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The quadratic Yang-Mills operator has zero directions along gauge orbits, so it has no propagator until one chooses a gauge fixing. The Faddeev-Popov determinant generated by this choice is represented by anticommuting Faddeev-Popov ghost fields, which cancel unphysical gauge-field contributions in loop calculations. A Nakanishi-Lautrup field imposes the gauge condition algebraically and lets the gauge-fixing plus ghost action be written as a BRST-exact term with off-shell nilpotency.

For the convention in the question, a finite Yang-Mills gauge transformation acts covariantly on the field strength:or with $U$ and $U^{†}$ exchanged if the opposite convention is used for $D_{\mu }$. Cyclicity of the matrix trace givesso the Yang-Mills theory Lagrangian is gauge invariant.

The quadratic gauge-field operator has zero directions $A_{\mu }\sim A_{\mu }+D_{\mu }\alpha$ along each gauge orbit. It therefore has no inverse on the full field space. Gauge fixing removes this degeneracy and produces a propagator, while the Faddeev-Popov determinant accounts for the corresponding Jacobian.