanif

3、olds.............................324QuantizationinCotangentBundles364.1Prequantization..................................364.2TheMaslovcorrection..............................414.3Phasefunctionsandlagrangiansubmanifolds..................464.4WKBquantization................

4、................565TheSymplecticCategory645.1Symplecticreduction...............................645.2Thesymplecticcategory.............................765.3Symplecticmanifoldsandmechanics......................796FourierIntegralOperators836.1Compositionsofsemi-classicalstat

5、es......................836.2WKBquantizationandcompositions......................877GeometricQuantization937.1Prequantization..................................937.2Polarizationsandthemetaplecticcorrection..................987.3Quantizationofsemi-classicalstates...........

6、............1098AlgebraicQuantization1118.1PoissonalgebrasandPoissonmanifolds.....................1118.2Deformationquantization............................1128.3Symplecticgroupoids...............................114ADensities119BThemethodofstationaryphase121CCechcohomology

7、ˇ124DPrincipalT~bundles1252PrefaceThesenotesarebasedonacourseentitled“Symplecticgeometryandgeometricquantization”taughtbyAlanWeinsteinattheUniversityofCalifornia,Berkeley,inthefallsemesterof1992andagainattheCentreEmileBorel(InstitutHenriPoincar´e)inthespringsemesterof1994

8、.Theonlyprerequisiteforthecourse(andforthesenotes)wasaknowledgeofthebasicnotionsfromthetheoryofd