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

Abstract

The dual model is generally factorized using Lorentz oscillators ${a}_{n}^{\ensuremath{\mu}}$ with ghost (or negativenorm) states arising from the indefinite metric ($[{a}_{n}^{0}, {a}_{n}^{0\ifmmode\dagger\else\textdagger\fi{}}]=\ensuremath{-}1$). Here all ghost states are proven to decouple for unit Regge intercept (${\ensuremath{\alpha}}_{0}=1$) as a consequence of the Virasoro gauges (${L}_{n}$). By reformulating vertices in light-cone variables and exploiting the local commutators (for ${Q}^{\ensuremath{\mu}}$, ${P}^{\ensuremath{\mu}}$) on the Koba-Nielson circle, the spectrum-generating algebra (${A}_{n}^{i}$, ${A}_{n}^{(+)}$) 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 $\sqrt{n}{a}_{n}^{i}$, $[{A}_{n}^{i}, {A}_{m}^{j}]=n{\ensuremath{\delta}}_{\mathrm{ij}}{\ensuremath{\delta}}_{n+m,0}$, and the isomorphism (up to $c$ numbers) of the longitudinal generators ${A}_{n}^{(+)}$ with the conformal group generators ${L}_{l}$, $[{A}_{n}^{(+)}, {A}_{m}^{(+)}]=(n\ensuremath{-}m){A}_{n+m}^{(+)}+2{n}^{3}{\ensuremath{\delta}}_{n+m,0}$. Increasing the number of spatial oscillators (${a}_{n}^{i}$, $i=1, \dots{}, D\ensuremath{-}1$), one observes a critical dimension $D=26$. For $D>26$ ghosts appear, for $D<26$ there are no ghosts, and ${A}_{1}^{(+)}$ gives the null states postulated by Brower and Thorn. But for $D=26$, all ${A}_{n}^{(+)}$ correspond to null states, so that the second-order Pomeranchukon is precisely a Regge pole (${\ensuremath{\alpha}}_{P}=\frac{1}{2}{\ensuremath{\alpha}}^{\ensuremath{'}}s+2$) as proposed by Lovelace.