arXiv:hep-th/9912205 v3 23 Aug 2005FIELDSWARRENSIEGELC.N.YangInstituteforTheoreticalPhysicsStateUniversityofNewYorkatStonyBrookStonyBrook,NewYork11794-3840 USAmailto:siegel@insti.physics.sunysb.eduhttp://insti.physics.sunysb.edu/siegel/plan.html2CONTENTSPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Someeldtheorytexts . . . . . . . . . . . 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . PARTONE:SYMMETRY. . . . . . . . . . . . . . . . . .I.GlobalA. Coordinates1. Nonrelativity. . . . . . . . . . . . . 392. Fermions . . . . . . . . . . . . . . . . . 463. Liealgebra . . . . . . . . . . . . . . . 514. Relativity . . . . . . . . . . . . . . . . 545. Discrete: C,P,T. . . . . . . . . 656. Conformal . . . . . . . . . . . . . . . 68B. Indices1. Matrices . . . . . . . . . . . . . . . . . 732. Representations . . . . . . . . . . 763. Determinants . . . . . . . . . . . . 814. Classicalgroups . . . . . . . . . . 845. Tensornotation . . . . . . . . . . 86C. Representations1. Morecoordinates . . . . . . . . . 922. Coordinatetensors . . . . . . . 943. Youngtableaux . . . . . . . . . . 994. Colorandavor . . . . . . . . . 1015. Coveringgroups . . . . . . . . . 107II.SpinA. Twocomponents1. 3-vectors . . . . . . . . . . . . . . . . 1102. Rotations . . . . . . . . . . . . . . . 1143. Spinors . . . . . . . . . . . . . . . . . 1154. Indices . . . . . . . . . . . . . . . . . . 1175. Lorentz . . . . . . . . . . . . . . . . . 1206. Dirac . . . . . . . . . . . . . . . . . . . 1267. Chirality/duality . . . . . . . . 128B. Poincare1. Fieldequations. . . . . . . . . . 1312. Examples . . . . . . . . . . . . . . . 1343. Solution. . . . . . . . . . . . . . . . . 1374. Mass. . . . . . . . . . . . . . . . . . . . 1415. Foldy-Wouthuysen . . . . . . 1446. Twistors . . . . . . . . . . . . . . . . 1487. Helicity . . . . . . . . . . . . . . . . . 151C. Supersymmetry1. Algebra . . . . . . . . . . . . . . . . . 1562. Supercoordinates . . . . . . . . 1573. Supergroups . . . . . . . . . . . . 1604. Superconformal . . . . . . . . . 1635. Supertwistors . . . . . . . . . . . 164III.LocalA. Actions1. General . . . . . . . . . . . . . . . . . 1692. Fermions . . . . . . . . . . . . . . . . 1743. Fields . . . . . . . . . . . . . . . . . . . 1764. Relativity . . . . . . . . . . . . . . . 1805. Constrainedsystems. . . . 186B. Particles1. Free . . . . . . . . . . . . . . . . . . . . 1912. Gauges . . . . . . . . . . . . . . . . . 1953. Coupling. . . . . . . . . . . . . . . . 1974. Conservation. . . . . . . . . . . . 1985. Paircreation . . . . . . . . . . . . 201C. Yang-Mills1. Nonabelian. . . . . . . . . . . . . . 2042. Lightcone . . . . . . . . . . . . . . . 2083. Planewaves . . . . . . . . . . . . . 2124. Self-duality . . . . . . . . . . . . . 2135. Twistors . . . . . . . . . . . . . . . . 2176. Instantons . . . . . . . . . . . . . . 2207. ADHM . . . . . . . . . . . . . . . . . 2248. Monopoles . . . . . . . . . . . . . . 226IV.MixedA. Hiddensymmetry1. Spontaneousbreakdown . 2322. Sigmamodels . . . . . . . . . . . 2343. Cosetspace . . . . . . . . . . . . . 2374. Chiralsymmetry . . . . . . . . 2405. St uckelberg . . . . . . . . . . . . . 2436. Higgs . . . . . . . . . . . . . . . . . . . 2457. Dilatoncosmology. . . . . . . 247B. Standardmodel1. Chromodynamics. . . . . . . . 2592. Electroweak. . . . . . . . . . . . . 2643. Families. . . . . . . . . . . . . . . . . 2674. GrandUniedTheories. . 269C. Supersymmetry1. Chiral . . . . . . . . . . . . . . . . . . 2752. Actions . . . . . . . . . . . . . . . . . 2773. Covariantderivatives . . . . 2804. Prepotential. . . . . . . . . . . . . 2825. Gaugeactions . . . . . . . . . . . 2846. Breaking . . . . . . . . . . . . . . . . 2877. Extended . . . . . . . . . . . . . . . 2893. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . PARTTWO:QUANTA. . . . . . . . . . . . . . . . . . . .V.QuantizationA. General1. Pathintegrals . . . . . . . . . . . 2982. Semiclassicalexpansion. . 3033. Propagators . . . . . . . . . . . . . 3074. S-matrices . . . . . . . . . . . . . . 3105. Wickrotation . . . . . . . . . . . 315B. Propagators1. Particles . . . . . . . . . . . . . . . . 3192. Properties. . . . . . . . . . . . . . . 3223. Generalizations. . . . . . . . . . 3264. Wickrotation . . . . . . . . . . . 329C. S-matrix1. Pathintegrals . . . . . . . . . . . 3342. Graphs . . . . . . . . . . . . . . . . . 3393. Semiclassicalexpansion. . 3444. Feynmanrules . . . . . . . . . . 3495. Semiclassicalunitarity. . . 3556. Cuttingrules. . . . . . . . . . . . 3587. Crosssections . . . . . . . . . . . 3618. Singularities. . . . . . . . . . . . . 3669. Grouptheory . . . . . . . . . . . 368VI.QuantumgaugetheoryA. Becchi-Rouet-Stora-Tyutin1. Hamiltonian . . . . . . . . . . . . 3732. Lagrangian. . . . . . . . . . . . . . 3783. Particles . . . . . . . . . . . . . . . . 3814. Fields . . . . . . . . . . . . . . . . . . . 382B. Gauges1. Radial . . . . . . . . . . . . . . . . . . 3862. Lorenz . . . . . . . . . . . . . . . . . . 3893. Massive . . . . . . . . . . . . . . . . . 3914. Gervais-Neveu. . . . . . . . . . . 3935. SuperGervais-Neveu . . . . 3966. Spacecone. . . . . . . . . . . . . . . 3997. Superspacecone . . . . . . . . . 4038. Background-eld . . . . . . . . 4069. Nielsen-Kallosh . . . . . . . . . 41210. Superbackground-eld . . 415C. Scattering1. Yang-Mills . . . . . . . . . . . . . . 4192. Recursion . . . . . . . . . . . . . . . 4233. Fermions . . . . . . . . . . . . . . . . 4264. Masses . . . . . . . . . . . . . . . . . . 4295. Supergraphs . . . . . . . . . . . . 435VII.LoopsA. General1. Dimensionalrenormalizn4402. Momentumintegration . . 4433. Modiedsubtractions . . . 4474. Opticaltheorem. . . . . . . . . 4515. Powercounting. . . . . . . . . . 4536. Infrareddivergences . . . . . 458B. Examples1. Tadpoles . . . . . . . . . . . . . . . . 4622. Eectivepotential . . . . . . . 4653. Dimensionaltransmutn . 4684. Masslesspropagators . . . . 4705. Bosonization . . . . . . . . . . . . 4736. Massivepropagators. . . . . 4787. Renormalizationgroup . . 4828. Overlappingdivergences . 485C. Resummation1. Improvedperturbation . . 4922. Renormalons . . . . . . . . . . . . 4973. Borel . . . . . . . . . . . . . . . . . . . 5004. 1/Nexpansion . . . . . . . . . . 504VIII.GaugeloopsA. Propagators1. Fermion. . . . . . . . . . . . . . . . . 5112. Photon . . . . . . . . . . . . . . . . . 5143. Gluon. . . . . . . . . . . . . . . . . . . 5154. GrandUniedTheories. . 5215. Supermatter . . . . . . . . . . . . 5246. Supergluon. . . . . . . . . . . . . . 5277. Schwingermodel . . . . . . . . 531B. Lowenergy1. JWKB. . . . . . . . . . . . . . . . . . 5372. Axialanomaly . . . . . . . . . . 5403. Anomalycancellation . . . 5444. 02. . . . . . . . . . . . . . . . 5465. Vertex . . . . . . . . . . . . . . . . . . 5486. NonrelativisticJWKB. . . 5517. Lattice. . . . . . . . . . . . . . . . . . 554C. Highenergy1. Conformalanomaly . . . . . 5612. e+e hadrons . . . . . . . . 5643. Partonmodel . . . . . . . . . . . 5664. Maximalsupersymmetry5735. Firstquantization . . . . . . . 5764. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . PARTTHREE: HIGHERSPIN. . . . . . . . . . . . . .IX.GeneralrelativityA. Actions1. Gaugeinvariance . . . . . . . . 5872. Covariantderivatives . . . . 5923. Conditions . . . . . . . . . . . . . . 5984. Integration. . . . . . . . . . . . . . 6015. Gravity . . . . . . . . . . . . . . . . . 6056. Energy-momentum. . . . . . 6097. Weylscale . . . . . . . . . . . . . . 611B. Gauges1. Lorenz . . . . . . . . . . . . . . . . . . 6202. Geodesics . . . . . . . . . . . . . . . 6223. Axial . . . . . . . . . . . . . . . . . . . 6254. Radial . . . . . . . . . . . . . . . . . . 6295. Weylscale . . . . . . . . . . . . . . 633C. Curvedspaces1. Self-duality . . . . . . . . . . . . . 6382. DeSitter . . . . . . . . . . . . . . . . 6403. Cosmology . . . . . . . . . . . . . . 6424. Redshift. . . . . . . . . . . . . . . . 6455. Schwarzschild . . . . . . . . . . . 6466. Experiments . . . . . . . . . . . . 6547. Blackholes. . . . . . . . . . . . . . 660X.SupergravityA. Superspace1. Covariantderivatives . . . . 6642. Fieldstrengths . . . . . . . . . . 6693. Compensators . . . . . . . . . . . 6724. Scalegauges . . . . . . . . . . . . 675B. Actions1. Integration. . . . . . . . . . . . . . 6812. Ectoplasm . . . . . . . . . . . . . . 6843. Componenttransformns6874. Componentapproach. . . . 6895. Duality . . . . . . . . . . . . . . . . . 6926. Superhiggs . . . . . . . . . . . . . . 6957. No-scale . . . . . . . . . . . . . . . . 698C. Higherdimensions1. Diracspinors. . . . . . . . . . . . 7012. Wickrotation . . . . . . . . . . . 7043. Otherspins . . . . . . . . . . . . . 7084. Supersymmetry . . . . . . . . . 7095. Theories . . . . . . . . . . . . . . . . 7136. ReductiontoD=4. . . . . . . 715XI.StringsA. Generalities1. Reggetheory. . . . . . . . . . . . 7242. Topology. . . . . . . . . . . . . . . . 7283. Classicalmechanics. . . . . 7334. Types . . . . . . . . . . . . . . . . . . . 7365. T-duality . . . . . . . . . . . . . . . 7406. Dilaton . . . . . . . . . . . . . . . . . 7427. Lattices . . . . . . . . . . . . . . . . . 747B. Quantization1. Gauges . . . . . . . . . . . . . . . . . 7562. Quantummechanics . . . . . 7613. Commutators . . . . . . . . . . . 7664. Conformal transformatns7695. Triality . . . . . . . . . . . . . . . . . 7736. Trees . . . . . . . . . . . . . . . . . . . 7787. Ghosts . . . . . . . . . . . . . . . . . . 785C. Loops1. Partitionfunction . . . . . . . 7912. JacobiThetafunction . . . 7943. Greenfunction . . . . . . . . . . 7974. Open . . . . . . . . . . . . . . . . . . . 8015. Closed . . . . . . . . . . . . . . . . . . 8066. Super . . . . . . . . . . . . . . . . . . . 8107. Anomalies . . . . . . . . . . . . . . 814XII.MechanicsA. OSp(1,1[2)1. Lightcone . . . . . . . . . . . . . . . 8192. Algebra . . . . . . . . . . . . . . . . . 8223. Action . . . . . . . . . . . . . . . . . . 8264. Spinors . . . . . . . . . . . . . . . . . 8275. Examples . . . . . . . . . . . . . . . 829B. IGL(1)1. Algebra . . . . . . . . . . . . . . . . . 8342. Innerproduct . . . . . . . . . . . 8353. Action . . . . . . . . . . . . . . . . . . 8374. Solution. . . . . . . . . . . . . . . . . 8405. Spinors . . . . . . . . . . . . . . . . . 8436. Masses . . . . . . . . . . . . . . . . . . 8447. Backgroundelds . . . . . . . 8458. Strings . . . . . . . . . . . . . . . . . . 8479. RelationtoOSp(1,1[2) . . 852C. Gaugexing1. Antibracket . . . . . . . . . . . . . 8552. ZJBV. . . . . . . . . . . . . . . . . . . 8583. BRST . . . . . . . . . . . . . . . . . . 862AfterMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866PARTONE:SYMMETRY 5OUTLINEInthisOutlinewegiveabrief descriptionof eachitemlistedintheContents.WhiletheContentsandIndexarequickwaystosearch, orlearnthegeneral layoutof the book, the Outline gives more detail for the uninitiated. (The PDF version alsoallowsuseoftheFindcommandinPDFreaders.)Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23general remarks onstyle, organization, focus, content, use, dierences fromothertexts,etc.Someeldtheorytexts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36recommendedalternativesorsupplements(butseePreface). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . PARTONE:SYMMETRY. . . . . . . . . . . . . . . . . .Relativisticquantummechanicsandclassical eldtheory. Poincaregroup=specialrelativity. Enlarged spacetime symmetries: conformal and supersymmetry. Equationsof motionandactionsforparticlesandelds/wavefunctions. Internal symmetries:global(classifyingparticles),local(eldinteractions).I.GlobalSpacetimeandinternalsymmetries.A. Coordinatesspacetimesymmetries1. Nonrelativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Poissonbracket,Einsteinsummationconvention,Galilean symmetry(in-troductoryexample)2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46statistics, anticommutator; anticommutingvariables, dierentiation, in-tegration3. Liealgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51general structureofsymmetries(includinginternal); Liebracket, group,structureconstants;briefsummaryofgrouptheory4. Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Minkowskispace, antiparticles,Lorentz and Poincare symmetries,propertime,Mandelstamvariables,lightconebases5. Discrete: C,P,T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65chargeconjugation,parity,timereversal,inclassicalmechanicsandeldtheory;Levi-Civitatensor66. Conformal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68broken,butuseful,enlargementofPoincare;projectivelightconeB. Indiceseasywaytogrouptheory1. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Hilbert-spacenotation2. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76adjoint, Cartanmetric, Dynkinindex, Casimir, (pseudo)reality, directsumandproduct3. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81withLevi-Civitatensors,Gaussianintegrals;Pfaan4. Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84andgeneralizations,viatensormethods5. Tensornotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86indexnotation,simplestbasesforsimplestrepresentationsC. Representationsusefulspecialcases1. Morecoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Diracgammamatricesascoordinatesfororthogonalgroups2. Coordinatetensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94formulationsofcoordinatetransformations;dierentialforms3. Youngtableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99picturesforrepresentations,theirsymmetries,sizes,directproducts4. Colorandavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101symmetriesofparticlesofStandardModelandobservedlighthadrons5. Coveringgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107relatingspinorsandvectorsII.SpinExtensionofspacetimesymmetrytoincludespin. Fieldequationsforeldstrengthsof all spins. Most ecientmethods for Lorentz indices in QuantumChromoDynamicsor pure Yang-Mills. Supersymmetry relates bosons and fermions, also useful for QCD.A. Twocomponents22matricesdescribethespacetimegroupsmoreeasily(2___0 :___timelikelightlike/nullspacelikeIn particular, the 4-momentum is timelike for massive particles (m2> 0) and lightlikefor massless ones (while tachyons, with spacelike momenta and m2< 0, do not exist,for reasons that are most clear from quantum eld theory). With respect to properLorentztransformations, thosethatcanbeobtainedcontinuouslyfromtheidentity,wecanfurtherclassifytimelikeandlightlikevectorsasforwardandbackward,sincethereisnowaytocontinuouslyrotateavector fromforwardtobackwardwithout it being spacelike(sideways),so only spacelikevectorscan havetheir timecomponentchangesigncontinuously.Thequantummechanicswill bedescribedlater, buttheresultisthatthiscon-straintcanbeusedasthewaveequation. Themainqualitativedistinctionfromthenonrelativisticcaseintheconstraintnonrelativistic : 2mE +p2= 0relativistic : E2+m2+p2= 0is that the equationfor the energyEp0is nowquadratic, andthus has twosolutions:p0= , =_(pi)2+m2Laterwellseehowthesecondsolutionisinterpretedasanantiparticle.A.COORDINATES 57ThetranslationsandLorentztransformationsmakeupthePoincaregroup, thesymmetrythatdenes special relativity. (TheLorentzgroupinD1spaceand1time dimensionis the orthogonal group O(D1,1). The properLorentz groupSO(D1,1), where the S is for special, transforms the coordinates by a matrixwhosedeterminantis1. ThePoincaregroupisISO(D1,1), wheretheIstandsfor inhomogeneous.) For thespinless particletheyaregeneratedbycoordinatetransformationsGI= (Pa, Jab):Pa= pa, Jab= x[apb](where alsoa, b =0, ..., 3). Thenthe fact that thephysics of the freeparticleisinvariantunderPoincaretransformationsisexpressedas[Pa, p2+m2] = [Jab, p2+m2] = 0Writingan arbitraryinnitesimaltransformation asalinearcombinationofthegen-erators,wendxm= xnnm+ m, mn= nmwherethesareconstants. Note that antisymmetryof mndoesnot implyantisym-metryofmn=mppn, becauseofadditional signs. (SimilarremarksapplytoJab.)Exponentiatingtondthenitetransformations,wehavexm= xnnm+m, mpnqpq= mnThesameLorentztransformationsapplytopm, butthetranslationsdonotaectit. Theconditiononfollowsfrompreservationof theMinkowski norm(orinnerproduct), butitisequivalenttotheantisymmetryofmnbyexponentiating=e(compareexerciseIA3.3).Since dxapa is invariant under the coordinate transformations dened by the Pois-sonbracket(thechainrule, sinceeectivelypa a), itfollowsthatthePoincareinvarianceofp2isequivalenttotheinvarianceofthelineelementds2= dxmdxnmnwhichdenesthepropertimes. SpacetimewiththisindenitemetriciscalledMinkowski space, in contrast to the Euclidean space with positive denite metricusedto describenonrelativisticlength measuredin just the three spatial dimensions.(Thesignatureofthemetricisthusthenumbersofspaceandtimedimensions.)58 I.GLOBALExerciseIA4.1Forgeneral variables(qm, pm)andgeneratorG, showfromthedenitionofthePoissonbracketthat(dqmpm) = d_GpmGpm_andthatthisvanishesforanycoordinatetransformation.Forthemassivecase,wealsohavepa= mdxadsFor the masslesscase ds = 0: Masslessparticles travelalong lightlikelines. However,wecandeneanewparametersuchthatpa=dxadiswell-denedinthemasslesscase. Ingeneral,wethenhaves = mWhilethisxes= s/m in themassivecase,in themasslesscaseit insteadrestrictss=0. Thus, propertimedoesnotprovideauseful parametrizationof theworldlineofaclassical masslessparticle, while does: Foranypieceofsuchaline, d isgivenin termsof (any componentof) paand dxa. Later wellsee how this parameterappearsinrelativisticclassical mechanics, andisuseful forquantummechanicsandeldtheory.ExerciseIA4.2Startingfromtheusual Lorentzforcelawforamassiveparticleintermsofproper time s (which doesnt apply to m = 0), rewrite it in terms of to ndaformwhichcanapplytom = 0.ExerciseIA4.3TherelationbetweenxandpiscloselyrelatedtothePoincareconservationlaws:a ShowthatdPa= dJab= 0 p[adxb]= 0andusethistoprovethatconservationofPandJimplytheexistenceofaparametersuchthatpa= dxa/d.bConsideramultiparticlesystem(butstill withoutspin)wheresomeof theparticles caninteract onlywhenat thesamepoint (i.e., bycollision; theyA.COORDINATES 59actasfreeparticlesotherwise). DenePa=

I pIaandJab=

I xI[apIb]asthesumof theindividual momentaandangularmomenta(wherewelabeltheparticlewithI ). Showthatmomentumconservationimpliesangularmomentumconservation,Pa= 0 Jab= 0wherereferstothechangefrombeforetoafterthecollision(s).Special relativitycanalsobestatedasthefactthattheonlyphysicallyobserv-able quantities are those that are Poincare invariant. (Other objects, such as vectors,dependonthechoiceof referenceframe.) Forexample, considertwospinless par-ticlesthatinteract bycollision, producingtwospinless particles(whichmaydierfromtheoriginals). Considerjustthemomenta. (Quantummechanically, thisisacomplete description.) All invariants can be expressed in terms of the masses and theMandelstamvariables(nottobeconfusedwithtimeandpropertime)s = (p1 +p2)2, t = (p1p3)2, u = (p1p4)2where we have usedmomentumconservation, whichshows that eventhese threequantitiesarenotindependent:p2I= m2I, p1 +p2= p3 +p4 s +t +u =4

I=1m2I(The explicit index now labels the particle, for the process 1+23+4.)The simplestreferenceframetodescribethisinteractionisthecenter-of-massframe(actuallythecenter of momentum, where the two 3-momenta cancel). In that Lorentz frame, usingalso rotational invariance, momentum conservation, and the mass-shell conditions, themomentacanbewrittenintermsoftheseinvariantsasp1=1s(12(s +m21m22), 12, 0, 0)p2=1s(12(s +m22m21), 12, 0, 0)p3=1s(12(s +m23m24), 34cos , 34sin, 0)p4=1s(12(s +m24m23), 34cos , 34sin, 0)cos =s2+ 2st (

m2I)s + (m21m22)(m23m24)412342IJ=14[s (mI+mJ)2][s (mI mJ)2]Thephysical regionof momentumspaceis thengivenbys (m1+ m2)2and(m3 +m4)2,and [cos[ 1.60 I.GLOBALExerciseIA4.4Derivetheaboveexpressionsforthemomentaintermsof invariantsinthecenter-of-massframe.ExerciseIA4.5Findtheconditionsons, tanduthatdenethephysical regioninthecasewhereallmassesareequal.Forsomepurposesitwillprovemoreconvenienttousealightconebasisp=12(p0p1) mn=______+ 2 3+ 0 1 0 0 1 0 0 02 0 0 1 03 0 0 0 1______, p2= 2p+p+(p2)2+(p3)2andsimilarlyforthelightconecoordinates(x, x2, x3). (Lightconeisanunfor-tunatebutcommonmisnomer, havingnothingtodowithconesinmostusages.) Inthisbasisthesolutiontothemass-shellconditionp2+m2= 0canbewrittenasp= p=(pi)2+m22p(wherenowi =2, 3), whichmorecloselyresembles thenonrelativisticexpression.(Notethechangeonindices+ uponraisingandlowering.) Aspeciallightconebasisisthenullbasis,p=12(p0p1), pt=12(p2ip3), pt=12(p2+ip3) mn=______+ tt+ 0 1 0 0 1 0 0 0t 0 0 0 1t 0 0 1 0______, p2= 2p+p + 2pt ptwherethesquareof avector is linear ineachcomponent. (Weoftenuse toindicatecomplexconjugation.)ExerciseIA4.6Showthatforp2+ m2=0(m20, pa,=0), thesignsof p+andparealwaysthesameasthesignofthecanonicalenergyp0.ExerciseIA4.7Consider the Poincare group in 1 extra space dimension (D space, 1 time) foramasslessparticle. Interpretp+asthemass,andpastheenergy.A.COORDINATES 61a Showthat theconstraintp2= 0givesthe usualnonrelativisticexpressionfortheenergy.bShow that the subgroup of the Poincare group generated by all generators thatcommute with p+is the Galilean group (in D1 space and 1 time dimensions).Nownonrelativistic mass conservationis part of momentumconservation,andall theGalileantransformationsarecoordinatetransformations. Also,positivity of the mass is related to positivity of the energy (see exercise IA4.4).There are two standard examples of relativistic eects on geometry. Without lossof generalitywecanconsider2dimensions, byconsideringmotioninjust1spatialdirection. One exampleiscalledLorentz-Fitzgerald contraction: Consideranite-sized object movingwith constantvelocity. In our 2D space, this looks like 2 parallellines,representingtheendpoints:(Inhigher dimensions, thisrepresents aone-spatial-dimensional object, likeathinruler, movinginthedirectionof itslength.) If wewereintherestframeof thisobject, thelineswouldbevertical. Inthatframe, thereisasimplephysical waytomeasurethelengthof theobject: Sendlightfromaclocksittingatoneendtoamirrorsittingattheotherend, andtimehowlongittakestomaketheroundtrip.Aclockmeasuressomethingphysical, namelythepropertimeT _ ds2alongitsworldline(thecurvedescribingitshistoryinspacetime). Sinceds2isbydenitionthesameinanyframe,wecancalculatethisquantityinourframe.In this 2D picture lightlikelinesare alwaysslanted at 45. The 2 linesrepresentingtheends of theobject are(inthis frame) x=vt andx=L + vt. Somesimple62 I.GLOBALgeometrythengivesT=2L1 v2 L =1 v2T/2ThismeansthatthelengthLwemeasurefortheobjectisshorter thanthelengthT/2 measured in the objects rest frame by a factor1 v2< 1. Unlike T, the L wehavedenedisnotaphysical propertyoftheobject: Itdependsonboththeobjectandourvelocitywithrespecttoit. Thereisadirectanalogyforrotations: Wecaneasilydene an innitestrip of constantwidthin terms of 2 parallel lines (the ends),wherethewidthisdenedbymeasuringalongalineperpendiculartotheends. Ifweinsteadmeasureatanarbitraryangletotheends, wewontndthewidth, butthewidthtimesafactordependingonthatangle.The most commonpoint of confusionabout relativityis that events that aresimultaneousin one referenceframe are not simultaneousin another (unlessthey areatthesameplace,inwhichcasetheyarethesameevent). Afrequentexampleisofthissort: Youhavetoomuchjunkinyourgarage,soyourcarwonttanymore. Soyourspouse/roommate/whateversays, Noproblem, justdriveitnearthespeedoflight,anditwillLorentzcontracttot.Soyoutryit,butinyourframeinsidethecar you nd it is the garage that has contracted, so your car ts even worse. The realquestion is, What happens to the car when it stops? The answer is, It depends onwhen the front end stops, and when the back end stops. You might expect that theystopatthesametime. Thatsprobablywrong, butassumingitstrue, wehave(atleast)twopossibilities: (1)Theystopatthesametimeasmeasuredinthegaragesreference frame. Then the car ts. However, in the cars frame (its initial fast frame),thefrontendhasstoppedrst, andthebackendkeepsgoinguntil itsmashesintothe front enough to make it t. (2) They stop at the same time in the cars frame. Inthegaragesframe,thebackendofthecarstopsrst,andthefrontendkeepsgoinguntilitsmashesoutthebackofthegarage.Theotherstandardexampleistimedilation: Considertwoclocks. Onemoveswithconstantvelocity, sowechoosetheframewhereitisatrest. Theothermovesatconstantspeedinthisframe,butitstartsatthepositionoftherstclock,movesaway, andthenreturns. (Itisusuallyconvenienttocomparetwoclockswhentheyareatthesamepointinspace,sincethatmakesitunambiguousthatoneisreadingthetwoclocksatthesametime.)A.COORDINATES 63Asimplecalculationshowsthatwhenthemovingclockreturnsitmeasuresatimethatisshorterbyafactorof1 v2. Ofcourse, thisalsohasaNewtoniananalog:Curves between two given points are longer than straight lines. For relativity, straightlinesarealwaysthelongest timelikecurvesbecauseof thefunnyminussigninthemetric.ExerciseIA4.8Youarestandingintheroad, andapolicecarcomestowardyou, ashingitslightsat regularintervals. Itrunsyoudownandkeeps right ongoing,asyouwatchitcontinuetoashitslightsatyouatthesameintervals(asmeasuredbytheclockinthecar). Treatthisasatwo-dimensional problem(onespace,onetime),andapproximatethecarsvelocityasconstant. DrawtheMinkowski-spacepicture(includingyou, thecar, andthelightrays). Ifthe car movesat speed vand ashes its lights at intervals t0(as measuredbytheclockscar), atwhatintervals(accordingtoyourwatch)doyouseethelightsashingwhenitisapproaching,andatwhatintervalsasitisleaving?Special relativity is so fundamental a part of physics that in some areas of physicseveryexperimentismoreevidenceforit, sothatthemanyearlyexperimental testsofitaremoreofhistoricalinterestthanscientic.TheGalileangroupisasymmetryofparticlesmovingatspeedssmallcomparedtolight, butelectromagnetismissymmetricunderthePoincaregroup(actuallytheconformal group). This caused some confusion historically: Since the two groups haveonly translations and rotations in common, it was assumed that nature was invariantunder no velocity transformation (neither Galilean nor Lorentz boost). In particular,thespeedoflightitselfwouldseemtodependonthereferenceframe,sincethelawsofnaturewouldbecorrectonlyinarestframe. Toexplainatrestwithrespecttowhat,physicistsinventedsomethingthatisinvariantunderrotationsandspaceand time translations, but not velocity transformations, and called this medium forwavepropagation the ether, probably becausethey wereonly semiconsciousat the64 I.GLOBALtime. (Theideawassupposedtobelikesoundtravelingthroughtheair, althoughnobodyhadeverfeltanetherealwind.)Many experiments were performed to test the existence of the ether, or at least toshow that the wave equation for light was correct only in references frames at rest. Soasnot tokeepyouinsuspense,wersttellyouthegeneralresultwasthat theethertheorywaswrong. Onthecontrary,onendsthatthespeedoflightinavacuumismeasuredascinbothoftworeferenceframesthataremovingatconstantvelocitywith respect to each other. This means that electromagnetism is right and Newtonianmechanicsiswrong(oratleastinaccurate),sinceMaxwellsequationsareconsistentwiththespeedoflightbeingthesameinall frames, whileNewtonianmechanicsisnotconsistentwithanyspeedbeingthesameinallframes.Therstsuchexperiment wasperformedbyA.A. MichelsonandE.W. Morleyin1887. TheymeasuredthespeedoflightinvariousdirectionsatvarioustimesofyeartotrytodetecttheeectoftheEarthsmotionaroundthesun. Theydetectednodierences, toanaccuracyof1/6ththeEarthsspeedaroundthesun( 104c).(The method was interferometry: seeing if a light beam split into perpendicular pathsofequallengthinterferedwithitself.)Another interestingexperiment was performedin1971byJ.C. Hafele andR.Keating, whocomparedsynchronizedatomicclocks, oneatrestwithrespecttotheEarthssurface,onecarriedbyplane(acommercialairliner)westaroundtheworld,one east. Afterwardsthe clocksdisagreed in a way predictedby the relativisticeectoftimedilation.Probablythemostconvincingevidenceof special relativitycomesfromexperi-ments relatedtoatomic, nuclear, andparticlephysics. Inatomsthespeedof theelectrons is of theorder of thenestructure constant (1/137) times c, andthecorrespondingeects on atomic energy levelsand suchis typicallyof the order of thesquareof that(104), well withintheaccuracyof suchexperiments. Inparticleaccelerators(andalsocosmicrays), variousparticlesareacceleratedtoover99%c,sorelativisticeectsareexaggeratedtothepointwhereparticlesactmorelikelightwavesthanNewtonianparticles. Innuclearphysicstherelativisticrelationbetweenmass and energy is demonstrated by nuclear decay where,unlike Newtonian mechan-ics, thesumof the(rest)massesisnotconserved; thustheatomicbombprovidesastrongproof of special relativity(althoughitseemslikearatherextremewaytoproveapoint).A.COORDINATES 655. Discrete: C,P,TBy considering only symmetries than can be obtained continuously from the iden-tity (Lie groups), we have missed some important symmetries: those that reect someof thecoordinates. Itssucient toconsiderasinglereectionof aspacelikeaxis,andoneofatimelikeaxis; all otherreectionscanbeobtainedbycombiningthesewiththecontinuous(proper, orthochronous)Lorentztransformations. (SpacelikeandtimelikevectorscantbeLorentztransformedintoeachother, andreectionofalightlikeaxiswontpreservep2+ m2.) Also, thereectionofonespatial axiscanbecombinedwitharotationaboutthat axis, resultinginreectionof all threespatial coordinates. (Similargeneralizationsholdforhigherdimensions. Notethatthe product of an even number of reections about dierent axes is a proper rotation;thus,forevennumbersofspatialdimensionsreectionsofallspatialcoordinatesareproper rotations, even though the reection of a single axis is not.)The reversal of thespatialcoordinatesiscalledparity(P),whilethatofthetimecoordinateiscalledtimereversal(T; actually, forhistorical reasons, tobeexplainedshortly, thisisusuallylabeledCT.) Thesetransformationshavethesameeectonthemomen-tum,sothat thedenitionofthePoissonbracketisalsopreserved. Thesediscretetransformations, unlikethe proper ones,are not symmetriesof nature (exceptin cer-tainapproximations): Theonlyexceptionisthetransformationthatreectsallaxes(CPT).Whilethemetricmnisinvariantunderall Lorentztransformations(bydeni-tion),theLevi-Civitatensormnpqtotallyantisymmetric, 0123= 0123= 1is invariant under onlyproper Lorentztransformations: It has anoddnumber ofspaceindicesandof timeindices, soitchangessignunderparityortimereversal.(Moreprecisely, under PorTtheLevi-Civitatensor doesnotsuer theexpectedsignchange, since its constant, sothere is anextrasigncomparedtothe oneexpectedfor a tensor.) Consequently,we can use it to dene pseudotensors: Givenpolar vectors, whose signs change as position or momentum under improper Lorentztransformations, andscalars, whichareinvariant, wecandeneaxialvectorsandpseudoscalarsasVa= abcdBbCcDd, = abcdAaBbCcDdwhichget an extrasign change undersuch transformations (P or CT, but not CPT).66 I.GLOBALThereisanothersuchdiscretetransformationthatisdenedonphasespace,butwhichdoesnotaectspacetime. Itchangesthesignof all componentsof themomentum, while leaving the spacetime coordinates unchanged. This transformationis calledchargeconjugation(C), andis alsoonlyanapproximate symmetryinnature. (Quantummechanically, complexconjugationof the position-space wavefunctionchangesthesignofthemomentum.) Furthermore,itdoesnotpreservethePoissonbracket, butchangesitbyanoverall sign. (ThemisnomerCTfortimereversalfollowshistoricallyfromthefactthatthecombinationofreversingthetimeaxisandchargeconjugationpreservesthesignoftheenergy.) Thephysicalmeaningofthistransformationisclearfromthespacetime-momentumrelationofrelativisticclassical mechanicsp=mdx/ds: Itisproper-timereversal, changingthesignofs.The relation to charge follows from minimal coupling: The covariant momentummdx/ds = p +qA(for chargeq)appearsintheconstraint(p +qA)2+m2= 0inanelectromagneticbackground;p pthenhasthesameeectasq q.Intheprevioussubsection, wementionedhownegativeenergieswereassociatedwithantiparticles. Nowwecanbetterseetherelationintermsofchargeconjuga-tion. Notethatchargeconjugation,sinceitonlychangesthesignof butdoesnoteectthecoordinates, doesnotchangethepathof theparticle, butonlyhowitisparametrized. Thisisalsotrueintermsofmomentum,sincethevelocityisgivenbypi/p0. Thus,theonlyobservablepropertythat ischangedischarge;spacetimeprop-erties(path, velocity, mass; alsospin, aswell seelater)remainthesame. AnotherwaytosaythisisthatchargeconjugationcommuteswiththePoincaregroup. Oneway to identify an antiparticle is that it has all the same kinematical properties (mass,spin)asthecorrespondingparticle,butoppositesignforinternalquantumnumbers(likecharge). (Anotherwayispaircreationandannihilation: SeesubsectionIIIB5below.)Allthesetransformationsaresummarizedinthetable:C CT P T CP PT CPTs + + +t + + + x + + + E + + + p + + + (Theupper-left33matrixcontainsthedenitions, therestisimplied.) Intermsofcomplexwavefunctions, weseethatCisjustcomplexconjugation(noeectonA.COORDINATES 67coordinates, but momentum and energy change sign because of the i in the Fouriertransform). Ontheotherhand,forCTandPthereisnocomplexconjugation,butchanges insignof thecoordinates that arearguments of thewave functions, andalsoonthecorrespondingindicestheorbitalandspinpartsofthesediscretetransformations. (Forexample,derivativesahavesignchangesbecausexadoes,soa vector wavefunction amust have the same sign changes on its indicesfor aatotransformasascalar.) Theothertransformationsfollowasproductsofthese.ExerciseIA5.1Find the eectof each of these7 transformations on wavefunctionsthat are:ascalars,bpseudoscalars,cvectors,daxialvectors.However,from the point of view of the particle there issome kind of kinematicchange, sincethepropertimehaschangedsign: If wethinkof themechanicsof aparticleasaone-dimensional theoryin space(theworldline), wherex()(aswellas any such variables describing spin or internal symmetry) is a wave function or eldonthatspace,then isTonthatone-dimensionalspace. (ThefactwedontgetCTcanbeseenwhenweaddadditional variables: Forexample, if wedescribeinternal U(N)symmetryintermsofcreationandannihilationoperatorsaiandai,thenCmixesthemonboththeworldlineandspacetime. So, ontheworldlinewehavethepureworldlinegeometricsymmetryCTtimesC=T.)Thus, intermsofzerothquantization,worldlineT spacetimeCOn the other hand, spacetimePand CTare simply internal symmetries with respecttotheworldline(asareproper,orthochronousPoincaretransformations).Quantummechanically, thereisagoodreasonforparticlesof negativeenergy:Theyappearincomplex-conjugatewavefunctions, since(eit)*=e+it. Sincewealwaysevaluateexpressionsoftheform f[i),itisnaturalforenergiesofbothsignstoappear.In classical eld theory, we can identify a particle with its antiparticle by requiringtheeldtobeinvariant under chargeconjugation: Forexample, forascalar eld(spinlessparticle),wehavetherealitycondition(x) = *(x)orinmomentumspace,byFouriertransformation,(p) = [(p)]*whichimpliestheparticlehaschargezero(neutral).68 I.GLOBAL6. ConformalPoincaretransformationsarethemostgeneral coordinatetransformationsthatpreservethemassconditionp2+ m2=0, butthereisalargergroup, theconfor-malgroup,thatpreservesthisconstraintinthemasslesscase. Althoughconformalsymmetryis not observedin nature,itis importantin allapproachestoeldtheory:(1) First of all, it is useful in the constructionof free theories (see subsectionsIIB1-4below). Allmassiveeldscanbedescribedconsistentlyinquantumeldtheoryintermsofcouplingmasslesselds. Masslesstheoriesareasubsetofconformaltheories, andsomeconditionsonmasslesstheoriescanbefoundmoreeasilybyndingtheappropriatesubsetofthoseonconformaltheories. Thisisrelatedtothefactthattheconformalgroup,unlikethePoincaregroup,issimple: Ithasno nontrivial subgroup that transforms into itself under the rest of the group (likethewaytranslationstransformintothemselvesunderLorentztransformations).(2) Ininteractingtheoriesattheclassical level, conformal symmetryisalsoimpor-tant in nding and classifying solutions, since at least some parts of the action areconformallyinvariant,socorrespondingsolutionsare relatedbyconformal trans-formations (seesubsectionsIIIC5-7). Furthermore, it is often convenientto treatarbitrarytheories as brokenconformal theories, introducingelds withwhichthebreakingis associated, andanalyze theconformal andconformal-breakingeldsseparately. Thisisparticularlytrueforthecaseofgravity(seesubsectionsIXA7,B5,C2-3,XA3-4,B5-7).(3) Within quantum eld theory at the perturbative level, the only physical quantumeld theories are ones that are conformal at high energies (see subsection VIIIC1).Thequantumcorrectionstoconformal invarianceathighenergyarerelativelysimple.(4) Beyond perturbation theory, the only quantum theories that are well dened maybejusttheoneswhosebreakingof conformal invarianceatlowenergyisonlyclassical (seesubsectionsVIIC2-3,VIIIA5-6). Furthermore, thelargestpossiblesymmetryof anontrivial S-matrixis conformal symmetry(or superconformalsymmetryifweincludefermionicgenerators).(5) Self-duality(ageneralizationof aconditionthatequateselectricandmagneticelds) isuseful forndingsolutionstoclassical eldequations aswell assim-plifyingperturbationtheory,andiscloselyrelatedtotwistors(seesubsectionsIIB6-7,C5,IIIC4-7). In general, self-duality is related to conformal invariance: ForA.COORDINATES 69example, itcanbeshownthatthefreeconformal theoriesinarbitraryevendi-mensionsarejustthosewith(on-mass-shell)eldstrengthsonwhichself-dualitycan be imposed. (In arbitrary odd dimensions the free conformal theories are justthescalarandspinor.)Transformationsthatsatisfy[a(x)pa, p2] = (x)p2for somealsopreservep2= 0, although theydontleavep2invariant. Equivalently,wecanlookforcoordinatetransformationsthatscaledx2= (x)dx2ExerciseIA6.1Findtheconformalgroupexplicitlyintwodimensions,andshowitsinnitedimensional (not just the SO(2,2) describedbelow). (Hint: Use lightconecoordinates.)Thissymmetrycanbemademanifestbystartingwithaspacewithoneextraspaceandtimedimension:yA= (ya, y+, y) y2= yAyBAB= (ya)22y+ywhere (ya)2=yaybabuses the usual D-dimensional Minkowski-space metric ab,andthetwoadditional dimensions have beenwritteninalightcone basis (not tobeconfusedforthesimilarbasisthatcanbeusedfortheMinkowski metricitself).Withrespect tothis metric, the original SO(D1,1) Lorentz symmetryhas beenenlargedtoSO(D,2). ThisistheconformalgroupinDdimensions. However, ratherthan also preserving (D+2)-dimensional translation invariance, we instead impose theconstraintandinvariancey2= 0, yA= (y)yAThisreduces theoriginal spacetotheprojective(invariantunder the scaling)lightcone(whichinthiscasereallyisacone).ThesetwoconditionscanbesolvedbyyA= ewA, wA= (xa, 1,12xaxa)Projectiveinvariancethen meansindependencefrom e (y+), whilethe lightconecon-ditionhas determinedy. y2=0implies y dy =0, sothe simplest conformalinvariantisdy2= (edw +wde)2= e2dw2= e2dx270 I.GLOBALwherewehaveusedw2= 0 wdw= 0. ThismeansanySO(D,2)transformationonyAwillsimplyscaledx2,andscalee2intheoppositeway:dx2=_e2e2_dx2inagreementwiththepreviousdenitionoftheconformalgroup.Theexplicitformofconformaltransformationsonxa= ya/y+nowfollowsfromtheirlinearformonyA,usingthegeneratorsGAB= y[ArB], [rA, yB] = iBAof SO(D,2) in terms of the momentum rA conjugate to yA. (These are dened the samewayastheLorentzgeneratorsJab= x[apb].) Forexample,G+justscalesxa. (Scaletransformationsarealsoknownasdilatations, orjustdilations.) Wecanalsorecognize G+aas generating translations on xa. The only complicated transformationsaregeneratedbyGa, knownasconformal boosts(accelerationtransformations).Sincetheycommutewitheachother(liketranslations), itseasytoexponentiatetondthenitetransformations:y = eGy, G = vay[a]forsomeconstantD-vectorva(whereA /yA). Sincetheconformalboostsactasloweringoperatorsforscaleweight(+ a ),onlytherstthreetermsintheexponentialsurvive:Gy = 0, Gya= vay, Gy+= vayay = y, ya= ya+vay, y+= y++vaya +12v2yxa=xa+12vax21 +vx +14v2x2usingxa= ya/y+,y/y+=12x2.ExerciseIA6.2Make thechange of variables toxa=ya/y+, e =y+, z =12y2. ExpressrAintermsofthemomenta(pa, n, s)conjugateto(xa, e, z). Showthattheconditionsy2= yArA= r2= 0becomez=en = p2= 0intermsofthenewvariables.ExerciseIA6.3Findthegeneratorofinnitesimalconformalboostsintermsofxaandpa.A.COORDINATES 71WeactuallyhavethefullO(D,2)symmetry: Besidesthecontinuoussymmetries,andthediscrete ones of SO(D1,1), we have asecondtimereversal (fromoursecondtimedimension):y+y xaxa12x2Thistransformationiscalledaninversion.ExerciseIA6.4Showthataniteconformalboostcanbeobtainedbyperformingatransla-tionsandwichedbetweentwoinversions.ExerciseIA6.5The conformal group for Euclidean space (or any spacetime signature) can beobtained by the same construction. Consider the special case of D=2 for theseSO(D+1,1)transformations. (Thisisasubgroupof the2Dsuperconformalgroup: SeeexerciseIA6.1.) Usecomplexcoordinatesforthetwophysicaldimensions:z=12(x1+ix2)a Showthattheinversionisz 1z*bShow that the conformal boost is (using a complex number also for the boostvector)z z1 +v*zExerciseIA6.6Anyparitytransformation(reectioninaspatialaxis)canbeobtainedfromanyother byarotationof the spatial coordinates. Similarly, whenthereis morethanonetimedimension, anytimereversal canbeobtainedfromanother (but time reversal cant be rotated into parity, since a timelike vectorcantberotatedintoaspacelikeone). Thus, thecompleteorthogonalgroupO(m,n) can be obtained from those transformations that are continuous fromthe identitybycombining themwith1paritytransformationand1timereversaltransformation(formn, =0).a For the conformal group, nd the rotation (in terms of an angle) that rotatesbetweenthetwotimedirections,andexpressitsactiononxa.bShow that for angle it produces a transformation that is the product of timereversalandinversion.72 I.GLOBALc Use this toshowthat inversionis relatedtotime reversal byndingthecontinuumofconformaltransformationsthatconnectthem.REFERENCES1 M. Planck, S.-B. Preuss. Akad. Wiss. (1899) 440, Ann. der Phys.1 (1900) 69, Physikal-ischeAbhandlungenundVortr age,Bd.I(Vieweg,Braunschweig,1958)560,614:naturalunits.2 I.M. Mills, P. J. Mohr, T. J. Quinn, B.N. Taylor, andE.R. Williams, Metrologia 42(2005)71:some arguments for xing the value of h by denition, so the kilogram is a derived unit.3 F.A.Berezin,Themethodofsecondquantization(Academic,1966):calculuswithanticommutingnumbers.4 P.A.M.Dirac,Proc.Roy.Soc. A126(1930)360:antiparticles.5 E.C.G.St uckelberg,Helv.Phys.Acta14(1941)588, 15(1942)23;J.A.Wheeler,1940,unpublished:therelationofantiparticlestopropertime.6 S.Mandelstam,Phys.Rev. 112(1958)1344.7 H.W.Brinkmann,Proc.Nat.Acad.Sci.(USA)9(1923)1:projectivelightconeasconformaltoatspace.8 P.A.M.Dirac,Ann.Math. 37(1936)429;H.A.Kastrup,Phys.Rev. 150(1966)1186;G.MackandA.Salam,Ann.Phys. 53(1969)174;S.Adler,Phys.Rev. D6(1972)3445;R.MarneliusandB.Nilsson,Phys.Rev. D22(1980)830:conformalsymmetry.9 S.ColemanandJ.Mandula,Phys.Rev. 159(1967)1251:conformalsymmetryasthelargest(bosonic)symmetryoftheS-matrix.10 W.Siegel,Int.J.Mod.Phys.A4(1989)2015:equivalencebetweenconformalinvarianceandself-dualityinalldimensions.B.INDICES 73. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.INDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Inthe previous sectionwe sawvarious spacetime groups (Galilean, Poincare,conformal)intermsof howtheyactedoncoordinates. Thisnotonlygavethemasimple physical interpretation, but alsoallowedadirect relationbetweenclassicalandquantumtheories. However, asweknowfromstudyingrotationsinquantumtheoryintermsofspin,wewilloftenneedtostudysymmetriesofquantumtheoriesforwhichtheclassicalanalogisnotsousefulorperhapsevennonexistent.Wethereforenowconsidersomegeneralresultsofgrouptheory,mostlyforcon-tinuousgroups. Weusetensormethods, ratherthantheslightlymorepowerful butgreatlylessconvenientCartan-Weyl-Dynkinmethods. Muchof thissectionshouldbe review,but is included here for completeness;it is not intended as a substitute foragrouptheorycourse, butasasummaryof thoseresultscommonlyuseful ineldtheory.1. MatricesMatricesaredenedbythewaytheyactonsomevectorspace; annnmatrixtakesonen-componentvectortoanother. Givensomegroup,anditsmultiplicationtable(whichdenesthegroupcompletely), thereismorethanonewaytorepresentit bymatrices. Anysetofmatriceswendthat hasthe samemultiplicationtableasthe group elements is called a representation of that group, and the vector space onwhichthosematricesactiscalledtherepresentationspace.Therepresentationofthe algebraorgroup in termsof explicitmatricesis givenby choosinga basisforthevector space. If we include innite-dimensional representations, then a representationof agroupissimplyawaytowriteitstransformationsthatislinear: =Mislinearin. Moregenerally, wecanalsohavearealizationofagroup, wherethetransformations can be nonlinear. Thesetend to be more cumbersome,so we usuallytrytomakeredenitionsofthevariablesthatmaketherealizationlinear. Aprecisedenitionofmanifestsymmetryisthatall therealizationsusedarelinear. (Onepossibleexceptionisaneorinhomogeneoustransformations=M1 + M2,suchastheusualcoordinaterepresentationofPoincaretransformations,sincethesetransformationsarestillverysimple, becausetheyarereallystilllinear, thoughnothomogeneous.)ExerciseIB1.1Consider a general real ane transformation = M+Von an n-component74 I.GLOBALvectorfor arbitrary real n n matricesMand real n-vectorsV . A generalgroupelementisthus(M, V ).a Perform2suchtransformationsconsecutively, andgivetheresultinggroupmultiplicationrulefor(M1, V1)(M2, V2)=(M3, V3).bFind the innitesimal form of this transformation. Dene the n2+n generatorsasoperatorson,intermsofaand/a.c Findthecommutationrelationsofthesegenerators.dComparealltheabovewith(nonrelativistic)rotationsandtranslations.ExerciseIB1.2Letsconsidersomepropertiesofmatrixinverses:a Show(AB)1=B1A1formatricesAandBthathaveinversesbutdontnecessarilycommutewitheachother.bShowthat1A+B=1A 1AB 1A+1AB 1AB 1A ...(Theremaybeotherassumptions; ignoreconvergencequestions. Hint: Mul-tiplybothsidesbyA +B.)For convenience,wewritematriceswith aHilbert-space-likenotation, but unlikeHilbert space we dont necessarilyassociate bras directlywithkets byHermitianconjugation, oreventransposition. Ingeneral, thetwospacescanevenbedierentsizes, todescribe matrices that are not square; however, for grouptheorywe areinterestedonlyinmatricesthattakeusfromsomevectorspaceintoitself, sotheyare square. Bras havean inner product with kets,but neither necessarilyhas a norm(innerproductwithitself): Ingeneral, if westartwithsomevectorspace, writtenaskets, wecanalwaysdenethedualspace, writtenasbras, bydeningsuchaninnerproduct. Inourcase, wemaystartwithsomerepresentationof agroup, interms of some vector space, and that will give us directly the dual representation. (Iftherepresentationisintermsofunitarymatrices, wehaveaHilbertspace, andthedualrepresentationisjustthecomplexconjugate.)So, wedenecolumnvectors [)withabasis [I), androwvectors [ withabasis I[,whereI=1, ...,ntodescribennmatrices. Thetwobaseshavearelativenormalizationdenedsothattheinnerproductgivestheusualcomponentsum:[) = [I)I, [ = II[; I[J) = JI [) = II; I[) = I, [I) = IThesebasesthendenenotonlythecomponentsofvectors,butalsomatrices:M= [I)MIJJ[, I[M[J) = MIJB.INDICES 75whereasusual theIonthecomponent(matrixelement)MIJlabelstherowofthematrixM, andJ thecolumn. Thisimplies theusual matrixmultiplicationrules,insertingtheidentityintermsofthebasis,I= [K)K[ (MN)IJ= I[M[K)K[N[J) = MIKNKJCloselyrelatedisthedenitionofthetrace,trM= I[M[I) = MII tr(MN) = tr(NM)(Welldiscussthedeterminantlater.)Thebra-ketnotationisreallyjustmatrixnotationwritteninawaytoclearlydistinguishcolumnvectors, rowvectors, andmatrices. Wecan, of course, alsousetheusualpictorialnotation[) =__12...__, [ = (12. . .)M=________1 2 . . . J . . .1 M11M12. . . M1J. . .2 M21M22. . . M2J. . ...................I MI1MI2. . . MIJ. . ...................________Thisisusefulonlywhenlistingindividualcomponents.Wecaneasilytranslatetransformationlawsfrommatrixnotationintoindexno-tationjustbyusingabasisfortherepresentationspace. WenowwritegandGtorefertoeithermatrixrepresentationsof thegroupandalgebraelements, ortotheabstractelements: i.e., eithertoaspecicrepresentation, orthemostgeneral one.Againwritingg= eiG,g[I) = [J)gJI, G[I) = [J)GJIG = iGi, [) = iG[) = [I)ii(Gi)IJJ I= ii(Gi)IJJThedual spaceisntneededforthispurpose. However, foranyrepresentationofagroup,thetranspose(MT)IJ= MJIoftheinverseofthosematricesalsogivesarepresentationofthegroup,sinceg1g2= g3 (g1)T1(g2)T1= (g3)T176 I.GLOBAL[G1, G2] = G3 [GT1, GT2 ] = GT3Thisisthedualrepresentation,whichfollowsfromdeningtheaboveinnerproducttobeinvariantunderthegroup:[) = 0 I= iJi(Gi)JIThe complex conjugate of a complex representation is also a representation, sinceg1g2= g3 g1*g2* = g3*[G1, G2] = G3 [G1*, G2*] = G3*Fromanygivenrepresentation, wecanthusndthreeothersfromtakingthedualandtheconjugate: Inmatrixandindexnotation,= g : I= gIJJ= (g1)T: I= g1JIJ= g* : .I= g*.I.J.J= (g1): .I= g*1.J.I.Jsince (g1)T, g*, and (g1)(but not gT, etc.) satisfy the same multiplication algebraasg, includingordering. Weuseup/downanddotted/undottedindicestodenotethetransformationlawofeachtypeofindex; contractingundottedupindiceswithundotted down indices preserves the transformation law as indicated by the remainingindices, and similarly for dotted indices. These four representations are not necessarilyindependent: Imposingrelationsamongthemishowtheclassicalgroupsaredened(seesubsectionsIB4-5below).2. RepresentationsFor example, we always have the adjoint representation of a Lie group/algebra,whichishowthealgebraactsonitsowngenerators:(1)adjointasoperator: G = iGi, A = iGi A = i[G, A] = jifijkGk i= ikj(Gj)ki, (Gi)jk= ifijkThisgivesustwowaystorepresenttheadjointrepresentationspace: aseithertheusual vector space, orintermsof thegenerators. Thus, weeither usethematrixB.INDICES 77A=iGi(for arbitraryrepresentationof thematrices Gi, or treatingGias justabstractgenerators),orwecanwriteAasarowvector:(2)adjointasvector: A[ = ii[ A[ = iA[G ii[ = ikj(Gj)kii[The adjoint representation also provides a convenient way to dene a (symmetric)groupmetricinvariantunderthegroup,theCartanmetric:ij= trA(GiGj) = fiklfjlk(trAreferstothetracetakenwithrespecttotherepresentationA; equivalently, wecouldtaketheGsinsidethetracetobeintheArepresentation.) ForAbeliangroupsthestructureconstantsvanish, andthussodoesthismetric. Semisimplegroups are those where the metric is invertible (no vanishing eigenvalues). A simplegrouphasnonontrivial subgroupthat transformsintoitself under therest of thegroup: Semisimple groups can be written as products of simple groups. Compactgroupsarethosewhereitispositivedenite(alleigenvaluespositive); theyarealsothose for which the invariant volume of the group space is nite. For simple, compactgroupsitsconvenienttochooseabasiswhereij= cAijforsomeconstantcA(theDynkinindexfortheadjointrepresentation). Forsomegeneral irreduciblerepresentationR of sucha group the normalization of the trace istrR(GiGj) = cRij=cRcAijNowtheproportionalityconstantcR/cAisxedbythechoiceofR(only), sincewehavealreadyxedthenormalizationofourbasis.ExerciseIB2.1WhatiscRforanAbeliangroup?(Hint: notjust1.)In general, the cyclicity property of the trace implies, for any representation, that0 = tr([Gi, Gj]) = ifijktr(Gk)sotr(Gi) = 0forsemisimplegroups. Similarly,wendfijk fijllk= itrA([Gi, Gj]Gk)78 I.GLOBAListotallyantisymmetric: Forsemisimplegroups,thisimpliesthetotal antisymmetryof the structure constants fijk, up to factors (which are absentfor compactgroups inabasiswhereij ij). Thisalsomeanstheadjointrepresentationisitsowndual.(Forexample, forthecompactgroupSO(3), wehaveij= ikljlk=2ij.) Thus,wecanwriteAinathirdway,asacolumnvector(3)adjointasdualvector: [A) = [i)i [i)jji [A) = iG[A)Wecan alsodothisforAbeliangroups,bydeningan invertiblemetricunrelatedtothe Cartan metric: This is trivial for Abelian groups, since the generators themselvesareinvariant,andthussoisanymetriconthem.AnidentityrelatedtothetraceoneisthenormalizationofthevaluekRof theCasimiroperatorforanyparticularrepresentation,ijGiGj= kRIItsproportionalitytotheidentityfollowsfromthefactthatitcommuteswitheachgenerator:[jkGjGk, Gi] = ifjikGj, Gk = 0usingtheantisymmetryofthestructureconstants. (Thusit takesthesamevalueonanycomponentofanirreduciblerepresentation, sincetheyareall relatedbygrouptransformations.) Bytracingthisidentity,andcontractingthetraceidentity,cRcAdA= trR(ijGiGj) = kRdR kR=cRdAcAdRwheredR trR(I)isthedimensionofthatrepresentation.Although quantum mechanics is dened on Hilbert space, which is a kind of com-plexvector space, moregenerallywewant toconsider real objects, likespacetimevectors. Thisrestrictstheformoflineartransformations: Specically, ifweabsorbisasg=eG, theninsuchrepresentations Gitself mustbereal. Theserepresen-tationsarethencalledreal representations, whileacomplexrepresentationisone whoserepresentationisntreal in anybasis. Acomplexrepresentationspacecanhavearealrepresentation,butarealrepresentationspacecanthaveacomplexrep-resentation. In particular, coordinate transformations (of real coordinates) have onlyreal representations, whichiswhyabsorbingtheisintothegeneratorsisausefulconventionthere. For semisimpleunitary groups, hermiticityof the generators of theadjointrepresentationimplies(usingtotal antisymmetryof thestructureconstantsB.INDICES 79andrealityoftheCartanmetric)thatthestructureconstantsarereal,andthustheadjointrepresentationisarealrepresentation. Moregenerally,anyrealunitaryrep-resentation will have antisymmetric generators (G = G* = G G = GT). If thecomplexconjugaterepresentationisthesameastheoriginal(samematricesuptoasimilaritytransformationg*=MgM1), buttherepresentationisnotreal, thenitis called pseudoreal. (An example is the spinor of SU(2), to be described in sectionIC.)Foranyrepresentationgof thegroup, atransformationg g0gg10oneverygroupelement gforsomeparticulargroupelement g0clearlymapsthealgebratoitself,andpreservesthemultiplicationrules. (Similarremarksapplytoapplyingthetransformation to the generators.)However, the same is true for complex conjugation,gg*: Not onlyare the multiplicationrules preserved, but for anyelement gof that representationof the group, g*is alsoanelement. (This canbe shown,e.g., bydeningrepresentationsintermsofthevaluesofall theCasimiroperators,contructedfromvariouspowers of thegenerators.) Inquantummechanics (wherethe representationsare unitary), the latter is called an antiunitary transformation.Althoughthis is asymmetryof the group, it cannot bereproducedbyaunitarytransformation,exceptwhentherepresentationis(pseudo)real.ExerciseIB2.2ShowhowthisworksfortheAbeliangroupU(1). Explainthisantiunitarytransformationinterms of two-dimensional rotations O(2). (U(1)=SO(2),theproperrotationsobtainedcontinuouslyfromtheidentity.)A very simple way to build a representation from others is by direct sum. If wehavetworepresentationsofagroup, ontwodierentspaces, thenwecantaketheirdirectsumbyjustputtingonecolumnvectorontopoftheother, creatingabiggervectorwhosesize(dimension)isthesumofthatoftheoriginaltwo. Explicitly, ifwestartwiththebasis [)fortherstrepresentationand [)forthesecond, thentheunion([), [))isthebasisforthedirectsum. (Wecanalsowrite [I) = ([), [)),where=1, ..., m; =1, ..., n; I=1, ..., m, m + 1, ..., m + n.) Thegroupthenactsoneachpartofthenewvectorintheobviousway:= [), = [) ; g[) = [)g, g[) = [)g [) = [)[) = [) [) or () =__g[) = [)g[)g or (g) =_g00 g_80 I.GLOBAL(Wecanreplacethe withanordinary+ifweunderstandthebasisvectorstobenowinabiggerspace, wheretheelementsof therstbasishavezerosforthenewcomponentsonthebottomwhilethoseofthesecondhavezerosforthenewcompo-nentsontop.) Theimportantpointisthatnogroupelementmixesthetwospaces:Thegrouprepresentationisblockdiagonal. Anyrepresentationthatcanbewrittenas a direct sum (after an appropriate choice of basis) is called reducible. For exam-ple,wecanbuildareduciblerealrepresentationfromanirreduciblecomplexonebyjust taking the direct sum of this complex representation with the complex conjugaterepresentation. Similarly,wecantakedirectsumsofmorethantworepresentations.A more useful way to build representations is by direct product. The idea thereistotakeacolummnvectorandarowvectorandusethemtoconstructamatrix,where the group element acts simultaneously on rows according to one representationandcolumnsaccordingtotheother. Ifthetwooriginal basesareagain [)and [),thenewbasiscanalsobewrittenas [I) = [)(I= 1, ..., mn). Explicitly,[) = [) [) , g([) [)) = [) [)gg g= ggorintermsofthealgebraG= G+GA familar example from quantum mechanics is rotations (or Lorentz transformations),wheretherstspaceispositionspace(soisthecontinuousindexx), actedonbythe orbital part of the generators, while the second space is nite-dimensional,and isacted on by the spin part of the generators. Direct product representations are usuallyreducible: Theythencanbewrittenalsoasdirectsums, inawaythatdependsontheparticularsofthegroupandtherepresentations.Considerarepresentationconstructedbydirectproduct: InmatrixnotationGi= GiI +I GiUsingtr(AB) = tr(A)tr(B),andassumingtr(Gi) = tr(Gi) = 0,wehavetr( Gi Gj) = tr(I)tr(GiGj) + tr(I)tr(GiGj)For example,for SU(N) (see subsectionIB4below)wecan constructthe adjoint rep-resentationfromthedirectproductoftheN-dimensional, deningrepresentationand itscomplexconjugate. (Wealsoget asinglet,butit willnot aect the resultfortheadjoint.) InthatcasewendtrA(GiGj) = 2N trD(GiGj) cDcA=12NFormostpurposes,weusetrD(GiGj) = ij(cD= 1)forSU(N),socA= 2N.B.INDICES 813. DeterminantsWenowreviewsomepropertiesofdeterminantsthatwill proveuseful forthegroupanalysisofthefollowingsubsections. DeterminantscanbedenedintermsoftheLevi-Civitatensor. Asaconsequenceofitsantisymmetry,totallyantisymmetric, 12...n= 12...n= 1 J1...JnI1...In= I1[J1 InJn]since eachpossible numerical indexvalue appears once ineach, sotheycanbematchedupwiths. Bysimilarreasoning,1m!K1...KmJ1...JnmK1...KmI1...Inm= I1[J1 InmJnm]wherethenormalizationcompensatesforthenumberoftermsinthesummation.ExerciseIB3.1Applytheseidentitiestorotationsinthreedimensions:a Givenonlythecommutationrelations[Jij, Jkl] =i[k[iJj]l]andthedenitionGi 12ijkJjk,derivefijk= ijk.bShowtheJacobiidentity[ijlk]lm= 0byexplicitevaluation.c FindtheCartanmetric,andthusthevalueofcA.Thistensorisusedtodenethedeterminant:detMIJ=1n!J1...JnI1...InMI1J1 MInJn J1...JnMI1J1 MInJn= I1...IndetMsince anything totally antisymmetric in n indices must be proportional to the tensor.Thisyieldsanexplicitexpressionfortheinverse:(M1)J1I1=1(n1)!J1...JnI1...InMI2J2 MInJn(detM)1Fromthisfollowsausefulexpressionforthevariationofthedeterminant:MIJdetM= (M1)JIdetMwhichisequivalenttolndetM= tr(M1M)ReplacingMwitheMgivestheoften-usedidentitylndeteM= tr(eMeM) = trM deteM= etr M82 I.GLOBALwherewehaveusedtheboundaryconditionfor M=0. Finally, replacingMinthelastidentitywithln(1 + L)andexpandingbothsidestoorderLngivesgeneralexpressionsfordeterminantsofn nmatricesintermsoftraces:det(1 +L) = etr ln(1+L) detL =1n!(trL)n12(n2)!(trL2)(trL)n2+ExerciseIB3.2Usethedenitionof thedeterminant(andnotitsrelationtothetrace)toshowdet(AB) = det(A)det(B)Theseidentities canalsobederivedbydeningthedeterminant intermsof aGaussianintegral. Werstcollectsomegeneral propertiesof (indenite)Gaussianintegrals. Thesimplestsuchintegralis_d2x2ex2/2=_20d2_0drrer2/2=_0dueu= 1_dDx(2)D/2ex2/2=__dx2ex2/2_D=__d2x2ex2/2_D/2= 1Thecomplexformofthisintegralis_dDz*dDz(2i)De[z[2= 1byreducingtoreal parametersasz =(x + iy)/2. Thesegeneralizetointegralsinvolvingareal,symmetricmatrixSoraHermitianmatrixHas_dDx(2)D/2exTSx/2= (detS)1/2,_dDz*dDz(2i)DezHz= (detH)1bydiagonalizingthe matrices, makingappropriate redenitions of the integrationvariables, andidentifyingthedeterminant of adiagonal matrix. Alternatively, wecanusetheseintegralstodenethedeterminant,andderivethepreviousdenition.TherelationforthesymmetricmatrixfollowsfromthatfortheHermitianonebyseparatingzintoitsreal and imaginaryparts forthespecialcaseH= S. Ifwetreatzandz*asindependentvariables, thedeterminantcanalsobeunderstoodastheJacobianforthe(dummy)variablechangez H1z, z* z*. Moregenerally, ifwedenetheintegralbyanappropriatelimitingprocedureoranalyticcontinuation(for convergence), wecanchoosez andz*tobeunrelated(or evenseparaterealvariables),andSandHtobecomplex.B.INDICES 83ExerciseIB3.3Other properties of determinants can also be derived directly from the integraldenition:a Find an integral expressionfor the inverseof a (complex)matrix Mby usingtheidentity0 =_zI(zJ ezMz)bDerivetheidentitylndetM=tr(M1M)byvaryingtheGaussiande-nitionofthe(complex)determinantwithrespecttoM.Anevenbetter denitionof thedeterminant is interms of ananticommutingintegral (seesubsectionIA2), sinceanticommutativityautomaticallygivestheanti-symmetryoftheLevi-Civitatensor,andwedonthavetoworryaboutconvergence.Wethenhave,foranymatrixM,_dDdDeM= detMwhere can be chosen as the Hermitian conjugate of or as an independent variable,whicheverisconvenient. Fromthedenitionofanticommutingintegration,theonlytermsintheTaylorexpansionoftheexponential thatcontributearethosewiththeproductofoneofeachanticommutingvariable. Total antisymmetryin andinthen yieldsthe determinant;wedenedDdD to givethe correctnormalization.(Thenormalizationisambiguousanywaybecauseofthesignsinorderingtheds.)This determinant can also be considered a Jacobian, but the inverse of the commutingresultfollowsfromthefactthattheintegralsarenowreallyderivatives.ExerciseIB3.4Divide up the range of a square matrix into two (not necessarily equal) parts:Inblockform,M=_A BC D_anddothesamefor the(commutingor anticommuting) variables usedindeningitsdeterminant. Showthatdet_A BC D_= detD det(A BD1C) = detA det(D CA1B)a byintegratingoveronepartof thevariablesrst(thisrequireso-diagonalchangesofvariablesoftheformy y +Ox,whichhaveunitJacobian),orbbyrstprovingtheidentity_A BC D_=_I BD10 I__ABD1C 00 D__I 0D1C I_84 I.GLOBALWethenhave,foranyanti symmetric(even-dimensional)matrixA,_d2DeTA/2= PfA, (PfA)2= detAby the same method as the commutingcase (again with appropriate denitionof thenormalizationofd2D;thedeterminantofanodd-dimensionalantisymmetricmatrixvanishes, since det M= det MT). However, there is now an important dierence: ThePfaanisnotmerelythesquarerootofthedeterminant, butitselfapolynomial,sincewecanevaluateitalsobyTaylorexpansion:PfAIJ=1D!2DI1...I2DAI1I2 AI2D1I2Dwhichcanbe usedas analternate denition. (Normalizationcanbe checkedbyexaminingaspecialcase;theoverallsignispartofthenormalizationconvention.)4. Classical groupsTherotationgroupinthreedimensionscanbeexpressedmostsimplyintermsof22matrices. Thisdescriptionisthemostconvenientfornotonlyspin1/2, butall spins. Thisresultcanbeextendedtoorthogonal groups(suchastherotation,Lorentz,andconformalgroups)inotherlowdimensions,includingallthoserelevanttospacetimesymmetriesinfourdimensions.Therearean innitenumberofLiegroups. Ofthecompactones,allbutanitenumberareamongtheclassicalLiegroups. Theseclassical groupscanbedenedeasily in terms of (real or complex) matrices satisfying a few simple constraints. (Theremainingexceptionalcompact groups can be dened in a similar way with a littleextra eort, but they are of rather specialized interest,so we wont cover them here.)These matrices are thus called the dening representation of the group. (Sometimesthis representation is also called the fundamental representation; however, this termhasbeenusedinslightlydierentwaysintheliterature,sowewillavoidit.) Theseconstraintsareasubsetof:volume: Special: det(g) = 1metric:___hermitian: Unitary:(anti)symmetric:_Orthogonal:Symplectic:gg = gTg= gTg= ( = )(T= )(T= )reality:_Real:pseudoreal(*):g* = g1g* = g1B.INDICES 85wheregisanymatrixinthedeningrepresentationofthegroup, while, , aregroupmetrics,deninginnerproducts(whilethedeterminantdenesthevolume,as in the Jacobian). For the compactcasesand can be chosento be the identity,but we will also consider some noncompact cases. (There are also some uninterestingvariations of Special for complex matrices, setting the determinant to be real or itsmagnitudetobe1.)ExerciseIB4.1Write all the deningconstraints of the classical groups (S, U, O, Sp, R,pseudoreal)intermsofthealgebraratherthanthegroup.Notethemodieddenitionofunitarity, etc. Suchthingsarealsoencounteredinquantummechanics withghosts, since theresultingHilbert space canhave anindenitemetric. Forexample, if wehaveanite-dimensional Hilbertspacewheretheinnerproductisrepresentedintermsofmatricesas[) = thenobservablessatisfyapseudohermiticitycondition[H) = H[) H= HandunitaritygeneralizestoU[U) = [) UU= SimilarremarksapplywhenreplacingtheHilbert-spacesesquilinear(vectortimescomplexconjugateofvector)innerproductwithasymmetric(orthogonal)oranti-symmetric(symplectic)bilinearinnerproduct. Animportantexampleiswhenthewave function carries a Lorentz vector index, as expected for a relativistic descriptionofspin1;thenclearlythetimecomponentisunphysical.Thegroupsofmatricesthatcanbeconstructedfromtheseconditionsarethen:GL(n,C)[SL(n,C)] U:[S]U(n+,n)O:[S]O(n,C)Sp: Sp(2n,C)R:GL(n)[SL(n)]*: [S]U*(2n)U R *O [S]O(n+,n) SO*(2n)Sp Sp(2n) USp(2n+,2n)Of the non-determinant constraints, in the rst column we applied none (GL meansgeneral linear, andCreferstothecomplexnumbers; thereal numbersRare86 I.GLOBALimplicit); in the second column we applied one; in the third column we applied three,since twoof thethree types (unitarity, symmetry, reality) implythe third. (Thecorrespondinggroupswithunitdeterminant, whendistinct, aregiveninbrackets.)Thesesquarematricesareofsizen, n++n, 2n, or2n++2n, asindicated. n+andnrefer tothenumber of positiveandnegativeeigenvalues of themetric or.O(n)diersfromSO(n)byincludingparity-typetransformations, whichcantbeobtainedcontinuouslyfromtheidentity. (SSp(2n)isthesameasSp(2n).) Forthisreason, andalsoforstudyingtopologicalproperties, fornitetransformationsitissometimes moreuseful toworkdirectlywiththegroupelements g, rather thanparametrizingthem in termsof algebra elementsas g= eiG. U(n)diers from SU(n)(andsimilarlyforGL(n)vs. SL(n))onlybyincludingaU(1)groupthatcommuteswiththeSU(n): AlthoughU(1)isnoncompact(itconsistsofjustphasetransforma-tions),acompactformofitcanbeusedbyrequiringthatallchargesareintegers(i.e.,all representationstransform as = eiqfor group parameter , whereqis anintegerdeningtherepresentation).Of these groups, the compact ones are just SU(n), SO(n) (and O(n)), and USp(2n)(allwithn=0). Thecompactgroupshaveaninterestinginterpretationintermsofvariousnumbersystems: SO(n)istheunitarygroupof nnmatricesovertherealnumbers, SU(n)isthesameforthecomplexnumbers, andUSp(2n)isthesameforthequaternions. (Similarinterpretationscanbemadeforsomeof thenoncompactgroups.)The remaining compact Lie groups that we didnt discuss, the exceptionalgroups, can be interpreted as unitary groups over the octonions. (Unlike the classicalgroups, whichforminniteseries, thereareonlyveexceptional compact groups,becauseoftherestrictionsfollowingfromthenonassociativityofoctonions.)5. TensornotationUsuallynonrelativisticphysicsiswritteninmatrixorGibbs notation. Thisisinsucient evenfor19thcenturyphysics: Wecanwriteacolumnorrowvectorpformomentum, andamatrixTformomentofinertia, buthowdowewriteinthatnotationmoregeneral objects? Thesearedierent representations of therotationgroup: Wecanwritehoweachtransformsunderrotations:p= pA, T = ATTATheproblemistowriteall representations.Onealternativeisusedfrequentlyinquantummechanics: Ascalarisspin0, avector is spin 1, etc. Spin s has 2s+1 components, so we can write a column vectorB.INDICES 87withthatmanycomponents. Forexample, momentof inertiaisasymmetric33matrix, andsohas6components. ItcanbeseparatedintoitstraceSandtracelesspiecesR,whichdontmixunderrotations:T= R +13SI, tr(T) = S, tr(R) = 0 tr(T) = tr(ATTA) = tr(AATT) = tr(T) tr(R) = 0, S= Susingthecyclicityof thetrace. Thustheirreduciblepartsof TarethescalarSandthespin-2(5components)R. ButifweweretowriteRasa5-vector, itwouldbeamesstorelatethe55matrixthatrotatesittothe33matrixA, andevenworse to write a scalar like pRpTin terms of 2 3-vectors and 1 5-vector. (In quantummechanics,thisisdonewithClebsch-Gordan-Wignercoecients.)The simplest solutionis touse indices. Thenits easytowrite anobject ofarbitraryinteger spinsasageneralizationof what wejust didforspins0,1,2: Ithass3-vectorindices, inwhichitistotally(forany2ofitsindices)symmetricandtraceless:Ti1...is: T...i...j...= T...j...i..., T...i...j...ij= 0andittransformsastheproductofvectors:Ti1...is= Tj1...jsAj1i1...AjsisSimilarremarksapplytogrouptheoryingeneral: Althoughhistoricallygrouprepresentations have usually been taught in the notation where an m-component rep-resentationof agroupdenedbynnmatricesisrepresentedbyanm-componentvector, carrying a single index with values 1 to m, a much more convenient and trans-parent methodis tensor notation, where ageneral representationcarries manyindicesrangingfrom 1to n,withcertainsymmetries(and perhapstracelessness)im-posedonthem. (Tensornotationforacoveringgroupisgenerallyknownasspinornotation for the corresponding orthogonal group: See subsection IC5.)This notationtakesadvantageofthepropertydescribedaboveforexpressingarbitraryrepresenta-tions in terms of direct products of vectors. In terms of transformation laws, it meansweneedtoknowonlythedeningrepresentation, sincethetransformationof thisrepresentationisappliedtoeachindex.Thereareatmostfourvectorrepresentations, bytakingthedual andcomplexconjugate; we use the correspondingindexnotation. Thenthe groupconstraintssimplystatetheinvarianceofthegroupmetrics(andtheircomplexconjugatesandinverses), whichthuscanbeusedtoraise,lower,andcontractindices:88 I.GLOBALvolume: Special: I1...Inmetric:_hermitian: Unitary:(anti)symmetric:_Orthogonal:Symplectic:.IJIJIJreality:_Real:pseudoreal(*):.IJ.IJAsaresult,wehaverelationssuchasI[J) = IJor IJ, .I[J) = .IJWealsodeneinversemetricssatisfyingKIKJ= KIKJ= .KI .KJ= IJ(andsimilarlyforcontractingthesecondindexof eachpair). Therefore, withuni-tarity/(pseudo)realitywecanignorecomplexconjugaterepresentations(anddottedindices),convertingthemintounconjugatedoneswiththemetric,whilefororthogo-nality/symplecticitywecandothesamewithrespecttoraising/loweringindices:Unitary: .I= .IJJOrthogonal: I= IJJSymplectic: I= IJJReal: .I= .IJJpseudoreal(*): .I= .IJJFor the real groups there is also the constraint of reality on the dening representation:.I (I)* = .I .IJJExerciseIB5.1As anexample of theadvantages of indexnotation, showthat SSpis thesameasSp. (Hint: Writeoneinthedenitionofthedeterminantintermsof sbytotal antisymmetrization, whichthencanbedroppedbecauseitis enforcedbytheother . Onecanignorenormalizationbyjust showingdetM= detI.)ForSO(n+,n), thereisaslightmodicationof asignconvention: Sincethenindicescanberaisedandloweredwiththemetric, I...isusuallydenedtobetheresultofraisingindicesonI...,whichmeans12...n= 1 12...n= det= (1)nB.INDICES 89ThenI...shouldbereplacedwith(1)nI...intheequationsofsubsectionIB3: Forexample,J1...JnI1...In= (1)nI1[J1 InJn]Wenowgivethesimplestexplicitformsforthedeningrepresentationsof theclassical groups. Themostconvenientnotationistolabel thegeneratorsbyapairof fundamental indices, sincetheadjointrepresentationisobtainedfromthedirectproductofthefundamental representationanditsdual (i.e., asamatrixlabeledbyrowandcolumn). ThesimplestexampleisGL(n), sincethegeneratorsarearbitrarymatrices. Wetherefore choose as abasis matrices witha1as oneentryand0severywhere else, and label that generator by the row and column where the 1 appears.Explicitly,GL(n) : (GIJ)KL= LI JK GIJ= [J)I[ThisbasisappliesforGL(n,C)aswell,theonlydierencebeingthatthecoecients in G = IJGJIare complex instead of real. The next simplest case is U(n): We canagainusethisbasis, althoughthematricesGIJarenotall hermitian, byrequiringthatIJbeahermitianmatrix. Thisturnsouttobemoreconvenientinpracticethanusingahermitianbasis forthegenerators. Awell knownexampleisSU(2),wherethetwogeneratorswiththe1asano-diagonal element(and0selsewhere)are knownas the raisingand loweringoperators J, and are more convenientthantheirhermitianpartsforpurposesof contructingrepresentations. (Thisgeneralizestoother unitarygroups, where all the generators ononesideof the diagonal areraising, all those on the other side are lowering, and those along the diagonal give themaximalAbeliansubalgebra,orCartansubalgebra.)Representationsfortheotherclassical groupsfollowfromapplyingtheirdeni-tionstotheGL(n)basis. WethusndSL(n) : (GIJ)KL= LI JK 1nJILK GIJ= [J)I[ 1nJI[K)K[SO(n) : (GIJ)KL= K[ILJ] GIJ= [[I)J][Sp(n) : (GIJ)KL= K(ILJ) GIJ= [(I)J)[As before, SL(n,C) and SU(n) use the same basis as SL(n), etc. For SO(n) and Sp(n)wehaveraisedandloweredindiceswiththeappropriatemetric(soSO(n)includesSO(n+,n)). For some purposes (especially for SL(n)), its more convenient to imposetracelessnessor(anti)symmetryonthematrix,andusethesimplerGL(n)basis.ExerciseIB5.2Ournormalizationforthegeneratorsoftheclassical groupsisthesimplest,andindependentofn(exceptforsubtractingouttraces):90 I.GLOBALa Find the commutation relations of the generators (structure constants) for thedeningrepresentationof GL(n) as givenin the text. Note that the values ofallthestructureconstantsare0, i. ShowthatcD= 1(seesubsectionIB2).bConsider the GL(m) subgroup of GL(n) (m