A treatise on quantum Clifford algebras

182 Pages·2002·1.062 MB·English

by  Fauser.

2 A Treatise on
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Quantum Clifford Algebras
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Habilitationsschrift
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t Dr. Bertfried Fauser
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Universita¨t 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:QuantumCliffordAlgebras(QCA),i.e. CliffordHopfgebrasbased
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-Cayleyco-gebras whichare very efficientand
maybeusedinRobotics,leftandrightcontractions,leftandrightco-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
toachoice. AfrequentlyusedKuperberglemmaisrevisitednecessitatingthedef-
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-trivialintegrals in low-dimensional Clifford co-gebras is shown.
Generalizedcliffordizationis discussedwhichis basedonnon-exponentiallygen-
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-Dysonhierarchiesof non-linear spinorfieldtheory andspinorelectro-
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) quantumfield theory.
MSC2000:
16W30Coalgebras,bialgebras, Hopfalgebras;
15-02 Researchexposition (monographs, surveyarticles);
15A66Cliffordalgebras,spinors;
15A75Exterior algebra,Grassmann algebra;
81T15Perturbativemethods of renormalization
II A Treatise onQuantumCliffordAlgebras
Contents
Abstract I
Table of Contents II
Preface VII
Acknowledgement XII
1 PeanoSpaceand Graßmann-CayleyAlgebra 1
1.1 Normed space–normed algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Hilbert space,quadratic space– classicalCliffordalgebra . . . . . . . . . . . . . 3
1.3 Weyl space–symplectic Cliffordalgebras (Weylalgebras) . . . . . . . . . . . . 4
1.4 Peanospace –Graßmann-Cayleyalgebras . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 The bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 The wedgeproduct –join . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 The vee-product– meet . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.4 Meet andjoinforhyperplanesand co-vectors . . . . . . . . . . . . . . . 11
2 BasicsonCliffordalgebras 15
2.1 Algebras recalled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Tensor algebra,Graßmann algebra,Quadratic forms . . . . . . . . . . . . . . . . 17
2.3 Cliffordalgebrasby generators andrelations . . . . . . . . . . . . . . . . . . . . 20
2.4 Cliffordalgebrasby factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Cliffordalgebrasby deformation –Quantum Cliffordalgebras . . . . . . . . . . 22
2.5.1 The Clifford(cid:0)(cid:2)(cid:1)(cid:4)m(cid:3)(cid:6)(cid:5)(cid:8)ap(cid:7)(cid:6)(cid:9)(cid:11).(cid:10) . . .(cid:0)(cid:2)(cid:1)(cid:4).(cid:3)(cid:6)(cid:5)(cid:8).(cid:7)(cid:13).(cid:12)(cid:14).(cid:10) . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Relationof and . . . . . . . . . . . . . . . . . . . . . 25
2.6 Cliffordalgebrasof multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Cliffordalgebrasby cliffordization . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Dottedand un-dottedbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8.1 Linearforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8.2 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8.3 Reversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
III
IV A Treatise onQuantumCliffordAlgebras
3 Graphicalcalculi 33
3.1 The Kuperberggraphical method . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Originof themethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Tensoralgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.3 Pictographicalnotation of tensoralgebra . . . . . . . . . . . . . . . . . 37
3.1.4 Some particulartensors andtensorequations . . . . . . . . . . . . . . . 38
3.1.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.6 Kuperberg’sLemma3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Commutativediagramsversus tangles . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Tangles forknottheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Tangles forconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Hopfalgebras 49
4.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
(cid:15)
4.1.2 -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Co-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
(cid:16)
4.2.2 -comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Hopf algebras i.e. antipodal bialgebras . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Morphismsofconnectedco-algebrasandconnectedalgebras: grouplike
convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.2 Hopf algebradefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Hopfgebras 65
5.1 Cup andcap tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.1 Evaluation andco-evaluation . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.2 Scalar andco-scalarproducts. . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.3 Induced gradedscalarand co-scalarproducts . . . . . . . . . . . . . . . 68
5.2 Product co-productduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.1 Byevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.2 Byscalar products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Cliffordizationof RotaandStein . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 Cliffordizationof products . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.2 Cliffordizationof co-products . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.3 Cliffordmaps for anygrade . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.4 Inversionformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Convolution algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
BERTFRIED FAUSER — UNIVERSITY OF KONSTANZ V
5.5 Crossing fromtheantipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Local versus non-local productsandco-products . . . . . . . . . . . . . . . . . 85
5.6.1 KuperbergLemma3.2. revisited . . . . . . . . . . . . . . . . . . . . . . 85
5.6.2 Interactingandnon-interacting Hopf gebras . . . . . . . . . . . . . . . . 87
6 Integrals, meet,join, unipotents,and ‘spinorial’antipode 91
6.1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Meet andjoin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Convolutiveunipotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4.1 Convolutive’adjoint’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4.2 Asquare rootof the antipode . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4.3 Symmetrizedproductco-procduct tangle . . . . . . . . . . . . . . . . . 100
7 Generalized cliffordizat(cid:5)(cid:19)ion(cid:18) (cid:5) 101
(cid:17) (cid:17)
7.1 Linear formson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Properties of generalizedCliffordproducts . . . . . . . . . . . . . . . . . . . . . 103
7.2.1 Unitsfor generalizedCliffordproducts . . . . . . . . . . . . . . . . . . 104
7.2.2 Associativityof generalizedCliffordproducts . . . . . . . . . . . . . . . 105
7.2.3 Commutationrelationsand generalizedCliffordproducts . . . . . . . . . 107
7.2.4 Laplaceexpansion i.e. product co-productduality impliesexponentially
generatedbilinearforms . . . . . . . . . . . . . . . . . . . . . . . . . . 108
(cid:20)
7.3 Renormalization groupand -pairing . . . . . . . . . . . . . . . . . . . . . . . 109
7.3.1 Renormalizationgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3.2 Renormalized time-orderedproductsas generalizedCliffordproducts . . 111
8 (Fermionic)quantumfield theoryand CliffordHopfgebra 115
8.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3 Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.4 Vertex renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.5 Time- andnormal-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.5.1 Spinorfieldtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.5.2 Spinorquantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . 125
8.5.3 Renormalized time-orderedproducts . . . . . . . . . . . . . . . . . . . . 127
8.6 On thevacuumstructure . . . . . . . .(cid:3)(cid:23)(cid:22)(cid:24).(cid:10) . . . . . . . . . . . . . . . . . . . . . 128
(cid:21)
8.6.1 Oneparticle Fermioscillator, (cid:3)(cid:26)(cid:25)(cid:27)(cid:10) . . . . . . . . . . . . . . . . . . . . 128
(cid:21)
8.6.2 Twoparticle Fermioscillator, . . . . . . . . . . . . . . . . . . . . 130
VI A Treatise onQuantumCliffordAlgebras
A CLIFFORDand BIGEBRApackagesforMaple 137
A.1 Computer algebraandMathematical physics . . . . . . . . . . . . . . . . . . . . 137
A.2 The CLIFFORDPackage –rudiments of version5 . . . . . . . . . . . . . . . . 139
A.3 The BIGEBRAPackage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.3.1 &cco–Cliffordco-product . . . . . . . . . . . . . . . . . . . . . . . . 144
A.3.2 &gco–Graßmann co-product . . . . . . . . . . . . . . . . . . . . . . . 144
A.3.3 &gco d –dottedGraßmann co-product . . . . . . . . . . . . . . . . . . 145
A.3.4 &gpl co– GraßmannPlu¨ckerco-product . . . . . . . . . . . . . . . . 146
A.3.5 &map–maps productsontotensorslots . . . . . . . . . . . . . . . . . . 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– contractionof tensorslots . . . . . . . . . . . . . . . . . . 148
A.3.10 define– Maple define,patched . . . . . . . . . . . . . . . . . . . . . 149
A.3.11 drop t –dropstensorsigns . . . . . . . . . . . . . . . . . . . . . . . . 149
A.3.12 EV–evaluation map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.3.13 gantipode–Graßmann antipode . . . . . . . . . . . . . . . . . . . . 149
A.3.14 gco unit– Graßmannco-unit . . . . . . . . . . . . . . . . . . . . . . 150
A.3.15 gswitch– graded(i.e. Graßmann)switch . . . . . . . . . . . . . . . . 151
A.3.16 help–mainhelp-page of BIGEBRApackage . . . . . . . . . . . . . . 151
A.3.17 init–init procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.3.18 linop/linop2–actionof a linear operatorona Cliffordpolynom . . 151
A.3.19 make BI Id–cup tangleneed for&cco . . . . . . . . . . . . . . . . . 152
A.3.20 mapop/mapop2–actionof an operatorona tensorslot . . . . . . . . . 152
A.3.21 meet–same as &v(vee-product) . . . . . . . . . . . . . . . . . . . . . 152
A.3.22 pairing– Apairing w.r.t. a bilinearform . . . . . . . . . . . . . . . . 152
A.3.23 peek–extract a tensorslot . . . . . . . . . . . . . . . . . . . . . . . . 152
A.3.24 poke–inserta tensorslot . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.3.25 remove eq– removestautological equations . . . . . . . . . . . . . . 153
A.3.26 switch– ungraded switch . . . . . . . . . . . . . . . . . . . . . . . . 153
A.3.27 tcollect– collectsw.r.t. the tensorbasis . . . . . . . . . . . . . . . . 153
A.3.28 tsolve1– tanglesolver . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.3.29 VERSION– showsthe versionof thepackage . . . . . . . . . . . . . . . 154
A.3.30 type/tensorbasmonom–new Mapletype . . . . . . . . . . . . . . 154
A.3.31 type/tensormonom– newMaple type . . . . . . . . . . . . . . . . 154
A.3.32 type/tensorpolynom–new Mapletype . . . . . . . . . . . . . . . 155
Bibliography 156