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The matter sector and the ghost sector commutes. The $L_1$ and $c_1$ annihilate the $|0;k\rangle$ state. Together with the commutation relations $\{c_n,b_m\}=\delta_{n+m}$ and $\left[\alpha_n^\mu,\alp …

If $|p\rangle$ is an eigenstate of $p^\mu$,there is no reason to make this state a ghost. Ghost would appear if you have polarization in this states. Is better to see this in a second quantization. No …

Actually the BRST-approach is far from be just pedagogical. … The gauge transformations of the field will be described by the generalization of the BRST-exact fields $\delta\Psi=Q\Lambda+g\left[\Psi,\Lambda\right]$ and the equation of motion will be: $$ Q\Psi + …

The second one is that we can use the BRST construction where we have a nilpotent fermionic conserved charge $Q^2=0$ generating a symmetry of the path integral (assuming no anomaly), given by $$ Q=c^{ … This BRST charge is useful to explore new "gauge fixings" as well as to obtain the gauge invariant spectrum in a covariant manner. …

To fix notation, the "b-ghost" $b_{\alpha}$ satisfy $[Q,b_{\alpha}]=\partial_{\alpha}$ and we have $$ [Q,\phi^{(0)}]=0,\qquad \phi_{\alpha}^{(1)}=[b_{\alpha},\phi^{(0)}],\qquad [Q,\phi^{(1)}_{\alpha}] …

The supercurrent is fixed by the superconformal algebra, which closes after adding the energy-momentum tensor $T_{B}$: $$ T_F(z)T_F(0)\sim\frac{2c}{3z^3}+\frac{2}{z}T_{B}(0)\,, $$ $$ T_{B}(z)T_F(0)\si …