Gauge Theories in Particle Physics 2026, 43 - Non-Abelian Global Symmetries in Lagrangian QFT

A Lagrangian of free fermions of equal mass m can be written in its Dirac form using vectors of fermionic wavefunctions, and it is visible from this form that the Lagrangians and the associated Hamiltonians are invariant under the global U(1) transformation associated with baryon number conservation, and under global SU(2) in the flavour-space. The conserved quantities are Hermitian operators T generating unitary matrices U and serving as representation of the generators of SU(2)_f (Eq. 12.101). It can be related to the conserved isospin current operator T^μ through

(Eqs. 12.109 - 110). which transforms as a 4-vector under Lorentz transformation and as a T = 1 under SU(2)_f. This can be streamlined through the infinitesimal transformation, under which L is invariant. This defines the equations of motion (Eqs. 12.111-115). In SU(3), the vector q contains the colour triplet, instead of the fermion duo (12.4.1).

Chiral symmetry is also a global, non-Abelian symmetry. Given the one-fermion case, in small mass regime, the representation of Dirac matrices and the general Dirac spinor is used. It gives energy terms Eϕ = σ_˙pϕ + mχ; Eχ = -σ_˙pχ + mϕ. We take the QFT γ-matrix (Eqs. 21.140-143). With the regime applied, m → 0, E → |p|, so (σ_˙/|p|)ϕ = ϕ; (σ_˙/|p|)χ = -χ (Eqs. 12.145-146), which decouples the helicity spinors, which reflects the Lorentz transformation's reversal of helicity on massless particles. The Noether current can be derived with the usual QFT formalism, resulting in a conserved chirality operator Q₅ = ∫(ϕ^†ϕ - χ^†χ) d³x (Eq. 12.156) and defining the parity operator as PQ₅ | + ⟩ = -Q₅ | + ⟩ (Eq. 12.160). Through inspection of the T-operators, which in this case raise/lower isospin, QCD proves itself flavour blind. Its interactions bind the quark into nucleons, but don't immediately explain mass degenerate configurations.