Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55794", author = "Ali Shojaei-Fard", title = "Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra", 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.", volume = "3", number = "11", pages = "976-6", }
