Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

  • Michael Dütsch
  • Karl-Henning Rehren
Open Access


The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field \({\varphi}\) on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field \({\phi}\) on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS4 by differential renormalization, and calculate the anomalous dimension of the composite boundary field \({\varphi^2}\) with bulk interaction \({\kappa \phi^4}\).


Anomalous Dimension Conformal Symmetry Boundary Limit Adiabatic Limit Renormalization Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



MD profitted from discussions with Günter Scharf and Raymond Stora during an early stage of this work. Extensive discussions with Klaus Fredenhagen clarified many conceptual issues. We thank the anonymous referee for insisting, by his very detailed and qualified inquiries, onmore detailed explanations in Sect. 2.3, and for raising the interesting issue of the structure of the OPE.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Abramovitz M., Stegun I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Avis S.J., Isham C.J., Storey D.: Quantum field theory in anti-De Sitter space-time. Phys. Rev. D 18, 3565–3576 (1978)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Bertola M., Bros J., Moschella U., Schaeffer R.: A general construction of conformal field theories from scalar anti-de Sitter quantum field theories. Nucl. Phys. B 587, 619–644 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Bertola M., Bros J., Gorini V., Moschella U., Schaeffer R.: Decomposing quantum fields on branes. Nucl. Phys. B 581, 575–603 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Bros J., Epstein H., Moschella U.: Towards a general theory of quantized fields on the anti-de Sitter space-time. Commun. Math. Phys. 231, 481–528 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Dütsch M., Fredenhagen K.: A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203, 71–105 (1999)ADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Dütsch M., Fredenhagen K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5–30 (2001)ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Dütsch M., Fredenhagen K.: Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity. Rev. Math. Phys. 16, 1291–1348 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dütsch M., Rehren K.-H.: A comment on the dual field in the AdS-CFT correspondence. Lett. Math. Phys. 62, 171–184 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dütsch M., Rehren K.-H.: Generalized free fields and the AdS-CFT correspondence. Ann. Henri Poincaré 4, 613–635 (2003)zbMATHCrossRefGoogle Scholar
  12. 12.
    Epstein H.: On the Borchers class of a free field. Nuovo Cim. 27, 886–893 (1963)zbMATHCrossRefGoogle Scholar
  13. 13.
    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19, 211–295 (1973)MathSciNetGoogle Scholar
  14. 14.
    Freedman D.Z., Johnson K., Latorre J.I.: Differential regularization and renormalization: a new method of calculation in quantum field theory. Nucl. Phys. B 371, 353–414 (1992)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Fronsdal C.: Elementary particles in a curved space. II. Phys. Rev. D 10, 589–598 (1974)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Hollands S., Wald R.M.: Local Wick polynomials and time-ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Hollands S., Wald R.M.: Existence of local covariant time-ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Hollands S., Wald R.M.: Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227–312 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Källen G.: Formal integration of the equations of quantum theory in the Heisenberg representation. Ark. Fysik 2, 371–410 (1950)Google Scholar
  21. 21.
    Lewin L.: Polylogarithms and Associated Functions. Elsevier North Holland, Amsterdam (1981)zbMATHGoogle Scholar
  22. 22.
    Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998)MathSciNetADSzbMATHGoogle Scholar
  23. 23.
    Moretti V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189–221 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Rehren K.-H.: Algebraic holography. Ann. Henri Poincaré 1, 607–623 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Rehren K.-H.: Local quantum observables in the AdS-CFT correspondence. Phys. Lett. B 493, 383–388 (2000)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Rehren, K.-H.: QFT lectures on AdS-CFT. In: Proceedings of the 3rd Summer School in Modern Mathematical Physics, Zlatibor, Serbia (2004), B. Dragovich (ed.), Belgrade, 2005, pp. 95–118Google Scholar
  27. 27.
    Rühl W.: Lifting a conformal field theory from D-dimensional flat space to (D + 1)-dimensional AdS space. Nucl. Phys. B705, 437–456 (2005)ADSCrossRefGoogle Scholar
  28. 28.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge Univ. Press, 1958 (2nd edition)Google Scholar
  29. 29.
    Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany
  2. 2.Courant Research Centre “Higher Order Structures in Mathematics”Universität GöttingenGöttingenGermany

Personalised recommendations