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A Treatise on Quantum Clifford Algebras Contents A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser arXiv:math/0202059v1 [math.QA] 7 Feb 2002 Universitat¨ Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al- ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Graßmann-Cayley algebras including co-meet and co-join for Graßmann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clif- ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A ‘spinorial’ generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor field theory and spinor electro- dynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 15-02 Research exposition (monographs, survey articles); 15A66 Clifford algebras, spinors; 15A75 Exterior algebra, Grassmann algebra; 81T15 Perturbative methods of renormalization II A Treatise on Quantum Clifford Algebras Contents Abstract I Table of Contents II Preface VII Acknowledgement XII 1 Peano Space and Graßmann-Cayley Algebra 1 1.1 Normedspace–normedalgebra . ... 2 1.2 Hilbert space, quadratic space – classical Clifford algebra............. 3 1.3 Weyl space – symplectic Clifford algebras (Weyl algebras) ............ 4 1.4 Peanospace–Graßmann-Cayleyalgebras . ....... 5 1.4.1 Thebracket................................. 6 1.4.2 Thewedgeproduct–join ......................... 7 1.4.3 Thevee-product–meet .......................... 8 1.4.4 Meetandjoinforhyperplanesandco-vectors . ....... 11 2 Basics on Clifford algebras 15 2.1 Algebrasrecalled ................................ .. 15 2.2 Tensor algebra, Graßmann algebra, Quadratic forms . ............ 17 2.3 Clifford algebras by generators and relations . ............ 20 2.4 Cliffordalgebrasbyfactorization . ......... 22 2.5 Clifford algebras by deformation – Quantum Clifford algebras .......... 22 2.5.1 TheCliffordmap.............................. 23 2.5.2 Relation of Cℓ(V,g) and Cℓ(V, B) ..................... 25 2.6 Cliffordalgebrasofmultivectors . ........ 25 2.7 Cliffordalgebrasbycliffordization . .......... 27 2.8 Dottedandun-dottedbases . .... 29 2.8.1 Linearforms ................................ 29 2.8.2 Conjugation................................. 30 2.8.3 Reversion.................................. 30 III IV A Treatise on Quantum Clifford Algebras 3 Graphical calculi 33 3.1 TheKuperberggraphicalmethod . ..... 33 3.1.1 Originofthemethod ............................ 33 3.1.2 Tensoralgebra ............................... 34 3.1.3 Pictographicalnotationoftensoralgebra . ........ 37 3.1.4 Someparticulartensorsandtensorequations . ........ 38 3.1.5 Duality ................................... 41 3.1.6 Kuperberg’sLemma3.1. 41 3.2 Commutativediagramsversustangles . ....... 42 3.2.1 Definitions ................................. 42 3.2.2 Tanglesforknottheory. 45 3.2.3 Tanglesforconvolution. .. 47 4 Hopf algebras 49 4.1 Algebras....................................... 50 4.1.1 Definitions ................................. 50 4.1.2 A-modules ................................. 54 4.2 Co-algebras ..................................... 55 4.2.1 Definitions ................................. 55 4.2.2 C-comodules................................ 57 4.3 Bialgebras...................................... 58 4.3.1 Definitions ................................. 58 4.4 Hopfalgebrasi.e.antipodalbialgebras . .......... 61 4.4.1 Morphisms of connected co-algebras and connected algebras : group like convolution................................. 61 4.4.2 Hopfalgebradefinition. 63 5 Hopf gebras 65 5.1 Cupandcaptangles................................ 66 5.1.1 Evaluationandco-evaluation. .... 66 5.1.2 Scalarandco-scalarproducts. .... 68 5.1.3 Induced graded scalar and co-scalar products . ........ 68 5.2 Productco-productduality . ..... 70 5.2.1 Byevaluation................................ 70 5.2.2 Byscalarproducts ............................. 71 5.3 CliffordizationofRotaandStein . ....... 75 5.3.1 Cliffordizationofproducts . .... 75 5.3.2 Cliffordizationofco-products . ..... 77 5.3.3 Cliffordmapsforanygrade . 78 5.3.4 Inversionformulas . .. .. .. .. .. .. .. .. 79 5.4 Convolutionalgebra. .. .. .. .. .. .. .. .. ... 80 BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ V 5.5 Crossingfromtheantipode . .... 82 5.6 Localversusnon-localproductsandco-products . ........... 85 5.6.1 KuperbergLemma3.2.revisited . ... 85 5.6.2 Interactingandnon-interactingHopfgebras . ......... 87 6 Integrals, meet, join, unipotents, and ‘spinorial’ antipode 91 6.1 Integrals....................................... 91 6.2 Meetandjoin .................................... 93 6.3 Crossings ...................................... 96 6.4 Convolutiveunipotents . .... 97 6.4.1 Convolutive’adjoint’. .. 98 6.4.2 Asquarerootoftheantipode. .. 99 6.4.3 Symmetrizedproductco-procducttangle . .. ..100 7 Generalized cliffordization 101 7.1 Linear forms on V × V ............................101 7.2 Properties of generalized Clifford products . ............103 V V 7.2.1 UnitsforgeneralizedCliffordproducts . .......104 7.2.2 Associativity of generalized Clifford products . ...........105 7.2.3 Commutation relations and generalized Clifford products. .107 7.2.4 Laplace expansion i.e. product co-product duality implies exponentially generatedbilinearforms . .108 7.3 Renormalization group and Z-pairing .......................109 7.3.1 Renormalizationgroup . 109 7.3.2 Renormalized time-ordered products as generalized Clifford products . 111 8 (Fermionic) quantum field theory and Clifford Hopf gebra 115 8.1 Fieldequations .................................. 116 8.2 Functionals ..................................... 117 8.3 Functionalequations . .. .. .. .. .. .. .. .. 121 8.4 Vertexrenormalization . .122 8.5 Time-andnormal-ordering . .123 8.5.1 Spinorfieldtheory .............................124 8.5.2 Spinorquantumelectrodynamics. .125 8.5.3 Renormalizedtime-orderedproducts. .. ..127 8.6 Onthevacuumstructure .. .. .. .. .. .. .. .. 128 8.6.1 One particle Fermi oscillator, U(1) ....................128 8.6.2 Two particle Fermi oscillator, U(2) ....................130 VI A Treatise on Quantum Clifford Algebras A CLIFFORDandBIGEBRApackagesforMaple 137 A.1 ComputeralgebraandMathematicalphysics . .........137 A.2 TheCLIFFORDPackage–rudimentsofversion5 . .. ..139 A.3 TheBIGEBRAPackage ..............................143 A.3.1 &cco –Cliffordco-product . .144 A.3.2 &gco –Graßmannco-product . .144 A.3.3 &gco d –dottedGraßmannco-product . 145 A.3.4 &gpl co –GraßmannPl¨uckerco-product . 146 A.3.5 &map –mapsproductsontotensorslots . 146 A.3.6 &t –tensorproduct.............................146 A.3.7 &v –vee-product,i.e.meet. .147 A.3.8 bracket –thePeanobracket. .148 A.3.9 contract –contractionoftensorslots. 148 A.3.10 define –Mapledefine,patched . .149 A.3.11 drop t –dropstensorsigns. .149 A.3.12 EV –evaluationmap ............................149 A.3.13 gantipode –Graßmannantipode . .149 A.3.14 gco unit –Graßmannco-unit . .150 A.3.15 gswitch –graded(i.e.Graßmann)switch . 151 A.3.16 help –mainhelp-pageofBIGEBRApackage . 151 A.3.17 init –initprocedure ...........................151 A.3.18 linop/linop2 – action of a linear operator on a Clifford polynom . 151 A.3.19 make BI Id – cup tangle need for &cco .................152 A.3.20 mapop/mapop2 –actionofanoperatoronatensorslot . 152 A.3.21 meet – same as &v (vee-product) . .152 A.3.22 pairing –Apairingw.r.t.abilinearform . 152 A.3.23 peek –extractatensorslot . .152 A.3.24 poke –insertatensorslot . .153 A.3.25 remove eq –removestautologicalequations . 153 A.3.26 switch –ungradedswitch . .153 A.3.27 tcollect –collectsw.r.t.thetensorbasis . 153 A.3.28 tsolve1 –tanglesolver .........................153 A.3.29 VERSION –showstheversionofthepackage. 154 A.3.30 type/tensorbasmonom –newMapletype . .154 A.3.31 type/tensormonom –newMapletype . .154 A.3.32 type/tensorpolynom –newMapletype . .155 Bibliography 156 BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ VII “Al-gebra and Co-gebra are brother and sister“ Zbigniew Oziewicz Seht Ihr den Mond dort stehen er ist nur halb zu sehen und ist doch rund und schon¨ so sind gar manche Sachen die wir getrost belachen weil unsre Augen sie nicht sehn. Matthias Claudius Preface This ‘Habilitationsschrift’
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