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 22
. Lagrangian and the Feynman integral 24
. The Hamiltonian and canonical quantization 25
. The simplest example 27
. Green’s functions 30
. Wick rotation and Euclidean Green’s functions 31
3. Feynman diagrams 34
. The simplest case 35
. The origins of renormalization 38
. Feynman graphs and rules 41
. Connected Green’s functions 45
. The effective action and one-particle irreducible graphs 46
. Physically observable parameters 49
. The physics idea of renormalization 51
4. Dimensional regularization 54
5. The graph by graph method of Bogoliubov–Parasiuk–Hepp–Zimmermann 59
. The simplest example of subdivergence 60
. Superficial degree of divergence 64
. Subdivergences and preparation 64
6. The Connes–Kreimer theory of perturbative renormalization 70
. Commutative Hopf algebras and affine group schemes 71
. The Hopf algebra of Feynman graphs: discrete part 74
. The Hopf algebra of Feynman graphs: full structure 81
. BPHZ as a Birkhoff factorization 83
. Diffeographisms and diffeomorphisms 88
. The renormalization group 89
7. Renormalization and the Riemann–Hilbert correspondence 94
. Counterterms and time-ordered exponentials 95
. Flat equisingular connections 100
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. Equivariant principal bundles and the group G = G o Gm 109
. Tannakian categories and representations of affine group schemes 114
. Differential Galois th