ABSTRACT
The gauge transformations of QED form an Abelian, i.e. commutative, group of transformations. Motivated by the non-commutative Isotopic Spin symmetry of nuclear physics, C. N. Yang and R. Mills formulated in 1953 a new category of field theories invariant under gauge transformations based on non-commutative symmetries, since then known as Yang-Mills theories. The initial part of the Chapter illustrates the fundamental concepts of a non abelian gauge theory: gauge transformations of matter and gauge fields, covariant derivatives, the gauge field tensor https://www.w3.org/1998/Math/MathML" display="inline"> G μ ν A (analogous to https://www.w3.org/1998/Math/MathML" display="inline"> F μ ν , the electromagnetic field tensor of Electrodynamics introduced in Chapter 5) and the invariant Lagrangian. The transformation properties of quark and lepton fields under gauge transformations of the symmetry group of ElectroWeak interactions, https://www.w3.org/1998/Math/MathML" display="inline"> S U ( 2 ) L ⊗ U ( 1 ) R , are illustrated as well as the transformation properties of quark fields under the colour group https://www.w3.org/1998/Math/MathML" display="inline"> S U ( 3 ) c o l o u r that generates the Strong Interactions. In the final part of the Chapter the quantisation of a non-abelian gauge theory is considered in the Faddeev-Popov scheme, with particular attention to the gauge-fixing procedure. This leads to the introduction of a set of non-physical Faddeev-Popov ghosts, anti commuting scalar fields, exchanged in closed loops in parallel to the exchange of the gauge fields. The need of ghost fields violating the spin-statistics theorem had been already pointed out by Feynman, to enforce the unitarity condition of diagrams containing loops of gauge fields, in his study of the quantisation of a Yang-Mills theory, see the following Chapter 15.
