A Combinatorial Perspective on Quantum Field Theory

✍ Scribed by Karen Yeats (auth.)

• Publisher: Springer International Publishing
• Year: 2017
• Tongue: English
• Leaves: 120
• Series: SpringerBriefs in Mathematical Physics 15
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-ix
Front Matter....Pages 1-1
Introduction....Pages 3-4
Quantum Field Theory Set Up....Pages 5-7
Combinatorial Classes and Rooted Trees....Pages 9-18
The Connes-Kreimer Hopf Algebra....Pages 19-34
Feynman Graphs....Pages 35-54
Front Matter....Pages 55-55
Introduction to Dyson-Schwinger Equations....Pages 57-59
Sub-Hopf Algebras from Dyson-Schwinger Equations....Pages 61-66
Tree Factorial and Leading Log Toys....Pages 67-70
Chord Diagram Expansions....Pages 71-80
Differential Equations and the (Next-To) ({}^{m}) Leading Log Expansion....Pages 81-84
Front Matter....Pages 85-85
Feynman Integrals and Feynman Periods....Pages 87-92
Period Preserving Graph Symmetries....Pages 93-96
An Invariant with These Symmetries....Pages 97-99
Weight....Pages 101-107
The (c_2) Invariant....Pages 109-111
Combinatorial Aspects of Some Integration Algorithms....Pages 113-115
Back Matter....Pages 117-120

✦ Subjects

Quantum Field Theories, String Theory;Mathematical Physics;Discrete Mathematics