Feynman graph generation and calculations in the Hopf algebra of Feynman graphs

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

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On Kreimer's Hopf algebra structure of Feynman graphs

The European Physical Journal C ◽

10.1007/s100520050439 ◽

1999 ◽

Vol 7 (4) ◽

pp. 697 ◽

Cited By ~ 1

Author(s):

T. Krajewski ◽

R. Wulkenhaar

Keyword(s):

Hopf Algebra ◽

Algebra Structure ◽

Hopf Algebra Structure ◽

Feynman Graphs

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Automatic Feynman Graph Generation

Journal of Computational Physics ◽

10.1006/jcph.1993.1074 ◽

1993 ◽

Vol 105 (2) ◽

pp. 279-289 ◽

Cited By ~ 674

Author(s):

P. Nogueira

Keyword(s):

Feynman Graph ◽

Graph Generation

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The Hopf Algebra of Feynman Graphs in Quantum Electrodynamics

Letters in Mathematical Physics ◽

10.1007/s11005-006-0092-4 ◽

2006 ◽

Vol 77 (3) ◽

pp. 265-281 ◽

Cited By ~ 18

Author(s):

Walter D. Van Suijlekom

Keyword(s):

Hopf Algebra ◽

Quantum Electrodynamics ◽

Feynman Graphs

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FEYNMAN GRAPH RULES FOR IINN-INTERACTION AND MATRIX ELEMENTS FOR A SIMPLIFIED pπ0 → pπ0 MODEL IN 5-DIMENSIONAL CONFORMAL QFT

International Journal of Modern Physics A ◽

10.1142/s0217751x89002314 ◽

1989 ◽

Vol 04 (20) ◽

pp. 5411-5432

Author(s):

G. A. LUNA-ACOSTA

Keyword(s):

Feynman Graph ◽

Matrix Elements ◽

Hadron Spectra ◽

Feynman Graphs ◽

Matter Interaction ◽

Interesting Possibility ◽

The Matrix ◽

The Individual ◽

Energy Dependent ◽

Infinite Sum

Recently we reported a successful fitting of the kinematic predictions of the Mass Theory of 5-dimensional Conformal Relativity to the Hadron Spectra. We use results obtained there to study the dynamical properties of the trilinear boson-matter interaction Lagrangian implied by isospin SU(2) gauge invariance. The isovector massive gauge boson allows for two interpretations: a J = 1 field and the 5-D derivative of a J = 0 field. We construct the Feynman Graphs for the ΠNN-interaction and show that (1) any given graph is actually an infinite sum of similar graphs, one for each member of the Conformal Family Field of the virtual Hadron and (2) that the individual coupling of each of these members depends on its mass, giving rise to an effectively energy-dependent coupling. Based on these features, we speculate on the interesting possibility that Perturbation Theory may be usable for hadron processes and calculate the matrix elements for a simplified model of pπ0 → pπ0 to second order. Useful comparisons are made with the analogous treatment in ordinary QFT.

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Random tensor models—the multi-orientable (MO) model

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0013 ◽

2021 ◽

pp. 209-233

Author(s):

Adrian Tanasa

Keyword(s):

Feynman Graph ◽

General Term ◽

Intermediate Step ◽

Tensor Model ◽

Invariant Model ◽

Tensor Models ◽

Feynman Graphs ◽

The One ◽

Definition Of

In its first section, this chapter presents the definition of the multi-orientable tensor model. The 1/N expansion and the large N limit of this model are exposed in the second section of the chapter. In the third section, a thorough enumerative combinatorial analysis of the general term of the 1/N expansion is presented. The implementation of the double scaling mechanism is then exhibited in the fourth section. This chapter presents the multi-orientable (MO) tensor model and it follows the review article. This rank three model, having O(N) U(N) O(N) symetry, can be seen as an intermediate step between the U(N) invariant model presented in the previous chapter, and the O(N) invariant model presented in the following chapter. The class of Feynman graph generated by perturbative expansion of MO model is strictly larger than the class of Feynman graphs of the U(N) invariant model and strictly smaller than the one of the O(N) invariant model.

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