Special Topics in Particle Physics (Robert Gero...

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1、SpecialTopicsinParticlePhysicsRobertGerochApril18,2005Contents1TheKlein-GordonEquation32HilbertSpaceandOperators63Positive-FrequencySolutionsoftheKlein-GordonEquation84ConstructingHilbertSpacesandOperators135HilbertSpaceandOperatorsfortheKlein-GordonEquation156TheDir

2、ectSumofHilbertSpaces187TheCompletionofanInner-ProductSpace208TheComplex-ConjugateSpaceofaHilbertSpace229TheTensorProductofHilbertSpaces2310FockSpace:TheSymmetricCase2811FockSpace:TheAnti-SymmetricCase3112Klein-GordonFieldsasOperators3413TheHilbertSpaceofSolutionsofM

3、axwell'sEquations3914MaxwellFieldsasOperators4415ThePoincareGroup4716RepresentationsofthePoincareGroup4917CasimirOperators:SpinandMass5418Spinors6119TheDiracEquation6520TheNeutrinoEquation7221ComplexKlein-GordonFields7522PositiveEnergy7523FieldsasOperators:Propagat

4、ors7924SpinandStatistics83125?-Algebras8526Scattering:TheS-Matrix9127TheHilbertSpaceofInteractingStates9428CalculatingtheS-Matrix:AnExample9629TheFormulafortheS-Matrix10030Dimensions10231ChargeReversal10432ParityandTimeReversal10733ExtendingOperatorstoTensorProductsa

5、ndDirectSums11134ElectromagneticInteractions11435TransitionAmplitudes11921TheKlein-GordonEquationWewanttowritedownsomekindofaquantumtheoryforafreerelativisticparticle.WearefamiliarwiththeoldSchr•odingerprescription,whichmoreorlessinstructsusastohowtowritedownaquantum

6、theoryforasimple,nonrel-ativisticclassicalsystem.Theideaistomimicasmuchatthatprescriptionaswecan.Indoingthis,anumberofdicultieswillbeencounteredwhich,how-ever,weshellbeabletoresolve.Thereisareasonableandconsistentquantumtheoryforafreerelativistic(spinzero)particle.R

7、ecalltheSchr•odingerprescription.Wehaveaclassicalsystem(e.g.,apendulum,oraballrollingonatable).Themanifoldofpossibleinstantaneouscongurationsofthissystemiscalledcongurationspace,andpointsofthismanifoldarelabeledbyletterssuchasx.However,inordertospecifycompletelythe

8、stateofthesystem(i.e.,inordertogiveenoughinformationtouniquelydetermineitsfutureevolution),wemustspecifyatsomeinitialtimebothitscon