Introduction to Relativistic Quantum Field Theory

IntroductiontoRelativisticQuantumFieldTheoryHendrikvanHees1TexasA&MUniversityCyclotronInstituteMS-3366CollegeStation,TX77483-3366USA10thMay20061e-mail:[e-mail]2Contents1PathIntegrals111.1QuantumMechanics....................................111.2ChoiceofthePicture....................................131.3FormalSolutionoftheEquationsofMotion.......................161.4Example:TheFreeParticle................................181.5TheFeynman-KacFormula................................201.6ThePathIntegralfortheHarmonicOscillator......................241.7SomeRulesforPathIntegrals...............................271.8TheSchr¨odingerWaveEquation.............................271.9PotentialScattering....................................291.10GeneratingfunctionalforVacuumExpectationValues.................351.11BosonsandFermions,andwhatelse?...........................372NonrelativisticMany-ParticleTheory392.1TheFockSpaceRepresentationofQuantumMechanics................393CanonicalFieldQuantisation433.1SpaceandTimeinSpecialRelativity...........................443.2TensorsandScalarFields.................................483.3Noether’sTheorem(ClassicalPart)............................533.4CanonicalQuantisation..................................583.5TheMostSimpleInteractingFieldTheory:φ4.....................633.6TheLSZReductionFormula...............................653.7TheDyson-WickSeries..................................673.8Wick’sTheorem......................................693.9TheFeynmanDiagrams..................................713Contents4RelativisticQuantumFields774.1CausalMassiveFields...................................784.1.1MassiveVectorFields...............................794.1.2MassiveSpin-1/2Fields..............................804.2CausalMasslessFields...................................844.2.1MasslessVectorField...............................844.2.2MasslessHelicity1/2Fields............................874.3QuantisationandtheSpin-StatisticsTheorem......................874.3.1Quantisationofthespin-1/2DiracField.....................884.4DiscreteSymmetriesandtheCPTTheorem.......................924.4.1ChargeConjugationforDiracspinors......................924.4.2TimeReversal...................................944.4.3Parity........................................964.4.4LorentzClassificationofBilinearForms.....................964.4.5TheCPTTheorem................................984.4.6RemarkonStrictlyNeutralSpin–1/2–Fermions.................1004.5PathIntegralFormulation.................................1014.5.1Example:TheFreeScalarField..........................1074.5.2TheFeynmanRulesforφ4revisited.......................1094.6GeneratingFunctionals...................................1114.6.1LSZReduction...................................1124.6.2Theequivalencetheorem.............................1134.6.3GeneratingFunctionalforConnectedGreen’sFunctions............1144.6.4EffectiveActionandVertexFunctions......................1174.6.5Noether’sTheorem(QuantumPart).......................1214.6.6~-Expansion.....................................1234.7ASimpleInteractingFieldTheorywithFermions....................1275Renormalisation1335.1Infinitiesandhowtocurethem..............................1335.1.1Overviewovertherenormalisationprocedure..................1375.2Wickrotation........................................1395.3Dimensionalregularisation.................................1435.3.1TheΓ-function...................................1445.3.2Sphericalcoordinatesinddimensions......................1524Contents5.3.3Standard-integralsforFeynmanintegrals....................1525.4The4-pointvertexcorrectionat1-looporder......................1555.5Powercounting.......................................1575.6Thesetting-sundiagram..................................1605.7Weinberg’sTheorem....................................1645.7.1ProofofWeinberg’stheorem...........................1675.7.2ProofoftheLemma................................1745.8ApplicationofWeinberg’sTheoremtoFeynmandiagrams...............1755.9BPH-Renormalisation...................................1795.9.1Someexamplesofthemethod...........................1805.9.2ThegeneralBPH-formalism............................1825.10Zimmermann’sforestformula...............................1845.11Globallinearsymmetriesandrenormalisation......................1875.11.1Example:1-looprenormalisation.........................1925.12Renormalisationgroupequations.............................1955.12.1HomogeneousRGEsandmodifiedBPHZrenormalisation...........1965.12.2ThehomogeneousRGEanddimensionalregularisation.............1995.12.3SolutionstothehomogeneousRGE.......................2005.12.4IndependenceoftheS-Matrixfromtherenormalisationscale.........2015.13Asymptoticbehaviourofvertexfunctions........................2025.13.1TheGell-Mann-Lowequation...........................2035.13.2TheCallan-Symanzikequation..........................2046QuantumElectrodynamics2096.1GaugeTheory........................................2096.2