Mass spectrum of strings in anti-de Sitter spacetime

Abstract

We perform string quantization in anti--de Sitter (AdS) spacetime. The string motion is stable, oscillatory in time with real frequencies ${\mathrm{w}}_{\mathrm{n}}$=${\mathrm{n}}^{2}$+${\mathrm{m}}^{2}$\ensuremath{\alpha}${\ensuremath{'}}^{2}$${\mathrm{H}}^{2}$, and the string size and energy are bounded. The string fluctuations around the center of mass are well behaved. We find the mass formula which is also well behaved in all regimes. There is an infinite number of states with arbitrarily high mass in AdS spacetime [in de Sitter (dS) spacetime there is a finite number of states only]. The critical dimension at which the graviton appears in D=25, as in de Sitter space. A cosmological constant \ensuremath{\Lambda}\ensuremath{\ne}0 (where its sign) introduces a fine structure effect (splittng of levels) in the mass spectrum at all states beyond the graviton. The high mass spectrum changes drastically with respect to flat Minkowski spacetime. For \ensuremath{\Lambda}0, we find 〈${\mathrm{m}}^{2}$〉\ensuremath{\sim}|\ensuremath{\Lambda}|${\mathrm{N}}^{2}$, independent of \ensuremath{\alpha}\ensuremath{'}, and the level spacing grows with the eigenvalue of the number operator N. The density of states \ensuremath{\rho}(m) grows like exp[(m/|\ensuremath{\Lambda}|${)}^{1/2}$] [instead of \ensuremath{\rho}(m)\ensuremath{\sim}exp(m\ensuremath{\alpha}\ensuremath{'}) as in Minkowski space], thus discarding the existence of a critical string temperature. For the sake of completeness, we also study the quantum strings in the black string background, where strings behave, in many respects, as in ordinary black hole backgrounds. The mass spectrum is equal to the mass spectrum in flat Minkowski space.