I have read about $bc$ ant $\beta\gamma$ system. These are the fermionic and bosonic ghost system respectively. My question is
Why these systems are called ghost system?
What is the spin for these two systems?
I mean is there spin integer or half-integer?
In $d=2$ Conformal Field Theory a holomorphic field $\mathcal{O}(z)$, i.e. $\partial_{\bar{z}}\mathcal{O}(z)=0$, with conformal weight $h$ has also spin $h$. Just apply the rotation $z\rightarrow e^{i\theta}z$ to convince yourself. The $bc$ ghost system has conformal weights $h_b=2$ and $h_c=-1$, while the $\beta\gamma$ ghost system has conformal weight $h_\beta=3/2$ and $h_\gamma= -1/2$. Note that those fermions has integer spin while the bosons has semi-integer spins, the opposite of the spin-statistic, characterizing them as ghosts (if you assume that your theory obey spin-statistic for the physical states).
