The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Application of Deformed Lie Algebras to Non-Perturbative Quantum Field Theory


Affiliations
1 Independent Scholar, Mathematician, Tehran, 1461863596, Marzdaran Blvd, Iran, Islamic Republic of
     

   Subscribe/Renew Journal


The manuscript implements Connes-Kreimer Hopf algebraic renormalization of Feynman diagrams and Dubois-Violette type noncommutative differential geometry to discover a new class of differential calculi with respect to infinite formal expansions of Feynman diagrams which are generated by Dyson-Schwinger equations.

Keywords

Hopf Algebraic Renormalization, Dyson-Schwinger Equations, Dubois-Violette Noncommutative Differential Forms, Non-Perturbative Renormalization Group.
Subscription Login to verify subscription
User
Notifications
Font Size


  • C. M. Bender, Perturbation Theory In Large Order, Adv. in Math., vol. 30 no. 3 (1978), 250-267.

  • C. Bergbauer, D. Kreimer, Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lectures in Mathematics and Theoretical Physics, 10 (2006), 133-164.

  • F. Brown, D. Kreimer, Angles, scales and parametric renormalization, Lett. Math. Phys., vol. 103 no. 9 (2013), 933-1007.

  • G. Baditoiu, S. Rosenberg, Lax pair equations and Connes-Kreimer renormalization, Commun. Math. Phys., vol. 296 no. 3 (2010), 655-680.

  • J. C. Collins, Renormalization: an introduction to renormalization, the renormalization group and the operator-product expansion, Cambridge University Press, 1986.

  • 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, Commun. Math. Phys., vol. 210 no. 1 (2000), 249-273.

  • A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The -function, diffeomorphisms and the renormalization group, Commun. Math. Phys., vol. 216 no. 1 (2001), 215-241.

  • J. F. Carinena, J. Grabowski, G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A, vol. 15 no. 30 (2000), 4797-4810.

  • A. Connes, M. Marcolli, Noncommutative geometry, quantum fields and motives, Colloquium Publications, American Mathematical Society, vol. 55 (2008).

  • A.E.F. Djemai, Introduction to Dubois-Violette's noncommutative differential geometry, Int. J. Theoret. Phys., vol. 34 no. 6 (1995), 801-887.

  • I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Nonlinear science: Theory and applications, John Wiley & Sons Ltd. (1993).

  • 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, Series: Mathematical Physics Studies, vol. 23 (2001).

  • K. Ebrahimi-Fard, On the associative Nijenhuis relation, Electron. J. Combin., vol. 11 no.1 (2004), Research Paper 38, 13 pp. (electronic).

  • K. Ebrahimi-Fard, L. Guo, Mixable shuffles, quasi-shuffles and Hopf algebras, J. Algebraic Combin., vol. 24 no. 1 (2006), 83-101.

  • K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras in renormalization of perturbative quantum field theory. universality and renormalization, Fields Inst. Commun., vol. 50 (2007), 47-105.

  • L. Foissy, Faa di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. in Math., vol. 218 (2008), 136-162.

  • L. Guo, An introduction to Rota-Baxter algebra, Surveys of modern mathematics, International Press, Somerville, MA; Higher Education Press, Beijing, Vol. 4 (2012).

  • S. Gra, V. Grecchi, B. Simon, Borel summability: application to the anharmonic oscillator, Phys. Lett. B, vol. 32B (1970), 631-634.

  • M. E. Ho man, Quasi-shuffle products, J. Algebraic Combin., vol. 11 no. 1 (2000), 49-68.

  • Z. Horvath, L. Palla, Non-perturbative QFT methods and their applications, Proceedings of the 24th Johns Hopkins Workshop, Bolyai College, Budapest, Hungary, 19-21 August 2000. World Scientific.

  • D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., vol. 2 no. 2 (1998), 303-334.

  • D. J. Broadhurst, D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Comput., vol. 27 no. 6 (1999), 581-600.

  • D. Kreimer, Structures in Feynman graphs - Hopf algebras and symmetries, Proc. Symp. Pure Math., vol. 73 (2005), 43-78.

  • D. Kreimer, Anatomy of a gauge theory, Ann. Phys., vol. 321 no. 12 (2006), 2757-2781.

  • D. Kreimer, Dyson-Schwinger equations: from Hopf algebras to number theory, Universality and renormalization, Fields Inst. Commun., vol. 50 (2007), 225-248.

  • Y. Kosmann-Schwarzbach, F. Magri, Poisson-Nijenhuis structures, Ann. Inst. Henri Poincare., vol. 53 no. 1 (1990), 35-81.

  • P.P. Kulish, E.K. Sklyanin, Solutions of the Yang-Baxter equations, J. Soviet Math., vol. 19 (1982), 1596-1620.

  • J. Madore, An introduction to noncommutative geometry and its physical applications, Cam- bridge University Press, 1999.

  • M. Marino, Non-perturbative effects and non-perturbative definitions in matrix models and topological strings, Journal of High Energy Physics, vol. 2008 (2008), JHEP12(2008).

  • A. Morozov, Integrability in non-perturbative QFT, AIP Proc., vol. 1562 (2013), 167-176.

  • M.E. Peskin, D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley Publishing Company, 1995

  • J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B, vol. 231 Issue 2 (1984), 269-295.

  • P. Ramond, Field Theory : A Modern Primer (Frontiers in Physics Series, Vol 74), Westview Press, 2001

  • M. Sakakibara, On the differential equations of the characters for the renormalization group, Mod. Phys. Lett. A., vol. 19 no. 19 (2004), 1453-1456.

  • M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl., vol. 17 (1983), 259-272.

  • M.A. Semenov-Tian-Shansky, Integrable Systems and factorization problems. Factorization and integrable systems, Oper. Theory Adv. Appl., vol. 141 (2003), 155-218.

  • A. Shojaei-Fard, From Dyson-Schwinger equations to the Riemann-Hilbert correspondence, Int. J. Geom. Methods Mod. Phys., vol. 7 no. 4 (2010), 519-538.

  • A. Shojaei-Fard, Riemann-Hilbert problem and Quantum Field Theory: integrable renormalization, Dyson-Schwinger equations, Monograph, Lambert Academic Publishing, ISBN 978-3-8473-4067-6, (2012), 136 pages.

  • A. Shojaei-Fard, What is a quantum equation of motion?, Electronic Journal of Theoretical Physics, vol. 10 no. 28 (2013), 1-8.

  • A. Shojaei-Fard, A geometric perspective on counterterms related to Dyson-Schwinger equations, Internat. J. Modern Phys. A, vol. 28 no. 32 (2013), 1350170 (15 pages).

  • A. Shojaei-Fard, The global beta-functions from solutions of Dyson-Schwinger equations, Modern Phys. Lett. A, vol. 28 no. 34 (2013), 1350152 (12 pages).

  • T. Schafer, E. V. Shuryak, Instantons in QCD, Rev. Modern Phys., vol. 70 no. 2 (1998), 323-425.

  • K. Yeats, Rearranging Dyson-Schwinger Equations, Memoirs of the American Mathematical Society, vol. 211 (2011).


Abstract Views: 502

PDF Views: 1




  • Application of Deformed Lie Algebras to Non-Perturbative Quantum Field Theory

Abstract Views: 502  |  PDF Views: 1

Authors

Ali Shojaei-Fard
Independent Scholar, Mathematician, Tehran, 1461863596, Marzdaran Blvd, Iran, Islamic Republic of

Abstract


The manuscript implements Connes-Kreimer Hopf algebraic renormalization of Feynman diagrams and Dubois-Violette type noncommutative differential geometry to discover a new class of differential calculi with respect to infinite formal expansions of Feynman diagrams which are generated by Dyson-Schwinger equations.

Keywords


Hopf Algebraic Renormalization, Dyson-Schwinger Equations, Dubois-Violette Noncommutative Differential Forms, Non-Perturbative Renormalization Group.

References