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Lessons from quantum field theory: Hopf algebras and spacetime geometries. (English) Zbl 0965.81046

In this survey letter the authors present their belief that in some way the true geometry of spacetime is actually dictated by quantum field theories as currently used by physicists in the calculation of radiative corrections. The main role plays the Hopf algebra of renormalization, i.e. the Hopf algebra of rooted trees decorated by Feynman diagrams. The antipode in this Hopf algebra generates the counterterms necessary for perturbative renormalization. The frequent occurrence of this or related Hopf algebras in mathematics provide surprising coincidences. Explicitly discussed are foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. The Butcher group is presented as a unifying structure in numerical integration of ordinary differential equations and applied in quantum field theory. The paper provides a thoroughly written access to the topic.

MSC:

81T15 Perturbative methods of renormalization applied to problems in quantum field theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
46L87 Noncommutative differential geometry
58B34 Noncommutative geometry (à la Connes)
81T18 Feynman diagrams
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