FIELDSWarrenSiegelC.N.YangInstituteforTheoreticalPhysicsStateUniversityofNewYorkatStonyBrookStonyBrook,NewYork11794-3840 USAmailto:siegel@insti.physics.sunysb.eduhttp://insti.physics.sunysb.edu/siegel/plan.html2CONTENTSPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Someeldtheorytexts . . . . . . . . . . . 35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . PARTONE:SYMMETRY. . . . . . . . . . . . . . . . . .I.GlobalA. Coordinates1. Nonrelativity. . . . . . . . . . . . . 392. Fermions . . . . . . . . . . . . . . . . . 443. Liealgebra . . . . . . . . . . . . . . . 484. Relativity . . . . . . . . . . . . . . . . 525. Discrete: C,P,T. . . . . . . . . 586. Conformal . . . . . . . . . . . . . . . 61B. Indices1. Matrices . . . . . . . . . . . . . . . . . 662. Representations . . . . . . . . . . 693. Determinants . . . . . . . . . . . . 744. Classicalgroups . . . . . . . . . . 775. Tensornotation . . . . . . . . . . 79C. Representations1. Morecoordinates . . . . . . . . . 842. Coordinatetensors . . . . . . . 863. Youngtableaux . . . . . . . . . . 914. Colorandavor . . . . . . . . . . 935. Coveringgroups . . . . . . . . . . 98II.SpinA. Twocomponents1. 3-vectors . . . . . . . . . . . . . . . . 1022. Rotations . . . . . . . . . . . . . . . 1053. Spinors . . . . . . . . . . . . . . . . . 1074. Indices . . . . . . . . . . . . . . . . . . 1095. Lorentz . . . . . . . . . . . . . . . . . 1116. Dirac . . . . . . . . . . . . . . . . . . . 1187. Chirality/duality . . . . . . . . 120B. Poincare1. Fieldequations. . . . . . . . . . 1232. Examples . . . . . . . . . . . . . . . 1263. Solution. . . . . . . . . . . . . . . . . 1294. Mass. . . . . . . . . . . . . . . . . . . . 1335. Foldy-Wouthuysen . . . . . . 1366. Twistors . . . . . . . . . . . . . . . . 1407. Helicity . . . . . . . . . . . . . . . . . 143C. Supersymmetry1. Algebra . . . . . . . . . . . . . . . . . 1472. Supercoordinates . . . . . . . . 1483. Supergroups . . . . . . . . . . . . 1504. Superconformal . . . . . . . . . 1545. Supertwistors . . . . . . . . . . . 155III.LocalA. Actions1. General . . . . . . . . . . . . . . . . . 1592. Fermions . . . . . . . . . . . . . . . . 1643. Fields . . . . . . . . . . . . . . . . . . . 1664. Relativity . . . . . . . . . . . . . . . 1695. Constrainedsystems . . . . 174B. Particles1. Free . . . . . . . . . . . . . . . . . . . . 1792. Gauges . . . . . . . . . . . . . . . . . 1833. Coupling. . . . . . . . . . . . . . . . 1844. Conservation. . . . . . . . . . . . 1855. Paircreation . . . . . . . . . . . . 188C. Yang-Mills1. Nonabelian. . . . . . . . . . . . . . 1912. Lightcone . . . . . . . . . . . . . . . 1953. Planewaves . . . . . . . . . . . . . 1994. Self-duality . . . . . . . . . . . . . 2005. Twistors . . . . . . . . . . . . . . . . 2046. Instantons . . . . . . . . . . . . . . 2077. ADHM . . . . . . . . . . . . . . . . . 2118. Monopoles . . . . . . . . . . . . . . 213IV.MixedA. Hiddensymmetry1. Spontaneousbreakdown . 2192. Sigmamodels . . . . . . . . . . . 2213. Cosetspace . . . . . . . . . . . . . 2244. Chiralsymmetry . . . . . . . . 2275. St uckelberg . . . . . . . . . . . . . 2306. Higgs . . . . . . . . . . . . . . . . . . . 232B. Standardmodel1. Chromodynamics. . . . . . . . 2352. Electroweak. . . . . . . . . . . . . 2403. Families. . . . . . . . . . . . . . . . . 2434. GrandUniedTheories. . 245C. Supersymmetry1. Chiral . . . . . . . . . . . . . . . . . . 2512. Actions . . . . . . . . . . . . . . . . . 2533. Covariantderivatives . . . . 2564. Prepotential . . . . . . . . . . . . . 2585. Gaugeactions . . . . . . . . . . . 2606. Breaking . . . . . . . . . . . . . . . . 2637. Extended . . . . . . . . . . . . . . . 2653. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . PARTTWO:QUANTA. . . . . . . . . . . . . . . . . . . .V.QuantizationA. General1. Pathintegrals . . . . . . . . . . . 2742. Semiclassicalexpansion. . 2793. Propagators . . . . . . . . . . . . . 2834. S-matrices . . . . . . . . . . . . . . 2865. Wickrotation . . . . . . . . . . . 291B. Propagators1. Particles . . . . . . . . . . . . . . . . 2952. Properties. . . . . . . . . . . . . . . 2983. Generalizations. . . . . . . . . . 3014. Wickrotation . . . . . . . . . . . 305C. S-matrix1. Pathintegrals . . . . . . . . . . . 3102. Graphs . . . . . . . . . . . . . . . . . 3153. Semiclassicalexpansion. . 3204. Feynmanrules . . . . . . . . . . 3255. Semiclassicalunitarity. . . 3316. Cuttingrules . . . . . . . . . . . . 3337. Crosssections . . . . . . . . . . . 3378. Singularities. . . . . . . . . . . . . 3419. Grouptheory . . . . . . . . . . . 343VI.QuantumgaugetheoryA. Becchi-Rouet-Stora-Tyutin1. Hamiltonian . . . . . . . . . . . . 3492. Lagrangian. . . . . . . . . . . . . . 3543. Particles . . . . . . . . . . . . . . . . 3574. Fields . . . . . . . . . . . . . . . . . . . 358B. Gauges1. Radial . . . . . . . . . . . . . . . . . . 3622. Lorentz . . . . . . . . . . . . . . . . . 3653. Massive . . . . . . . . . . . . . . . . . 3674. Gervais-Neveu. . . . . . . . . . . 3695. SuperGervais-Neveu . . . . 3726. Spacecone. . . . . . . . . . . . . . . 3757. Superspacecone . . . . . . . . . 3798. Background-eld . . . . . . . . 3829. Nielsen-Kallosh . . . . . . . . . 38710. Superbackground-eld . . 390C. Scattering1. Yang-Mills . . . . . . . . . . . . . . 3942. Recursion . . . . . . . . . . . . . . . 3983. Fermions . . . . . . . . . . . . . . . . 4014. Masses . . . . . . . . . . . . . . . . . . 4035. Supergraphs . . . . . . . . . . . . 408VII.LoopsA. General1. Dimensionalrenormalizn4142. Momentumintegration . . 4173. Modiedsubtractions . . . 4214. Opticaltheorem. . . . . . . . . 4255. Powercounting. . . . . . . . . . 4276. Infrareddivergences . . . . . 432B. Examples1. Tadpoles . . . . . . . . . . . . . . . . 4352. Eectivepotential . . . . . . . 4383. Dimensionaltransmutn . 4414. Masslesspropagators . . . . 4435. Massivepropagators. . . . . 4466. Renormalizationgroup . . 4517. Overlappingdivergences . 453C. Resummation1. Improvedperturbation . . 4602. Renormalons . . . . . . . . . . . . 4653. Borel . . . . . . . . . . . . . . . . . . . 4684. 1/Nexpansion . . . . . . . . . . 471VIII.GaugeloopsA. Propagators1. Fermion. . . . . . . . . . . . . . . . . 4782. Photon . . . . . . . . . . . . . . . . . 4813. Gluon. . . . . . . . . . . . . . . . . . . 4824. GrandUniedTheories. . 4885. Supermatter . . . . . . . . . . . . 4916. Supergluon. . . . . . . . . . . . . . 4937. Bosonization . . . . . . . . . . . . 4988. Schwingermodel . . . . . . . . 501B. Lowenergy1. JWKB. . . . . . . . . . . . . . . . . . 5072. Axialanomaly . . . . . . . . . . 5103. Anomalycancelation . . . . 5144. 02. . . . . . . . . . . . . . . . 5165. Vertex . . . . . . . . . . . . . . . . . . 5186. NonrelativisticJWKB. . . 521C. Highenergy1. Conformalanomaly . . . . . 5262. e+e hadrons . . . . . . . . 5293. Partonmodel . . . . . . . . . . . 5314. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . PARTTHREE:HIGHERSPIN. . . . . . . . . . . . . .IX.GeneralrelativityA. Actions1. Gaugeinvariance . . . . . . . . 5402. Covariantderivatives . . . . 5453. Conditions . . . . . . . . . . . . . . 5504. Integration. . . . . . . . . . . . . . 5545. Gravity . . . . . . . . . . . . . . . . . 5586. Energy-momentum. . . . . . 5617. Weylscale . . . . . . . . . . . . . . 564B. Gauges1. Lorentz . . . . . . . . . . . . . . . . . 5722. Geodesics . . . . . . . . . . . . . . . 5743. Axial . . . . . . . . . . . . . . . . . . . 5774. Radial . . . . . . . . . . . . . . . . . . 5815. Weylscale . . . . . . . . . . . . . . 585C. Curvedspaces1. Self-duality . . . . . . . . . . . . . 5902. DeSitter . . . . . . . . . . . . . . . . 5923. Cosmology . . . . . . . . . . . . . . 5944. Redshift . . . . . . . . . . . . . . . . 5975. Schwarzschild . . . . . . . . . . . 5996. Experiments . . . . . . . . . . . . 6077. Blackholes. . . . . . . . . . . . . . 611X.SupergravityA. Superspace1. Covariantderivatives . . . . 6152. Fieldstrengths . . . . . . . . . . 6203. Compensators . . . . . . . . . . . 6234. Scalegauges . . . . . . . . . . . . 626B. Actions1. Integration. . . . . . . . . . . . . . 6322. Ectoplasm . . . . . . . . . . . . . . 6353. Componenttransformns6384. Componentapproach. . . . 6405. Duality . . . . . . . . . . . . . . . . . 6436. Superhiggs . . . . . . . . . . . . . . 6467. No-scale . . . . . . . . . . . . . . . . 649C. Higherdimensions1. Diracspinors . . . . . . . . . . . . 6522. Wickrotation . . . . . . . . . . . 6553. Otherspins . . . . . . . . . . . . . 6594. Supersymmetry . . . . . . . . . 6605. Theories . . . . . . . . . . . . . . . . 6646. ReductiontoD=4. . . . . . . 666XI.StringsA. Scattering1. Reggetheory. . . . . . . . . . . . 6742. Classicalmechanics . . . . . 6783. Gauges . . . . . . . . . . . . . . . . . 6804. Quantummechanics . . . . . 6855. Anomaly. . . . . . . . . . . . . . . . 6886. Treeamplitudes . . . . . . . . . 690B. Symmetries1. Masslessspectrum. . . . . . . 6972. Realityandorientation . . 6993. Supergravity . . . . . . . . . . . . 7004. T-duality . . . . . . . . . . . . . . . 7015. Dilaton . . . . . . . . . . . . . . . . . 7036. Superdilaton . . . . . . . . . . . . 7067. Conformaleldtheory . . 7088. Triality . . . . . . . . . . . . . . . . . 712C. Lattices1. Spacetimelattice . . . . . . . . 7172. Worldsheetlattice . . . . . . . 7213. QCDstrings . . . . . . . . . . . . 724XII.MechanicsA. OSp(1,1[2)1. Lightcone . . . . . . . . . . . . . . . 7302. Algebra . . . . . . . . . . . . . . . . . 7333. Action . . . . . . . . . . . . . . . . . . 7364. Spinors . . . . . . . . . . . . . . . . . 7385. Examples . . . . . . . . . . . . . . . 740B. IGL(1)1. Algebra . . . . . . . . . . . . . . . . . 7452. Innerproduct . . . . . . . . . . . 7463. Action . . . . . . . . . . . . . . . . . . 7484. Solution. . . . . . . . . . . . . . . . . 7515. Spinors . . . . . . . . . . . . . . . . . 7546. Masses . . . . . . . . . . . . . . . . . . 7557. Backgroundelds . . . . . . . 7568. Strings . . . . . . . . . . . . . . . . . . 7589. RelationtoOSp(1,1[2) . . 763C. Gaugexing1. Antibracket . . . . . . . . . . . . . 7662. ZJBV. . . . . . . . . . . . . . . . . . . 7693. BRST . . . . . . . . . . . . . . . . . . 773AfterMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777PART ONE: SYMMETRY 5OUTLINEInthisOutlinewegiveabrief descriptionof eachitemlistedintheContents.WhiletheContentsandIndexarequickwaystosearch, orlearnthegeneral layoutof the book, the Outline gives more detail for the uninitiated. (The PDF version alsoallowsuseofAcrobatReadersFindcommand.)Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23general remarks onstyle, organization, focus, content, use, dierences fromothertexts,etc.Someeldtheorytexts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35recommendedalternativesorsupplements(butseePreface). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . PARTONE:SYMMETRY. . . . . . . . . . . . . . . . . .Relativisticquantummechanicsandclassicaleldtheory. Poincaregroup=specialrelativity. Enlarged spacetime symmetries: conformal and supersymmetry. Equationsof motionandactionsforparticlesandelds/wavefunctions. Internal symmetries:global(classifyingparticles),local(eldinteractions).I.GlobalSpacetimeandinternalsymmetries.A. Coordinatesspacetimesymmetries1. Nonrelativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Poisson bracket, Einstein summation convention, Galilean symmetry (in-troductoryexample)2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44statistics, anticommutator; anticommutingvariables, dierentiation, in-tegration3. Liealgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48general structureofsymmetries(includinginternal); Liebracket, group,structureconstants;briefsummaryofgrouptheory4. Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Minkowski space, antiparticles, Lorentz and Poincare symmetries, propertime,Mandelstamvariables,lightconebases5. Discrete: C,P,T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58chargeconjugation,parity,timereversal,inclassicalmechanicsandeldtheory;Levi-Civitatensor66. Conformal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61broken,butuseful,enlargementofPoincare;projectivelightconeB. Indiceseasywaytogrouptheory1. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Hilbert-spacenotation2. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69adjoint, Cartanmetric, Dynkinindex, Casimir, (pseudo)reality, directsumandproduct3. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74withLevi-Civitatensors,Gaussianintegrals;Pfaan4. Classicalgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77andgeneralizations,viatensormethods5. Tensornotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79indexnotation,simplestbasesforsimplestrepresentationsC. Representationsusefulspecialcases1. Morecoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Diracgammamatricesascoordinatesfororthogonalgroups2. Coordinatetensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86formulationsofcoordinatetransformations;dierentialforms3. Youngtableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91picturesforrepresentations,theirsymmetries,sizes,directproducts4. Colorandavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93symmetriesofparticlesofStandardModelandobservedlighthadrons5. Coveringgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98relatingspinorsandvectorsII.SpinExtensionofspacetimesymmetrytoincludespin. Fieldequationsforeldstrengthsof all spins. Most ecient methods 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,forreasonsthataremostclearfromquantumeldtheory).Thequantummechanicswill bedescribedlater, buttheresultisthatthiscon-straintcanbeusedasthewaveequation. Themainqualitativedistinctionfromthenonrelativisticcaseintheconstraintnonrelativistic : 2mE +p2= 0relativistic : E2+m2+p2= 0is that the equationfor the energyEp0is nowquadratic, andthus has twosolutions:p0= , =

(pi)2+m2Laterwell seehowthesecondsolutionisinterpretedasanantiparticle. Wealsouse(natural/Planck)unitsc=1, solengthanddurationaremeasuredinthesameunits;cthenappearsonlyasaparameterfordeningnonrelativisticexpansionsandlimits.ThetranslationsandLorentztransformationsmakeupthePoincaregroup, thesymmetrythatdenesspecial relativity. (TheLorentzgroupinD1spaceand1time dimension is the orthogonal group O(D1,1). The proper Lorentz groupSO(D1,1), where the S is for special, transforms the coordinates by a matrixwhosedeterminantis1. ThePoincaregroupisISO(D1,1), wheretheIstands54 I. GLOBALfor inhomogeneous.) For thespinless particletheyaregeneratedbycoordinatetransformationsGI= (Pa, Jab):Pa= pa, Jab= x[apb](wherealsoa, b =0, ..., 3). Thenthefact that thephysics of thefreeparticleisinvariantunderPoincaretransformationsisexpressedas[Pa, p2+m2] = [Jab, p2+m2] = 0Writinganarbitraryinnitesimaltransformationasalinearcombinationofthegen-erators,wendxm= xn

nm+ m, mn= nmwherethesareconstants. Notethatantisymmetryofmndoesnotimplyantisym-metryofmn=mppn, becauseofadditionalsigns. (SimilarremarksapplytoJab.)Exponentiatingtondthenitetransformations,wehavextm= 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 metricused to describe nonrelativistic length measured in just the three spatial dimensions.(Thesignatureofthemetricisthusthenumbersofspaceandtimedimensions.)ExerciseIA4.1Forgeneral variables(qm, pm)andgeneratorG, showfromthedenitionofthePoissonbracketthat(dqmpm) = d

GpmGpm

andthatthisvanishesforanycoordinatetransformation.A. COORDINATES 55Forthemassivecase,wealsohavepa= mdxadsFor the massless case ds = 0: Massless particles travel along lightlike lines. However,wecandeneanewparametersuchthatpa=dxadiswell-denedinthemasslesscase. Ingeneral,wethenhaves = mWhilethisxes= s/minthemassivecase,inthemasslesscaseitinsteadrestrictss=0. Thus, proper timedoesnot provideauseful parametrizationof theworldlineofaclassicalmasslessparticle, while does: Foranypieceofsuchaline, d isgiven in terms of (any component of) paand dxa. Later well see how this parameterappearsinrelativisticclassicalmechanics, andisusefulforquantummechanicsandeldtheory.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; theyactasfreeparticlesotherwise). DenePa=I pIaandJab=I xI[apIb]asthesumof theindividual momentaandangularmomenta(wherewelabeltheparticlewithI). Showthatmomentumconservationimpliesangularmomentumconservation,Pa= 0 Jab= 056 I. GLOBALwherereferstothechangefrombeforetoafterthecollision(s).Special relativitycanalsobestatedasthefactthattheonlyphysicallyobserv-able quantities are those that are Poincare invariant. (Other objects, such as vectors,dependonthechoiceof referenceframe.) Forexample, considertwospinlesspar-ticlesthatinteractbycollision, producingtwospinlessparticles(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 =4I=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.ExerciseIA4.4Derivetheaboveexpressionsforthemomentaintermsof invariantsinthecenter-of-massframe.ExerciseIA4.5Findtheconditionsons, tanduthatdenethephysical regioninthecasewhereallmassesareequal.A. COORDINATES 57Forsomepurposesitwillprovemoreconvenienttousealightconebasisp=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 Showthatthe constraintp2= 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).58 I. GLOBAL5. 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. Itssucienttoconsiderasinglereectionof aspacelikeaxis,andoneofatimelikeaxis; all otherreectionscanbeobtainedbycombiningthesewiththecontinuous(proper, orthochronous)Lorentztransformations. (SpacelikeandtimelikevectorscantbeLorentztransformedintoeachother, andreectionofalightlikeaxiswontpreservep2+ m2.) Also, thereectionofonespatial axiscanbecombinedwitharotationabout that 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, forhistoricalreasons,tobeexplainedshortly, thisisusuallylabeledCT.) Thesetransformationshavethesameeectonthemomen-tum,sothatthedenitionofthePoissonbracketisalsopreserved. Thesediscretetransformations, unlike the proper ones, are not symmetries of nature (except in cer-tainapproximations): Theonlyexceptionisthetransformationthatreectsallaxes(CPT).Whilethemetricmnisinvariantunderall Lorentztransformations(bydeni-tion),theLevi-Civitatensor

mnpqtotallyantisymmetric, 0123= 0123= 1is invariant under onlyproper Lorentztransformations: It has anoddnumber ofspaceindicesandof timeindices, soitchangessignunderparityortimereversal.(Moreprecisely, underPorTtheLevi-Civitatensordoesnotsuertheexpectedsignchange, since its constant, sothere is anextrasigncomparedtothe oneexpected for 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, = abcdAaBbCcDdwhich get an extra sign change under such transformations (P or CT, but not CPT).A. COORDINATES 59Thereisanothersuchdiscretetransformationthatisdenedonphasespace,butwhichdoesnotaectspacetime. Itchangesthesignof all componentsof themomentum, while leaving the spacetime coordinates unchanged. This transformationis calledchargeconjugation(C), andis alsoonlyanapproximatesymmetryinnature. (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(forchargeq)appearsintheconstraint(p +qA)2+m2= 0inanelectromagneticbackground;p pthenhasthesameeectasq q.Intheprevioussubsection,wementionedhownegativeenergieswereassociatedwithantiparticles. Nowwecanbetterseetherelationintermsofchargeconjuga-tion. Notethatchargeconjugation,sinceitonlychangesthesignof butdoesnoteectthecoordinates, doesnotchangethepathof theparticle, butonlyhowitisparametrized. Thisisalsotrueintermsofmomentum,sincethevelocityisgivenbypi/p0. Thus,theonlyobservablepropertythatischangedischarge;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(noeecton60 I. GLOBALcoordinates, but momentum and energy change sign because of the i in the Fouriertransform). Ontheotherhand,forCTandPthereisnocomplexconjugation,butchanges insignof thecoordinates that arearguments of thewavefunctions, andalsoonthecorrespondingindicestheorbitalandspinpartsofthesediscretetransformations. (Forexample,derivativesahavesignchangesbecausexadoes,soa vector wave function amust have the same sign changes on its indices for aatotransformasascalar.) Theothertransformationsfollowasproductsofthese.ExerciseIA5.1Find the eect of each of these 7 transformations on wave functions that 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: For example, for ascalar eld(spinlessparticle),wehavetherealitycondition(x) = *(x)orinmomentumspace,byFouriertransformation,(p) = [(p)]*whichimpliestheparticlehaschargezero(neutral).A. COORDINATES 616. ConformalPoincaretransformationsarethemostgeneral coordinatetransformationsthatpreservethemassconditionp2+ m2=0, butthereisalargergroup, theconfor-malgroup,thatpreservesthisconstraintinthemasslesscase. Althoughconformalsymmetryisnotobservedinnature,itisimportantinallapproachestoeldtheory:(1) First of all, it is useful in the construction of free theories (see subsections IIB1-4below). 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 areconformally invariant,so corresponding solutions are relatedby conformaltrans-formations (see subsections IIIC5-7). Furthermore, it is often convenient to treatarbitrarytheories as brokenconformal theories, introducingelds withwhichthebreakingis associated, andanalyzetheconformal 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 maybejust theoneswhosebreakingof conformal invarianceat lowenergyisonlyclassical (seesubsectionsVIIC2-3,VIIIA5-6). Furthermore, thelargestpossiblesymmetryof anontrivial S-matrixis conformal symmetry(or superconformalsymmetryifweincludefermionicgenerators).(5) Self-duality(ageneralizationof aconditionthatequateselectricandmagneticelds) isuseful for ndingsolutionstoclassical eldequationsaswell assim-plifyingperturbationtheory,andiscloselyrelatedtotwistors(seesubsectionsIIB6-7,C5,IIIC4-7). In general, self-duality is related to conformal invariance: For62 I. GLOBALexample, 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 some also preserve p2= 0, although they dont leave p2invariant. Equivalently,wecanlookforcoordinatetransformationsthatscaledxt2= (x)dx2ExerciseIA6.1Findtheconformalgroupexplicitlyintwodimensions,andshowitsinnitedimensional (not just theSO(2,2) describedbelow). (Hint: Uselightconecoordinates.)Thissymmetrycanbemademanifestbystartingwithaspacewithoneextraspaceandtimedimension:yA= (ya, y+, y) y2= yAyBAB= (ya)22y+ywhere (ya)2=yaybabuses the usual D-dimensional Minkowski-space metric ab,andthetwoadditional dimensions havebeenwritteninalightconebasis (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)yAThisreducestheoriginal spacetotheprojective(invariantunderthe scaling)lightcone(whichinthiscasereallyisacone).ThesetwoconditionscanbesolvedbyyA= ewA, wA= (xa, 1,12xaxa)Projective invariance then means independence from e (y+), while the lightcone con-ditionhas determinedy. y2=0implies ydy =0, sothe simplest conformalinvariantisdy2= (edw +wde)2= e2dw2= e2dx2A. COORDINATES 63wherewehaveusedw2= 0 wdw= 0. ThismeansanySO(D,2)transformationonyAwillsimplyscaledx2,andscalee2intheoppositeway:dxt2=

e2et2

dx2inagreementwiththepreviousdenitionoftheconformalgroup.Theexplicitformofconformaltransformationsonxa= ya/y+nowfollowsfromtheirlinearformonyA,usingthegeneratorsGAB= y[ArB], [rA, yB] = iBAof SO(D,2)intermsof themomentumrAconjugatetoyA. (ThesearedenedthesamewayastheLorentzgeneratorsJab=x[apb].) Forexample, G+justscalesxa.(Scale transformations are also known as dilatations.)We can also recognize G+aasgeneratingtranslationsonxa. TheonlycomplicatedtransformationsaregeneratedbyGa, knownas conformal boosts(accelerationtransformations). Since theycommutewitheachother (liketranslations), its easytoexponentiatetondthenitetransformations:yt= eGy, G = vay[a]forsomeconstantD-vectorva(whereA /yA). Sincetheconformalboostsactasloweringoperatorsforscaleweight(+ a ),onlytherstthreetermsintheexponentialsurvive:Gy= 0, Gya= vay, Gy+= vayayt= y, yta= ya+vay, yt+= y++vaya +12v2yxta=xa+12vax21 +vx +14v2x2usingxa= ya/y+,y/y+=12x2.ExerciseIA6.2Makethechangeof 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.64 I. GLOBALWeactuallyhavethefullO(D,2)symmetry: Besidesthecontinuoussymmetries,andthediscreteones of SO(D1,1), wehaveasecondtimereversal (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 identitybycombiningthemwith1paritytransformationand1timereversaltransformation(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.A. COORDINATES 65c Use this toshowthat inversionis relatedtotime reversal byndingthecontinuumofconformaltransformationsthatconnectthem.REFERENCES1 F.A. Berezin, The method of second quantization(Academic, 1966):calculus with anticommuting numbers.2 P.A.M. Dirac, Proc. Roy. Soc. A126 (1930) 360:antiparticles.3 E.C.G. St uckelberg, Helv. Phys. Acta14 (1941) 588, 15 (1942) 23;J.A. Wheeler, 1940, unpublished:the relation of antiparticles to proper time.4 S. Mandelstam, Phys. Rev. 112 (1958) 1344.5 H.W. Brinkmann, Proc. Nat. Acad. Sci. (USA) 9 (1923) 1:projective lightcone as conformal to at space.6 P.A.M. Dirac, Ann. Math. 37 (1936) 429;H.A. Kastrup, Phys. Rev. 150 (1966) 1186;G. Mack and A. Salam, Ann. Phys. 53 (1969) 174;S. Adler, Phys. Rev. D6 (1972) 3445;R. Marnelius and B. Nilsson, Phys. Rev. D22 (1980) 830:conformal symmetry.7 S. Coleman and J. Mandula, Phys. Rev. 159 (1967) 1251:conformal symmetry as the largest (bosonic) symmetry of the S-matrix.8 W. Siegel, Int. J. Mod. Phys. A 4 (1989) 2015:equivalence between conformal invariance and self-duality in all dimensions.66 I. GLOBAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.INDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Inthe previous sectionwe sawvarious spacetime groups (Galilean, Poincare,conformal)intermsof howtheyactedoncoordinates. Thisnotonlygavethemasimplephysical interpretation, but alsoallowedadirect relationbetweenclassicalandquantumtheories. However, asweknowfromstudyingrotationsinquantumtheoryintermsofspin,wewilloftenneedtostudysymmetriesofquantumtheoriesforwhichtheclassicalanalogisnotsousefulorperhapsevennonexistent.Wethereforenowconsidersomegeneralresultsofgrouptheory,mostlyforcon-tinuousgroups. Weusetensormethods, ratherthantheslightlymorepowerfulbutgreatlylessconvenientCartan-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),thereismorethanonewaytorepresentitbymatrices. Anysetofmatriceswendthathasthesamemultiplicationtableasthe group elements is called a representation of that group, and the vector space onwhichthosematricesactiscalledtherepresentationspace.Therepresentationofthe algebra or group in terms of explicit matrices is given by choosing a basis for thevector space. If we include innite-dimensional representations, then a representationof agroupissimplyawaytowriteitstransformationsthatislinear: t=Mislinearin. Moregenerally, wecanalsohavearealizationofagroup, wherethetransformations can be nonlinear. These tend to be more cumbersome, so we usuallytrytomakeredenitionsofthevariablesthatmaketherealizationlinear. Aprecisedenitionof manifestsymmetryisthatall therealizationsusedarelinear. (Onepossibleexceptionisaneorinhomogeneoustransformationst=M1 + M2,suchastheusualcoordinaterepresentationofPoincaretransformations,sincethesetransformationsarestillverysimple,becausetheyarereallystilllinear,thoughnothomogeneous.)ExerciseIB1.1Consider a general real ane transformation t= M+Von an n-componentB. INDICES 67vector for arbitrary real n n matrices Mand real n-vectors V . 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.ExerciseIB1.2Show(AB)1=B1A1formatricesAandBthathaveinversesbutdontnecessarilycommutewitheachother. Useittoshowthat1A +B=1A 1AB 1A+1AB 1AB 1A ...(ThesemaybedierentAandB. Theremaybeotherassumptions; ignoreconvergencequestions.)For convenience,we write matrices with a Hilbert-space-like notation,but unlikeHilbert space we dont necessarilyassociate bras directlywithkets byHermitianconjugation, oreventransposition. Ingeneral, thetwospacescanevenbedierentsizes, todescribematrices that arenot square; however, for grouptheoryweareinterestedonlyinmatricesthattakeusfromsomevectorspaceintoitself, sotheyare square. Bras have an inner product with kets, but neither necessarily has 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, '[ = I'I[; 'I[J` = JI '[` = II; 'I[` = I, '[I` = IThesebasesthendenenotonlythecomponentsofvectors,butalsomatrices:M= [I`MIJ'J[, 'I[M[J` = MIJ68 I. GLOBALwhereasusualtheIonthecomponent(matrixelement)MIJlabelstherowofthematrixM, andJthecolumn. Thisimpliestheusual 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, foranyrepresentationof agroup,thetranspose(MT)IJ= MJIoftheinverseofthosematricesalsogivesarepresentationofthegroup,sinceg1g2= g3 (g1)T1(g2)T1= (g3)T1B. INDICES 69[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,t= g: tI= gIJJt= (g1)T: tI= g1JIJt= g*: t.I= g*.I.J.Jt= (g1): t.I= g*1.J.I.Jsince (g1)T, g*, and (g1)(but not gT, etc.) satisfy the same multiplication algebraasg, includingordering. Weuseup/downanddotted/undottedindicestodenotethetransformationlawof eachtypeof index; 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 vectorspace, orintermsof thegenerators. Thus, weeitherusethematrix70 I. GLOBALA=iGi(for arbitraryrepresentationof thematrices Gi, or treatingGias justabstractgenerators),orwecanwriteAasarowvector:(2)adjointasvector: 'A[ = i'i[ 'A[ = i'A[G i'i[ = ikj(Gj)ki'i[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 simplegrouphas nonontrivial subgroupthat transforms intoitself 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 irreducible representation Rof such a 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)B. INDICES 71istotallyantisymmetric: Forsemisimplegroups,thisimpliesthetotalantisymmetryof the structure constants fijk, up to factors (which are absent for compact groups inabasiswhereij ij). Thisalsomeanstheadjointrepresentationisitsowndual.(Forexample, forthecompactgroupSO(3), wehaveij= ikl

jlk=2ij.) Thus,wecanwriteAinathirdway,asacolumnvector(3)adjointasdualvector: [A` = [i`i [i`jji [A` = iG[A`WecanalsodothisforAbeliangroups,bydeninganinvertiblemetricunrelatedtothe Cartan metric: This is trivial for Abelian groups, since the generators themselvesareinvariant,andthussoisanymetriconthem.Anidentityrelatedtothetraceoneisthenormalizationof thevaluekRof theCasimiroperatorforanyparticularrepresentation,ijGiGj= kRIItsproportionalitytotheidentityfollowsfromthefactthatitcommuteswitheachgenerator:[jkGjGk, Gi] = ifjikGj, Gk = 0usingtheantisymmetryofthestructureconstants. (Thusittakesthesamevalueonanycomponentof anirreduciblerepresentation, 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, if weabsorbisasg=eG, theninsuchrepresentationsGitself mustbereal. Theserepresen-tations arethencalledreal representations, whileacomplexrepresentationisone whose representation isnt real in any basis. A complex representation space canhavearealrepresentation,butarealrepresentationspacecanthaveacomplexrep-resentation. In particular, coordinate transformations (of real coordinates) have onlyreal representations, whichiswhyabsorbingtheisintothegeneratorsisausefulconvention there. For semisimple unitary groups, hermiticity of the generators of theadjointrepresentationimplies(usingtotal antisymmetryof thestructureconstants72 I. GLOBALandrealityoftheCartanmetric)thatthestructureconstantsarereal,andthustheadjointrepresentationisarealrepresentation. Moregenerally,anyrealunitaryrep-resentationwill haveantisymmetricgenerators(G=G*= G G= GT). Ifthecomplexconjugaterepresentationisthesameastheoriginal (samematricesupto a similarity transformation g* = MgM1), but the representation is not real, thenitiscalledpseudoreal. (AnexampleisthespinorofSU(2),tobedescribedinthenextsection.)For anyrepresentationgof thegroup, atransformationg g0gg10oneverygroupelementgforsomeparticulargroupelementg0clearlymapsthealgebratoitself,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,contructedfromvariouspowersof thegenerators.) Inquantummechanics(wherethe representations are unitary), the latter is called an antiunitary transformation.Althoughthis is asymmetryof thegroup, 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; t=1, ..., n; I=1, ..., m, m + 1, ..., m + n.) Thegroupthenactsoneachpartofthenewvectorintheobviousway:= [`, = [

` ; g[` = [`g, g[

` = [

`g

[` = [`[

` = [` [` or () =

g[` = [`g[

`g

or (g) =

g00 g

B. INDICES 73(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

= gg

orintermsofthealgebraG

= G

+G

A 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= GiIt +I GtiUsingtr(A B) = tr(A)tr(B),andassumingtr(Gi) = tr(Gti) = 0,wehavetr( Gi Gj) = tr(It)tr(GiGj) +tr(I)tr(GtiGtj)For example, for SU(N) (see subsection IB4 below) we can construct the adjoint rep-resentationfromthedirectproductoftheN-dimensional, deningrepresentationand its complex conjugate. (We also get asinglet,but itwill not aect the result fortheadjoint.) InthatcasewendtrA(GiGj) = 2N trD(GiGj) cDcA=12NFormostpurposes,weusetrD(GiGj) = ij(cD= 1)forSU(N),socA= 2N.74 I. GLOBAL3. DeterminantsWenowreviewsomepropertiesofdeterminantsthatwillproveusefulforthegroupanalysisofthefollowingsubsections. DeterminantscanbedenedintermsoftheLevi-Civitatensor. Asaconsequenceofitsantisymmetry,totallyantisymmetric, 12...n= 12...n= 1 J1...Jn

I1...In= I1[J1 InJn]sinceeachpossiblenumerical indexvalueappears onceineach, sotheycanbematchedupwiths. Bysimilarreasoning,1m!

K1...KmJ1...Jnm

K1...KmI1...Inm= I1[J1 InmJnm]wherethenormalizationcompensatesforthenumberoftermsinthesummation.ExerciseIB3.1Applytheseidentitiestorotationsinthreedimensions:a Givenonlythecommutationrelations[Jij, Jkl]=i[k[iJj]l]andthedenitionGi 12

ijkJjk,derivefijk= ijk.bShowtheJacobiidentity[ijl

k]lm= 0byexplicitevaluation.c FindtheCartanmetric,andthusthevalueofcA.Thistensorisusedtodenethedeterminant:detMIJ=1n!

J1...Jn

I1...InMI1J1 MInJn J1...JnMI1J1 MInJn= I1...IndetMsince anything totally antisymmetric in n indices must be proportional to thetensor.Thisyieldsanexplicitexpressionfortheinverse:(M1)J1I1=1(n1)!

J1...Jn

I1...InMI2J2 MInJn(detM)1Fromthisfollowsausefulexpressionforthevariationofthedeterminant:MIJdetM= (M1)JIdetMwhichisequivalenttolndetM= tr(M1M)ReplacingMwitheMgivestheoften-usedidentitylndeteM= tr(eMeM) = trM deteM= etr MB. INDICES 75wherewehaveusedtheboundaryconditionfor M=0. Finally, replacingMinthelastidentitywithln(1 + L)andexpandingbothsidestoorderLngivesgeneralexpressionsfordeterminantsofn nmatricesintermsoftraces:det(1 +L) = etr ln(1+L) detL =1n!(trL)n12(n2)!(trL2)(trL)n2+ExerciseIB3.1Usethedenitionof thedeterminant(andnotitsrelationtothetrace)toshowdet(AB) = det(A)det(B)Theseidentitiescanalsobederivedbydeningthedeterminantintermsof 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, andidentifyingthedeterminantof adiagonal matrix. Alternatively, wecanusetheseintegralstodenethedeterminant,andderivethepreviousdenition.TherelationforthesymmetricmatrixfollowsfromthatfortheHermitianonebyseparatingzintoitsrealandimaginarypartsforthespecialcaseH= S. Ifwetreatzandz*asindependentvariables, thedeterminantcanalsobeunderstoodastheJacobianforthe(dummy)variablechangez H1z, z* z*. Moregenerally, ifwedenetheintegralbyanappropriatelimitingprocedureoranalyticcontinuation(for convergence), wecanchoosez andz*tobeunrelated(or evenseparaterealvariables),andSandHtobecomplex.76 I. GLOBALExerciseIB3.2Other properties of determinants can also be derived directly from the integraldenition:a Find an integral expression for the inverse of 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,theonlytermsintheTaylorexpansionoftheexponentialthatcontributearethosewiththeproductof oneof eachanticommutingvariable. Total antisymmetryin andinthen yields the determinant;we dene dDdD to give the correct normalization.(Thenormalizationisambiguousanywaybecauseofthesignsinorderingtheds.)This determinant can also be considered a Jacobian, but the inverse of the commutingresultfollowsfromthefactthattheintegralsarenowreallyderivatives.ExerciseIB3.3Divide 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

= detDdet(A BD1C) = detAdet(D CA1B)a byintegratingoveronepartof thevariablesrst(thisrequireso-diagonalchangesofvariablesoftheformy y +Ox,whichhaveunitJacobian),orbbyrstprovingtheidentity

A BC D

=

I BD10 I

A BD1C 00 D

I 0D1C I

B. INDICES 77Wethenhave,foranyanti symmetric(even-dimensional)matrixA,

d2DeTA/2= PfA, (PfA)2= detAby the same method as the commuting case (again with appropriate denition of thenormalizationofd2D;thedeterminantofanodd-dimensionalantisymmetricmatrixvanishes, since det M= det MT). However, there is now an important dierence: ThePfaanisnotmerelythesquarerootofthedeterminant,butitselfapolynomial,sincewecanevaluateitalsobyTaylorexpansion:PfAIJ=1D!2D

I1...I2DAI1I2 AI2D1I2Dwhichcanbe usedas analternate denition. (Normalizationcanbe checkedbyexaminingaspecialcase;theoverallsignispartofthenormalizationconvention.)4. ClassicalgroupsTherotationgroupinthreedimensionscanbeexpressedmostsimplyintermsof22matrices. Thisdescriptionisthemostconvenientfornotonlyspin1/2, butall spins. Thisresultcanbeextendedtoorthogonal groups(suchastherotation,Lorentz,andconformalgroups)inotherlowdimensions,includingallthoserelevanttospacetimesymmetriesinfourdimensions.ThereareaninnitenumberofLiegroups. Ofthecompactones,allbutanitenumberareamongtheclassicalLiegroups. Theseclassicalgroupscanbedenedeasily in terms of (real or complex) matrices satisfying a few simple constraints. (Theremaining exceptional compact 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* = g178 I. GLOBALwheregisanymatrixinthedeningrepresentationof thegroup, while, , aregroupmetrics,deninginnerproducts(whilethedeterminantdenesthevolume,as in the Jacobian). For the compact cases and can be chosen to 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.Notethemodieddenitionof unitarity, etc. Suchthingsarealsoencounteredinquantummechanics withghosts, sincetheresultingHilbert spacecanhaveanindenitemetric. Forexample, if wehaveanite-dimensional Hilbertspacewheretheinnerproductisrepresentedintermsofmatricesas'[` = thenobservablessatisfyapseudohermiticitycondition'[H` = 'H[` H= Handunitaritygeneralizesto'U[U` = '[` UU= SimilarremarksapplywhenreplacingtheHilbert-spacesesquilinear(vectortimescomplexconjugateof vector)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 meansgenerallinear, andCreferstothecomplexnumbers; therealnumbersRareB. INDICES 79implicit); in the second column we applied one; in the third column we applied three,sincetwoof thethreetypes (unitarity, symmetry, reality) implythethird. (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, fornitetransformationsitissometimesmoreuseful toworkdirectlywiththegroupelementsg, rather thanparametrizing them in terms of algebra elements as 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 representations transform as t= eiqfor group parameter , where qis 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. TensornotationAlthoughhistoricallygrouprepresentationshaveusuallybeentaughtintheno-tationwhereanm-componentrepresentationofagroupdenedbynnmatricesisrepresentedbyanm-componentvector, carryingasingleindexwithvalues1tom,amuchmoreconvenientandtransparentmethodistensornotation,whereagen-eral representation carries many indices ranging from 1 to n, with certain symmetries(andperhapstracelessness)imposedonthem. (Tensornotationforacoveringgroupisgenerallyknownasspinornotationforthecorrespondingorthogonalgroup: SeesubsectionIC5.) Thisnotationtakesadvantageofthepropertydescribedaboveforexpressingarbitraryrepresentationsintermsofdirectproductsofvectors. Intermsoftransformationlaws, itmeansweneedtoknowonlythedeningrepresentation,80 I. GLOBALsincethetransformationof thisrepresentationisappliedtoeachindex. Thereareatmostfourvectorrepresentations, bytakingthedual andcomplexconjugate; weusethecorrespondingindexnotation. Thenthegroupconstraintssimplystatetheinvarianceof thegroupmetrics(andtheircomplexconjugatesandinverses), whichthuscanbeusedtoraise,lower,andcontractindices:volume: Special: I1...Inmetric:

hermitian: Unitary:(anti)symmetric:

Orthogonal:Symplectic:.IJIJIJreality:

Real:pseudoreal(*):.IJ.IJAsaresult,wehaverelationssuchas'I[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 anexampleof theadvantages of indexnotation, showthat SSpis thesameasSp. (Hint: Writeoneinthedenitionofthedeterminantintermsof sbytotal antisymmetrization, whichthencanbedroppedbecauseitis enforcedbytheother . Onecanignorenormalizationbyjust showingdetM= detI.)B. INDICES 81ForSO(n+,n), thereisaslightmodicationof asignconvention: Sincethenindicescanberaisedandloweredwiththemetric, I...isusuallydenedtobetheresultofraisingindicesonI...,whichmeans

12...n= 1 12...n= det= (1)nThenI...shouldbereplacedwith(1)n

I...intheequationsofsubsectionIB3: Forexample,

J1...Jn

I1...In= (1)nI1[J1 InJn]Wenowgivethesimplestexplicitformsforthedeningrepresentationsof theclassical groups. Themostconvenientnotationistolabel thegeneratorsbyapairof fundamental indices, sincetheadjointrepresentationisobtainedfromthedirectproductofthefundamentalrepresentationanditsdual(i.e., asamatrixlabeledbyrowandcolumn). ThesimplestexampleisGL(n),sincethegeneratorsarearbitrarymatrices. Wethereforechooseas 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 for thegenerators. Awell knownexampleis SU(2),wherethetwogeneratorswiththe1asano-diagonal element(and0selsewhere)are known as the raising and lowering operators J,and are more convenient thantheirhermitianpartsforpurposesof contructingrepresentations. (Thisgeneralizestoother unitarygroups, whereall thegenerators ononesideof thediagonal 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)includes82 I. GLOBALSO(n+,n)). For some purposes (especially for SL(n)), its more convenient to imposetracelessnessor(anti)symmetryonthematrix,andusethesimplerGL(n)basis.ExerciseIB5.2Ournormalizationforthegeneratorsoftheclassical groupsisthesimplest,andindependentofn(exceptforsubtractingouttraces):a Find the commutation relations of the generators (structure constants) for thedening representation of GL(n) as given in the text. Note that the values ofallthestructureconstantsare0, i. ShowthatcD= 1(seesubsectionIB2).bConsider the GL(m) subgroup of GL(n) (m4 means that dropping them willgiveresultsinadimension-independentform.) Therefore,only (++and (+acanbedenedingeneral masslesstheories, butwell seethatthesearesucienttodenethe kinematics. The formeris justthe masslessness condition,whichwe used topicktheconstraintsintherstplace.As we saw earlier, just scales xa: We can therefore write the relevant generatorsasPa= a, Jab= x[ab]+Sab, =12xa, a +w 1 = xaa +w +D22(We have used the antihermitian form of the generators.)The scale weight w+ D22isthereal spinpartof , justasSabisthespinpartof theangularmomentumJab. Topreservethealgebrait must commutewitheverything, andthus wecanset it equal toaconstant onanirreduciblerepresentation. Well seeshortlythatitsvalueisactuallydeterminedbythespinSab. Itistheengineeringdimensionofthecorrespondingeld. Ithasbeennormalizedforlaterconvenience; thevalueofwdepends ontherepresentationof Sab, but is independent of D. ThedilatationgeneratorisnotexactlyantihermitianbecausetheintegrationmeasuredDxisntinvariantunderscaling. Thisisanotherreasonwisdetermined, bythefreeaction.The form we have given preserves reality of elds. The commutation relations for thespinparts,andthetotalgenerators,arethesameasthosefortheorbitalparts;e.g.,[Sab, Scd] = [c[aSb]d](Aconvenientmnemonicforevaluatingthiscommutatoringeneral istouseSabx[ab]instead.)ExerciseIIB1.2We canalsouse this methodtondthe stronger conditions for the fullyconformalcase:a Findanexpressionfor Kainterms of x, , S, andwthat preserves thecommutationrelations.bEvaluatealltheconstraints (,andexpresstheindependentonesintermsofjust,S,andw(nox).126 II. SPINSubstituting the explicit representation of the generators into the constraint (+a,and using the former constraint P2= 0 (when acting on wave functions on the right),wendthatallxdependencedropsout,leavingfor (+atheconditionSabb +wa= 0(payingcarefulattentiontoquantummechanicalordering).ExerciseIIB1.3Dene spin for the conformal group by starting in D+2 dimensions: In termsofthe(D+2)-dimensionalcoordinatesyAandtheirderivativesA,GAB= y[AB] +SABBesidesthepreviousconditionsy2= 2= yA, A = 0imposetheconstraints,inanalogytotheD-dimensionaleldequations,andtakingintoaccountthesymmetrybetweenyand,SAByB +wyA= SABB +wA= 0a Showthatthealgebraofconstraintscloses,ifweincludetheadditionalcon-straint12S(ACSB)C +w(w +D2 )AB= 0bSolveall theconstraintswithexplicitysforeverythingwithanupperindex, reducing the manifest symmetry to SO(D1,1), in analogy to the wayy2= 0wassolvedtondy.c Writealltheconformalgeneratorsintermsofxa,a,Sab,andw.2. ExamplesWenowexaminetheconstraintsSabb + wa=0inmoredetail. Webeginbylookingatsomesimple(butuseful)examples. Thesimplestcaseisspin0:Sab= 0 w = 0Thenextsimplestcase(forarbitrarydimension)istheDiracspinor(seesubsectionsIC1andIIA6):Sab= 12[a, b] Sabb +wa= abb + (w 12)aB. POINCARE 127 aa= 0, w =12wherewehaveseparatedoutthepiecesof theconstraintthatareirreduciblewithrespecttotheLorentzgroup(e.g., bymultiplyingontheleftwitha). Thisgivesthe (massless) Dirac equation /= 0. The next case is the vector: In terms of thebasis [V ` = Va[a`,thespinis(seesubsectionIB5)Sab= [[a`'b][However,thevectoryieldsjustanotherdescriptionofthescalar:ExerciseIIB2.1Applytheeldequationsforgeneral eldstrengthstothecaseof avectoreldstrength.a Findtheindependenteldequations(assumingtheeldstrengthisnotjustaconstant)[aFb]= 0, aFa= 0, w = 1Note that solving the rst equation determines the vector in terms of a scalar,whilethesecondthengivestheKlein-Gordonequationforthatscalar, andthe third xes the weight of the scalar to be the same as that found by startingwithascalareldstrength.bLorentzcovariantlysolvethesecondequationrsttondagaugeeldthatisnotascalar.All other representations can be built up from the spinor and vector. As our nalexample, weconsiderthecasewheretheeldisa2nd-rankantisymmetrictensor:Using the direct product representation (applied as in subsection IB2 given the vectorrepresentation)F= Fab[a` [b`, Sab([c` [d`) = (Sab[c`) [d` +[c` Sab[d` (SabF)cd= [c[aFb]d]wendtheequations(Sabb +wa)Fcd=12[aFcd]a[cbFd]b + (w 1)aFcd [aFbc]= bFab= 0, w = 1which are Maxwells equations, again separating out irreducible pieces (e.g., by tracingandantisymmetrizing).128 II. SPINExerciseIIB2.2Verifytherepresentationof LorentzspingivenaboveforFabbyndingthecommutationrelationsimpliedbythisrepresentation.ExerciseIIB2.3Use the denition of the action of the Lorentz generators on a vector in vectorandspinornotations,Sab= [[a`'b][, [a` = [` [.`S= [(`')[, S..= [(.`'.)[,toderiveS.,.= 12(C S.. +C..S)ExerciseIIB2.4Consider the eld equations in 4D spinor notation for a general eld strength,totallysymmetricinitsmundottedindicesandndottedindices,S.m.=S...n.= 0, w =12(m+n)a Showthisimplies........= ........= 0bTranslatetheeldequationsintovectornotation(intermsof Sab), ndingSabb +wa= 0andanaxialvectorequation.c Show that the two equations are equivalent by deriving the equations of parta from Sabb+wa= 0 alone, and from the axial equation alone (except thattheaxialequationdoesntworkforthecasesm = n).Ineachcase, choosingthewrongscaleweightwwouldimplytheeldwascon-stant. NotethatwechosetheeldstrengthFabtodescribeelectromagnetism: Theargumentsweusedtoderiveeldequationswerebasedonphysical degreesoffree-dom, anddidnottakegaugeinvarianceintoaccount. InchapterXIIweusemorepowerful methods to nd the gauge covariant eld equations for the gauge elds, andtheiractions.B. POINCARE 1293. SolutionFreeeldequationscanbesolvedeasilyinmomentumspace. Thenthesimplestway to do the algebra is in the lightcone frame. This is a reference frame, obtainedbyaLorentztransformation,whereamasslessmomentumtakesthesimpleformpa= a+p+(usingonlyrotations), or the evensimpler formpa= a+(usingalsoaLorentzboost), where again is the signof the energy. Inthat frame the general eldequationSabb +wa= 0reducestoSi= 0, w = S+TheconstraintSi=0determinesS+totakeitsmaximumpossiblevaluewithinthatirreduciblerepresentation, sinceSiistheraisingoperatorsforS+: ForanyeigenstateofS+,S+[h` = h[h` S+(Si[h`) = (SiS+ + [S+, Si])[h` = (h + 1)(Si[h`)Theremainingconstraintthendeterminesw: Itisthemaximumvalueof S+forthatrepresentation. Byparity(+ ), wistheminimum,sow 0; w = 0 Sab= 0sinceifS+=0forall statesthenSab=0byLorentztransformation. Aswehaveseenbyothermethods(butcaneasilybederivedbythismethod), w=12fortheDirac spinor and w = 1 for the vector; since general representations can be built fromreducingdirectproductsof these, weseethatwisanintegerforbosonsandhalf-integerforfermions. Ifwedescribeageneral irreduciblerepresentationbyaYoungtableaufor SO(D1,1) (withtracelessness imposed), or aYoungtableautimes aspinor(withalso-tracelessnessaa...b=0), thenitiseasytoseefromtheresultsforthespinorandvector, andantisymmetryinrows, thatwissimplythenumberof columnsof thetableau(itswidth), countingaspinorindexashalf acolumn:S+just counts the maximum number of indices that can be stuck in the boxesdescribingthebasis elements. (Infact, Diracspinor Diracspinor gives just allpossible1-columnrepresentations.)ThisleavesundeterminedonlySijandS+i. However,S+i(creationoperator)iscanonicallyconjugatetoSi(annihilationoperator),soitsactionhasalsobeenxed:[Si, S+j] = ijS+ +Sij130 II. SPIN(Sijvanishes for i = j, so S+iand Siare conjugate, though not orthonormal. TheconstantS+wasxedabovetobenonvanishing,exceptforthetrivialcaseofspin0.) Equivalently, Sijpreserves Si= 0, while S+idoesnt: Sijare the only nontrivialspinoperatorsactingwithinthesubspacesatisfyingtheconstraint.ThusonlythelittlegroupSO(D2)spinSijremainsnontrivial: Theoriginalirreducible representation of SO(D1,1) Lorentz spin Sabwas a reducible representa-tionofSO(D2)spinSij; theirreducibleSO(D2)representationwiththehighestvalue of S+is picked out of this SO(D1,1) representation. This solution also givestheeldstrengthintermsof thegaugeeld: Workingwithjustthehighest-S+-weightstatesisequivalenttoworkingwiththegaugeeld,uptofactorsof+.As an explicit example, for spin 1/2 we have simply = 0, which kills half thecomponents,leavingthehalfgivenby+. Forspin1,wendpbFab= 0 Fa= 0p[aFbc]= 0 onlyF+a= 0

onlyF+i= 0InthelightconegaugeA+= 0,wehaveF+i= +Ai,sothehighest-weightpartofFabisthetransversepartof thegaugeeld. Thegeneral pattern, intermsof eldstrengths,isthentokeeponlypieceswithasmanyaspossibleupper+indicesandnoupper indices(andthushighestS+weight). Intermsofthevectorpotential,wehaveFab p[aAb] onlyAi= 0Thegeneral rulefor thegaugeeldis todrop indices, sotheeldbecomes anirreducible representation of SO(D2). All + indices on the eld strength are pickedup by the momenta, which also account for the scale weight of the eld strength: Allgaugeeldshavew = 0forbosonsandw =12forfermions.ExerciseIIB3.1Usingonlytheanticommutationrelations a, b = ab,constructprojec-tionoperatorsfrom: TheseareoperatorsIthatsatisfyIJ= IJI

no

,I= 1Becauseof timereversal symmetry+ (orparity+), theseprojectontotwosubspacesequalinsize.Amethodequivalenttousingthelightconeframeistoperformaunitarytrans-formationUonthe spinthat is the inverse of the transformationonthe coordi-nates/momentumthat wouldtakeus tothelightconeframe: Wewant aLorentztransformationabontheeldequations,whichareoftheformOabpb= 0, Oab= Sab+wbaB. POINCARE 131thathastheeectUOabU1= acOcdbd, bapb= pta, pta= a+p+0 = UOabpbU1= acOcdbdpb= acOcdptd Oabptb= 0If [`satisestheoriginalconstraint,thenU[`willsatisfythenewone. Ifwelike,we can always transform back at the end. This is equivalent to a gauge transformationintheeldtheory.ItiseasytocheckthattheappropriateoperatorisU= eS+ipi/p+AnyoperatorVathattransformsasavectorunderSab,[Sab, Vc] = V[ab]cbutcommuteswithp,istransformedbyUintoUV U1= VtasVt+= V+, Vti= Vi+V+ pip+, Vt= V +Vi pip++V+(pi)22(p+)2asfollowsfromexplicitTaylorexpansion, whichterminatesbecauseS+iactaslow-eringoperators(asforconformal boostsinsubsectionIA6). ThisyieldsthedesiredresultVtapa= Vapta +V+2p+p2whenweimposetheeldequationp2= 0.ExerciseIIB3.2Checkthis result byperformingthe transformationexplicitlyonthe con-straint. Beforethetransformation, thelightconedecompositionof thecon-straintis(S+ +w)p++S+ipi= 0Sip++Sijpj+wpiSi+p= 0Sipi+ (S++w)p= 0Showthatafterthistransformation,theconstraintbecomes(S+ +w)p+= 0Sip++ (S+ +w)pi12S+i p2p+= 0132 II. SPINSipi+ (S+ +w)pS+ p2p+ 12S+ipip2p+2= 0Clearlytheseimplyw = S+, Si= 0withp2= 0.On the other hand, if instead of using the lightcone identication of x+as time,we choose to use the usual x0for purposes of nding the evolution of the system, thenwewanttoconsidertransformationsthatdonotinvolvep0,insteadofnotinvolvingtheenergyp. Thus, byp0-independentrotationsalone, thebestwecandoistochoosepi= 0, p1= i.e., wecanxthevalueofthespatialmomentum, butnotinawaythatrelatestothesignoftheenergy. Theresultisthenp0> 0 : pa= a+p+p0< 0 : pa= apThe result is similar to before, but now the positive and negative energy solutions areseparated: Inthisframetheeldequationsreducetop0> 0 : Si= 0, S+= wp0< 0 : S+i= 0, S+= wThus, whilewtakes thesamevalueas before, nowthepositive-energystates areassociatedwiththehighestweightof S+, whilethenegative-energyonesgowiththelowestweight(andnothingbetween). TheunitarytransformationthatachievesthisresultisaspinrotationthatrotatesSabintheeldequationswiththesameeectasanorbitaltransformationthatwouldrotate(p1, pi) (, 0). Bylookingatthespecial caseD=3(wherethereisonlyonerotationgenerator), weeasilyndtheexplicittransformationU= exptan1

[pi[p1

S1ipi[pi[

ExerciseIIB3.3Performthistransformation:a Find the action of the above transformation on an arbitrary vector Va. (Hint:LookatD=3togetthetransformationonthelongitudinalpartof thevector.) Inparticular,showthatVtapa= Vapta, pta= a0p0+a1B. POINCARE 133bShowtheeldequationsaretransformedasS0apa +wp0 S10+wp0= p0(w p0 S10) 1S10p2S1apa +wp11pi(S1ip0S0i) +p1(w p0 S10)Siapa +wpi [ij1(+p1)pipj](S1jp0S0j) +pi(w p0 S10)Notethattherstequationgivesthetime-dependentSchrodingerequation,withHamiltonianH=1w(S10p1S0ipi) 1wS10ThisdiagonalizestheHamiltonianH(inarepresentationwhereS10isdiag-onal). Thustheonlyindependentequationsarep2= 0, S10= (p0)w, S1i(p0)S0i= 0leadingtotheadvertisedresult.c Findthetransformationthatrotatestothepidirectioninsteadofthe1di-rection,soH 1wS0ipi[pj[4. MassSofar we have consideredonlymassless theories. We nowintroduce massesbydimensional reduction, identifyingmasswiththecomponentof momentuminanextradimension. As withthe extradimensions usedfor describingconformalsymmetry, this extra dimension is just a mathematical construct used to give a simplederivation. (Theorieshavebeenpostulatedwithextra, unseendimensionsthatarehiddenbycompactication: Spacecurlsupinthosedirectionstoasizetoosmalltodetectwithpresentexperiments. However, nocompellingreasonhasbeengivenforwhytheextradimensionsshouldwanttocompactify.)Themethodisto:(1) extendtherangeofvectorindicesbyoneadditional spatial direction, whichwecall1;(2) setthecorrespondingcomponentofmomentumtoequalthemass,p1= m134 II. SPINand(3) introduce extra factors of i to restore reality, since 1= ip1= im, by a unitarytransformation.Sinceall representations canbeconstructedbydirect products of thevector andspinor, itssucienttodenethislaststeponthem. Forthescalarthismethodistrivial, since then simply p2p2+m2. Except for the last step, the other constraintbecomesSabb +Sa,1im+wa= 0, S1aa +wim = 0Forthespinor, sinceanytransformationonthespinorindexcanbewrittenintermsofthegammamatrices,andthetransformationmustaectonlythe 1direc-tion, wecanuseonly1. (Forevendimensions, wecanidentifythe1ofdimen-sional reductionwiththeonecomingfromtheproductofall theothers, sinceinodddimensionstheproductofallthesisproportionaltotheidentity.) WendU= exp(1/22) : 1 1, a 21aWeperformthistransformationdirectlyonthespinoperatorsappearinginthecon-straints, or the inverse transformation on the states. Dimensional reduction, followedbythistransformation,thenmodiesthemasslessequationofmotionasi/ i/ m1 21(i/ +m2)soi/= 0 (i/ +m2)= 0.TheprescriptionforthevectorisU= exp(12i[1`'1[) : [1` i[1`, '1[ i'1[ ('1[1` = 1)withtheotherbasisstatesunchanged. Thishastheeectofgivingeachelda iforeach(1)-index. Forexample,forMaxwellsequations[aFbc]

[aFbc][aFb]1 +imFab

[aFbc](redundant)i([aFb]1mFab)bFab

bFab +imFa1aF1a

bFab +mFa1iaF1a(redundant)Notethatonlythemass-independentequationsareredundant. Also, Fa1appearsexplicitlyasthepotential forFab, butwithoutgaugeinvariance. Alternatively, wecankeepthegaugepotential:Fab= [aAb]

Fab= [aAb]Fa1= aA1imAa

Fab= [aAb]iFa1= i(aA1 +mAa)B. POINCARE 135ThisisknownastheSt uckelbergformalismforamassivevector,whichmaintainsgaugeinvariancebyhavingascalarA1inadditiontothevector: Thegaugetrans-formationsarenowAa= a

Aa= aA1= im

Aa= aiA1= imExerciseIIB4.1Consider the general massive eldequations that followfromthe generalmasslessonesbydimensionalreduction. OneoftheseisS1aa +wim = 0(beforerestoringreality). Thisscalarequationalonegivesthecompleteeldequationsforw=1/2and1(antisymmetrictensors),0beingtrivial.a Showthatforw=1/2itgivesthe(massive)Diracequation.bExpandingthestateoverexplicitelds, ndthecovarianteldequationsitimpliesforw=1. Showthesearesucienttodescribespins0(vectoreldstrength: seeexerciseIIB2.1)and1(FabandFa1). NotethatS1aactasgeneralizedmatrices(the Dirac matrices for spin1/2,the Dun-Kemmermatricesforw=1),whereSab= [S1a, S1b]c ShowthatthesecovarianteldequationsimplytheKlein-Gordonequationfor arbitraryantisymmetrictensors. Showthat inD=4all antisymmetrictensors(comingfrom0-5indicesinD=5)areequivalenttoeitherspin0orspin1,ortrivial. (Hint: Useabcd.)dConsiderthereduciblerepresentationcomingfromthedirectproductoftwoDiracspinors,andrepresentthewavefunctionitselfasamatrix:Sij=Sij+ Sijwhere i = (1, a) andSijis the usual Dirac-spinor representation. Using thefact that any 44 (in D=4) matrix can be written as a linear combination ofproductsof-matrices(antisymmetricproducts,sincesymmetrizationyieldsanticommutators), ndtheirreduciblerepresentationsof SO(4,1)in, andrelatetopartc.ExerciseIIB4.2Solvetheeldequationsformassivespins1/2and1inmomentumspacebygoingtotherestframe.136 II. SPINThe solution to the general massive eld equations can also be found by going totherestframe(p0=m): Thecombinationofthatanddimensional reductionis, intermsofthemassiveanalogoflightconecomponents,p+=12(p0+p1) =2m, p=12(p0p1) = 0, pi= 0where piare now the other D1 (spatial) components. This xing of the momentumis thesameas thelightconeframeexcept that p1has beenreplacedbyp1, andthuspinowhasD1componentsinsteadofD2. Thesolutiontotheconstraintsisthusalsothesame,exceptthatweareleftwithanirreduciblerepresentationofthelittlegroupSO(D1)asfoundintherestframeforthemassiveparticle, vs. oneofSO(D2)foundinthelightconeframeforthemasslesscase.5. Foldy-WouthuysenTheotherframeweusedforthemasslessanalysis, whichinvolvedonlyenergy-independent rotations, can also be applied to the massive case by dimensional reduc-tion. The result is known as the Foldy-Wouthuysen transformation, and is useful foranalyzinginteractingmassiveeldequationsinthenonrelativisticlimit. Replacingp1p1= minourpreviousresult,wehaveforthefreecaseU= exptan1

[ p [m

S1ipi[ p [

, UHU1=1wS10For purposes of generalization to interactions, it was important that in the free trans-formation(1)weusedonlythespinpartof arotation, sincetheorbital partcouldintroduce explicit xdependence, and(2) we usedonlyrotations, since aLorentzboost wouldintroduce p0dependence inthe parametersof the transformation,whichcouldgenerateadditionalp0(timederivative)termsintheeldequation.ExerciseIIB5.1Performthis transformationforthe Dirac spinor,andthenapplythereality-restoringtransformationtoobtainH20Wethencanusethediagonal representation0=

I00I

/2. (Wecande-nethisrepresentation, uptophases, byswitching0and1of theusualrepresentation.)In general the reality-restoring transformation will be unnec-essaryforanyspin,sinceapplyingtheeldequationS10= wpicksoutarepresentationofthelittlegroupSO(D1).B. POINCARE 137In the interacting case the result generally cant be obtained in closed form, so it isderived perturbatively in 1/m. The goal is again a Hamiltonian diagonal with respectto S10,to preserve the separation of positive and negative energies;we thencan setS10=wtodescribejustpositiveenergies. WethuschoosethetransformationtocancelanytermsinHthatareo-diagonal,whichcomefromoddtotalnumbersof1and0indicesfromthespinfactorsinanyterm: i.e., oddnumbersof S0iandS1i(e.g., theS0ipitermintheoriginal H). Forexample, forcouplingtoanelectromagneticeld, theexponentofUisgeneralizedbycovariantizingderivatives(minimal coupling = +iA), but also requires eld-strength (Eand B) termstocancel certainonesof thosegeneratedfromcommutatorsof thesederivativesinthetransformation:a= a+iAa [a, b] = iFabBeforeperformingthistransformationexplicitlyfortherstfeworders,wecon-sidersomegeneral propertiesthatwill allowustocollectsimilartermsinadvance.(Fewduplicateterms wouldappear totheorder weconsider, but theybreedlikerabbits at higher orders.)We start with a eld equation Tthat can be separated intoeven terms cand odd ones O, each of which can be expanded in powers of 1/m:T= c +O : c=n=1mncn, O =n=0mnOnNotethat theleading(m+1) termis even; thus wechooseonlyoddgenerators totransformawaytheoddtermsin T,perturbativelyfromthisleadingterm:Tt= eGTeG, G =n=1mnGnSince TtisevenwhileGisodd,wecanseparatethisequationintoitsevenandoddpartsasTt= cosh(LG)c +sinh(LG)O0 = sinh(LG)c +cosh(LG)O(with LG=[G, ] as insubsectionIA3). Sincewecanperturbativelyinvert anyTaylor-expandable function of LGthat begins with 1, we can use the second equationtogivearecursionrelationforGn: Separatingtheleadingtermof T,c= mc1 +c, m[G, c1] = [G, c] +LGcoth(LG)O138 II. SPINwhich we can expand in 1/m [after Taylor expanding LGcoth(LG)] to give an expres-sionfor[Gn, c1] tosolveforGn. Wecanalsousetheimplicitsolutionfor[G, c]directlytosimplifytheexpressionfor Tt:Tt= c +tanh(12LG)OForexample,toorder1/m2wehavefor TtTt1= c1, Tt0= c0, Tt1= c1 +12[G1, O0]Tt2= c2 +12[G2, O0] +12[G1, O1]Tothisorderwethereforeneedtosolve[G1, c1] = O0, [G2, c1] = O1 + [G1, c0]Forourapplicationswewillalwayshavec1= 1wS10unchangedbyinteractions. Wehaveoversimpliedthingsabitintheabovederiva-tion: Forgeneral spinweneedtoconsidermorethanjustevenandoddterms; weneedtoconsideralleigenvaluesofS10:[S10, Ts] = sTsandndthetransformationthatmakes Ttcommutewithit(s = 0). Theprocedureis to rst divide into even and odd values of s, as above, then to divide the remainingeventermsin Ttintotwiceevenvaluesofs(multiplesof4)asthenew ctandtwiceoddasthenew Ot, whicharetransformedawaywiththenewtwiceoddGt, andsoon. Thisveryrapidlyremovesthelowernonzerovaluesof [s[ (1 2 4 ...),whichhas amaximum value of 2w(fromthe operators thatmix the maximumvalueS10= wwith the minimum S10= w). For example, for the case of most interest,theDiracspinor, theonlyeigenvalues(foroperators)are0and 1, sotheoriginalevenpartdoescommutewithS10, andtheprocedureneedbeappliedonlyonce.Furthermore, terms in Tof eigenvaluescanbegeneratedonlyat order m1sorhigher; soatanygivenordertheprocedurerapidlyremovesall undesiredtermsforanyspin.Sincethetermswewanttocancelareexactlytheoneswithnonvanishingeigen-valuesofS10,theycanalwaysbewrittenas[G, S10]forsomeG,sowecanalwaysndatransformationtoeliminatethem:[S10, Gsn] = sGsn Gsn= ws[G, c] +LGcoth(LG)OsnB. POINCARE 139(ThisisjustdiagonalizationofaHermitianmatrixinoperatorlanguage.) Inpartic-ularfortheDiracspinor,since c1hasonly 1eigenvalues,itseasytoseethatnotonly doallevenoperators commute with it,butalloddoperators anticommute withit. (Considerthediagonalrepresentationof c1:

1001

,

0ab0

= 0.) Wethenhavesimplyw =12 (c1)2= 1 [c1, c] = c1, O = c1, G = 0 mG = 12[G, c] +LGcoth(LG)Oc1Asanalstep,wecanapplytheusualtransformationU0= eimtS10/wwhichcommuteswithall butthep0termin c0tohavethesoleeectof cancelingc1, eliminatingtherest-masstermfromthenonrelativistic-styleexpressionfortheenergy.Fortheminimalelectromagneticcouplingdescribedabove,wehavebesides c1c0= 0, O0=1wS0iiwhere we have written a= pa+Aa(instead of a= ia,to save some is). Thereare no additional terms in Tfor minimal coupling for spin 1/2, but later well need toinclude nonminimal eective couplings coming from quantum (eld theoretic) eects.There are also extra terms for spins 0 and 1 because the eld strength is not the sameasthefundamentaleld, sowelltreatonlyspin1/2here, butwellcontinuetousethegeneralnotationtoillustratetheprocedure. Usingtheaboveresults,wendtoorder1/m2for TtG1= S1ii, G2= wS0iiF0iinagreementwithwiththefreecaseuptoeldstrengthterms. ThediagonalizedSchrodingerequationisthentothisorder,includingtheeectofU0,Tt1= 0, Tt0= 0, Tt1= 12w[12S1i, S0jiFij+S10(i)2]Tt2= 14[S0i, S0j(iF0j) SijiF0i, j]For spin 1/2 we are done, but for other spins we would need a further transformation(beforeU0)topickoutthepartof Tt2thatcommuteswithS10(byeliminatingthetwiceoddpart);thenalresultisTt2=14[12(S1i, S1j S0i, S0j)(iF0j) +SijiF0i, j]140 II. SPINItcanalsobeconvenienttotranslateinto notation(asforthemasslesscase, butwithindex1 1): WethenwriteTt1= 0, Tt0= 0, Tt1= 12w[12S+i, SjiFij+S+(i)2]Tt2= 14[12S+(i, Sj)(iF0j) SijiF0i, j]InthisnotationtheeigenvalueofS+= S10foranycombinationofspinoperatorscanbesimplyreadoasthenumberof indicesminusthenumberof+.ExerciseIIB5.2Find the Hamiltonian for spin 1/2 in background electromagnetism, expandednonrelativisticallytothisorder, bysubstitutingtheappropriateexpressionsforthespinoperatorsintermsof matrices, andapplyingS+= wontherightforpositive/negativeenergy. (Ignorethereality-restoringtransfor-mation.)-matrix algebra can be performed directly with the spin operators:FortheDiracspinorwehavetheidentitiesS(a(bSc)d)=12b(adc)acbd S+i, Sj =12ij2SijS+6. TwistorsBesidesdescribingspin1/2,spinorsprovideaconvenientwaytosolvethecondi-tion p2= 0 covariantly: Any hermitian matrix with vanishing determinant must haveazeroeigenvalue(considerthediagonalizedmatrix), andsosucha22matrixcanbesimplyexpressedintermsofitsothereigenvector. Absorbingall butthesignofthenontrivialeigenvalueintothenormalizationoftheeigenvector,wehavep2= 0 p.= pp.forsomespinorp(wherep.(p)*). Sincep0isthe(canonical) energy, the isthesignoftheenergy. Thisexplainswhytimereversal(actuallyCTintheusualterminology) is not a linear transformation. Note that pis a commuting object, whilemostspinorsarefermionic, andthusanticommuting(atleastinquantumtheory).Suchcommutingspinorsarecalledtwistors.ExerciseIIB6.1Showthat, intermsof itsenergyEandtheangulardirection(, )(withrespecttothe1axis)ofitsvelocity,amasslessparticleisdescribedbythetwistorp= 21/4

[E[(cos2ei/2, sin2ei/2)B. POINCARE 141Oneusefulwaytothinkoftwistorsisintermsofthelightconeframe. Inspinornotation,themomentumisp.=

1000

IfwewriteanarbitrarymasslessmomentumasaLorentztransformationfromthislightconeframe, thenthetwistor