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Tables of Contents for Knots and Feynman Diagrams
Chapter/Section Title
Page #
Page Count
Acknowledgements
xi
Introduction
1
6
Motivation
1
6
Perturbative quantum field theory
7
29
pQFT
7
14
Canonical quantization
7
3
Perturbation theory
10
5
Feynman rules
15
2
Schwinger-Dyson equations
17
4
Regularization
21
4
Dimensional regularization
23
2
Basic facts about renormalization
25
11
Multiplicative renormalization
26
3
Power counting
29
7
The Hopf algebra structure of renormalization
36
61
Preliminaries
36
5
Vertex corrections
41
7
The first iteration
41
3
The factorization
44
4
Overlapping divergences
48
4
A toy model
49
3
Technicalities
52
4
Form factors
53
1
Other degrees of divergence
54
2
Towards a Hopf algebra
56
5
Feynman diagrams as a realization
61
6
The Hopf algebra
67
13
The coproduct
72
3
The antipode
75
5
Realizations of A
80
13
Toy models
80
7
Quantum field theories
87
3
Once more: overlapping divergences
90
3
An ultimate example
93
1
Remarks
94
3
Rationality: no knots, no transcendentals
97
9
A combinatorial approach
99
3
Delbourgo's argument
102
1
Toy models versus QFT
103
3
The simplest link diagrams
106
12
Link diagrams from ladder diagrams
106
8
Disentangling the link diagram
108
4
Gauss codes
112
2
Links and ladders - the overlapping case
114
4
Necessary topics from knot theory
118
12
Basics
118
3
Torus knots
121
2
Braids
123
2
Knot polynomials
125
5
Knots to numbers: (2, 2n - 3) torus knots and ζ (2n - 3)
130
13
ζ (3) from a counterterm
130
4
The (2, q) torus knots and ζ(q)
134
5
Factor knots
139
2
Gauss codes
141
2
One-loop words
143
20
Definitions
144
3
An elementary example
147
7
Continuation to a dressed two-loop graph
154
6
Higher order dressing
160
3
Euler-Zagier sums
163
11
Relations coming from the Drinfeld associator
165
3
Shuffle algebras
168
1
Euler-Zagier sums and MZVs
169
5
Knots and transcendentals
174
40
The (3,4) torus knot and the first Euler double sum
176
7
The knot 8 19 and its knot-number
177
2
A momentum routing analysis
179
3
A Gauss code analysis
182
1
Chord diagrams, knots and numbers to five loops
183
1
&phis;4-theory: more knots and numbers
183
6
Rationality and the β-function of quenched QED
189
4
Euler double sums
193
3
Field theory, knot theory, number theory
196
3
From knots to numbers
199
15
How many knot numbers?
200
1
Knot-numbers from evaluations of Feynman diagrams
201
4
Positive knots associated with irreducible MZVs
205
9
The four-term relation
214
22
Introduction
214
4
The 4TR between primitive graphs
218
14
A test
232
2
Hopf algebra and 4TR
234
2
Hopf algebras, non-commutative geometry, and what else?
236
17
The Hopf algebra HR of rooted trees
236
7
Formulae for renormalization
241
2
The relation between HR and HT
243
4
And what else?
247
6
References
253
6
Index
259