Hopf algebra and renormalization: A brief review (original) (raw)
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Arxiv preprint hep-th/0307112, 2003
The relation between Connes-Kreimer Hopf algebra approach to renormalization and deformation quantization is investigated. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is conjectured that the *-product obtained by Lie bialgebra deformation quantization and the *-product corresponding in the sense of Kontsevich-Cattaneo to the QFT obtained via renormalization, correspond.
On Hopf algebra deformation approach to renormalization
2003
The relation between Connes-Kreimer Hopf algebra approach to renormalization and deformation quantization is investigated. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is conjectured that the *-product obtained by Lie bial- gebra deformation quantization and the *-product corresponding in the sense of Kontsevich-Cattaneo to the QFT obtained via renormalization, correspond.
Communications in Mathematical Physics, 2000
We study the Connes-Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Ward identities and the Slavnov-Taylor identities (in the abelian and non-abelian case respectively) are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a purely combinatorial and rigorous proof of compatibility of the Slavnov-Taylor identities with renormalization.
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We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurences of this or closely related Hopf algebras in other mathematical domains, such as foliations,
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From Quantum Mechanics to Quantum Field Theory: The Hopf route
Journal of Physics: Conference Series, 2011
We show that the combinatorial numbers known as Bell numbers are generic in quantum physics. This is because they arise in the procedure known as Normal ordering of bosons, a procedure which is involved in the evaluation of quantum functions such as the canonical partition function of quantum statistical physics, inter alia. In fact, we shall show that an evaluation of the non-interacting partition function for a single boson system is identical to integrating the exponential generating function of the Bell numbers, which is a device for encapsulating a combinatorial sequence in a single function. We then introduce a remarkable equality, the Dobinski relation, and use it to indicate why renormalisation is necessary in even the simplest of perturbation expansions for a partition function. Finally we introduce a global algebraic description of this simple model, giving a Hopf algebra, which provides a starting point for extensions to more complex physical systems.
COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I
Reviews in Mathematical Physics, 2005
This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faà di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section 10 we describe the first. Then in Section 11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.
Renormalization Hopf algebras and combinatorial groups
Geometric and Topological Methods for Quantum Field Theory
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Hopf algebras and Quantum groups with their treatments in particle physics
In the recent years’ Hopf algebras have been introduced to describe certain combinatorial properties of quantum field theories.I have a short review of Hopf algebras and Quantum groups in this lecture. I will give a basic introduction to these algebras and objects and review some occurrences in particle physics and explain our conclude and ideas in this matter with some examples.
A generic Hopf algebra for quantum statistical mechanics
Physica Scripta, 2010
In this note we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is the following. Recent studies have revealed that perturbative Quantum Field Theory (pQFT) displays an astonishing interplay between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, thus far not exactly solvable and replete with divergences, the essential simplicity of the relationships between these areas can be somewhat obscured. The intention here is to display some of pevious-mentioned structures in the context of a simple bosonic quantum 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 quantum canonical partition function for a boson gas.