When quantizing Yang-Mills theory, we introduce the ghosts as a way to gauge-fix the path integral and make sure that we "count" only one contribution from each gauge-orbit of the gauge field $A_\mu\,^a$, because physically only the orbits themselves correspond to distinct physical configurations whereas the motion within the gauge-orbit should not contribute to the path-integral.
How come we don't run into this problem when we quantize the Fermions, which also have gauge transformations, and also have a gauge orbit? Shouldn't we include a gauge-fixing term for the Fermions as well, or does the term introduced for the Boson fields already pick out the gauge orbit for the Fermions as well? How does this technically come to be?
So far I introduce a gauge fixing term into the Lagrangian as $$ 1 = \int d\left[\alpha\right]\det\left(\frac{\delta G\left[A_{\mu}\left[\alpha\right]\right]}{\delta\alpha}\right)\delta\left(G\left[A\left[\alpha\right]\right]\right) $$ where $\alpha(x)$ are the gauge functions, and $G[]$ is a functional which is non-zero only for a unique gauge-representative in each gauge-orbit, where we have the transformations as: $$ \begin{cases} \psi_{c_{i}} & \mapsto\left(1+i\alpha^{a}t^{a}\right)_{c_{i}c_{j}}\psi_{c_{j}}+\mathcal{O}\left(\left(\alpha^{a}\right)^{2}\right)\\ A_{\mu}\,^{a} & \mapsto A_{\mu}\,^{a}+\frac{1}{g}D_{\mu}\,^{ab}\alpha^{b}+\mathcal{O}\left(\left(\alpha^{a}\right)^{2}\right) \end{cases} $$
Answer
Why do we gauge-fix the path integral in the first place? If we were doing lattice gauge theory, we didn't need to gauge-fix. But in the continuum case, (the Hessian of) the action for a generalized$^1$ gauge theory has zero-directions that lead to infinite factors when performing the path integral over gauge orbits. In a BRST formulation (such as, e.g., the Batalin-Vilkovisky formulation) of a generalized gauge theory, the gauge-fixing conditions can in principle depend on gauge fields, matter fields, ghost fields, anti-ghost fields, Lagrange multipliers, etc. Perturbatively, a necessary condition for a good gauge-fixing procedure is that the gauge-fixed Hessian is non-degenerate (in the extended field-configuration space). Generically, the number of gauge-fixing conditions should match the number of gauge symmetries.
For Yang-Mills theory with Lie group $G$, one needs ${\rm dim}(G)$ gauge-fixing conditions. One may check that for various standard gauges that only involve the gauge fields, it is not necessary to gauge-fix matter fields to achieve a non-degenerate Hessian.
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$^1$ By the word generalized gauge theories, we mean gauge theories that are not necessarily of Yang-Mills type.
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