From BRST symmetry to the Zinn-Justin equation

Abstract Chapter 14 contains a general discussion of the quantization and renormalization of non–Abelian gauge theories. The quantization necessitates gauge fixing and introduces the Faddeev–Popov determinant. Slavnov–Taylor identities for vertex (one–particle–irreducible (1PI)) functions, the basis of a first proof of renormalizability, follow. The Faddeev–Popov determinant leads to a non–local action. A local form is generated by introducing Faddeev–Popov ghost fields. The new local action has an important new symmetry, the BRST symmetry. However, the explicit realization of the symmetry is not stable under renormalization. By contrast, a quadratic equation that is satisfied by the action and generating functional of 1PI functions, the Zinn–Justin equation, is stable and at the basis of a general proof of the renormalizability of non–Abelian gauge theories. The proof involves some simple elements of BRST cohomology. The renormalized form of BRST symmetry then makes it possible to prove gauge independence and unitarity.