A Combinatorial Perspective on Quantum Field Theory

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.

• Dyson-Schwinger equations
• graph theory
• Feynman graphs
• Feynman periods
• Connes-Kreimer Hopf algebra
• Schnetz twist
• c2 invariant
• the zigzag result
• rooted trees
• combinatorial classes
• combinatorial Hopf algebras
• sub Hopf algebras
• chord diagram expansion
• leading log expansion

• Department of Mathematics, Simon Fraser University, Burnaby, Canada

  Karen Yeats

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