Quantization of Non-Abelian Gauge Fields

Abstract

The begin of non-Abelian gauge theories originates from the problem of a generalization of the Abelian group of gauge transformations of the electromagnetic potential of Chap. 9 . The solution involves a non-linear coupling of gauge field components. The coupling of gauge fields to matter fields together with the Higgs mechanism of mass generation led to the standard (Weinberg-Salam) model of particle physics which is the basis of the contemporary particle physics describing in a unified way electromagnetic, weak and strong interactions. We discuss briefly the mechanism of mass generation in classical gauge theory as it is the basis of applications to the Standard Model. Then, the quantization of the scalar field in an external gauge field is discussed. The quantization of gauge fields requires the gauge fixing which is achieved in the Faddeev-Popov formalism. The renormalization of gauge theories is discussed at one-loop level. We calculate the one loop effective action in the theory with scalar and gauge fields. The one-loop formula involves the heat kernel representation of differential operators discussed earlier in Chap. 5 . We discuss the stability of non-Abelian gauge theories (asymptotic freedom) on the basis of one-loop effective action. A mathematically precise gauge fixing and perturbation expansion in gauge theories is possible after lattice regularization which is discussed in Chap. 12 .