Skip to main content
SpringerLink
Log in

Systems of Linear Dyson–Schwinger Equations

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

Systems of Dyson–Schwinger equation represent the equations of motion in quantum field theory. In this paper, we follow the combinatorial approach and consider Dyson–Schwinger equations as fixed point equations that determine the perturbation series by usage of graph insertion operators. We discuss their properties under the renormalization flow, prove that fixed points are scheme independent, and construct solutions for coupled systems with linearized arguments of the insertion operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kreimer, D.: Anatomy of a gauge theory. Annals Phys. 321, 2757–2781 (2006). arXiv:hep-th/0509135, https://doi.org/10.1016/j.aop.2006.01.004

    Article  ADS  MathSciNet  Google Scholar 

  2. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys 2, 303–334 (1998). arXiv:q-alg/9707029

    Article  MathSciNet  Google Scholar 

  3. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. 1. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000). arXiv:hep-th/9912092, https://doi.org/10.1007/s002200050779

    Article  ADS  MathSciNet  Google Scholar 

  4. Kreimer, D., Yeats, K.: An Etude in non-linear Dyson-Schwinger Equations. Nucl. Phys. Proc. Suppl 160, 116–121 (2006). arXiv:hep-th/0605096. https://doi.org/10.1016/j.nuclphysbps.2006.09.036

    Article  ADS  Google Scholar 

  5. Kreimer, D., Yeats, K.: Recursion and growth estimates in renormalizable quantum field theory. Commun. Math. Phys 279, 401–427 (2008). arXiv:hep-th/0612179, https://doi.org/10.1007/s00220-008-0431-7

    Article  ADS  MathSciNet  Google Scholar 

  6. Foissy, L.: Systems of Dyson-Schwinger equations, arXiv:0909.0358 (2009)

  7. Foissy, L.: General Dyson–Schwinger equations and systems. Commun. Math. Phys 327, 151–179 ((2014)). arXiv:1112.2606, https://doi.org/10.1007/s00220-014-1941-0

    Article  ADS  MathSciNet  Google Scholar 

  8. Foissy, L.: Mulitgraded Dyson-Schwinger systems, arXiv:1511.06859 (2015)

  9. Kreimer, D.: Etude for linear dyson-schwinger equations, Tech. Rep. IHES-P-2006-23, Inst. Hautes Etud. Sci., Bures-sur-Yvette (2006)

  10. Marie, N., Yeats, K.: A chord diagram expansion coming from some Dyson-Schwinger equations. Commun. Num. Theor. Phys 07, 251–291 (2013). arXiv:1210.5457, https://doi.org/10.4310/CNTP.2013.v7.n2.a2

    Article  MathSciNet  Google Scholar 

  11. Hihn, M., Yeats, K.: Generalized chord diagram expansions of Dyson-Schwinger equations, arXiv:1602.02550

  12. Kißler, H.: Hopf-algebraic Renormalization of QED in the linear covariant Gauge. Annals Phys 372, 159–174 (2016). arXiv:1602.07003, https://doi.org/10.1016/j.aop.2016.05.008

    Article  ADS  MathSciNet  Google Scholar 

  13. Kißler, H.: Computational and diagrammatic techniques for perturbative quantum electrodynamics. Ph.D. thesis, Humboldt-Universität zu Berlin (2017)

  14. Bergbauer, C., Kreimer, D.: Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations from Hochschild cohomology. IRMA Lect. Math. Theor. Phys 10, 133–164 (2006). arXiv:hep-th/0506190, https://doi.org/10.4171/028-1/4

    MathSciNet  MATH  Google Scholar 

  15. van Suijlekom, W.D.: Renormalization of gauge fields: a Hopf algebra approach. Commun. Math. Phys 276, 773–798 (2007). arXiv:hep-th/0610137, https://doi.org/10.1007/s00220-007-0353-9

    Article  ADS  MathSciNet  Google Scholar 

  16. Collins, J.C.: Renormalization, vol. 26 of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  17. Brown, F., Kreimer, D.: Angles, Scales and Parametric Renormalization. Lett. Math. Phys 103, 933–1007 (2013). arXiv:1112.1180, https://doi.org/10.1007/s11005-013-0625-6

    Article  ADS  MathSciNet  Google Scholar 

  18. Broadhurst, D.J., Kreimer, D.: Combinatoric explosion of renormalization tamed by Hopf algebra: thirty loop Pade-Borel resummation. Phys. Lett. B475, 63–70 (2000). arXiv:hep-th/9912093, https://doi.org/10.1016/S0370-2693(00)00051-4

    Article  ADS  Google Scholar 

  19. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. 2. The beta function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215–241 (2001). arXiv:hep-th/0003188, https://doi.org/10.1007/PL00005547

    Article  ADS  Google Scholar 

  20. Manchon, D.: Hopf algebras, from basics to applications to renormalization. In: 5th Mathematical Meeting of Glanon: Algebra, Geometry and Applications to Physics Glanon, Burgundy, France, July 2-6 2001. arXiv:math/0408405(2001)

  21. Gracey, J.A.: Six dimensional QCD at two loops. Phys. Rev. D93(2), 025025 (2016). arXiv:1512.04443, https://doi.org/10.1103/PhysRevD.93.025025

    ADS  MathSciNet  Google Scholar 

  22. Gracey, J.A., Simms, R.M.: Higher dimensional higher derivative ϕ 4 theory. Phys. Rev. D96 (2), 025022 (2017). arXiv:1705.06983, https://doi.org/10.1103/PhysRevD.96.025022

    ADS  MathSciNet  Google Scholar 

  23. Gracey, J.A.: Renormalization of scalar field theories in rational spacetime dimensions, arXiv:1703.09685

  24. Gracey, J.A., Simms, R.M.: Six dimensional Landau-Ginzburg-Wilson theory. Phys. Rev. D95(2), 025029 (2017). arXiv:1701.03618, https://doi.org/10.1103/PhysRevD.95.025029

    ADS  MathSciNet  Google Scholar 

  25. Kißler, H.: On linear systems of Dyson–Schwinger equations, master’s thesis, Humboldt-Universität zu Berlin (2012)

Download references

Acknowledgments

The author likes to thank Dirk Kreimer, Karen Yeats, Eric Panzer, and Marco Berghoff for many valuable discussions pertaining to this work. It is acknowledged that parts of the results of this work were discussed in the master’s thesis by the author of this paper [25]. This research was supported by Deutsche Forschungsgemeinschaft (DFG) through the grant KR 1401/5-1.

Author information

Authors and Affiliations

  1. Humboldt-Universität zu Berlin, Berlin, Germany

    Henry Kißler

Authors
  1. Henry Kißler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henry Kißler.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kißler, H. Systems of Linear Dyson–Schwinger Equations. Math Phys Anal Geom 22, 20 (2019). https://doi.org/10.1007/s11040-019-9320-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-019-9320-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation