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Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

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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 occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.

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  1. Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 route de Chartres, F-91440, Bures-sur-Yvette, France

    Alain Connes & Dirk Kreimer

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  1. Alain Connes
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  2. Dirk Kreimer
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Connes, A., Kreimer, D. Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries. Letters in Mathematical Physics 48, 85–96 (1999). https://doi.org/10.1023/A:1007523409317

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