Yang–Mills Equations, Constraints, and Observables

Yang–Mills evolution is nonlinear, but not every component of the field equation evolves initial data. The time component is Gauss’s law: a first-class constraint that generates gauge transformations. After that redundancy is removed, a massless Yang–Mills field has two local physical polarizations per Lie-algebra generator, while physical observables are gauge-invariant local composites or extended operators such as Wilson loops.

Required background. The Yang–Mills action and gauge self-interaction supplies the curvature, action, and sign conventions. Constraints, Dirac brackets, and symplectic reduction supplies first-class constraints and the phase-space degree count.

Helpful background. Gauge orbits, Gauss constraints, and stabilizers explains the geometric quotient and boundary-sensitive meaning of a gauge transformation.

Covariant dynamics and kinematic identity

With $D_\mu=\partial_\mu-igA_\mu$ and $F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+gf^{abc}A^b_\mu A^c_\nu$, variation of the action gives

$$(D_\mu F^{\mu\nu})^a=-g j^{a\nu}, \qquad j^{a\nu}=\bar\psi\gamma^\nu T_R^a\psi.$$

The Bianchi identity instead follows from the definition of curvature:

$$D_{[\lambda}F_{\mu\nu]}=0, \qquad D_\mu\widetilde F^{\mu\nu}=0.$$

Applying $D_\nu$ to the equation of motion gives $D_\nu j^\nu=0$. The apparent second derivative of $F$ vanishes because antisymmetry reduces it to a commutator proportional to $[F_{\mu\nu},F^{\mu\nu}]=0$. This is covariant color-current conservation; a component color current is not a gauge-invariant local observable. The distinction among field equation, Bianchi identity, and covariant conservation is developed in Schwartz 2014, §§ 25.2–25.3, pp. 488–495, with a compatible component derivation in Srednicki 2007, § 69, pp. 407–411.

For pure Yang–Mills theory in Lorenz gauge, $\partial_\mu A^{a\mu}=0$, the equation can be displayed as a nonlinear wave equation:

$$\begin{aligned} 0={}&\square A^{a\nu} +g f^{abc}\left( 2A^b_\mu\partial^\mu A^{c\nu} -A^b_\mu\partial^\nu A^{c\mu} \right)\\ &+g^2 f^{abc}f^{cde} A^b_\mu A^{d\mu}A^{e\nu}. \end{aligned}$$

The gauge condition simplifies the principal part but does not turn $A_\mu$ into an observable. This local hyperbolic form is an orientation to the classical initial-value problem, not a catalog of global solutions.

Canonical split and Gauss constraint

Define

$$E^{ai}=F^{ai0}, \qquad B^{ai}=\frac12\epsilon^{ijk}F^a_{jk}.$$

The field $A_0^a$ has no time derivative, so its conjugate momentum vanishes. Its Euler–Lagrange equation is the secondary Gauss constraint

$$G^a(x)=(D_iE^i)^a+g j^{a0}=0.$$

For pure Yang–Mills theory the Hamiltonian, up to the multiplier enforcing $G^a=0$ and possible boundary terms, is

$$H=\frac12\int \mathrm d^3x\, \left(E_i^aE_i^a+B_i^aB_i^a\right).$$

Using $\{A_i^a(\mathbf x),E_j^b(\mathbf y)\}=\delta^{ab}\delta_{ij}\delta^3(\mathbf x-\mathbf y)$, the constraints close:

$$\{G^a(\mathbf x),G^b(\mathbf y)\} =g f^{abc}G^c(\mathbf x)\delta^3(\mathbf x-\mathbf y).$$

Thus they are first class. With the present sign convention, $-\int\mathrm d^3x\,\alpha^aG^a$ generates $\delta_\alpha A_i^a=(D_i\alpha)^a$ together with the corresponding adjoint rotation of $E_i$. The bracket closes because the commutator of two gauge transformations is another gauge transformation; the same Jacobi identity also underlies the Bianchi identity.

The local degree count can be made either before or after eliminating $A_0$. The three $A_i^a$ and three $E_i^a$ components give six phase-space coordinates per generator. One first-class Gauss constraint and its gauge orbit remove two, leaving four phase-space coordinates: two configuration-space polarizations. In the unreduced description, the primary constraint conjugate to $A_0$ and Gauss’s law remove the same total number.

Gauss’s law is preserved by the evolution equations. It therefore constrains admissible initial data but need only be imposed on one Cauchy surface. Numerical or analytic evolution that develops a nonzero $G^a$ has violated either the equations, the discretization’s constraint preservation, or the boundary prescription.

Generators, charges, and observables

A gauge parameter that vanishes suitably at the spatial boundary generates a redundancy. If it approaches a nonzero boundary value, the generator must generally be supplemented by a surface term; that term can define a genuine conserved charge. Calling every Gauss transformation “pure gauge” without specifying boundary conditions erases this distinction.

Gauge-invariant local observables include

$$\operatorname{tr}(F_{\mu\nu}F^{\mu\nu}), \qquad \operatorname{tr}(F_{\mu\nu}\widetilde F^{\mu\nu}), \qquad T^{\mu\nu}.$$

The basic extended observable is the Wilson loop in a representation $R$,

$$W_R(C)=\operatorname{tr}_R\, \mathcal P\exp\left(ig\oint_C A_\mu\,\mathrm dx^\mu\right).$$

For a closed curve, endpoint transformations cancel inside the trace. An open Wilson line instead transforms at its two ends and becomes gauge invariant only after suitable charged endpoint operators or boundary data are supplied. Wilson loops diagnose transport and global gauge information; by themselves they do not prove confinement in a theory with dynamical matter.

The stress tensor

$$T^{\mu\nu} =-F^{a\mu\rho}F^{a\nu}{}_{\rho} +\frac14\eta^{\mu\nu}F^a_{\rho\sigma}F^{a\rho\sigma}$$

is gauge invariant and conserved on the pure equations of motion. By contrast, $A_\mu^a$, $E_i^a$, a fixed color component of $j_\mu^a$, and a gauge-fixed propagator are representation-dependent intermediate quantities.

Independent checks and limits

1. Constraint algebra: evaluating the Poisson bracket of two smeared Gauss constraints must return a Gauss constraint with the Lie bracket of the smearing functions.
2. Propagation: take a covariant divergence of the equations and verify that no independent evolution equation for $G^a$ appears.
3. Polarization count: one first-class constraint per generator removes one coordinate and its conjugate momentum, leaving two local polarizations.
4. Abelian limit: $f^{abc}\to0$ makes the constraint algebra Abelian, turns $D_iE^i$ into $\partial_iE^i$, and reduces $W_R$ to an ordinary exponential.

Common pitfalls

Confusing Gauss’s law with a gauge choice. Gauss’s law restricts physical initial data; a gauge condition chooses one representative of each orbit. They play different roles and form different brackets.

Counting four components minus one constraint. A first-class constraint also generates a gauge orbit, so it removes two phase-space dimensions, not one configuration component.

Using a gauge-fixed field as an observable. A solution for $A_\mu^a$ can be useful, but physical statements must be expressed through gauge-invariant composites, dressed charged operators, boundary charges, or extended observables.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), §§ 25.1–25.3, DOI.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press (2007), § 69, DOI.
• C. N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Physical Review 96 (1954), 191–195, DOI.