Gauge Theories in Particle Physics 2026, 44 - Local Non-Abelian Gauge Symmetries

The global SU(2) isospinor transformation for an isospin doublet wavefunction has a positional dependence in the wavefunction, but not in the parameter α, so it is a global transformation. Invariance under this transformation implies free choice of base states. Any non-Abelian phase transformation on chosen pairs produces equally good pairs, but the choice is dependent on the space-time coordinates (p. 39). From here, Yang-Mills theory applies to get a form of the wavefunction that is analogous to the electromagnetic U(1) and using the coupling strength constant g (p. 40).

The electromagnetic covariant derivative is the familiar D^μ = ∂^μ = iqA^μ(x) (Eq. 13.4). Applied to the derivative of the wave function, it introduces an extra term due to the gauge field A^μ, which means that the quantum field case means that a globally U(1)-invariant Lagrangian is immediately to a locally U(1)-invariant through moving from the partial derivative to the covariant derivative. The key property for the SU(2) covariant derivative is that it transform as ψ^(1/2).

(Eq. 13.9), generalizing to D^μ = ∂^μ + igτ_˙W^μ(x)/2 (Eq. 13.10) with W^μ(x) a tensor of three independent gauge fields generalizing the single electromagnetic gauge field A^μ. The W-tensor consists of 3 independent Yang-Mills fields. It can be shown that δW^μ = -∂^με(x) - g[ε(x) × W^μ] (Eq. 13.23). With these, one can derive the transformation law for gauge fields

(Eqs. 13.33-34). Finally, the covariant derivative for the gauge field should be written D^μ = ∂^μ + igT^(t)_˙W^μ (Eq. 13.44). The gauge field interacts with isomultiplets universally, which follows the local covariance requirements (13.1.1). The commutator of covariant derivatives is [D^μ, D^ν]ψ = ieF^μνψ (Eq. 13.45) and [D^μ, D^ν]ψ^(1/2) = igτ/2 _˙ (∂^μW^ν - ∂^νW^μ - gW^μ × W^ν)ψ^(1/2) (Eq. 13.48), which defines the field strength tensor (13.1.2). The SU(3) local symmetry needs to be transitioned into the global one for QCD considerations. For this, introduce a gauge fields for gluons that couples with the constant g_s. Its covariant derivative is D^μ = ∂^μ + ig_sλ/2 _˙ A^μ (Eq. 13.53). The free Dirac equation remains the same, and applied to the covariant derivative leads to the one-gluon emission amplitude, and the usualy field-strength equations (13.2).