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Gauge invariance of the classical pure SU(2) Yang–Mills action

Physical scope

Classical gauge invariance of the adopted source-free pure SU(2) Yang–Mills action on oriented, time-oriented flat Minkowski spacetime, for the selected smooth right-principal bundle sector. No quantum or empirical conclusion is asserted.

Statement

Let $M = R^{1, 3}$ have orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ and metric $η = diag (- 1, 1, 1, 1)$ . Fix the pure $S U (2)$ model with a smooth right principal bundle $P \to M$ , no matter fields, the form $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ on $s u (2)$ , and fixed $g_{Y M} > 0$ . For a smooth connection with local curvature $F_{i}$ , the adopted action density is $- κ (F_{i} \land *_{η} F_{i}) / (2 g_{Y M}^{2})$ .

For every smooth gauge transformation $Φ : P \to P$ satisfying $π \circ Φ = π$ and $Φ (p g) = Φ (p) g$ , let $Φ_{*} H = d Φ (H)$ be its action on a principal connection $H$ . Then the local curvature density is unchanged in every section, so every defined local action and every convergent global action are invariant under $H \mapsto Φ_{*} H$ . More precisely, for any local section $s_{i} : U_{i} \to P$ , define the smooth local map $v_{i} : U_{i} \to S U (2)$ by $Φ^{- 1} \circ s_{i} = s_{i} v_{i}$ . The transformed curvature components satisfy $F_{i}^{Φ} = {Ad}_{v_{i}^{- 1}} F_{i}$ . The $v_{i}$ depend on the chosen local sections. Because $M = R^{1, 3} ≅ R^{4}$ , the bundle admits a global section; relative to any such chosen section, $Φ$ is represented by a global map $M \to S U (2)$ . No global section is fixed or needed here.

Statement references

Matthijs Vákár, Principal Bundles and Gauge Theories

Dale Husemöller, Fibre Bundles, §§9.2 and 10.3

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 $S U (2)$ model, a smooth principal connection $H$ on its selected bundle $P \to M$ , and a smooth gauge transformation $Φ$ .

[A1]

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

[A2]

Classical pure SU(2) Yang–Mills model: the global group is $S U (2)$ , $P$ is the selected right principal bundle, there are no matter fields, and $κ$ is the normalized positive $Ad$ -invariant form.

[A3]

Classical pure SU(2) Yang–Mills model: the adopted action density is $- κ (F \land *_{η} F) / (2 g_{Y M}^{2})$ with fixed $g_{Y M} > 0$ .

[F1]

Compactness and normalized invariant form for SU(2): $κ$ is positive definite and satisfies $κ ({Ad}_{g} X, {Ad}_{g} Y) = κ (X, Y)$ for every $g \in S U (2)$ .

[F2]

Gauge group of a smooth principal bundle: a gauge transformation is a smooth diffeomorphism $Φ$ of $P$ with $π \circ Φ = π$ and $Φ (p g) = Φ (p) g$ .

[F3]

Gauge transformations act on principal connections and carry curvature: $H^{Φ} = d Φ (H)$ is a principal connection and its vertical-valued curvature obeys $K_{H^{Φ}} (d Φ X, d Φ Y) = d Φ (K_{H} (X, Y))$ .

[F4]

Principal connection 1-form from a horizontal distribution: a connection form reproduces fundamental vertical vectors, has kernel $H$ , and converts the vertical-valued curvature to the Lie-algebra-valued curvature.

[F5]

Principal curvature, Cartan structure equation, and Bianchi identity: the Cartan curvature is horizontal and equivariant, equals $ω \circ K_{H}$ , and its local components obey the stated section-transition law.

[F7]

The differential of Ad is ad: the derivative of the adjoint representation at the identity is $d (Ad)_{e} (Z) = {ad}_{Z}$ .

[A4]

Classical pure SU(2) Yang–Mills model: $ad (P)$ is the real associated bundle $P \times_{Ad} s u (2)$ with transition $η_{j} = {Ad}_{g_{i j}^{- 1}} η_{i}$ , and $κ (F_{i} \land *_{η} F_{i})$ is a global scalar four-form.

[F6]

Covariant derivative on an associated representation bundle: for a finite-dimensional complex representation, the associated local derivative is $d + ρ_{*} (A_{i})$ on sections. Extend it to complexified-adjoint-valued $p$ -forms by $d + ρ_{*} (A_{i}) \land \cdot$ . Complexifying the real adjoint representation gives such a representation on $s u (2)_{C}$ ; by [F7] this extension preserves the real adjoint subbundle and restricts to $D_{A_{i}} η = d η + [A_{i} \land η]$ .

Proof

1.1A2F2F4construct

Put $H^{Φ} = d Φ (H)$ and let $ω^{Φ}$ be its connection form. Equivariance of $Φ$ gives $d Φ_{p} (Z_{P} (p)) = Z_{P} (Φ (p))$ for every $Z \in s u (2)$ . Since $d Φ$ transports both the vertical and horizontal summands, their projections commute with $d Φ$ ; the defining formula $ω = ζ^{- 1} v$ therefore gives $ω_{Φ (p)}^{Φ} (d Φ_{p} X) = ω_{p} (X)$ for every $X \in T_{p} P$ .

2.1F3F4F5step 1.1algebra

By [F3] and [F5], $Ω_{H^{Φ}, Φ (p)} (d Φ_{p} X, d Φ_{p} Y) = ω_{Φ (p)}^{Φ} (K_{H^{Φ}} (d Φ_{p} X, d Φ_{p} Y)) = ω_{p} (K_{H} (X, Y)) = Ω_{H, p} (X, Y)$ . Thus $Ω_{H^{Φ}} = (Φ^{- 1})^{*} Ω_{H}$ .

3.1F2F5step 2.1construct

On a local trivializing set, $Φ^{- 1} \circ s_{i}$ is another smooth section, so there is a unique smooth $v_{i} : U_{i} \to S U (2)$ with $Φ^{- 1} \circ s_{i} = s_{i} v_{i}$ . Pulling the curvature identity of step 2.1 back along $s_{i}$ gives $F_{i}^{Φ} = (Φ^{- 1} \circ s_{i})^{*} Ω_{H}$ . Apply the section-transition formula in [F5] to the two sections $s_{i}$ and $s_{i} v_{i}$ ; it yields $F_{i}^{Φ} = {Ad}_{v_{i}^{- 1}} F_{i}$ . This computation uses only the local transition function; it neither selects a global section nor uses the global representative.

4.1A1A3A4F1F6F7step 3.1algebra∎

The Hodge operator acts on the spacetime-form factors and commutes pointwise with the adjoint action. By [F1], $κ (F_{i}^{Φ} \land *_{η} F_{i}^{Φ}) = κ ({Ad}_{v_{i}^{- 1}} F_{i} \land *_{η} {Ad}_{v_{i}^{- 1}} F_{i}) = κ (F_{i} \land *_{η} F_{i})$ on every trivializing set. These equal local scalar four-forms define the same global action density by the adjoint-bundle pairing in [A4], so their integrals over every region where the action is defined agree. No boundary term is introduced because the density is unchanged pointwise.

Gauge invariance of the classical pure SU(2) Yang–Mills action

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources