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. On a regular local part of the constraint surface, removing that redundancy leaves two physical polarizations per Lie-algebra generator. Gauge-invariant local composites and extended operators such as Wilson loops describe the resulting physics; boundary charges require a separate boundary prescription.

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 field equation and matter-current conservation appear in Schwartz 2014, §25.3, pp.493–494, Eqs.(25.75), (25.82)–(25.83). His named matter current is the negative of the one defined here. The action variation and Jacobi argument explain separately why the field equation is dynamical and the Bianchi identity is kinematic.

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}.

On a fixed flat spatial slice, the canonical spatial labels on $A_i$ , $E_i$ , and $B_i$ are contracted with $\delta_{ij}$ ; in this notation $E_i=\delta_{ij}E^j$ , rather than a Lorentz lowering of the index on $F^{i0}$ . 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).

For the following canonical calculation, take pure Yang–Mills theory, so $j^{a0}=0$ , and use field-independent, compactly supported gauge parameters. This removes surface terms; boundary transformations are discussed below. 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 solved calculation below derives both statements. The geometric generator is the same smeared functional. With dynamical matter, the Gauss constraint includes the displayed charge density and its closure also uses the appropriate matter brackets; the gauge-field bracket alone is not the full calculation.

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. This is a regular local reduction; stabilizers, global zero modes, and boundaries require their own analysis.

Gauss’s law is preserved by the evolution equations: covariantly differentiating the field equations leaves only the time derivative of the constraint once the spatial equations hold. With matter this argument also requires its covariantly conserved current. Gauss’s law therefore constrains admissible initial data but need only be imposed on one Cauchy surface in a compatible evolution problem. 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; see Schwartz 2014, §25.2.2, pp.491–492, Eqs.(25.57)–(25.63). 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

Constraint algebra: evaluating the Poisson bracket of two smeared Gauss constraints must return a Gauss constraint with the Lie bracket of the smearing functions.
Propagation: take a covariant divergence of the equations and verify that no independent evolution equation for $G^a$ appears.
Polarization count: one first-class constraint per generator removes one coordinate and its conjugate momentum, leaving two local polarizations.
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.

Exercise: the smeared Gauss algebra

For pure Yang–Mills theory, smear $G^a=(D_iE^i)^a$ with compactly supported, field-independent functions $\alpha^a$ and $\beta^a$ . Show explicitly that the Poisson bracket closes, and determine which sign of the smeared constraint generates $\delta_\alpha A_i=D_i\alpha$ .

Solution

Write $(X\times Y)^a=f^{abc}X^bY^c$ and use the invariant color dot product. Covariant integration by parts gives

\mathcal G[\alpha] =\int d^3x\,\alpha\cdot D_iE_i =-\int d^3x\,E_i\cdot D_i\alpha.

Since the smearing functions are independent of the fields, its functional derivatives are

\frac{\delta\mathcal G[\alpha]}{\delta E_i^a} =-(D_i\alpha)^a, \qquad \frac{\delta\mathcal G[\alpha]}{\delta A_i^a} =-g(\alpha\times E_i)^a.

Substitution in the canonical Poisson bracket, invariance of the color dot product, and the derivation rule for $D_i$ yield

\begin{aligned} \{\mathcal G[\alpha],\mathcal G[\beta]\} &=g\int d^3x\,\bigl[ (\alpha\times E_i)\cdot D_i\beta -D_i\alpha\cdot(\beta\times E_i)\bigr]\\ &=-g\int d^3x\,E_i\cdot D_i(\alpha\times\beta) =g\mathcal G[\alpha\times\beta]. \end{aligned}

This is the smeared form of the local bracket in the text. The field variations follow from the same derivatives:

\{A_i,-\mathcal G[\alpha]\}=D_i\alpha, \qquad \{E_i,-\mathcal G[\alpha]\}=g(E_i\times\alpha).

The minus sign in the generator is therefore fixed by the canonical bracket and by the chosen definition of $G^a$ . Compact support was essential to the integration by parts. For parameters that reach the boundary, the omitted electric-flux surface term must be restored and its allowed variation examined before calling the transformation a redundancy.

References

Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), §§ 25.1–25.3, DOI.