Hopf Algebra Structure of a Model Quantum Field Theory

Abstract

Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra.