research32 min studyLesson 701 in Physics

Yang–Mills curvature and covariant derivatives

Fix a non-Abelian convention and compute curvature arising from both spatial variation and noncommuting potentials.

Before you begin

On this page

The question for today

What you’ll understand

Derive curvature from a covariant-derivative commutator with consistent signs.
Compute a nonzero field strength for constant noncommuting potentials.
Distinguish gauge-dependent components from invariant contractions and dynamical solutions.

A non-Abelian gauge connection compares internal states at nearby spacetime points. Its components are matrices, so performing two infinitesimal comparisons in opposite orders can give different answers. Curvature measures that mismatch. The matrix algebra is as important as ordinary spatial variation: a constant potential can still have nonzero curvature. Before calculating, fix where the coupling and factors of i live; otherwise formulas copied from two correct conventions can combine into an incorrect field strength.

Declare the algebra and the derivative

Use Hermitian generators $T^a$ in a unitary representation, with commutator $[T^a,T^b]=if^{abc}T^c$ . For the fundamental representation of SU(N), normalize their trace to half the Kronecker delta. Let $A_\mu=A_\mu^aT^a$ and define the matter covariant derivative as ordinary derivative minus i times g times A. These conventions will be retained throughout this module; some sources instead absorb g into the connection or use anti-Hermitian generators.

Work in four spacetime dimensions and natural units with hbar and light speed one. The coupling is dimensionless, the connection has inverse-length units, and curvature has inverse-length-squared units. The representation specifies how the generators act on a matter multiplet. An adjoint-valued object instead has a commutator derivative, so applying the fundamental matrix formula to each color component as if it were independent would omit the interaction between internal directions.

D_\mu=\partial_\mu-igA_\mu,\qquad D_\mu X=\partial_\mu X-ig[A_\mu,X],\qquad [T^a,T^b]=if^{abc}T^c,\quad \operatorname{tr}(T^aT^b)=\tfrac12\delta^{ab}

The first derivative acts on matter in the chosen representation; the second acts on an adjoint matrix X. The trace normalization stated here is fundamental SU(N) normalization.↔ Scroll wide equations horizontally.

Derive the commutator curvature

Expand two covariant derivatives acting on a test matter field and subtract them in the opposite order. Terms containing derivatives of the test field cancel. The remaining derivative-of-connection terms and the connection commutator define F with an overall minus i g. Expanding its matrix commutator in generators produces a plus-g structure-constant term in components. The relative signs are fixed by the original derivative definition rather than by a separate convention for the magnetic field.

The Abelian limit removes the commutator, but it is not obtained merely by making the numerical potentials spatially constant. Constant matrices along different noncommuting generators can retain curvature. Conversely, a connection restricted to one fixed commuting generator behaves like an embedded Abelian field. This distinction identifies which nonlinear terms are genuinely present before imposing any gauge condition or using a perturbative approximation.

[D_\mu,D_\nu]=-igF_{\mu\nu},\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu],\qquad F^a_{\mu\nu}=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c

These are exact classical definitions in the declared convention. They do not assume small g or a particular gauge; the physical field units follow from the covariant derivative.↔ Scroll wide equations horizontally.

Use invariants without declaring a solution

Under a local gauge transformation the curvature is conjugated at its spacetime point. Individual color components change, whereas traces of products at that same point are invariant. For SU(2), choose $T^a=\sigma^a/2$ using the Pauli matrices. The trace of a squared single curvature component then provides a simple nonnegative Euclidean or magnetic diagnostic. It must not be confused with the full Lorentz contraction, whose electric and magnetic terms have different signs in Minkowski signature.

A specified connection is a field configuration, not automatically a source-free solution of the Yang-Mills equations. Constant noncommuting spatial potentials can have magnetic energy while failing their nonlinear equations of motion. Computing curvature establishes geometry; varying the action establishes dynamics. Quantization adds a further layer of assumptions about measures, gauge fixing, and physical observables. None of these steps can be inferred from an internal-vector picture alone.

2\operatorname{tr}(F_{12}^2)=\sum_a(F_{12}^a)^2,\qquad \mathcal E_{\rm magnetic}=\tfrac12\sum_a(F_{12}^a)^2\quad\text{if this is the sole magnetic component}

The energy-density expression uses canonically normalized gauge fields and no electric field. Both quantities have inverse-length-to-the-fourth units; the first expression is not the full spacetime contraction.↔ Scroll wide equations horizontally.

Work it through

Constant potentials with nonzero curvature

For SU(2) take constant spatial potentials $A_1=2M T^1$ , $A_2=3M T^2$ , other components zero, and $g=0.5$ . Here M is a fixed inverse-length scale. Find F-twelve and its magnetic energy density.

Spatial derivatives vanish, but $[T^1,T^2]=iT^3$ gives $[A_1,A_2]=6iM^2T^3$ . Constancy has not removed the non-Abelian matrix commutator.
Multiplying by $-ig$ yields $F_{12}=-i(0.5)(6i)M^2T^3=3M^2T^3$ . Thus its sole color coefficient is $F_{12}^3=3M^2$ , with the positive sign fixed by the chosen derivative.
The invariant is $2\operatorname{tr}(F_{12}^2)=9M^4$ , and magnetic energy density is $9M^4/2=4.5M^4$ . These are properties of this configuration, without claiming it solves the vacuum equations.

The result

The constant noncommuting potentials produce curvature three M-squared T-three and magnetic energy density 4.5 M-to-the-fourth.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Internal direction is only an analogy

Reset and set vector angle zero, magnitude one; predict x projection one. Compare a fresh Reset run at ninety degrees, magnitude one; predict y projection one. Distinct directions can label distinct internal generators, but the lab performs ordinary planar vector geometry and contains no matrix commutator, gauge transformation, or Yang-Mills dynamics.

Vectorx / y projectionsTrace marker

x component: 1.393
y component: 0.975
Magnitude: 1.7

Angle35°Magnitude1.7

t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA direction picture can organize labels, but non-Abelian curvature requires the declared matrix algebra.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The field strength contains minus i g times the connection commutator. Constant noncommuting matrices retain that term even when all ordinary derivatives vanish.

Your turn

Try it yourself

Instead set $A_1=0$ and $A_2=Bx^1T^3$ , with $B=0.4M^2$ . Compute F-twelve, twice the trace of its square, and magnetic energy density. Explain why its origin differs from the worked example.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The surviving contribution is an ordinary derivative; no pair of nonzero connection matrices is available for a commutator.

Compare with the solution

The derivative $\partial_1 A_2$ equals $BT^3$ , while the other derivative and commutator vanish. Therefore $F_{12}=0.4M^2T^3$ , the invariant is $B^2=0.16M^4$ , and magnetic energy density is $0.08M^4$ . This is an embedded Abelian configuration whose curvature comes from spatial variation, unlike the constant noncommuting potentials. Both calculations use the same generator normalization.

Keep the idea

Take it with you

Covariant-derivative conventions determine every curvature sign and coupling factor.
Noncommuting constant potentials can carry curvature.
Geometric invariants do not by themselves establish a dynamical solution or quantum prediction.

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