Old covariant string quantization ($(L_0-a)|\psi ⟩=0,L_{n>0}|\psi ⟩=0$)

Old covariant quantization retains all target-space string oscillators and imposes $(L_0-a)|\psi ⟩=0$ and $L_n|\psi ⟩=0$ for $n>0$. Negative-mode constraints are adjoints, not further annihilation conditions. Quotienting physical null string states removes redundant polarizations.

At $a=1$, let $k=\sqrt{2{\alpha }^{\prime }}p$, $k^2=-2$. The physical scalar $[{\alpha }_{-1}\cdot {\alpha }_{-1}+(D+4)(k\cdot {\alpha }_{-1}{)}^2/10+(D-1)k\cdot {\alpha }_{-2}/5]|p⟩$ has norm $2(D-1)(26-D)/25$. It is positive below 26, null at 26, and a string ghost state above 26. Matching the ordinary transverse spectrum removes this extra scalar at 26.

A polarization obeys $p\cdot ϵ=0$ and $p^2=(a-1)/{\alpha }^{\prime }$. If $a>1$, spacelike momentum allows a negative-norm timelike polarization. Thus ghost absence requires $a\le 1$. At $a=1$ the null string state identification $ϵ\sim ϵ+\lambda p$ leaves $D-2$ polarizations.