Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 2 / i / Solution

The Faddeev-Popov ghost fields $c$ and $\bar{c}$ are spin-zero Grassmann fields in the Adjoint representation of $G$; their statistics are anticommuting, despite their scalar transformation under spacetime rotations. The Nakanishi-Lautrup field $h$ is an auxiliary commuting spin-zero field, also in the Adjoint representation. ThusThese Faddeev-Popov ghost fields are not physical spin-zero fermion states; they represent the gauge-fixing Jacobian, so their scalar spin does not contradict the physical Spin-statistics theorem.