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Source-free Euler–Lagrange equation for the classical SU(2) Yang–Mills action

Physical scope

The conditional source-free Euler–Lagrange equation for the adopted classical pure SU(2) Yang–Mills action on oriented, time-oriented flat Minkowski spacetime, for smooth connections and compactly supported variations in the selected bundle sector. No matter current, quantum conclusion, or empirical claim is asserted.

Statement

Use the adopted pure $S U (2)$ model on oriented, time-oriented Minkowski $R^{1, 3}$ with $η = diag (- 1, 1, 1, 1)$ , no matter fields, $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ , and fixed $g_{Y M} > 0$ . For a local section $s_{i} : U_{i} \to P$ , write $A_{i} = s_{i}^{*} ω$ , $F_{i} = d A_{i} + \frac{1}{2} [A_{i} \land A_{i}]$ , and $D_{A_{i}} η = d η + [A_{i} \land η]$ for an $s u (2)$ -valued differential form $η$ . A smooth local variation is any $a \in Ω^{1} (U_{i}, s u (2))$ with compact support in $U_{i}$ ; it represents a global adjoint-bundle-valued connection variation by extension by zero. Define its dimensionless relative action difference by

Δ (S / ℏ)_{i} (ε; a) = - \frac{1}{2 g_{Y M}^{2}} \int_{U_{i}} (κ (F_{A_{i} + ε a} \land *_{η} F_{A_{i} + ε a}) - κ (F_{i} \land *_{η} F_{i})) .

The integrand difference is compactly supported, so this relative integral is defined even if the unvaried action over all of $M$ diverges.

The connection is stationary when the derivative at $ε = 0$ of $Δ (S / ℏ)_{i} (ε; a)$ vanishes for every such variation in every local trivialization. It is stationary if and only if $D_{A_{i}} (*_{η} F_{i}) = 0$ in every local section. The local equations transform by the adjoint action on overlaps and therefore define the global, source-free Yang–Mills equation on the selected bundle sector. This equivalence is a conditional classical deduction from the adopted action and compact-support boundary convention.

Statement references

Matthijs Vákár, Principal Bundles and Gauge Theories

Karl-Hermann Neeb, Differential Topology of Fiber Bundles (full lecture notes, 2024)

David Tong, Gauge Theory, Chapter 2, Yang-Mills Theory

Facts & Assumptions

Given: The adopted pure $S U (2)$ action, a smooth connection, and a smooth compactly supported local variation.

[A1]

Classical pure SU(2) Yang–Mills model: $M$ has the fixed Minkowski metric and orientation in the Statement.

[A3]

Classical pure SU(2) Yang–Mills model: the adopted action density is $- κ (F \land *_{η} F) / (2 g_{Y M}^{2})$ .

[A4]

Classical pure SU(2) Yang–Mills model: variations are smooth and compactly supported; no boundary values are imposed at infinity, and finite-region variations vanish near the boundary.

[F1]

Compactness and normalized invariant form for SU(2): $T_{1}, T_{2}, T_{3}$ are a basis of $s u (2)$ with $κ (T_{a}, T_{b}) = δ_{a b}$ and $κ$ is positive and $Ad$ -invariant.

[F2]

Principal connection 1-form from a horizontal distribution and Principal curvature, Cartan structure equation, and Bianchi identity: local curvature is $F_{i} = d A_{i} + \frac{1}{2} [A_{i} \land A_{i}]$ , and the exterior covariant derivative is $D_{A_{i}} η = d η + [A_{i} \land η]$ with the graded bracket convention in the definition.

[F3]

Graded bracket of Lie-algebra-valued differential forms: the bracket is graded skew and satisfies the graded derivation identity for the exterior derivative.

[A5]

Classical pure SU(2) Yang–Mills model defines the real adjoint bundle and its local transition law.

[F4]

Covariant derivative on an associated representation bundle defines the induced derivative $d + ρ_{*} (A_{i})$ on sections of a finite-dimensional complex representation bundle. Extend it to form-valued sections by $d + ρ_{*} (A_{i}) \land \cdot$ . Complexifying $s u (2)$ and extending $Ad$ gives such a representation; by [F6], this extension restricts on real forms to $D_{A_{i}} η = d η + [A_{i} \land η]$ .

[F5]

Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign: on an open Minkowski coordinate region, compactly supported $C^{1}$ coefficients admit differentiation under the integral sign and integration by parts; if a continuous scalar coefficient integrates to zero against every smooth compactly supported scalar test, it vanishes pointwise.

[F6]

Adjoint is a smooth Lie-group representation and The differential of Ad is ad: $Ad$ is a smooth representation and its differential is $ad$ , so a varying local transition map differentiates by its Lie-algebra-valued Maurer–Cartan form.

Proof

1.1F2F3algebra

In one local section, expand the Cartan formula at $A_{i} + ε a$ : $F_{A_{i} + ε a} = F_{i} + ε (d a + [A_{i} \land a]) + \frac{ε^{2}}{2} [a \land a] = F_{i} + ε D_{A_{i}} a + \frac{ε^{2}}{2} [a \land a]$ . Here $[A_{i} \land a] = [a \land A_{i}]$ because both inputs have degree one and the graded bracket is symmetric in that degree.

1.2F1F2F3algebra

In a Lie-algebra basis, differentiate the scalar-valued form $κ (a \land *_{η} F_{i})$ . The ordinary graded product rule gives $κ (d a \land *_{η} F_{i}) - κ (a \land d *_{η} F_{i})$ . Invariance of $κ$ , together with moving the degree-one factors $A_{i}$ and $a$ past one another, gives $κ ([A_{i} \land a] \land *_{η} F_{i}) = κ (a \land [A_{i} \land *_{η} F_{i}])$ . Therefore $d κ (a \land *_{η} F_{i}) = κ (D_{A_{i}} a \land *_{η} F_{i}) - κ (a \land D_{A_{i}} *_{η} F_{i})$ .

2.1A1A3A4F1F5step 1.1algebra

The difference of the two action densities vanishes outside the compact support of $a$ and its derivatives. Apply the parameter-differentiation clause of [F5] to its smooth coordinate coefficients. Symmetry of $κ$ and the even degree of $F_{i}$ then give ${\frac{d}{d ε} ∣}_{0} Δ (S / ℏ)_{i} (ε; a) = - \frac{1}{g_{Y M}^{2}} \int κ (D_{A_{i}} a \land *_{η} F_{i})$ .

3.1A1A4F5step 2.1step 1.2

Write the compactly supported scalar three-form as $κ (a \land *_{η} F_{i}) = \sum_{μ = 0}^{3} h_{μ} \hat{d x^{μ}}$ with $h_{μ} \in C_{c}^{1} (U_{i})$ . Its exterior derivative is $\sum_{μ} (\partial_{μ} h_{μ}) d^{4} x$ . Apply the integration-by-parts clause of [F5] with the constant function $1$ to each coefficient $h_{μ}$ ; each integral is zero, so $\int d κ (a \land *_{η} F_{i}) = 0$ . Substitution in step 2.1 yields ${\frac{d}{d ε} ∣}_{0} Δ (S / ℏ)_{i} (ε; a) = - \frac{1}{g_{Y M}^{2}} \int κ (a \land D_{A_{i}} *_{η} F_{i})$ .

4.1A3step 3.1algebra

If $D_{A_{i}} *_{η} F_{i} = 0$ , step 3.1 shows that the first variation vanishes for every compactly supported $a$ in every chart. Thus the connection is stationary.

4.2A4F1F5step 3.1algebra

Conversely, assume stationarity. In a local coordinate chart and the orthonormal basis from [F1], write $J_{i} = D_{A_{i}} *_{η} F_{i} = \sum_{μ = 0}^{3} \sum_{b = 1}^{3} J_{μ b} T_{b} \hat{d x^{μ}}$ , where $\hat{d x^{μ}}$ is the oriented coordinate three-form satisfying $d x^{μ} \land \hat{d x^{μ}} = d^{4} x$ . For fixed $μ, b$ , choose $a = ψ T_{b} d x^{μ}$ with arbitrary smooth compactly supported real $ψ$ . Stationarity and step 3.1 give $\int ψ J_{μ b} d^{4} x = 0$ for every such $ψ$ . The fundamental-lemma clause of [F5] implies $J_{μ b} = 0$ throughout the chart. All coefficients vanish, so the local equation holds.

5.1A5F2F4F6step 4.2algebra∎

It remains to check that the local equations agree on overlaps. If $s_{j} = s_{i} g_{i j}$ , then [F2] gives $A_{j} = {Ad}_{g_{i j}^{- 1}} A_{i} + θ_{L} (g_{i j})$ and $(*_{η} F_{j}) = {Ad}_{g_{i j}^{- 1}} (*_{η} F_{i})$ . For $X \in T_{x} (U_{i} \cap U_{j})$ , put $Z = θ_{L} (g_{i j}) (X)$ and use the path $g_{i j} \exp (t Z)$ with initial tangent $X$ . Differentiating the adjoint representation by [F6] gives $d ({Ad}_{g_{i j}^{- 1}}) (X) = - {ad}_{Z} {Ad}_{g_{i j}^{- 1}}$ . Hence $d ({Ad}_{g_{i j}^{- 1}} *_{η} F_{i}) = {Ad}_{g_{i j}^{- 1}} d (*_{η} F_{i}) - [θ_{L} (g_{i j}) \land {Ad}_{g_{i j}^{- 1}} (*_{η} F_{i})]$ . The added Maurer–Cartan term cancels the corresponding term in $[A_{j} \land *_{η} F_{j}]$ , leaving $D_{A_{j}} *_{η} F_{j} = {Ad}_{g_{i j}^{- 1}} (D_{A_{i}} *_{η} F_{i})$ . Thus the local zero equations define one global adjoint-bundle equation.

Source-free Euler–Lagrange equation for the classical SU(2) Yang–Mills action

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources