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We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator B+.submitted 29/03/04, accepted 01/06/04 相似文献
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We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated
to the NCG computations of tranverse index theory for foliations.
Received: 14 August 1998/ Accepted: 5 October 1998 相似文献
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Spencer Bloch Hélène Esnault Dirk Kreimer 《Communications in Mathematical Physics》2006,267(1):181-225
The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. 相似文献
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D.J. Broadhurst J.A. Gracey D. Kreimer 《Zeitschrift fur Physik C Particles and Fields》1997,75(3):559-574
Counterterms that are not reducible to ζ n are generated by 3 F 2 hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots (4, 3) = 819 and (5, 3) = 10124, are found in anomalous dimensions at O(1/N 3) in the large-N limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pade resummations of e-expansions, which are compared with analytical results in 3 dimensions. The O(1/N 3) results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in 1/N one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots 10139 and 10152 are not generated by such self-energy insertions. 相似文献
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We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the double Rota-Baxter construction, respectively Atkinsons theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.submitted 16/03/04, accepted 09/09/04This revised version was published online in May 2005 with correction to the addresses. 相似文献
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We discuss the structure of renormalized Feynman rules. Regarding them as maps from the Hopf algebra of Feynman graphs to ${\mathbb{C}}$ originating from the evaluation of graphs by Feynman rules, they are elements of a group ${G=\mathrm{Spec}_{\mathrm{Feyn}}(H)}$ . We study the kinematics of scale and angle-dependence to decompose G into subgroups ${G_{\mathrm{\makebox{1-s}}}}$ and ${G_{\mathrm{fin}}}$ . Using parametric representations of Feynman integrals, renormalizability and the renormalization group underlying the scale dependence of Feynman amplitudes are derived and proven in the context of algebraic geometry. 相似文献
9.
Dirk Kreimer 《Communications in Mathematical Physics》1999,204(3):669-689
Using set-theoretic considerations, we show that the forest formula for overlapping divergences comes from the Hopf algebra
of rooted trees.
Received: 14 October 1998/ Accepted: 12 February 1999 相似文献
10.
We review the algebraic structures imposed on the renormalization procedure in terms of Hopf and Lie algebras of Feynman graphs, and exhibit the connection to diffeomorphisms of physical observables. 相似文献