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Geometric and Topological Methods for Quantum Field Theory
These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2-20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar φ 3 theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series. The last lecture presents the Connes-Kreimer Hopf algebra of renormalization for the scalar theory and its associated proalgebraic group of formal series.
Hopf algebra and renormalization: A brief review
2015
We briefly review the Hopf algebra structure arising in the renormalization of quantum field theories. We construct the Hopf algebra explicitly for a simple toy model and show how renormalization is achieved for this particular model.
A Hopf algebra deformation approach to renormalization
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.
Mathematical renormalization in quantum electrodynamics via noncommutative generating series
In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics (The present work is part of a series of papers devoted to the study of the renormalization of divergent polyzetas (at positive and at negative indices) via the factorization of the non commutative generating series of polylogarithms and of harmonic sums and via the effective construction of pairs of bases in duality in ϕ-deformed shuffle algebras. It is a sequel of [6] and its content was presented in several seminars and meetings, including the 66th and 74th Séminaire Lotharingien de Combinatoire.), we focus on combinatorial aspects of their renormalization at {0, 1, +∞}.
Systematic Differential Renormalization to All Orders
Annals of Physics, 1994
We present a systematic implementation of differential renormalization to all orders in perturbation theory. The method is applied to individual Feynman graphs written in coordinate space. After isolating every singularity which appears in a bare diagram, we define a subtraction procedure which consists in replacing the core of the singularity by its renormalized form given by a differential formula. The organization of subtractions in subgraphs relies on Bogoliubov's formula, fulfilling the requirements of locality, unitarity and Lorentz invariance. Our method bypasses the use of an intermediate regularization and automatically delivers renormalized amplitudes which obey renormalization group equations.
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.
A Lie Theoretic Approach to Renormalization
Communications in Mathematical Physics, 2007
Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.
Renormalization group made clearer
I attempt to explain the use of renormalization group in quantum field theory from an elementary point of view. I review an elementary quantum-mechanical problem involving renormalization as a pedestrian example of a theory which is inherently ill-defined without a cutoff. After introducing a cutoff, one usually obtains a perturbative expansion that becomes invalid when the cutoff is removed. The renormalization group approach is treated as a purely mathematical technique (the Woodruff-Goldenfeld method) that improves the behavior of non-uniform perturbative expansions. By means of renormalization, one derives a perturbative expansion which is uniform in the cutoff, and therefore valid in the limit of infinite cutoff. I illustrate the application of this method to singular perturbation problems in ordinary differential equations.
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.