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1、ATreatiseonQuantumCliffordAlgebrasHabilitationsschriftDr.BertfriedFauserarXiv:math.QA/0202059v17Feb2002Universitat¨KonstanzFachbereichPhysikFachM67878457KonstanzJanuary25,2002ToDorotheaIdaandRudolfEugenFauserBERTFRIEDFAUSER—UNIVERSITYOFKONSTANZIABSTRACT:QuantumCliffordAlgebras(Q
2、CA),i.e.CliffordHopfgebrasbasedonbilinearformsofarbitrarysymmetry,aretreatedinabroadsense.Fiveal-ternativeconstructionsofQCAsareexhibited.GradefreeHopfgebraicproductformulasarederivedformeetandjoinofGraßmann-Cayleyalgebrasincludingco-meetandco-joinforGraßmann-Cayleyco-gebraswhic
3、hareveryefficientandmaybeusedinRobotics,leftandrightcontractions,leftandrightco-contractions,Cliffordandco-Cliffordproducts,etc.TheChevalleydeformation,usingaClif-fordmap,arisesasaspecialcase.WediscussHopfalgebraversusHopfgebra,thelatteremergingnaturallyfromabi-convolution.Antipo
4、deandcrossingareconsequencesoftheproductandco-productstructuretensorsandnotsubjectabletoachoice.AfrequentlyusedKuperberglemmaisrevisitednecessitatingthedef-initionofnon-localproductsandinteractingHopfgebraswhicharegenericallynon-perturbative.A`spinorial'generalizationoftheantipo
5、deisgiven.Thenon-existenceofnon-trivialintegralsinlow-dimensionalCliffordco-gebrasisshown.Generalizedcliffordizationisdiscussedwhichisbasedonnon-exponentiallygen-eratedbilinearformsingeneralresultinginnonunital,non-associativeproducts.Reasonableassumptionsleadtobilinearformsbase
6、don2-cocycles.Cliffordiza-tionisusedtoderivetime-andnormal-orderedgeneratingfunctionalsfortheSchwinger-Dysonhierarchiesofnon-linearspinorfieldtheoryandspinorelectro-dynamics.Therelationbetweenthevacuumstructure,theoperatorordering,andtheHopfgebraiccounitisdiscussed.QCAsarepropose
7、dasthenaturallanguagefor(fermionic)quantumfieldtheory.MSC2000:16W30Coalgebras,bialgebras,Hopfalgebras;15-02Researchexposition(monographs,surveyarticles);15A66Cliffordalgebras,spinors;15A75Exterioralgebra,Grassmannalgebra;81T15PerturbativemethodsofrenormalizationIIATreatiseonQuant
8、umCliffordAlgebrasContentsAbstractITableofConte
