Non-Abelian Gauge Theory (Yang–Mills)

The vector field and QED gauge the abelian group $U(1)$: the photon does not carry charge and does not self-interact. Promoting the internal symmetry to a non-abelian group $SU(N)$ changes this qualitatively — the gauge field carries the charge it mediates and couples to itself. This is Yang–Mills theory, the backbone of QCD and the electroweak sector. This page builds the classical theory; its quantization needs the Faddeev–Popov machinery.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$. Group generators $T^a$ satisfy $[T^a,T^b]=i{f}^{abc}{T}^c$; group theory background is in math/group-theory/00-README.md.

The gauge principle, non-abelian version

Start from matter fields $\psi$ in a representation of a compact Lie group $G$ (take $G=SU(N)$), with global symmetry $\psi \to U\psi$, $U=e^{i{\alpha }^a{T}^a}$. As in the abelian case, promoting the parameters to spacetime-dependent ${\alpha }^a(x)$ spoils invariance because ${\partial }_{\mu }\psi$ does not transform covariantly. The cure is again a covariant derivative with a gauge field, but now the gauge field is matrix-valued (Lie-algebra-valued), $A_{\mu }=A_{\mu }^a{T}^a$:

$$D_{\mu }={\partial }_{\mu }-ig{A}_{\mu },D_{\mu }\psi \to U(D_{\mu }\psi ),$$

which fixes the gauge transformation of $A_{\mu }$:

$$\boxed{A_{\mu }\to U{A}_{\mu }{U}^{-1}-\frac{i}{g}({\partial }_{\mu }U)U^{-1}.}$$

The infinitesimal form $\delta A_{\mu }^a=\frac{1}{g}{\partial }_{\mu }{\alpha }^a+f^{abc}{A}_{\mu }^b{\alpha }^c$ shows the new feature: the homogeneous term $f^{abc}{A}_{\mu }^b{\alpha }^c$ means the gauge field itself transforms in the adjoint representation — it carries charge. This is the matrix generalization of preliminaries § Gauge Fields.

The field strength and self-interaction

The field strength is defined so that it transforms covariantly, $F_{\mu \nu }\to U{F}_{\mu \nu }{U}^{-1}$; equivalently $F_{\mu \nu }=\frac{i}{g}[D_{\mu },D_{\nu }]$:

$$\boxed{F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c.}$$

The extra quadratic term $g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$ — absent in QED, where $f^{abc}=0$ — is the origin of everything distinctive about non-abelian theories. Unlike the abelian $F_{\mu \nu }$, it is not gauge-invariant (only covariant), so the invariant Lagrangian uses a trace:

$$\boxed{{\mathcal{L}}_{\text{YM}}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }=-\frac{1}{2}\mathrm{Tr}{F}_{\mu \nu }{F}^{\mu \nu }.}$$

Expanding ${\mathcal{L}}_{\text{YM}}$ in powers of $A$ reveals three- and four-gluon self-interactions ($\sim g{A}^2\partial A$ and $\sim g^2{A}^4$): the gauge bosons scatter off one another directly. The photon has no such vertices; the gluon does. These vertices are the entries in the Feynman rules that distinguish QCD from QED.

Geometric picture: connection and curvature

The structure is precisely that of a principal fiber bundle with gauge group $G$ (developed rigorously in math/differential-geometry — principal bundles, connections, and curvature):

$F_{\mu \nu }=0$ means the connection is flat (pure gauge); non-zero curvature is genuine gauge field. This geometric reading unifies gauge theory with general relativity (where the Christoffel connection has Riemann curvature) and underlies the topological objects — instantons and monopoles — classified by bundle invariants.

Coupling to matter and the full Lagrangian

Adding Dirac matter in a representation $R$ with generators $T_R^a$ gives the complete Yang–Mills–matter Lagrangian:

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }+\bar{\psi }(i/D-m)\psi ,D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }^a{T}_R^a.$$

For QCD, $G=SU(3{)}_{\text{color}}$, $R$ is the fundamental (quarks), and $A_{\mu }^a$ are the eight gluons. For the electroweak theory, $G=SU(2{)}_L\times U(1{)}_Y$ acting chirally on left-handed doublets. In both, the single coupling $g$ controls all interactions (matter–gauge and gauge self-couplings), a rigidity that gives non-abelian theories their strong predictive power.

The two decisive consequences

The gauge-field self-interaction drives the two phenomena that make non-abelian gauge theory the language of the strong and weak forces:

1. Asymptotic freedom — the self-coupling makes the $\beta$-function negative, so the coupling weakens at high energy (opposite to QED). This is why quarks behave as nearly free at short distance yet are confined at long distance.
2. The need for ghosts — quantizing the self-interacting gauge field covariantly requires Faddeev–Popov ghost fields to maintain unitarity, with BRST symmetry as the organizing principle.

Summary

• Non-abelian gauging: matrix-valued $A_{\mu }=A_{\mu }^a{T}^a$, covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$.
• Field strength $F_{\mu \nu }^a$ has an extra $g{f}^{abc}{A}^b{A}^c$ term ⇒ gauge-boson self-interaction (3- and 4-point vertices).
• Geometrically: a connection with curvature on a principal bundle.
• Consequences: asymptotic freedom and the need for Faddeev–Popov ghosts.

Where this leads

• Quantizing the theory: gauge fixing and Faddeev–Popov ghosts.
• Unitarity and BRST: BRST symmetry.
• The negative $\beta$-function: asymptotic freedom.
• The physical theories: QCD, electroweak.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 15.
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 15.
• Yang & Mills, Phys. Rev. 96, 191 (1954).
• Srednicki, Quantum Field Theory, Ch. 69–70.