= Solution
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 transformation>[BRST-exact] term with off-shell nilpotency.
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