Graph Theory and Feynman Integrals (Mathematics and Its Applications)

Title Page......Page 2
Preface......Page 4
Contents......Page 6
CHAPTER 1 GRAPH THEORY......Page 10
1-1 Set-theoretical concepts......Page 12
1-2 Graphs......Page 13
1-3 Paths, circuits, cut-sets, trees......Page 16
1-4 Feynman graphs......Page 17
2-1 Properties independent of orientation......Page 19
2-2 Fundamental sets of circuits and cut-sets......Page 23
2-3 Duality......Page 29
3-1 Matrices associated with a graph......Page 30
3-2 Topological formulas......Page 33
4-1 Characterizations of the planar graph......Page 41
4-2 Dual graphs......Page 42
5 Transport Problem......Page 46
References......Page 52
CHAPTER 2 FEYNMAN-PARAMETRIC FORMULA......Page 53
6-1 Physical background......Page 55
6-2 Definition of the Feynman Integral......Page 61
7-1 Derivation of the Feynman-parametric integral......Page 65
7-2 Topological formulas for the V function......Page 68
7-3 Remarks......Page 73
8-1 Inverse-Feynman-parametric integral......Page 77
8-2 Position-space Feynman-parametric integral......Page 80
9-1 Properties of U and V......Page 84
9-2 Presence of nonzero-spin particles......Page 85
10-1 Proof of Dyson\'s power-counting theorem......Page 90
10-2 Renormalization......Page 93
References......Page 97
CHAPTER 3 SINGULARITIES OF THE FEYNMAN INTEGRAL......Page 99
11 Feynman Function......Page 101
12-1 Landau equations......Page 103
12-2 Dual-graph analysis......Page 107
12-3 Behavior near the singularity......Page 108
13-1 General consideration on real singularities......Page 111
13-2 Normal and anomalous thresholds......Page 114
14-1 Multiple scattering......Page 121
14-2 Discontinuity formula......Page 123
14-3 Infrared Divergence......Page 124
15 Existence Region of Singularities......Page 127
16-1 Pure second-type singularities......Page 131
16-2 Mixed second-type singularities......Page 135
References......Page 136
CHAPTER 4 PERTURBATION-THEORETICAL INTEGRAL REPRESENTATIONS......Page 138
17-1 General Remarks......Page 140
17-2 Derivation of PTIR......Page 142
17-3 Position-space PTIR......Page 148
18-1 Analyticity......Page 149
18-2 Uniqueness......Page 150
18-3 Other properties......Page 154
19-1 General Method......Page 156
19-2 Vertex function......Page 158
19-3 Extended scattering amplitude......Page 160
20-1 Models......Page 167
20-2 Majorization rules......Page 169
20-3 Majorization procedure......Page 174
21 W Fimctioti Inequalities......Page 178
22-1 Vertex function......Page 184
22-2 Scattering amplitude......Page 185
22-3 Proof of dispersion relations......Page 189
22-4 One-particle production amplitude......Page 190
References......Page 191
CHAPTER 5 MISCELLANEOUS TOPICS......Page 192
23 Number of Feynman Graphs......Page 193
24 Divergence of the Perturbation Series......Page 198
25 Cut-Set Properties of Feynntan Graphs......Page 202
26 High-Energy Asymptotic Behavior of the Feynman Integral......Page 205
References......Page 208
A-1 Distributions......Page 209
A-2 Analytic functions......Page 212
References......Page 216
B-1 General remarks and Self-energy......Page 217
B-2 Vertex function......Page 219
B-4 Production Amplitudes......Page 222
References......Page 224
AUTHOR INDEX......Page 226
SUBJECT INDEX......Page 228