Skip to main content
SpringerLink
Log in

From Slavnov—Taylor Identities to the Renormalization of Gauge Theories

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

An important, and highly non-trivial, problem is proving the renormalizability and unitarity of quantized non-Abelian gauge theories. Lee and Zinn-Justin have given the first proof of the renormalizability of non-Abelian gauge theories in the spontaneously broken phase. An essential ingredient in the proof has been the observation, by Slavnov and Taylor, of a non-linear, non-local symmetry of the quantized theory, a direct consequence of Faddeev and Popov’s quantization procedure. After the introduction of non-physical fermions to represent the Faddeev-Popov determinant, this symmetry has led to the Becchi-Rouet-Stora-Tyutin fermionic symmetry of the quantized action and, finally, to the resulting Zinn-Justin equation, which makes it possible to solve the renormalization and unitarity problems in their full generality. For an elementary introduction to the discussion of quantum non-Abelian gauge field theories in the spirit of the article, see, for example, L. D. Faddeev, “Faddeev-Popov ghosts,” Scholarpedia 4 (4), 7389 (2009); A. A. Slavnov, “Slavnov-Taylor identities,” Scholarpedia 3 (10), 7119 (2008); C. M. Becchi and C. Imbimbo, “Becchi-Rouet-Stora-Tyutin symmetry,” Scholarpedia 3 (10), 7135 (2008); J. Zinn-Justin, “Zinn-Justin equation,” Scholarpedia 4 (1), 7120 (2009).

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. G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in the antifield formalism. I: General theorems,” Commun. Math. Phys. 174 (1), 57–91 (1995).

    MathSciNet  MATH  Google Scholar 

  2. G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in the antifield formalism. II: Application to Yang-Mills theory,” Commun. Math. Phys. 174 (1), 93–116 (1995).

    MathSciNet  MATH  Google Scholar 

  3. G. Barnich and M. Henneaux, “Renormalization of gauge invariant operators and anomalies in Yang-Mills theory,” Phys. Rev. Lett. 72 (11), 1588–1591 (1994).

    MathSciNet  MATH  Google Scholar 

  4. C. Becchi, A. Rouet, and R. Stora, “Renormalization of the abelian Higgs-Kibble model,” Commun. Math. Phys. 42 (2), 127–162 (1975).

    MathSciNet  Google Scholar 

  5. C. Becchi, A. Rouet, and R. Stora, “Renormalization of gauge theories,” Ann. Phys. 98 (2), 287–321 (1976).

    MathSciNet  Google Scholar 

  6. F. A. Berezin, The Method of Second Quantization (Nauka, Moscow, 1965; Academic, New York, 1966).

    MATH  Google Scholar 

  7. L. D. Faddeev and V. N. Popov, “Feynman diagrams for the Yang-Mills field,” Phys. Lett. B 25 (1), 29–30 (1967).

    Google Scholar 

  8. H. Kluberg-Stern and J. B. Zuber, “Ward identities and some clues to the renormalization of gauge-invariant operators,” Phys. Rev. D 12 (2), 467–481 (1975).

    Google Scholar 

  9. B. W. Lee and J. Zinn-Justin, “Spontaneously broken gauge symmetries. I: Preliminaries,” Phys. Rev. D 5 (12), 3121–3137 (1972).

    Google Scholar 

  10. B. W. Lee and J. Zinn-Justin, “Spontaneously broken gauge symmetries. II: Perturbation theory and renormalization,” Phys. Rev. D 5 (12), 3137–3155 (1972).

    Google Scholar 

  11. B. W. Lee and J. Zinn-Justin, “Spontaneously broken gauge symmetries. III: Equivalence,” Phys. Rev. D 5 (12), 3155–3160 (1972).

    Google Scholar 

  12. B. W. Lee and J. Zinn-Justin, “Spontaneously broken gauge symmetries. IV: General gauge formulation,” Phys. Rev. D 7 (4), 1049–1056 (1973).

    Google Scholar 

  13. V. N. Popov and L. D. Faddeev, “Perturbation theory for gauge-invariant fields,” Preprint ITP 67-36 (Inst. Theor. Phys. Acad. Sci. Ukr. SSR, Kiev, 1967). Engl. transl.: Preprint NAL-THY-57 (Natl. Accelerator Lab., Batavia, IL, 1972); repr. in 50 Years of Yang-Mills Theory, Ed. by G.’ t Hooft (World Scientific, Singapore, 2005), pp. 40–60.

  14. Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions, Ed. by C. H. Lai (World Scientific, Singapore, 1981).

    Google Scholar 

  15. A. A. Slavnov, “Ward identities in gauge theories,” Theor. Math. Phys. 10 (2), 99–104 (1972) [transl. from Teor. Mat. Fiz. 10 (2), 153–161 (1972)].

    Google Scholar 

  16. A. A. Slavnov, “Invariant regularization of gauge theories,” Theor. Math. Phys. 13 (2), 1064–1066 (1972) [transl. from Teor. Mat. Fiz. 13 (2), 174–177 (1972)].

    Google Scholar 

  17. A. A. Slavnov and L. D. Faddeev, Introduction to Quantum Theory of Gauge Fields (Nauka, Moscow, 1988). Engl. transl.: L. D. Faddeev and A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory (Addison-Wesley, Redwood City, CA, 1991), Front. Phys. 83.

    MATH  Google Scholar 

  18. J. C. Taylor, “Ward identities and charge renormalization of the Yang-Mills field,” Nucl. Phys. B 33 (2), 436–444 (1971).

    MathSciNet  Google Scholar 

  19. I. V. Tyutin, “Gauge invariance in field theory and statistical physics in operator formalism,” arXiv: 0812.0580 [hep-th].

  20. J. Zinn-Justin, “Renormalization of gauge theories,” in Trends in Elementary Particle Theory, Ed. by H. Rollnik and K. Dietz (Springer, Berlin, 1975), Lect. Notes Phys. 37, pp. 1–39.

    Google Scholar 

  21. J. Zinn-Justin, “Renormalization problems in gauge theories,” in Functional and Probabilistic Methods in Quantum Field Theory: Proc. 12th Winter School of Theoretical Physics in Karpacz, 1975 (Wydawn. Uniw. Wrocław., Wrocław, 1976), Vol. 1, Acta Univ. Wratislav. 368, pp. 433–453.

    Google Scholar 

  22. J. Zinn-Justin, “Renormalization and stochastic quantization,” Nucl. Phys. B 275 (1), 135–159 (1986).

    MathSciNet  Google Scholar 

  23. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, Oxford, 1989), Ch. 16; 4th ed. (Oxford Univ. Press, Oxford, 2002), Int. Ser. Monogr. Phys. 113.

    MATH  Google Scholar 

  24. J. Zinn-Justin, “Renormalization of gauge theories and master equation,” Mod. Phys. Lett. A 14 (19), 1227–1235 (1999).

    MathSciNet  Google Scholar 

  25. J. Zinn-Justin, “From Slavnov-Taylor identities to the ZJ equation,” Proc. Steklov Inst. Math. 272, 288–292 (2011).

    MathSciNet  MATH  Google Scholar 

  26. J. Zinn-Justin and R. Guida, “Gauge invariance,” Scholarpedia 3 (12), 8287 (2008).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Research into the Fundamental Laws of the Universe (IRFU/CEA), Paris-Saclay University, 91191, Gif-sur-Yvette Cedex, France

    Jean Zinn-Justin

  2. French Academy of Sciences, 23 quai de Conti, 75006, Paris, France

    Jean Zinn-Justin

Authors
  1. Jean Zinn-Justin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Zinn-Justin.

Additional information

In honour of the 80th birthday of my very esteemed colleague and friend A. A. Slavnov

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Vol. 309, pp. 338–345.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zinn-Justin, J. From Slavnov—Taylor Identities to the Renormalization of Gauge Theories. Proc. Steklov Inst. Math. 309, 317–324 (2020). https://doi.org/10.1134/S0081543820030232

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543820030232

Search

Navigation