From Slavnov–Taylor Identities to the Renormalization of Gauge Theories

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).