Covariant quantization of the bosonic string

Covariant quantization keeps target-spacetime Lorentz covariance manifest, imposes canonical commutation relations on every embedding coordinate and removes unphysical states with the Virasoro constraints.

A physical string state obeys the Virasoro mass-shell and positive-mode constraints. States differing by a null state represent the same physical polarization.

A spurious string state is a positive-level Virasoro descendant orthogonal to all physical string states.

A null string state is both physical and spurious. It has zero norm and is quotiented from the physical state space.