Representations of U(1,q) and Constructive Quatern

arXiv:hep-th/9308142v130Aug1993RepresentationsofU(1,q)andConstructiveQuaternionTensorProductsbyS.DeLeoandP.RotelliDipartimentodiFisica,Universit`adiLecceINFN-SezionediLecceAbstractTherepresentationtheoryofthegroupU(1,q)isdiscussedinde-tailbecauseofitspossibleapplicationinaquaternionversionoftheSalam-Weinbergtheory.Asaconsequence,frompurelygrouptheo-reticalargumentswedemonstratethattheeigenvaluesmustberight-eigenvaluesandthattheonlyconsistentscalarproductsarethecom-plexones.Wealsodefineanexplicitquaterniontensorproductwhichleadstoasetofadditionalgrouprepresentationsforinteger“spin”.1IIntroductionQuaternionshavebeensomewhatofanenigmaforPhysicistssincetheirdiscoverybyHamilton[1]in1843.NotwithstandingHamilton’sconvictionthatquaternionswouldsoonplayarolecomparable,ifnotgreaterthanthatofcomplexnumbers,theuseofquaternionsinPhysicsisverylimited.Amongstthecontributionstoquaternionquantummechanicswedrawatten-tiontothefundamentalworksofFinkelsteinetal.[3,4,5,6](onfoundationsofquaternionquantummechanics,onquaternionicrepresentationsofcom-pactgroups,etc),ofHorwitzandBiedenharn[7](onquaternionquantummechanics,secondquantizationandgaugefields)andtothemanystimulat-ingpapersofAdler[8,9,10](onquaternionpotentialsandCPviolation,onquaternionfieldtheory,etc).ComplexnumbersinPhysicshaveplayedadualrole,firstasatechnicaltoolinresolvingdifferentialequations(e.g.inclassicaloptics)orviathethe-oryofanalyticfunctionsforperformingrealintegrations,summingseriesetc,secondly,inamoreessentialwayinthedevelopmentofquantummechan-ics(andlaterfieldtheory)characterizedbycomplexwavefunctionsandforfermionsbycomplexwaveequations.Withquaternions,forthefirsttypeofapplication,i.e.asameanstosimplifycalculations,wecanquotetheoriginalworkofHamilton,butthisonlybecauseofthelatedevelopmentofvectoralgebra.EvenMaxwell[11]usedquaternionsasatoolinhiscalculations.Themoreexcitingpossibilitythatquaternionequationswilleventuallyplayasignificantroleissynonymous,forsomephysicists(butnotforthepresentauthors),withtheadventofarevolutioninPhysicscomparabletothatofquantummechanics.Ourownparticularpointofviewisthatevenifquaternionsdonotsim-plifycalculations,itwouldbeverystrangeifstandardquantummechanicsdidnotpermitaquaterniondescriptionotherthaninthetrivialsensethatcomplexnumbersarecontainedwithinthequaternions.Inotherwords,giventhevalidityofquantummechanicsattheelementaryparticlelevel,2wepredicttheexistenceevenatthislevelofquaternionversionsofallstan-dardtheories.Oneoftheauthors[12,13]hasindeedsucceededinthiswithaquaternionversionoftheDiracequation.Thisequation,thankstotheuseofthecomplexscalarproduct,reproducesthestandardresultsnotwithstandingthetwo-componentnatureofthewavefunctionsduetotheexistenceoftwo-dimensionalquaterniongammamatrices.ThesamedoublingofsolutionsimpliesthattheSchr¨odingerequationhasautomaticallytwoplanewaveso-lutionscorrespondingtospinupandspindown.Thisdoublingofsolutionscontinuesevenforbosonicequations.Asaresult,twophotonicsolutionsexist(onecalledanomalous),twoscalarsolutionsoftheKlein-Gordonequa-tionexistandsoforth.Ithasalsobeendemonstrated[14]thatthesenewanomaloussolutionscanbeassociatedwithcorrespondinganomalousfields.Weobservethattheexistenceofthesenewsolutionsimpliesthattheuseofquaternionsisnotwithoutpredictivepower,atleastwiththeformalismdescribedabove.Coherentwithourpointofview,itappearstousdesirabletodevelopaquaternionversionoftheSalam-Weinbergtheoryofelectro-weakinterac-tions.Thistheorymayalsoprovideaprovinggroundfortheanomalousparticles.Foritispossiblethattheanomalousphotonbeidentifiedwithoneofthemasslessneutralintermediatevectorfields,priortothecreationofmassviaspontaneoussymmetrybreaking.Inotherwordsithasbeensuggested[14]thattheanomalousphotoncouldbeidentifiedwiththeZ0.AsapreliminarytothisnontrivialobjectiveonemustdecideupontheappropriatequaternionversionoftheGlashowgroupSU(2,c)×U(1,c).Thecherespecifiescomplexgroupandimpliesonlycomplexmatrixelements,i.e.standardgrouptheory.Aqwithinagroupnamewillimplyaquaterniongroupwithingeneralquaternionmatrixelements,evenifthisdoesnotex-cludetheappearanceofpurelycomplexorevenrealgrouprepresentations.Surprisingly,thecomplexgroupU(1,c)remainsassuchevenforaquater-nionversionofSalam-Weinberg.Thisisnotdifficulttojustify,butweleavethisexplanationtoasubsequentarticle.ThegroupSU(2,c)isparticularly3interesting,firstbecausethisLiegroupisnotonlytheweakisospingroupofSalam-Weinbergbutisverycommoninparticlephysics(spin,isospin,etc)andsecondbecausewedoindeedhaveanalternativechoiceinthequaternionunitarygroupU(1,q),alsoreferredtoasthesymplecticgroupSp(1,q).ItiswellknownthatthesegroupsareisomorphictoSU(2,c)[2].However,asweshalldemonstrateinthispaper,thisdoesnotguaranteeidenticalphysicalcontent.Forexample,withthecomplexgroupSU(2,c)allrepresentationsareobtainablefromthespinorrepresentationwiththeaidoftensorproducts.ThiswillnotbethecaseforU(1,q),andindeedthedefinitionofasuitablequaterniontensorproductisstillofprimaryinterest[15,16,17].InthenextSectionweshalldeveloptherepresentationtheoryofU(1,q)inanalogywiththatofSU(2,c)(weshallhenceforthuseth