Action Ward Identity and the Stückelberg-Petermann Renormalization Group

  • Michael Dütsch
  • Klaus Fredenhagen
Part of the Progress in Mathematics book series (PM, volume 251)


A fresh look at the renormalization group (in the sense of Stückelberg-Petermann) from the point of view of algebraic quantum field theory is given, and it is shown that a consistent definition of local algebras of observables and of interacting fields in renormalized perturbative quantum field theory can be given in terms of retarded products. The dependence on the Lagrangian enters this construction only through the classical action. This amounts to the commutativity of retarded products with derivatives, a property named Action Ward Identity by Stora.


Renormalization Group Ward Identity Formal Power Series Local Functional Local Algebra 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N.N. Bogoliubov and D.V. Shirkov: Introduction to the Theory of Quantized Fields. Interscience, New York, 1959.Google Scholar
  2. 2.
    J. Bros and D. Buchholz: Towards a Relativistic KMS Condition. Nucl. Phys. B 429:291–318 (1994).ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Brunetti, M. Dütsch and K. Fredenhagen: work in progress.Google Scholar
  4. 4.
    R. Brunetti and K. Fredenhagen: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208:623 (2000).ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. Brunetti, K. Fredenhagen and R. Verch: The generally covariant locality principle — A new paradigm for local quantum physics. Commun. Math. Phys. 237:31–68.Google Scholar
  6. 6.
    D. Buchholz, I. Ojima and H. Roos: Thermodynamic Properties of Non-Equilibrium States in Quantum Field Theory. Annals Phys. 297:219–242 (2002).ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. Dütsch and K. Fredenhagen: Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion. Commun. Math. Phys. 219:5 (2001).ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. Dütsch and K. Fredenhagen: The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory. Commun. Math. Phys. 243:275–314 (2003).ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Dütsch and K. Fredenhagen: Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity. To appear in Rev. Math. Phys. Google Scholar
  10. 10.
    M. Dütsch and F.-M. Boas: The Master Ward Identity. Rev. Math. Phys. 14:977–1049 (2002).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    H. Epstein and V. Glaser: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19:211 (1973).MathSciNetMATHGoogle Scholar
  12. 12.
    V. Glaser, H. Lehmann and W. Zimmermann: Field Operators and Retarded Functions. Nuovo Cimen. 6:1122 (1957).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    R. Haag: Local Quantum Physics: Fields, particles and algebras. Springer-Verlag, Berlin, 2nd ed., 1996.CrossRefMATHGoogle Scholar
  14. 14.
    S. Hollands and R.M. Wald: Local Wick Polynomials and Time-Ordered-Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 223:289 (2001).ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    S. Hollands and R.M. Wald: Existence of Local Covariant Time-Ordered-Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 231:309–345 (2002).ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    S. Hollands and R.M. Wald: On the Renormalization Group in Curved Spacetime. Commun. Math. Phys. 237:123–160 (2003).ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    S. Hollands and R.M. Wald: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. [arXiv gr-qc/0404074].Google Scholar
  18. 18.
    P. Marecki: Quantum Electrodynamics on background external fields. [arXiv hep-th/0312304].Google Scholar
  19. 19.
    G. Pinter: Finite Renormalizations in the Epstein-Glaser Framework and Renormalization of the S-Matrix of Φ 4-Theory. Ann. Phys. (Leipzig) 10:333 (2001).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    G. Scharf: Finite Quantum Electrodynamics. The causal approach. Springer-Verlag, 2nd ed., 1995.Google Scholar
  21. 21.
    G. Scharf: Quantum Gauge Theories — A True Ghost Story. John Wiley and Sons, 2001.Google Scholar
  22. 22.
    O. Steinmann: Perturbation expansions in axiomatic field theory. Lecture Notes in Physics 11, Springer-Verlag, Berlin, Heidelberg New York, 1971.Google Scholar
  23. 23.
    G. Popineau and R. Stora: A pedagogical remark on the main theorem of perturbative renormalization theory. Unpublished preprint, 1982.Google Scholar
  24. 24.
    R. Stora: Pedagogical Experiments in Renormalized Perturbation Theory. Contribution to the conference Theory of Renormalization and Regularization, Hesselberg, Germany (2002). and private communication.Google Scholar
  25. 25.
    E.C.G. Stueckelberg and A. Petermann: La normalisation des constantes dans la theorie des quanta. Helv. Phys. Acta 26:499–520 (1953).MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 2007

Authors and Affiliations

  • Michael Dütsch
    • 1
  • Klaus Fredenhagen
    • 2
  1. 1.Institut für Theoretische PhysikUniversität ZürichZürichSwitzerland
  2. 2.Institut für Theoretische PhysikUniversität HamburgHamburgGermany

Personalised recommendations