Renormalization of Gauge Theories

Becchi, C.; Rouet, A.; Stora, R.

Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Exposés de : H. Araki, H.J. Borchers, J.P. Ferrier, P. Krée, J.F. Pommaret, D. Ruelle, R. Stora et A. Voros, Tome 22 (1975), Exposé no. 10, 57 p.

[1] C. Becchi, A. Rouet, R. Stora The Abelian Higgs Kibble Model, Unitarity of the S Operator Phys. Lett. 52B, 344 (1974).

C. Becchi, A. Rouet, R. Stora Renormalization of the Abelian Higgs Kibble Model Commun. math. Phys., to be published.

[2] W. Zimmermann Ann. Phys. 77, 536 (1973) | MR

W. Zimmermann Ann. Phys. 77, 570 (1973) | MR

[3] J. H. Lowenstein Commun. math. Phys. 24, 1 (1971) | MR | Zbl

[4] Y. M. P. Lam Phys. Rev. D6, 2145 (1972)

Y. M. P. Lam Phys. Rev. D7, 2943 (1973)

M. Bergere, Y. M. P. Lam to be published

[5] H. Epstein, V. Glaser : Ann. Inst. Henri Poincaré, XIX, n° 3, p. 211 (1973) | Numdam | MR

H. Epstein, V. Glaser : in Statistical Mechanics and Quantum Field Theory. Les Houches Summer School of Theoretical Physics, 1970, Gordon and Breach New York 1971, C. de Witt, R. Stora Ed. | Zbl

H. Epstein, V. Glaser : Colloquium on Renormalization Theory, CNRS, Centre de Physique Théorique, Marseille Juno 1971, CERN TH 1344, June 1971

[6] M. Gomes, J. H. Lowenstein Phys. Rev. D7, 550 (1973)

[7] J. H. Lowenstein, B. Schroer Phys. Rev. D7. 1929 (1973)

C. Becchi Commun. math. Phys. 33-97 (1973) | MR

[8] A complete bibliography about the theory of gauge fields can be found for instance in : E. S. Abers, B. V. Lee Phys. Reports 9C , n° 1 (Nov. 1973)

M. Veltman Invited talk presented at the International Symposium on Electron and Photon Interactions at High Energies, Bonn 2 / - 31, August 1973.

M. Veltman The algebraic structure referred to here is however along the lines of [1].

M. Veltman The results of the présent paper have been announced by C. Becchi, A. Rouet, R. Stora, CNRS Centre de Physique Théorique, Marseille, Colloquium on Recent Progress in Lagrangian Field Theory, June 1974 and summarized in

J. Iliopoulos Progress in gauge théories, Rapporteur's talk, 17th International Conference on High Energy Physics, London 1974.

J. Iliopoulos The symmetry property described in [1] is applied to traditional renoraalization procédures in :

J. Zinn Justin Lectures given at the International Summer Institute for Theoretical Physics, Bonn 1974.

[9] L. C. Biedenharn in Colloquium on Group Theoretical Methods in Physics. CNRS Marseille June 5-9 (1972) (where however the implications of renormalizability thanks to which all algebraic problems reduce to essentially finite dimensional ones, have not been investigated).

C. M. Lifwtilynn Smith Phys. Lett. 468, 233 (1973) and private discussions

J. Wess, B. Zumino Phys. Lett. 378, 95 (1971) | MR

[10] S. L. Adler Phys. Rev. 177, 2426 (1969) Lectures on Elementao Particles and Quantum Field Theory, 1970 Brandeis University Summer Institute of Theoretical Physics, S. Deser, M. Grisaru, H. Pendleton Ed., M.I.T. Press, Cambridge, Mass. 1970

W. Bardeen Phys. Rev. 184, 1848 (1969)

[11] J. H. Lowenstein, B. Schroer Phys. Rev. D6, 1553 (1972)

[12] J. H. Lowenstein, to be published

C. Becchi, to be published

T. Clark, A. Rouet, to be published

[13] J. H. Lowenstein, W. Zimmermann, to be published

Ph. Blanchard, R. Seneor : CERN/TH 1420 preprint, Nov. 1971 CNRS Centre de Physique Théorique Marseille, Colloquium on Renormalization Theory, June 1971, and to be published in Ann. Inst. Henri Poincaré

[14] A. A. Slavnov TMΦ 10, 153 (1972)

[15] L. D. Faddeev, V. N. Popov Phys. Lett. 25B, 29 (1967)

[16] G. 'T Hooft Nuclear Physics B35, 167 (1971)

G. 'T Hooft, H. Veltman CNRS Marseille, Colloquium on Yang Mills Flelds, June 1972 CERN/TH 1571

[17] Séminaire Sophus Lie, Ecole Normale Supérieure Paris 1954 : Théorie des Algèbres de Lie. in [9], the relevance of cohomology theory in the présent context can be detected.

[18] G. 'T Hooft, M. Veltman CERN Yellow Report TH/73/9 "Diagrammar" (1973) ;

G. 'T Hooft, M. Veltman Nuclear Physics B50, 318, (1972).

[19] The treatment of non linear gauges would require the introduction of two more multiplets of external fields, one coupled to the renormalized gauge function, one coupled to the renormalized Slavnov variation of the gauge function, with appropriate dimensions and quantum numbers. cf.Ref[1].

[20] C. Becchi, A. Rouet, R. Stora Lectures given at the International School of Elementary Particle Physics, Basko Polje, Yugoslavia 14-29 Sept. 1974 and to be published

[21] Using the LSZ formula, it is enough to compute $\partial {𝒵}_c\left(𝒥\right)/\partial \lambda$, $(\lambda =\kappa ,m_{4-\pi }^2)$ and, from the Legendre transform structure of ${𝒵}_c\left(𝒥\right)$. to show that the matrix elements of the gauge function between physical states vanish, as a consequence of the Slavnov identity. More generally, one can check that physical matrix elements of time ordered products of gauge functions are disconnected.

[22] G. 'T Hooft Nuclear Physics, B35, 167 (1971)

[23] The treatment of double poles given in [1] can easily be adapted modulo the modifications due to the non hermiticity of the Lagrangian described here.

[24] The main référence is : "Séminaire Sophus Lie 1. 1954/1955, "Théorie des Algèbres de Lie", Ecole Normale Supérieure, Secrétariat Mathématique, 11, rue Pierre Curie, Paris 5e. | Zbl

A sunmary can be found in N. Bourbaki, Groupes et Algèbres de Lie, Ch. I, §3, Problem n° 12, Hermann Paris (1960). | Zbl