Yang-Mills equations ( $D_{μ} F^{μν} = 0$ )

Stationarity of the Yang-Mills action in the absence of sources gives

D_{μ} F^{μν} = 0

, using the adjoint gauge covariant derivative. This dynamical equation differs from the geometric gauge-theory Bianchi identity.

Table of contents
- Covariant wave equation for the Yang-Mills field strength Yang-Mills equations

Covariant wave equation for the Yang-Mills field strength ( $D^{ρ} D_{ρ} F_{μν} = 2 [F_{μ}^{ρ}, F_{ν ρ}]$ )

Apply

D^{ρ}

to the gauge-theory Bianchi identity and commute it through the other derivatives using

[D_{μ}, D_{ν}] X = [F_{μν}, X]

. The Yang-Mills equations remove the differentiated divergences. The two remaining commutator terms both equal

[F_{μ}^{ρ}, F_{ν ρ}]

, proving the displayed nonlinear covariant wave equation for the Yang-Mills field strength on flat spacetime.

// Target: special-relativity.bigb

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The Yang–Mills equations are a set of partial differential equations that describe the behavior of gauge fields in the context of gauge theory, which is a fundamental aspect of modern theoretical physics. Named after physicists Chen-Ning Yang and Robert Mills, who formulated them in 1954, these equations generalize Maxwell's equations of electromagnetism to non-Abelian gauge groups, which are groups that do not necessarily commute.

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