Logarithmic derivatives and generalized Dynkin operators

Menous, Frédéric; Patras, Frédéric

doi:10.1007/s10801-013-0431-3

Cited by 10 publications

(11 citation statements)

References 23 publications

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“…It captures nicely certain phenomena related to Lie theory and the behaviour of the Dynkin operators: for example, the structure of certain renormalization group equations and the algebraic properties of beta functions (see the original article by Connes and Marcolli [9] and the detailed algebraic and combinatorial analysis of these phenomena in [36]. Further insights on the role of (generalized) Dynkin operators in the theory of differential equations can be found in [33]). However, the group and the descent algebra act on Feynman diagrams and do not encode operations that occur at the level of the target algebra of amplitudes.…”

Section: Introductionmentioning

confidence: 84%

Renormalization: A Quasi-shuffle Approach

Menous

Patras

2018

Computation and Combinatorics in Dynamics, Stochastics and Control

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In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semigroup (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process.

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Section: Introductionmentioning

confidence: 84%

Renormalization: A Quasi-shuffle Approach

Menous

Patras

2018

Computation and Combinatorics in Dynamics, Stochastics and Control

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“…(our hypotheses ensure that the summation in (20) has finitely-many non-zero terms and is therefore well defined). In this way H is a connected, commutative, graded Hopf algebra.…”

Section: Constructing the Hopf Algebramentioning

confidence: 99%

“…i.e. the product ⋆ in H * corresponds via duality to the coproduct (20) in H . The group of characters of H and the Lie algebra of infinitesimal characters are…”

Section: Constructing the Hopf Algebramentioning

confidence: 99%

Hopf Algebra Techniques to Handle Dynamical Systems and Numerical Integrators

Murua¹,

Sanz‐Serna

2018

Abel Symposia

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In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, universal version of it which is solved algebraically; then, the results are transferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.

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“…We briefly present the crucial properties of D which are employed in this text and further recommend in particular section 4 of [11] as well as [31]. Theorem C.9 (scattering formula from [10] in the form of [30] Observe that Ψ n (e) vanishes on any element x ∈ H of coradical degree less than n, wherefore this series is pointwise finite.…”

Section: Appendix C the Dynkin Operator D = S Ymentioning

confidence: 99%

Renormalization, Hopf algebras and Mellin transforms

Panzer¹

2015

Feynman Amplitudes, Periods and Motives

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Abstract. This article aims to give a short introduction into Hopf-algebraic aspects of renormalization, enjoying growing attention for more than a decade by now. As most available literature is concerned with the minimal subtraction scheme, we like to point out properties of the kinematic subtraction scheme which is also widely used in physics (under the names of MOM or BPHZ).In particular we relate renormalized Feynman rules φ R in this scheme to the universal property of the Hopf algebra H R of rooted trees, exhibiting a refined renormalization group equation which is equivalent to φ R :being a morphism of Hopf algebras to the polynomials in one indeterminate.Upon introduction of analytic regularization this results in efficient combinatorial recursions to calculate φ R in terms of the Mellin transform. We find that different Feynman rules are related by a distinguished class of Hopf algebra automorphisms of H R that arise naturally from Hochschild cohomology.Also we recall the known results for the minimal subtraction scheme and shed light on the interrelationship of both schemes.Finally we incorporate combinatorial Dyson-Schwinger equations to study the effects of renormalization on the physical meaningful correlation functions. This yields a precise formulation of the equivalence of the two different renormalization prescriptions mentioned before and allows for non-perturbative definitions of quantum field theories in special cases.

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Logarithmic derivatives and generalized Dynkin operators

Cited by 10 publications

References 23 publications

Renormalization: A Quasi-shuffle Approach

Renormalization: A Quasi-shuffle Approach

Hopf Algebra Techniques to Handle Dynamical Systems and Numerical Integrators

Renormalization, Hopf algebras and Mellin transforms

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