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.

For integer target dimension $2\le d\le 26$, intercept $a=1$ and ordinary nonzero momentum, the old covariant string quantization conditions leave a positive-semidefinite physical inner product. Quotienting its physical null string states removes gauge degeneracy. A DDF operator basis displays $d-2$ positive transverse oscillators; below dimension 26 the remaining physical longitudinal sector can be organized as a unitary Virasoro algebra sector with central charge $26-d$ and highest weight zero. At $d=26$ this extra sector becomes null and the quotient reduces to the 24 transverse oscillators. The result concerns negative-norm states, not the removal of the bosonic tachyon, and it does not itself cancel the worldsheet Weyl anomaly of a subcritical matter theory.

DDF operators are transverse physical string oscillators constructed as contour integrals of weight-one string vertex operators. Choose a null reference momentum $q$ and transverse polarizations ${ϵ}_i$ orthogonal to it and to a reference on-shell momentum. The integrand is proportional to ${ϵ}_i\cdot \partial X{e}^{inq\cdot X}$; its normalization and $q\cdot p$ are fixed so the operators obey the oscillator algebra in the title and commute with the positive-mode Virasoro constraints. For an open-string tachyon seed, one can use $2{\alpha }^{\prime }q\cdot p=1$. Acting with negative modes shifts momentum and raises oscillator level while preserving the physical mass-shell condition. Their transverse Fock space has positive norm and makes the physical content of the no-ghost theorem for the critical bosonic string explicit.

In flat 26-dimensional bosonic string theory, at ordinary nonzero momentum, take physical string states satisfying $L_{n>0}|\psi ⟩=0$ and $(L_0-1)|\psi ⟩=0$. The inner product restricted to these states is positive semidefinite. Quotienting by physical null string states removes the degeneracy; the resulting Hilbert space is represented by the 24 transverse string oscillators. This is a statement about negative-norm matter states, not about removing the auxiliary Faddeev-Popov ghosts. It does not remove the bosonic tachyon.

For $d=26$, $q^2=-2$ and the physical string state $[{\zeta }_{\mu \nu }{\alpha }_{-1}^{\mu }{\alpha }_{-1}^{\nu }+b_{\mu }{\alpha }_{-2}^{\mu }]|p⟩$, the Virasoro constraints give $b_{\nu }=-q^{\mu }{\zeta }_{\mu \nu }$ and $t={\zeta }^{\mu }{}_{\mu }=-2q\cdot b$. Every solution has the displayed decomposition, where $q\cdot A=0$ and $c$ is symmetric, traceless and transverse to $q$. The vector piece is $L_{-1}(2A\cdot {\alpha }_{-1}|p⟩)$; the scalar piece is $t(L_{-2}+3L_{-1}^2/2)|p⟩/10$. These are null string states, leaving the transverse traceless massive tensor as the physical polarization.

Time-coordinate oscillators produce negative norms in the covariant indefinite Fock space. A physical negative-norm state would violate probabilistic unitarity. Such ghosts differ from the anticommuting worldsheet ghost fields introduced by gauge fixing.

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.

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.

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.

For $L_0|p⟩=-|p⟩$, the descendant $Z=(L_{-2}+\frac{3}{2}{L}_{-1}^2)|p⟩$ obeys $L_{1}Z=0$ and $L_{2}Z=(D-26)|p⟩/2$. It becomes physical and null at 26 and supplies the scalar quotient needed for the massive spin-two light-cone gauge in string theory spectrum.