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1、ANINFORMALSUMMARYOFANEWFORMALISMFORCLASSIFYINGSPIN-ORBITSYSTEMSUSINGTOOLSDISTILLEDFROMTHETHEORYOFBUNDLES1K.HEINEMANNDepartmentofMathematicsandStatistics,UniversityofNewMexico,Albuquerque,NewMexico,USAheineman@math.unm.eduD.P.BARBERDeutschesElektronen–SynchrotronDESY,Hamburg,Germanympybar@mail.
2、desy.deJ.A.ELLISONDepartmentofMathematicsandStatistics,UniversityofNewMexico,Albuquerque,NewMexico,USAellison@math.unm.eduM.VOGTDeutschesElektronen–SynchrotronDESY,Hamburg,Germanyvogtm@mail.desy.de(January30,2015)AbstractWegiveaninformalsummaryofongoingworkwhichusestoolsdis-tilledfromthetheory
3、offibrebundlestoclassifyandconnectinvariantfieldsassociatedwithspinmotioninstoragerings.Wementionfourmajortheorems.Onetiesinvariantfieldswiththenotionofnormalform,thesecondallowscomparisonofdifferentinvariantfieldsandthetwootherstietheexistenceofinvariantfieldstotheexistenceofcertaininvariantsets.We
4、explainhowthetheoremsapplytothespinarXiv:1502.00538v1[physics.acc-ph]2Feb2015dynamicsofspin-1/2andspin-1particles.Ourapproachelegantlyunifiesthespin-vectordynamicsfromtheT-BMTequationwiththespin-tensordynamicsandotherdynamicsandsuggestsanavenueforaddressingthequestionoftheexistenceoftheinvarian
5、tspinfield.1BasedonapresentationatSpin2014,The21stInternationalSymposiumonSpinPhysics,Beijing,China,October2014.TobepublishedintheInternationalJournalofModernPhysics,ConferenceSeries.1IntroductionThepolarizationofabeamofspin-1/2orspin-1particlescirculatinginastorageringisbestsystematizedinterms
6、ofinvariantspinfields(ISF’s)andinvariantpolarization-tensorfields(ITF’s).Theyareessentialfordescribingtheequilibriumpolarizationstateofproton,deuteronandelectronbeams.WehavealreadytreatedtheconceptoftheISFindepthinRef.[1].Nu-mericalevidenceindicatesthatISF’scanberathercomplexentities.More-overth
7、equestionofexistence,althoughtrivialinsomesimplecases,e.g.,onorbitalresonance,isuptothisday,unsolvedand,asevidencedbyuseofstroboscopicaveraging[2]modelsituationscanoccurofforbitalresonancewhereISF’smightnotexist.Inotherwordsthereisnogood
