Hopf Algebra Structure of Renormalizable Quantum Field Theory

Abstract

Renormalization theory is a venerable subject put to daily use in many branches of physics. Here, we focus on its applications in quantum field theory, where a standard perturbative approach is provided through an expansion in Feynman diagrams. Whilst the combinatorics of the Bogoliubov recursion, solved by suitable forest formulas, has been known for a long time, the subject regained interest on the conceptual side with the discovery of an underlying Hopf algebra structure behind these recursions.Perturbative expansions in quantum field theory are organized in terms of one particle irreducible (1PI) Feynman graphs. The goal is to calculate the corresponding one-particle irreducible Green functions order by order in the coupling constants of the theory, by applying Feynman rules to these 1PI graphs of a renormalizable theory under consideration. This allows to disentangle the problem into an algebraic part and an analytic part.For the former one studies Feynman graphs as combinatorial objects which lead to the Lie and Hopf algebras discussed below. Feynman rules then assign analytic expressions to these graphs, with the analytic structure of finite renormalized quantum field theory largely dictated by the underlying algebra.The objects of interest in quantum field theory are the 1PI Green functions. They are parameterized by the quantum numbers, -masses, momenta, spin and such-, of the particles participating in the scattering process under consideration. We call a set of such quantum numbers an external leg structure r. For 1

Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees

“…In the case of the overlapping divergences, at the first step they should be reduced to a linear combination of disjoint and nested divergences and next we have a sum of rooted trees [21,26]. Depends on a fixed theory, all rooted trees are equipped with specific decorations (which store physical information such as (sub)-divergences) [12,20,22,23]. Renormalization Hopf algebra of a given theory can be made based on a decorated version of an interesting Hopf algebra structure on rooted trees and in this part this Hopf algebra is studied.…”

“…Therefore the lack of a practical mathematical basement for this interesting physical technique in QFT is covered. [1,20,21,22,23,24,25] Connes and Kreimer proved that perturbative renormalization can be explained by a general mathematical procedure namely, extraction of finite values based on the Riemann-Hilbert problem and in this way they showed that one can obtain the important physical data of a renormalizable QFT for instance renormalized values and counterterms from the Birkhoff decomposition of characters of the related Hopf algebra to the theory. In other words, they associated to each theory an infinite dimensional Lie group and proved that in dimensional regularization passing from unrenormalized to the renormalized value is equivalent to the replacement of a given loop (with values in the Lie group) with the value of the positive component of its Birkhoff factorization at the critical integral dimension D. In fact, in [2,3] an algebraic reconstruction from the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) method in renormalization is initiated.…”

“…[5,6,7] The combinatorial nature of the Hopf algebra of renormalization is observed by its relation with the Connes-Kreimer Hopf algebra of rooted trees [20,22]. This Hopf algebra on rooted trees has universal property with respect to the Hochschild cohomology theory [1,23]. On the other hand, we know that the universal Hopf algebra of renormalization has universal property with respect to Hopf algebras (related to all renormalizable physical theories) [5].…”

Dyson-Schwinger equations: From Hopf algebra to number theory

“…From a strictly perturbative viewpoint, this is summarized in this volume in the paper by Ebrahimi-Fard and Guo [3], with emphasis given to Rota-Baxter algebras. Our own summary below follows [4]. We hence will be short in our discussion of the perturbative viewpoint, and aim at a qualitative discussion of the above points, exemplified in the much stressed example of the propagator in massless Yukawa theory [5,6].…”