Spectrum-Generating Algebra and No-Ghost Theorem for the Dual Model

The dual model is generally factorized using Lorentz oscillators $a_n^{\mu }$ with ghost (or negativenorm) states arising from the indefinite metric ($[a_n^0, a_n^{0†}]=-1\mathrm{}$). Here all ghost states are proven to decouple for unit Regge intercept (${\alpha }_0=1\mathrm{}$) as a consequence of the Virasoro gauges ($L_n$). By reformulating vertices in light-cone variables and exploiting the local commutators (for $Q^{\mu }$, $P^{\mu }$) on the Koba-Nielson circle, the spectrum-generating algebra ($A_n^i$, $A_n^{\mathrm{}(+)\mathrm{}}$) is found that commutes with all the gauges $L_n$. All physical states are explicitly constructed. The noghost theorem follows from the remarkable isomorphism of the transverse generators $A_n^i$ ($i=1, 2$) of Del Giudice, Di Vecchia, and Fubini to the original oscillators $\sqrt{n}{a}_n^i$, $[A_n^i, A_m^j]=n{\delta }_{\mathrm{ij}}{\delta }_{n+m,0\mathrm{}}$, and the isomorphism (up to $c$ numbers) of the longitudinal generators $A_n^{\mathrm{}(+)\mathrm{}}$ with the conformal group generators $L_l$, $[A_n^{\mathrm{}(+)\mathrm{}}, A_m^{\mathrm{}(+)\mathrm{}}]=(n-m)\mathrm{}{A}_{n+m}^{\mathrm{}(+)\mathrm{}}+2\mathrm{}{n}^3{\delta }_{n+m,0\mathrm{}}$. Increasing the number of spatial oscillators ($a_n^i$, $i=1, \dots , D-1\mathrm{}$), one observes a critical dimension $D=26\mathrm{}$. For $D>\mathrm{}26\mathrm{}$ ghosts appear, for $D<\mathrm{}26\mathrm{}$ there are no ghosts, and $A_1^{\mathrm{}(+)\mathrm{}}$ gives the null states postulated by Brower and Thorn. But for $D=26\mathrm{}$, all $A_n^{\mathrm{}(+)\mathrm{}}$ correspond to null states, so that the second-order Pomeranchukon is precisely a Regge pole (${\alpha }_P=\frac{1}{2}{\alpha }^{\prime }s+2\mathrm{}$) as proposed by Lovelace.