Stetz--Lie Groups in Modern Physics 2011 .pdf

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1、LieGroupsinModernPhysicsA.W.StetzSeptember21,20112Contents1Introduction51.1TheDefinitionofaGroupandSomeExamples........71.2IsomorphismandHomomorphism................181.3TheLieAlgebraoftheRotationGroup............201.4FormalDefinitions........................231.5Reconstruc

2、tingtheGroup....................252Representations292.1TheClassicalMatrixGroups..................302.2TheExponentialofaMatrix..................392.3RepresentationsinQuantumMechanics............432.4EquivalentRepresentations...................462.5ReducibleandIrreducibleRe

3、presentation...........483FormalPropertiesofLieGroups533.1TheCompositionFunction...................533.2GlobalStructure.........................623.3Invariantintegrationandcompactgroups...........723.3.1Compactgroups.....................753.4TheLieAlgebra.................

4、........803.4.1LocalTransformationGroups..............813.4.2LocalLinearLieGroups.................853.4.3Parametertransformations...............884Thecatalogofalgebras934.1Representations..........................984.1.1TheAdjointRepresentation...............994.1.2TheJor

5、dancanonicalform...............1034.1.3SimultaneouseigenvectorsandLie’stheorem.....11034CONTENTS4.2TheSecularEquation......................1174.3TheCartanMetricandtheKillingForm............1304.4PropertiesoftheRoots.....................1354.5Classificationofsemisimplealgeb

6、ras..............1454.5.1Simpleroots.......................1544.5.2Dynkindiagrams.....................1625RealAlgebrasandTheirGroups1715.1Compactgroupsandcompactalgebras.............1725.2TheWeylcanonicalformandthecompactrealform.....1765.3Automorphism....................

7、......1815.4TheCatalogofAutomorphisms.................1855.4.1Innerautomorphisms...................1855.4.2Dynkindiagramsandouterautomorphisms......1865.4.3Summaryofalgebras...................190Chapter1IntroductionNewdevelopmentsinphysicsareoftenbasedonrecentdevelopments

8、inmathematics.Forexample,generalrelativityisanapplicationofnon-Euclideandifferentialgeometry.Quan