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Semi-Classical Quantization of Circular Strings in De Sitter and Anti De Sitter Spacetimes

We compute the {\it 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 semi-classically quantize the oscillating circular strings. The string mass is $m=\sqrt{C}/(πHα'),\;C$ being the Casimir operator, $C=-L_{μν}L^{μν},$ of the $O(3,1)$-dS [$O(2,2)$-AdS] group, and $H$ is the Hubble constant. We find $α'm^2_{\mbox{dS}}\approx 5.9n,\;(n\in N_0),$ and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.17/(H^2α')$ in de Sitter spacetime; $m^2_{\mbox{AdS}}\approx 4H^2n^2$ (large $n\in N_0$) and $N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows with $n$ in AdS spacetime, while is approximately constant (although larger than in Minkowski spacetime) in dS spacetime. The massive states in dS spacetime decay through tunnel effect and the semi-classical decay probability is computed. The semi-classical quantization of {\it exact} (circular) strings and the canonical quantization of generic string perturbations around the string center of mass strongly agree.