A Combinatorial Perspective on Quantum Field Theory

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Springer, Nov 23, 2016 - Science - 120 pages

This book explores combinatorial problems and insights in quantum field theory. It is not comprehensive, but rather takes a tour, shaped by the author’s biases, through some of the important ways that a combinatorial perspective can be brought to bear on quantum field theory. Among the outcomes are both physical insights and interesting mathematics.

The book begins by thinking of perturbative expansions as kinds of generating functions and then introduces renormalization Hopf algebras. The remainder is broken into two parts. The first part looks at Dyson-Schwinger equations, stepping gradually from the purely combinatorial to the more physical. The second part looks at Feynman graphs and their periods.

The flavour of the book will appeal to mathematicians with a combinatorics background as well as mathematical physicists and other mathematicians.

 

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Contents

1 Introduction
4
2 Quantum Field Theory Set Up
6
3 Combinatorial Classes and Rooted Trees
9
4 The ConnesKreimer Hopf Algebra
19
5 Feynman Graphs
35
Part II DysonSchwinger Equations
55
6 Introduction to DysonSchwinger Equations
57
7 SubHopf Algebras from DysonSchwinger Equations
61
10 Differential Equations and the NextTom Leading Log Expansion
81
Part III Feynman Periods
85
11 Feynman Integrals and Feynman Periods
87
12 Period Preserving Graph Symmetries
93
13 An Invariant with These Symmetries
97
14 Weight
100
15 The c2 Invariant
109
16 Combinatorial Aspects of Some Integration Algorithms
112

8 Tree Factorial and Leading Log Toys
67
9 Chord Diagram Expansions
71

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