Semiclassical quantization of circular strings in de Sitter and anti–de Sitter spacetimes

We compute the exact equation of state of circular strings in the (2+1)–dimensional de Sitter (dS) and anti–de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting, and expanding) strings. The string equation of state has the perfect fluid form P=(γ-1)E, with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient γ depending on the elliptic modulus. We semiclassically quantize the oscillating circular strings. The string mass is m= √C /(πHα’), C being the Casimir operator, C=-${\mathit{L}}_{\mathrm{\mu }\nu }$ ${\mathit{L}}^{\mathrm{\mu }\nu }$, of the O(3,1)-dS [O(2,2)-AdS] group, and H is the Hubble constant. We find α’ ${\mathit{m}}_{\mathrm{dS}}^2$≊4n-5${\mathit{H}}^2$α’ ${\mathit{n}}^2$ (n∈${\mathit{N}}_0$), and a finite number of states ${\mathit{N}}_{\mathrm{dS}}$≊0.34/(${\mathit{H}}^2$α’) in de Sitter spacetime; ${\mathit{m}}_{\mathrm{AdS}}^2$≊${\mathit{H}}^2$ ${\mathit{n}}^2$ (large n∈${\mathit{N}}_0$) and ${\mathit{N}}_{\mathrm{AdS}}$=∞ in anti–de Sitter spacetime. The level spacing grows with n in AdS spacetime, while it is approximately constant (although smaller than in Minkowski spacetime and slightly decreasing) in dS spacetime. The massive states in dS spacetime decay through the tunnel effect and the semiclassical decay probability is computed. The semiclassical quantization of exact (circular) strings and the canonical quantization of generic string perturbations around the string center of mass qualitatively agree.