Sequences of Trees and Higher-Order Renormalization Group Equations
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Date
2019-08-27
Authors
Dugan, William
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
In 1998, Connes and Kreimer introduced a combinatorial Hopf algebra HCK on the vector space of
forests of rooted trees that precisely explains the phenomenon of renormalization in quantum field theory.
This Hopf algebra has been of great interest since its inception, as it connects the disciplines of algebra,
combinatorics, and physics, providing interesting questions in each.
In this thesis we introduce the notion of higher-order renormalization group equations, which generalize
the usual renormalization group equation of quantum field theory, and further define a corresponding
notion of order on certain sequences of trees constituting elements of the completion of HCK. We also
give an explication of a result, due to Foissy, that characterizes which sequences of linear combinations of
trees with one generator in each degree generate Hopf subalgebras of HCK. We conclude with some results
towards classifying these sequences by their order (when such an order is admitted), and discuss the place
of the Connes-Moscovici Hopf subalgebra in the context of this new framework.
Description
Keywords
mathematics, algebraic combinatorics, Hopf algebras, quantum field theory, prelie algebras