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A nonzero null plane-wave solution of the pure SU(2) model

Physical scope

An exact smooth source-free classical solution of the adopted pure SU(2) model on the trivial bundle over Minkowski spacetime. It is a mathematical model solution, not an observed field or claim about nature.

Example

Specialize the adopted model to the trivial bundle $P = R^{1, 3} \times S U (2)$ with its canonical section. Let $q = x^{0} - x^{3}$ , choose a length $ℓ > 0$ and a nonzero constant $b$ with unit $m^{- 1}$ , and set $A = T_{3} b \sin (q / ℓ) d x^{1}$ . Its curvature is $F = T_{3} (b / ℓ) \cos (q / ℓ) (d x^{0} - d x^{3}) \land d x^{1}$ , which is not identically zero. With orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ and metric $η = diag (- 1, 1, 1, 1)$ , this connection satisfies $D_{A} (*_{η} F) = 0$ and $κ (F \land *_{η} F) = 0$ . It is therefore a stationary source-free solution of the adopted classical model; no uniqueness, quantum, or empirical claim is made.

Example references

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

Matthijs Vákár, Principal Bundles and Gauge Theories

Facts & Assumptions

Given: The trivial-bundle specialization of the adopted pure $S U (2)$ model, $b \neq 0$ , and $ℓ > 0$ .

[F1]

Classical pure SU(2) Yang–Mills model: the metric is $diag (- 1, 1, 1, 1)$ , the orientation is $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , and the action density uses the positive invariant form $κ$ .

[F2]

Compactness and normalized invariant form for SU(2): $T_{3} \in s u (2)$ and $κ (T_{3}, T_{3}) = 1$ .

[F3]

Graded bracket of Lie-algebra-valued differential forms and Principal curvature, Cartan structure equation, and Bianchi identity: $F = d A + \frac{1}{2} [A \land A]$ and $D_{A} η = d η + [A \land η]$ .

[F4]

Source-free Euler–Lagrange equation for the classical SU(2) Yang–Mills action: within the adopted model, the source-free equation $D_{A} (*_{η} F) = 0$ is equivalent to stationarity under all smooth compactly supported connection variations.

[A1]

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

Verification

1.1F1F2F3algebra

Write $q = x^{0} - x^{3}$ . Since the coefficient depends only on $q$ , $d A = T_{3} (b / ℓ) \cos (q / ℓ) (d x^{0} - d x^{3}) \land d x^{1}$ . The bracket term vanishes because $[T_{3}, T_{3}] = 0$ , so this is $F$ ; it is not identically zero because $b \neq 0$ ; if $b = 0$ , both the potential and curvature vanish.

2.1A1F1F2F3step 1.1algebra

The Lorentzian Hodge operator for the stated orientation gives $*_{η} ((d x^{0} - d x^{3}) \land d x^{1}) = - (d x^{0} - d x^{3}) \land d x^{2}$ . Hence $*_{η} F = - T_{3} (b / ℓ) \cos (q / ℓ) (d x^{0} - d x^{3}) \land d x^{2}$ . Its exterior derivative vanishes because differentiating its coefficient contributes a factor $(d x^{0} - d x^{3})$ already present in the wedge product; the covariant bracket term also vanishes because both Lie-algebra factors are $T_{3}$ . Thus $D_{A} (*_{η} F) = 0$ . The same repeated null one-form gives $F \land *_{η} F = 0$ , so the action density vanishes.

3.1F1F4step 1.1step 2.1∎

By [F4] and step 2.1, the connection is stationary under every admissible compactly supported variation. The curvature is nonzero by step 1.1, so this is a nontrivial classical solution with zero Lorentzian quadratic density.

A nonzero null plane-wave solution of the pure SU(2) model

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources