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Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra

Authors: Ali Shojaei-Fard


In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and Baker-Campbell-Hausdorff (BCH) formula, we show that how motion integral condition on components of Birkhoff factorization of a Feynman rules character on Connes- Kreimer Hopf algebra of rooted trees can determine a family of fixed point equations.

Keywords: Birkhoff Factorization, Connes-Kreimer Hopf Algebra of Rooted Trees, Integral Renormalization, Lax Pair Equation, Rota- Baxter Algebras

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[1] C. Bergbauer, D. Kreimer, Hopf algebras in renormalization theory: Locality and Dyson-Schinger equations from Hochschild cohomology, IRMA Lect. Math. Theor. Phys., 10, 133-164, 2006.
[2] G. Baditoiu, S. Rosenberg, Feynman diagrams and Lax pair equations, arXiv:math-ph/0611014v1, 2006.
[3] J.F. Carinena, J. Grabowski, G. Marmo, Quantum Bi-Hamiltonian systems, International Journal of Modern Physics A, 15, No.30, 4797-4810, 2000.
[4] A. Connes, D. Kreimer D, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199, 203-242, 1998.
[5] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 210, No.1, 249-273, 2000.
[6] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 216, No.1, 215-241, 2001.
[7] A. Connes, M. Marcolli, Renormalization, the Riemann-Hilbert correspondence and motivic Galois theory, Frontiers in number theory, physics and geometry. II, 617-713, Springer, Berlin, 2007.
[8] V.G. Drinfel-d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Doklady 27, 68-71, 1983.
[9] K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras in Renormalization of Perturbative Quantum Field Theory. Universality and renormalization, Fields Inst. Commun., 50, 47-105, 2007.
[10] K. Ebrahimi-Fard, L. Guo, D. Kreimer, Integrable renormalization I: the ladder case, J. Math. Phys., 45, No.10, 3758-3769, 2004.
[11] K. Ebrahimi-Fard, L. Guo L, D. Kreimer, Integrable renormalization II: the general case, Ann. Henri Poincare, 6, No.2, 369-395, 2005.
[12] V. Ginzburg, Lectures on Noncommutative Geometry, arXiv:math.AG/0506603 v1, 2005.
[13] L. Guo, Algebraic Birkhoff decomposition and its application, International school and conference of noncommutative geometry, China 2007, arXiv:0807.2266v1.
[14] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., No.2, 303-334, 1998.
[15] D. Kreimer, Renormalization automated by Hopf algebra, J. Symb. Comput., 27 (1999), 581.
[16] D. Kreimer, Structures in Feynman graphs-Hopf algebras and symmetries, Proc. Symp. Pure Math., 73, 43-78, 2005.
[17] D. Kreimer, Anatomy of a gauge theory, Annals Phys., 321, 2757-2781, 2006.
[18] Y. Kosmann-Schwarzbach, F. Magri, Poisson-Nijenhuis structures, Ann. Inst. Henri Poincare, Vol. 53, No. 1, 35-81, 1990.
[19] P.P. Kulish, E.K. Sklyanin, Solutions of the Yang-Baxter equations, J. Soviet Math. 19, 1596-1620, 1982.
[20] M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl. 17, 259-272, 1984.
[21] M.A. Semenov-Tian-Shansky , Integrable Systems and factorization problems. Factorization and integrable systems, Oper. Theory Adv. Appl., 141, 155-218, 2003.
[22] M. Sakakibara, On the differential equations of the characters for the renormalization group, Modern Phys. Lett. A, 19, 1453-1456, 2004.
[23] M. Dubois-Violette, Some aspects of noncommutative differential geometry, ESI-preprint, L.P.T.H.E.-ORSAY 95/78, 1995.
[24] M. Dubois-Violette, Lectures on graded differential algebras and noncommutative geometry, Proceedings of the workshop on noncommutative differential geometry and its application to physics, Shonan-Kokusaimura, 1999.
[25] W.D. van Suijlekom, Hopf algebra of Feynman graphs for gauge theories, Conference quantum fields, periods and polylogarithms II, IHES, June 2009.