文档介绍：mutative Geometry, Quantum Fields
and Motives
Alain Connes
Matilde Marcolli
A. Connes: College` de France, 3, rue d’Ulm, Paris, F-75005 France,
. and Vanderbilt University
E-mail address: alain@
M. Marcolli: Max–Planck Institut fur¨ Mathematik, Vivatsgasse 7,
Bonn, D-53111 Germany
E-mail address: ******@mpim-
Contents
Preface 9
Chapter 1. Quantum fields, mutative spaces, and motives 17
1. Introduction 17
2. Basics of perturbative QFT 22
. Lagrangian and Hamiltonian formalisms 23
. Lagrangian and the Feynman integral 25
. The Hamiltonian and canonical quantization 26
. The simplest example 28
. Green’s functions 31
. Wick rotation and Euclidean Green’s functions 32
3. Feynman diagrams 35
. The simplest case 36
. The origins of renormalization 40
. Feynman graphs and rules 43
. Connected Green’s functions 47
. The effective action and one-particle irreducible graphs 48
. Physically observable parameters 52
. The physics idea of renormalization 54
4. Dimensional regularization 56
5. The graph by graph method of Bogoliubov–Parasiuk–Hepp–Zimmermann 62
. The simplest example of subdivergence 63
. Superficial degree of divergence 67
. Subdivergences and preparation 68
6. The Connes–Kreimer theory of perturbative renormalization 74
. Commutative Hopf algebras and affine group schemes 75
. The Hopf algebra of Feynman graphs: discrete part 78
. The Hopf algebra of Feynman graphs: full structure 85
. BPHZ as a Birkhoff factorization 87
. Diffeographisms and diffeomorphisms 93
. The renormalization group 95
7. Renormalization and the Riemann–Hilbert correspondence 100
. Counterterms and time-ordered exponentials 100
. Flat equisingular connections 107
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. Equivariant principal bundles and the group G = G o Gm 116
. Tannakian categories and affine group schemes 120
. Differential Galois theory and the local