Without gauge fixing, every gauge orbit is integrated infinitely many times and the quadratic gauge-field operator has a zero mode in field theory for every pure-gauge direction. It therefore has no inverse and no propagator. A gauge condition selects one representative per orbit, up to global transformations and the possible nonperturbative Gribov ambiguity, and makes perturbative Gaussian integration well defined.
The anticommuting fields and represent the field-dependent Faddeev-Popov determinant. They are Lorentz-scalar Grassmann fields, appear only on internal lines, and contribute a minus sign for each closed ghost loop. Their diagrams cancel unphysical gauge-polarization contributions and are required for gauge-independent, unitary amplitudes in a non-Abelian covariant gauge.
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