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GROUP THEORY
Finite Discrete Groups and Applications
Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd.
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Preface Theroleplayedbysymmetryintheunderstandingofthephysicalworldis wellknown.Initially,mainlyintheeraofancientGreekphilosophersand mathematicians,itinvolvedthestudyandgeometricpropertiesof:
(a)planeshapes.
Notonlythecommonones,suchasregulartriangles,squaresandhexagons, butevenmoreinvolvedones,suchasthedodecagonandicosagon;and (b)three-dimensionalobjects.
Regularpolyhedra,includingthecommonones,suchasthecubeandtetrahedron,aswellasthosethataremoreinvolved,suchasthedodecahedron andicosahedron.Therecognitionoftheexistenceofsymmetriesledto thenotionoftransformations,whichleadfromonestateofthesystemto another.Itwasthenrealizedthatsuchtransformations,undertheoperationofmultiplication,constituteasetcalleda group bymathematicians, asystempossessingveryinterestingmathematicalproperties.Thus,group theorywasdeveloped.Thistheorybecamemuchmoreinterestingandled tosomeadditionalapplicationswiththeemergenceofquantummechanics.Soon,theinternaldegreesoffreedomwererecognizedandputintothe realmofsymmetries,andthus,grouptheoryisnolongerpurelygeometric.
Forpracticalaswellaspedagogicalreasons,grouptheoryissplitinto twodifferentparts.Thefirstdealswith discretegroups andthesecond with continuousgroups.Inthefirstcase,theelementsofthegroupare countableandusuallyfiniteinnumber.Thispartdealsmorewithgeometricsymmetries.Thesecondischaracterizedbygroupelements,which dependoncontinuousparameters.Wewillnotconsiderthecaseofcontinuousgroupsinthecurrentvolumesincethereexistmanybooksonthissubject,including,inparticular,atextcoveringsuchtopicsbythefirstauthor,
recentlypublishedbyWSPC,whichcanbeconsideredacompanionvolume tothisbook.
Thecurrentbookdealswithdiscretegroups.Itisatranslationfrom abookinGreekwiththesametitlebythefirstauthor,suitablyupdated andextended.Itwasintendedtocoverthematerialgiveninthefirstof atwo-semestercoursegivenattheUniversityofIoannina.Partsofthis materialhavealsobeendeliveredtosenior-levelphysicsstudentsatNanjing University,Nanjing,China,in2012.
Inthelongpast,thissubjectwastaughtinacourseonalgebra,withthe applicationsconsideredapartofmathematicalmethodsinphysics,witha focusonapplicationsinaspecificresearchareaofphysics.Wehavecomea longwayfromthat,andthissubjectisnowtaughteverywhere,asoutlined above,i.e.ina bonafide grouptheorycourse.Inmanycases,however,after abriefpresentationofthebasictheoreticalideas,manyinstructorsdevise specialrulesandrushtousetheminspecificapplications.Webelieve, however,thatgrouptheoryshouldbepresentedfromaunifiedperspective involvingboththebeautyofsymmetries,resultinginelegantmathematical expressions,andusingitasatoolforapplications,eithercurrentlypopular orexpectedtoemergeinthefuture.Thisapproachisnotrecommended fortheEpicureantypewhowouldliketolookupaformulaorusethe resultsofsometablewithoutunderstandingtheirderivation.Weexpect, however,thatinthelongrun,ourapproachwillbemorebeneficialfor thestudentwhowillhavethepatiencetogothroughthematerial.To thisend,wemadeeveryefforttoincludeillustrativeexamples,whichare thesimplestpossibletoclarifythebasicconceptsintroduced.Somesimple applications,especiallythosethatmaycontributetoabetterunderstanding ofsomerelevantconceptsofthetheory,havebeenincludedasillustrative examples.Asetofusefultableshasbeenincluded,withthededicated studentexpectedtoconstructsomesegmentsofthemaspartofhomework. Moredetailedapplicationsformthematerialofawholechapter,whichmay beskippedatfirstreading.Thisapproachresultedinamuchlargervolume forthebook.Ihopethiswillnothaveanadverseeffectonthestudent’s decisiontoregisterfora one-semestercourse.
Onthisoccasion,Iwouldliketothankformerstudentofphysics,MiltosChristoulakis,currentlythepresidentofHappy–Box,forpreparingthe coveraswellasmostofthefiguresinthebook.
J.D.Vergados,V-IVergadou-Remediaki Ioannina,October2022,
Preface vii
1.TheRoleofSymmetriesinPhysics—APrelude1
1.1Geometricsymmetries....................2
1.2Symmetryinthedynamicsofclassicalmechanics....5
1.3Symmetryinquantummechanics.............6
2.IntroductiontoDiscreteGroups13
2.1Definitionsandbasicconcepts...............13
2.2Thesymmetricgroup Sn ..................18
2.2.1Briefoverviewofpermutations..........18
2.3Afirstexaminationof Sn ..................20
2.3.1The A4 groupasasubgroupof S4 ........22
2.3.2Applicationtotheconstructionof(fully) symmetricandantisymmetricfunctions.....22
2.4Pointsymmetrygroups...................25
2.4.1Thesymmetryofammonia............27
2.4.2Somecategoriesofpointgroups..........30
2.5Extendedpointsymmetrygroups.............37
2.5.1Timereversal....................37
2.5.2Centralinversion..................38
2.6Problems...........................39
3.DiscreteGroups—BasicTheorems43
3.1Cayley’sreorderingtheorem................43
3.2Conjugateelementsandclassesofconjugate elements...........................47
3.3Theconjugateclassesof Sn .................48
3.4Subgroupsandcosets....................51
3.5Regularsubgroupsandsimplegroups— Thefactorgroup.......................55
3.6Possiblegroupsofagivenorder g .............59
3.7Directgroupproduct....................61
3.8Problems...........................62
4.ElementsofRepresentationTheory67
4.1Grouprepresentations:Illustrativeexamples.......67
4.2Representationsofgroupoperators............68
4.3Theregularrepresentationofagroup...........70
4.4Variouswaysofconstructingrepresentations.......73
4.5Directproductofrepresentations.............74
4.6Equivalentrepresentations.................78
4.7Representationconstructioninthespace offunctions.........................79
5.RepresentationReduction—Schur’sLemmas— CharacterTables91
5.1Representationreduction..................91
5.2Unitaryrepresentations...................96
5.3Charactersofrepresentations................99
5.4ThetwolemmasofSchur..................100
5.5Orthogonalitytheorem...................101
5.5.1Characterorthogonalityrelations.........102
5.5.2Acriterionofirreducibilityofgroup representations...................103
5.6Moreonrepresentationreduction.............105
5.7Someexamplesofrepresentationreduction........107
5.8Reductionofthedirectproductofgroup representations........................114
5.9Reductionofthedirectproductofgroups.........117
5.10Doublecoveringgroupsofdiscretegroups.........122
5.11Problems...........................123
6.SimpleApplicationsinQuantumMechanics127
6.1Behaviorofaquantumsystemacteduponbygroup operators...........................127
6.2Wigner’stheorem......................130
6.3Usefulexamples.......................134
6.4Problems...........................142
7.SymmetriesandNormalModesofOscillation145
7.1Introduction.........................145
7.2Normalmodesofoscillation................147
7.2.1Summary......................148
7.3Someapplicationsinvolvingtheuseofsymmetry.....149
7.4Problems...........................157
8.SpaceGroups161
8.1Introductorynotions....................161
8.2Thegroupoftranslationsinspace.............163
8.3Theallowedcrystalsymmetries..............164
8.4Spacegroups.........................167
8.4.1Crystalsystemsin3D...............170
8.4.2Crystalsystemsintwodimensions........171
8.5Charactertablesoftheirreduciblerepresentations ofcrystalgroups.......................172
8.6Generatorsofthecrystalgroups..............173
8.7Problems...........................174
9.IrreducibleRepresentationsofSpaceGroups175
9.1Irreduciblerepresentationsofspacegroups........175
9.2Thegroupofvector k ofthereducedBrillouinzone...180
9.3Constructionoftheirreduciblerepresentationsofthe spacegroup.........................183
GroupTheory:FiniteDiscreteGroups
9.4Graphene—Aremarkableexampleofa2Dcrystal...192
9.4.1Theirreduciblerepresentationsofthespacegroup connectedwithpointsymmetry C6v .......194
9.5Problems...........................196
10.NormalModesinCrystals199
10.1Introduction.........................199
10.2Normalmodesfor1Dcrystalwithonekindof particle............................200
10.3Normalmodesin1Dcrystalwithtwokindsof molecules...........................203
10.4Normaloscillationmodesofgraphene...........205
10.4.1Theclassicalequationofmotionapproach: Aprelude......................206
10.4.2Themethodofdiagonalizingthepotential....207
Specialtopicsandapplications
215
11.TheSymmetricGroup S n 217
11.1Theroleofthesymmetricgroup Sn inquantum mechanics..........................217
11.2Theirreduciblerepresentationsofthesymmetric group Sn ...........................220
11.3Constructionoffunctionsofagiven Sn symmetry....221
11.3.1TheYoungoperators................222
11.3.2Anexample—TheIrreduciblerepresentations of S3 .........................226
11.4Constructionofsymmetricandantisymmetricfunctions intwospaces.........................230
11.4.1Constructionofantisymmetricfunctions.....230
11.4.2Constructionofsymmetricfunctions.......232
11.5Kroneckerproductsandtheemergenceoftensors....232
11.6Constructionoftensorsofagivensymmetry.......233
11.7Kroneckerproducts[f ] ⊗ [f ]................238
11.8Problems...........................242
12.FurtherApplicationsinMolecularPhysicsand CrystalStructure—CrystalHarmonics245
12.1Introductorynotions....................245
12.2Reductionof SO (3)representationsunderdiscrete symmetries..........................246
12.2.1Reductionofsingle-valuedrepresentationsof SO (3)underdiscretesymmetries.........247
12.2.2Thereductionofdouble-valuedrepresentationsof SO (3)underdiscretesymmetries.........255
12.3Somereductiontablesofrepresentationsof SO (3)under discretesymmetries.....................259
12.4Crystalharmonics......................260 12.5Problems...........................267
13.ApplicationsofDiscreteGroupsinParticlePhysics269
13.1Thegroup A4 asasubgroupof S4 .............269
13.2Thestructureof A4 .....................270
13.3Thecharactermatrixof A4 .................275
13.4ReductionoftheKroneckerproductoftherepresentations of A4 .............................276
13.5Anapplicationinparticlephysics.............279
13.5.1Thescalarpotentialintheframeworkof A4 ...280
13.5.2Theleptonmassesinasimple A4 model.....281
13.5.3Asemirealisticmodelfortheneutrinomasses..286 13.6Problems...........................288
14.ExoticDiscreteGroupsforQuantum Mechanics—FieldTheory293
14.1Matrixgroupswithelementsintegers mod(p).......293
14.2Themodulargroup.....................295
14.3TheHeisenberg–Weylgroup................298
14.3.1Afaithfulrepresentationofthediscrete Heisenberggroup..................302
14.3.2Magnetictranslations...............305
14.3.3ThemetaplecticrepresentationbyWeyl.....306
14.4Theconstructionofthe SL2 (p)generators........310
14.5Examples:Thecases
14.5.1Thecase
14.5.2Thecase
15.AppendixI:ProofsofVariousTheorems319
16.AppendixII:RepresentationReduction ViaaChainofGroupOperators325
16.1Chainsofsubspacesofbasisvectors............325
17.AppendixIII:GeneratorsandCharacterTables ofPointGroups331
Chapter1 TheRoleofSymmetriesinPhysics— APrelude Thenotionofsymmetryisnot,ofcourse,anewone.Ithasbeenknown sincetheveryancienttimesthatthehumanbodypossessesright/leftsymmetry,a“mirrorsymmetry”,i.e.areflectionwithrespecttoaplane;thata cubehasahighdegreeofsymmetry,bothwithrespecttorotationsaround someaxesaswellasreflectionswith respecttocertainplanes;thatthe sphereisthemostsymmetricofallbodies.Theideaofgeometricsymmetryaffecteddeeplythethoughtofmany ancientGreekphilosophers,includingPythagoras,Platoandothers.Indeed,Platoattemptedtodescribethe motionoftheheavenlybodiesintermsofcirclesorcirclesovercircles (circlesonepicenters).
Inspiteofitsbeautyasatheory,however,grouptheory(GT)didnot becomeatrulyusefultoolinphysicsuntilafterthefoundationofquantum mechanicsinthe1920s.Tothisend,acrucialrolewasplayedbyWigner, Weyl,Gelfand,Racahandothers.Inotherwords,GTbecameveryuseful tothephysicalscienceswhenitwasrealizedthatthesetoflineartransformations,whichleadfromonestateofasystemtoanother,constitutea group.
Asoutlinedabove,theancientnotionofsymmetrybecameveryuseful tothephysicalsciences.Whatisa symmetry?Roughly,asymmetryis apropertyofsomeobjectwhichremainsinvariantundersomeoperations. Notethatthenotionofsymmetryrequiresbothanobjectandtheoperation whichactsoroperatesontheobject.Invariancemeansthattherelevant propertyoftheobjectremainsthesamebeforeandaftertheoperationhas beenacteduponit.
1.1Geometricsymmetries Aswehavealreadymentioned,thesesymmetrieswerethefirsttoberecognizedbytheancientGreeks.Theyrelatetotransformationswhichmaintainthedistancebetweentwopointsofabodyandmapitontoitself.Such transformationsareessentiallyofthreetypes:
(i)Rotationsaboutanaxisofsymmetrybyagivenangle. Asaconcreteexample,letusconsider asquare.Welabelthefourcornersof thesquare A,B,C,D .Intuitively,ifwerotatethesquareby90◦ clockwise, wegetthepictureshowninFig.1.1(a).Letuscallthisoperation S .Then, thesquareremainsthesamewiththeoriginalorientation.Whatdowe meanby“thesame”?Itmeansthat,ifwedropthelabels A,B,C,D ,you donotseeanydifference.Now,arotationof180◦ alsoleavesthesquare invariant.Letuscallthisoperation T.Clearly,wecanachievethesame resultbyapplyingtheoperator S twice.Thus,wewrite T = S S .Similarly, arotationof270◦ ,indicatedas U ,leavesthesquareinvariant,and U = (S S )S .Clearly,arotationof360◦ doesnotchangeanything,andwe identifythiswiththeidentityoperatoranddesignateitas E . Wecalltheoperation S arotationofthesystembyanangle π/4around anaxisperpendiculartotheplaneofthesquare,passingthroughitscenter.Inthestandardnotation,weindicatetheoperationassociatedwith S as C1 .Ifweapplythisoperationfourtimessuccessively,thesystem returnstoitsoriginalposition.Wesaythatthisaxisisoffourthorder, 2π π/2 = 360 90 =4.Weindicatethesuccessiveoperationsas C1 ,C 2 1 ,C 3 1 ,C 4 1 Obviously, C 4 1 = E ,where E istheidentityoperator
Wecaneasilyseethatarotationby π/2intheoppositesense,i.e. counterclockwise,whichisindicatedby π/2,givesasimilareffectasa clockwiserotationby3π/2.Indicatingthisas C 1 ,weverifythat C 1 = (a) (b)
Fig.1.1. Symmetryoperationsonasquare:(a)rotationaroundafour-foldaxis, (b)reflectionthroughaplanewhichisperpendiculartothesquareandpassesthrough themiddlesofABandCD.
C 3 1 .Thus, C 1 C1 = E ,i.e. C 1 istheinverseof C1 .Similarly, C 2 isthe inverseof C2 .Furthermore,theseoperatorscommute,i.e. C n 1 C k 1 = C k 1 C n 1 , andthesystem {C1 ,C 2 1 ,C 3 1 ,C 4 1 = E } isclosedundermultiplication.
(ii)Reflections.
Anotheroperationis reflection,i.e.amirrorimageofthesquarewith respecttoaplanethatcontainstheaboveaxisandisperpendiculartothe planeofthesquarepassingthroughthemiddleofitstwooppositesides, e.g.throughthemiddleof AB and CD indicatedas m1 ,seeFig.1.1(b). Therelevantoperatorwillbeindicatedby σx .Thereisanotherreflection correspondingtoasimilarreflectionplanepassingthroughthemiddleof AC and BD,withtherelevantoperatordesignatedas σy .Thereexisttwo additionalsimilarreflectionplanespassingthroughtheoppositecornersof thesquare,onethrough A and C andtheotherthrough B and D ;thesewill bedesignatedas σ1 and σ2 ,respectively.Obviously, σ 2 i = E , i = x,y, 1, 2, i.e.eachoftheseelementsisitsowninverse.
(iii)Inversion(withrespecttoacenter).
An inversion causesatransformationofthecoordinates (x,y,z ) → ( x, y, z ). (1.1)
Onecanseethatinourexample,inversionisnotanewoperation,but itcoincideswitharotationby π .Bothoftheseoperationshavethesame effect:theyinterchange A ↔ C and B ↔ D .Furthermore,onecanshow thattheset
{C1 ,C 2 1 ,C 3 1 ,C 4 1 = E,σx ,σy ,σ1 ,σ2 }
isaclosedsetundermultiplication.Notethattwoelements, X and Y ,do notalwayscommute,i.e.insomecases, X Y = Y X .Anyway,theabove elementsconstituteagroupassociatedwiththesymmetryknownas C4V Itwas,ofcourse,natural,inviewoftheideasoftheancientphilosophers, thatthegeometricsymmetrieswouldberecognizedasthefirstapplications ofgrouptheoryinphysics,inparticularincrystallographyandsolid-state structures.
Nextcomesthenotionofaspacegroup.Bythis,weunderstandthe symmetrygroupwhichischaracteristicofagivenperiodicsystem,asfor exampleanidealcrystal.Thisconsistsofthesetoftransformationswhich carryonepointofthesystemtoanother.Thus,thespacegroupsshould containpointgroups,suchastheones wehaveconsideredaboveaswellas translationtransformations.Thisimposesconditionsonhowtheelements ofthecrystalrepeatthemselvessothattheygeneratethewholecrystal.As
aresult,onlyafractionofthepointgroupsymmetriesarecompatiblewith therequiredspacesymmetry.Thisisbecausethestructureelementsof thepointgroupsymmetries,whenrepeated,mustcoverthewholecrystal. Weallknowthatthewholefloorcanbecoveredbyplacingsidebyside tilesofcertainshapes,e.g.triangles,parallelograms(squares,diamonds)or regularhexagons,seeFig.1.2.Thiscannotbedonebyplacingcirclesor
Fig.1.2. Variousshapesthatcancoverthewholeplane:(a)scalenetrianglescombinedtoformaparallelogram(second-orderaxis),(b)equilateraltriangles(third-order axis),(c)squares(fourth-orderaxis)and(d)hexagons(sixth-orderaxis).Thiscannot beaccomplishedbyusing:(e)regularpentagonsand(f)circlesorellipses.Notethatin eachshape,wehavetwogeometricquantities:twosizeparameters a and b aswellasthe angle φ betweenthem.Thus,(a)wehave a = b and φ arbitrary,andintheothercases, wehave a = b,whiletheanglesare2π/3, π/2and π/3for(b),(c)and(d),respectively.
(a)
(b)
(c)
(d)
(e)
(f )
regularpentagons.Thewholethree-dimensional(3D)spacecanbecovered byproperlyplacingprismswithbasesoftheaboveshapes.
1.2Symmetryinthedynamicsofclassicalmechanics Theroleofthedynamicsymmetries,i.e.thosethatleaveinvariantthe equationsofmotion,wasrecognizedlater.Afirstandveryinteresting examplewasthemotionofaplanetaroundthesun,Kepler’sproblem.The solutiontothisproblemwas,ofcourse,firstobtainedbysolvingNewton’s equationsofmotion,usingtheworldgravitationalattraction,alsoinvented byNewton.Thepredictedorbitswereinverygoodagreementwithobservation.Itisworthwhile,however,tolookatKepler’sproblemfromadifferentperspectiveandmorecarefully(seeFig.1.3).Themainpointsare asfollows:
• Theorbitliesonaplane.
• Theorbitisanellipse(oringeneralaconicsection)withafixedaxis length.
• Thedirectionofthevectorconnectingthetwofocioftheellipseremains fixed.Inotherwords,thepositionoftheperihelion(pointoftheorbit closesttothesun)isfixed.
TheplanarityoftheorbitstemsfromthefactthatNewton’sforceiscentral,whichleadstotheconservationofangularmomentum.Astheangular momentumcannotchangeandtheorbitmustalwayslieatrightanglestoit,
Fig.1.3. InKepler’sdescription,themotionofaplanetaroundthesunisexhibited. Inparticular,theroleoftheR¨ unge–Lenzvector,indictedinthepictureby A,guarantees thestabilityofperihelion,point1inthefigure.
GroupTheory:FiniteDiscreteGroups
theorbitmustbeplanar.Therefore,thesymmetry(centrality)oftheforce leadstotheplanarityoftheorbit.Theconservationofangularmomentumisrelatedtothefactthatthesystemwouldremainunchangedaftera rotationofspace(isotropy).
Thelengthofthelongeraxisisdeterminedbytheenergyofthesystem, whichisconservedbythespecificformofinteractionbetweentwobodies, knownas conservativeforce.Theeccentricityofthe motionisconstant becauseitisdeterminedbytheenergyandangularmomentumofthesystem,bothofwhichareconstant.Butwhyshouldtheperihelionremain constant?
Theanswertothisquestionislessobvious.Ithastodowiththefact thattheattractiveforceisinverselyproportionaltothesquareofthedistancebetweenthetwoobjects, F ∝ 1 r 2 .Asmalldeviationfromthis,even when F ∝ 1 r 2+ ,nomatterhowsmall mightbe,destroysthestabilityof theperihelion.Thisiswhatactuallyoccurswhenwetakeintoaccountthe presenceofotherplanetsorgeneralizeNewton’stheoryusingthegeneral theoryofrelativitydevelopedbyEinstein.1
Theimportantconclusiontobedrawnfromthisisthatbyrecognizingandinvokingsymmetry,theproblemposedbyKeplercanbesolved withoutrequiringthesolutionofadifferentialequation.Wewillnotdiscussthistopichere,butithasbeentreatedelsewhere,e.g.inthebook Vergados(2017).Evenmoreimportantly,theexistenceofsymmetryina systemresultsintheconservationofsomequantitythatcharacterizesthe system.Conversely,theconservationofsomequantitycharacterizingasystemdictatestheexistenceofasymmetrycharacterizingthesystem.This isknownasNoether’stheorem.
1.3Symmetryinquantummechanics Ashasalreadybeenmentioned,grouptheorybecameamoreusefultoolin solvingproblemsinphysicsafterthedevelopmentofquantummechanics. Here,theresultsweresurprising,especiallyinthecasewherethegroup transformationsleavethesystem’sHamiltonoperatorinvariant.TheanalogueinquantummechanicstoKepler’sproblemishydrogen-likeatoms. Thequantumnumbersthatdescribethestatesofthesystemareexpected
1 Indeed,thefirstindicationthatthegeneraltheoryofrelativityholdstruecamefrom studyingthemovementoftheperihelionofMercury,asthephenomenoncouldnotbe explainedsolelybytheinfluenceoftheotherplanetsonMercury’sorbit.
todependontheorbitalangularmomentumquantumnumber, ,but theymustbeindependentof m,i.e.,theprojectionoftheorbitalangularmomentumonthequantizationaxis.Thisisduetothefactthatthe Hamiltonoperatorremainsunchangedunderrotationin3Dspace.However,theeigenenergiesofthesystemareindependentoftheazimuthal, ; thisisindicativeofagreatersymmetry,whichisrotationin4Dspace, (Vergados,2017).Recognitionofthissymmetryallowsustocalculate theeigenenergiesofthehydrogen-likeatomwithouthavingtosolvethe Schr¨oedingerequation.
Thearrivalofquantummechanicswasthemainreasonforthefamiliarityofphysicistswithgrouptheory,whichhadalreadybeendevelopedby mathematicians.
Inquantumtheory,invarianceprinciplespermitevenmorefar-reaching conclusionsthaninclassicalmechanics.Inquantummechanics,thestate ofaphysicalsystemisdescribedbyarayinaHilbertspace, |Ψ .Asymmetrytransformationgivesrisetoalinearoperator, R ,thatactsonthese statesandtransformsthemintonewstates.Justasinclassicalphysics,the symmetrycanbeusedtogeneratenewallowedstatesofthesystem.However,inquantummechanics,thereisanewandpowerfultwistduetothe linearityofthesymmetrytransformationandthesuperpositionprinciple. Thus,if |Ψ isanallowedstate,thensois |R Ψ ,where R istheoperator intheHilbertspacecorrespondingtothesymmetrytransformation R.So far,thisissimilartoclassicalmechanics.However,wecannowsuperpose thesestates,i.e.constructanewallowedstate: |Ψ + |R Ψ .(Thereisno classicalanalogueforsuchasuperposition,e.g.thesuperpositionoftwo orbitsoftheEarth.)
Quantummechanicsalsorevealedanewkindofsymmetry:thatofthe exchangeofidenticalparticles.Thisledtoaclassificationofallelementary particlesas(a)Bosons,characterizedbyintegralspin.Then,thewavefunctiondescribingamany-identical-particlessystemmustbeinvariantunder theinterchangeofanytwoparticles;(b)Fermions,withhalf-integralspin, whosewavefunctionchangessignwhenanytwoparticlesareinterchanged. Thequantumstatisticsofacollectionofsuchparticlesisdifferent,with profoundimplicationsfortheirbehaviorinaggregate.
Atthisstage,EugeneWignerplayedaverycrucialrole,connecting grouptheorywithquantummechanics.Inaddition,agreatcontribution towardthefamiliarityofphysicistswithgrouptheorycanbeattributedto HeisenbergandWeyl,mainlyforshowingtheequivalenceofSchr¨odinger’s versionofquantummechanicswiththatofHeisenberg.
GroupTheory:FiniteDiscreteGroups
Onecaninterpretasymmetrytransformationasachangeinour pointofviewwhenlookingatasystemthatdoesnotaltertheresults ofpossibleexperiments.Inquantummechanics,weknowthat(pure) statesaredenotedbywavefunctions |Ψ intheHilbertspace.Thesefunctionsaremembersofavectorspaceequippedwithascalarproduct,2 i.e.
Ψf |Ψi = Ψi |Ψf ∗ .Themeasurablequantitiesarenotthewavefunctions themselves,buttheexpectationvaluesofanyobservablequantity.The probabilityofobtaininganexpectationvalueaftermeasurementdepends onthetransitionprobabilitiesbetweenstates,whichisgivenbythesquare ofthemodulusoftheoverlapoftwowavefunctions, | Ψf |Ψi 2 |.Hence, asymmetrytransformshouldkeeptheseprobabilitiesinvariant.So,ina quantumscenario,wedefineassymmetrytransformationsthosewhichpreservetransitionprobabilitiesbetweenthestates.
Sofar,thediscussionhasbeenaboutthetransitionprobabilities,but howdotheindividualstatesthemselvestransformunderasymmetrytransformation?TheanswertothisparticularquestionwasprovidedbyWigner. Wigner’stheorem:Anysymmetrytransformationcanberepresentedon theHilbertspaceofphysicalstates3 byanoperatorwhichiseitherlinear andunitaryorantilinearandantiunitary.
Forasymmetryoperator U ,thetheoremcanbestatedas:
or
Thetheoremwasstatedandprovedforthefirsttimein1931byWigner himselfinhisbook(Wigner,1959).Ithasthereafterbeenprovedbymany
2 Thescalarproductisdefinedthroughanintegral,e.g.inonedimension,as f |g = +∞ −∞ f ∗ (x)g (x)dx.Thesefunctionsmustsatisfythecondition f |f < ∞ andsimilarly for g .Thus,theycanbenormalized: f |f = g |g =1.Theserelationscaneasilybe generalizedin3D.
3 Mathematically,however,physicalstatesaredenotedby“rays”intheHilbertspace. Asetofnormalizedstateswhoseelementsdifferonlybyacomplexphasearecalled rays,i.e.,thestates ψ and eiθ ψ aremembersofthesameray R forsomereal θ .They correspondtothesamephysicalstate.Hence,asymmetrytransformationisaraytransformation T suchthatif T : R1 → TR1 and T : R2 → TR2 ,itfollowsthat
U ΨA |U ΨB = ΨA |ΨB ,U =unitary(1.2)
U ΨA |U ΨB
people,themostprominentbeingBargmann,Ulhornand,morerecently, (Weinberg,1996).
Unitarytransformationsarecommonlyusedanddiscussedextensively inthisbook.Theantiunitaryoperatorsareencounteredinthecaseoftime reversal,seeSection2.5.Onecanshowthatanantiunitaryoperatorcan bewrittenasaproductofaunitaryoperatorandthecomplexconjugation operation(Vergados,2018,Section12.4.2).
WealsoowetoWigneranotherinterestingapproachtoquantum mechanics:
Ifwedenoteas x allthecoordinatesthatdescribeasystem,thetimeindependentSchr¨ odingerdifferentialequationmaytakethefollowingform:
where H (x)istheHamiltoniandescribingthesystem.Inthisspace,we consideralineartransformation,whichisanelementofagroup, A ∈ G:
Thisimpliesatransformation TA infunctionalspace:
aswasobtainedbyWigner.Then,Eq.(1.4)becomes
Thus,Eq.(1.4(remainsunchangedaslongas
Thisisequivalentto
Infact,fromEqs.(1.8)and(1.7),onefinds
Now,consideringa representationof TA , TA → T (A),insomebasis,we showthat
, where T (H )istherepresentationoftheoperator H inthesamebasis.
GroupTheory:FiniteDiscreteGroups
Let ψi (x)beanorthogonalbasisinfunctionalspace.Then,
T (H )ij = ψi (x)|H |ψj (x) ≡ Hij ,T (A)ij = ψi (x)|TA |ψj (x) ,
Hij = T (H )ij = dxψ ∗ i (
)|H |ψj (x)= dxψ ∗ i (A 1 x)|H |ψj (A 1 x) = dx ψ ∗ i (x )|H (x )|ψj (x )= Hij
ij =
(
)
T (A)=(T (A))+ T (H )T (A)
FollowingWigner,wemakethereasonableassumptionthattherepresentationisunique,then T (A) ⇒ Γ(A), Γ+ (A)=Γ 1 (A)=Γ(A 1 ) ⇔ T (H ) =Γ 1 (A)T (H )Γ(A) ⇒ Γ(A)T (H )= T (H )Γ(A). (1.12)
InordertosolveEq.(1.4)thefollowingstrategyisused:Weexpand thefunction ψ inthebasis ψi , ψ = i αi ψi
Equation(1.4)nowbecomes:
.
Takingthescalarproductofbothsidesoftheequationwith ψj |,we find i αi ψj |H |ψi = i αi E ψj ψi = Eαj ⇒ i αi (Hji Eδij )=0
or,inmatrixform,
Eq.(1.13)isequivalentto(1.4).
Itissufficient,therefore,tosolveEq.(1.13).If,byutilizingthesymmetry,wehitthebull’seyewhenselectingthebasis,wewillbeabletoreduce thematrix(H ),i.e.
sincetheHamiltoniancannotmixstatesofdifferentsymmetry.Thisholds truethankstoanotherfamoustheorembyWigner,whichwillemergeas ourgrouptheoryevolves,seeSection6.2.Please,bepatient!
Forthemoment,justbesatisfiedthatallwehavetodoisdiagonalize matricesofsmallerdimensions.
Wemaythereforeconcludethatgrouptheoryisrecognizedasauseful toolforthecomprehensionandstudyofvarioussystems.Recognitionof symmetryinasystemrelievesusofthenecessitytosolvecomplexequationsinordertodiscoverthesystem’scharacteristics.Thisholdstrueeven whenthesymmetryisnotabsolute,butapproximate.Inthatcase,the symmetrywillhelpusfindanapproximatesolution,whichmaythenbe usedasabasisforfindingamoreaccuratesolutionwithperturbationtheory.Thematriceswhichmustbediagonalizedinthatcaseareconsiderably smaller.Moreover,theexistenceofsymmetryallowsustofindtheformof theHamiltonoperator,whenthatisnotalreadyknown.Inthisbook,we payspecialattentionto“geometricsymmetries”andtheiruseinquantum mechanicsproblems(normalmodes,etc.).
Somepeoplemayarguethatphysicistsneednotconcernthemselves withtheabstractgrouptheorythatamathematicianmightdealwith,but therealizationinsomewayoftheabstractgroupelements,i.etheirrepresentation,whichisnothingbutmatricesthatfollowthesamemultiplication ruleswiththeabstractelements.Weconsideritnecessary,however,todiscussthestructureofabstractgroups,despitethedifficultyposedforthose whoarenotmathematicallyinclined.Itis,afterall,necessarytointroduce aminimumofmathematicalconceptsbeforewegoontotheirapplications. ThisiswhatwewilldonextinChapter2.
