Null-state quotient of a string (${\mathcal{H}}_{\mathrm{phys}}/\mathrm{rad}{\mathcal{H}}_{\mathrm{phys}}$)

The physical state space of old covariant quantization is quotiented by its physical spurious states, which are orthogonal to all physical states. In a ghost-free sector these form the radical of the semidefinite inner product. The quotient retains positive physical polarizations.