δ A_{μ}^{a} = \frac{1}{g} (D_{μ} α)^{a}, (D_{μ} c)^{a} = \partial_{μ} c^{a} + g f^{ab c} A_{μ}^{b} c^{c} .

G^{a} [A] = \partial^{μ} A_{μ}^{a}

\frac{δ G ^{a} [ A ^{α} ] ( x )}{δ α ^{b} ( y )} = \frac{1}{g} \partial^{μ} D_{μ}^{ab} δ (x - y) .

The field-independent factor

1/ g

may be absorbed into normalization. The Grassmann Gaussian integral exponentiates the Faddeev-Popov determinant with anticommuting Faddeev-Popov ghost fields:

det (\partial^{μ} D_{μ}) = \int D \overset{c}{ˉ} D c e^{- \int d^{4} x \overset{c}{ˉ}^{a} \partial^{μ} (D_{μ} c)^{a}} .

A Gaussian average over the gauge condition supplies the covariant gauge-fixing term, so

S = S_{g} + \int d^{4} x [\frac{1}{2 ξ} (\partial^{μ} A_{μ}^{a})^{2} + \overset{c}{ˉ}^{a} \partial^{μ} (D_{μ} c)^{a}], D_{μ}^{a c} = δ^{a c} \partial_{μ} + g f^{ab c} A_{μ}^{b} .

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