NON-ABELIAN GAUGE THEORIES

Abstract

The quantum field theories that have proved successful in describing the real world are all non-Abelian gauge theories, theories based on principles of gauge invariance more general than the simple U (1) gauge invariance of quantum electrodynamics. These theories share with electrodynamics the attractive feature, outlined at the end of Section 8.1, that the existence and some of the properties of the gauge fields follow from a principle of invariance under local gauge transformations. In electrodynamics, fields ψ n ( x ) of charge e n undergo the gauge transformation ψ n ( x ) → exp( ie n Λ( x ))ψ n ( x ) with arbitrary Λ( x ). Since ∂ μ ψ n ( x ) does not transform like ψ n ( x ), we must introduce a field A μ ( x ) with the gauge transformation property A μ ( x ) → A μ ( x )+∂ μ Λ( x ), and use it to construct a gauge-covariant derivative ∂ μ ψ n ( x )–ie n A μ ( x )ψ n ( x ), which transforms just like ψ n ( x ) and can therefore be used with ψ n ( x ) to construct a gauge-invariant Lagrangian. In a similar way, the existence and some of the properties of the gravitational field g μ n ( x ) in general relativity follow from a symmetry principle, under general coordinate transformations. Given these distinguished precedents, it was natural that local gauge invariance should be extended to invariance under local non-Abelian gauge transformations. In the original 1954 work of Yang and Mills, the non-Abelian gauge group was taken to be the SU (2) group of isotopic spin rotations, and the vector fields analogous to the photon field were interpreted as the fields of strongly-interacting vector mesons of isotopic spin unity.