Letters in Mathematical Physics

, Volume 103, Issue 2, pp 171–181 | Cite as

Field Diffeomorphisms and the Algebraic Structure of Perturbative Expansions

  • Dirk Kreimer
  • Andrea Velenich


We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree-level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.

Mathematics Subject Classification (2010)

81S99 81T99 


diffeomorphism invariance Hopf ideals BCFW relations tadpoles renormalization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bern, Z.: Perturbative quantum gravity and its relation to Gauge theory. Living Rev. Relativ. 5, 5 (2002, cited on 03/01/12).
  2. 2.
    Nakai S.: Point transformation and its application. Prog. Theor. Phys. 13, 380 (1955)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Chisholm J.S.R.: Change of variables in quantum field theories. Nucl. Phys. 26, 469–479 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kamefuchi S., O’Raifeartaigh L., Salam A.: Change of variables and equivalence theorems in quantum field theories. Nucl. Phys. 28, 529–549 (1961)MathSciNetGoogle Scholar
  5. 5.
    Salam A., Strathdee J.: Equivalent formulations of massive vector field theories. Phys. Rev. D 2, 2869 (1970)ADSCrossRefGoogle Scholar
  6. 6.
    Keck B.W., Taylor J.G.: On the equivalence theorem for S-matrix elements. J. Phys. A Gen. Phys. 4, 291 (1971)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Suzuki T., Hattori C.: Relativistically covariant formulation in non-linear Lagrangian theories and factor ordering problems. Prog. Theor. Phys. 47, 1722 (1972)ADSCrossRefGoogle Scholar
  8. 8.
    Suzuki T., Hirshfeld A.C., Leschke H.: The role of operator ordering in quantum field theory. Prog. Theor. Phys. 63, 287 (1980)ADSCrossRefGoogle Scholar
  9. 9.
    Weinberg S.: The Quantum Theory of Fields, vol. 1, Chap. 7.7. Cambridge University Press, Cambridge (1995)Google Scholar
  10. 10.
    de Castro A.S.: Point transformations are canonical transformations. Eur. J. Phys. 20, L11 (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Apfeldorf K.M., Camblong H.E., Ordóñez C.R.: Field redefinition invariance in quantum field theory. Mod. Phys. Lett. A 16, 103 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    Weinberg S.: Quantum contributions to cosmological correlations. Phys. Rev. D 72, 043514 (2005)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Kreimer, D.: The core Hopf algebra. In: Proceedings for Alain Connes’ 60th birthday, in “Quanta of Maths”. Clay Mathematics Proceedings, vol. 11 (2011), pp. 313–322 (2009)Google Scholar
  14. 14.
    Kreimer D., van Suijlekom W.D.: Recursive relations in the core Hopf algebra. Nucl. Phys. B 820, 682 (2009)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Kreimer, D.: Not so non-renormalizable gravity. In: Fauser, B., Tolksdorf, J., Zeidlers, E. (eds.) Published in Quantum Field Theory: Competitive Models. Birkhauser (2009). Also in Leipzig 2007, Quantum field theory, pp. 155–162 (2008)Google Scholar
  16. 16.
    Cachazo, F., Mason, L., Skinner, D.: Gravity in Twistor Space and its Grassmannian Formulation. arXiv:1207.4712 (2012)Google Scholar
  17. 17.
    Kreimer D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Britto R., Cachazo F., Feng B.: New Recursion Relations for Tree Amplitudes of Gluons. Nucl. Phys. B 715, 499 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Britto R., Cachazo F., Feng B., Witten E.: Direct proof of tree-level recursion relation in Yang–Mills theory. Phys. Rev. Lett. 94, 181602 (2005)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Brown, F., Kreimer, D.: Angles, Scales and Parametric Renormalization. arXiv: 1112.1180 (2011)Google Scholar
  21. 21.
    Kreimer D.: A remark on quantum gravity. Ann. Phys. 323, 49 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Cutkosky R.E.: Singularities and discontinuities of Feynman amplitudes. J. Math. Phys. 1, 429 (1960)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Departments of Physics and of MathematicsHumboldt UniversityBerlinGermany
  2. 2.Department of PhysicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations