Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra

Authors: Ali Shojaei-Fard

Abstract:

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.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1063427

References:

[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.