Infinitely Many Strings in De Sitter Spacetime: Expanding and Oscillating Elliptic Function Solutions

The exact general evolution of circular strings in $2+1$ dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter $b$, related to the string energy, and falls into three classes depending on whether $b<1/4$ (oscillatory motion), $b=1/4$ (degenerated, hyperbolic motion) or $b>1/4$ (unbounded motion). The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. The world-sheet time $\tau$ is an infinite-valued function of the string physical time, each branch yields a different string. This has no analogue in flat spacetime. We compute the string energy $E$ as a function of the string proper size $S$, and analyze it for the expanding and oscillating strings. For expanding strings $(\dot{S}>0)$: $E\neq 0$ even at $S=0$, $E$ decreases for small $S$ and increases $\propto\hspace*{-1mm}S$ for large $S$. For an oscillating string $(0\leq S\leq S_{max})$, the average energy $$ over one oscillation period is expressed as a function of $S_{max}$ as a complete elliptic integral of the third kind.Comment: 32 pages, Latex file, figures available from the authors under request. LPTHE-PAR 93-5

On the Key-Uncertainty of Quantum Ciphers and the Computational Security of One-way Quantum Transmission

We consider the scenario where Alice wants to send a secret (classical) $n$-bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then $n$. We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns $n$ bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain- and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer.Comment: 19 pages, 2 figures. This is a revised version of an earlier version that appeared in the proc. of Eucrocrypt'04:LNCS3027, 200

Kinematical and Physical properties of a 700 pc large bubble in NGC 6946

The galaxy NGC 6946 contains a gas-star complex of 700 pc in diameter which appears populated by tens of young stellar clusters, and a Super Star Cluster (SSC) as massive as 10$^6$ M$\odot$. The ionized gas, as drawn by the H$\alpha$ emission, delineates an almost circular shape which we show here to be in expansion. Previous studies have analyzed the stellar component of the complex, as well as the structure of the atomic and ionized gas; these analyses were restricted to the blueshifted component along the whole extent of the bubble or to a smaller inner region where both sides of an expanding bubble were seen. In this work we present a complete spectroscopic study of this object for two position angles crossing each other close to the young massive SSC. We have obtained new data with a spectral resolution six times better than previous spectroscopic studies, taken under atmospheric conditions better than those previously reported, allowing us to detect the approaching and receding walls of one the largest bubbles in external galaxies ever studied in detail. The kinematical analysis shows a large expanding bubble, whose walls appear to be highly structured with superposed smaller shells, likely originated as the result of star forming events occurring at the edges of the larger scale shell, a la Huygens. We also study some diagnostic diagrams of the ionized gas and conclude that most of the observed ionization is originated by photons from hot stars, but with clear evidence that some of the gas is shock ionized. This peculiar complex is an excellent laboratory for the analysis of the interaction and feedback between the gas where the stars were formed and the young and massive generation of new born stars.Comment: 41 pages, 12 figure

New Coherent String States and Minimal Uncertainty in WZWN Models

We study the properties of {\bf exact} (all level $k$) quantum coherent states in the context of string theory on a group manifold (WZWN models). Coherent states of WZWN models may help to solve the unitarity problem: Having positive norm, they consistently describe the very massive string states (otherwise excluded by the spin-level condition). These states can be constructed by (at least) two alternative procedures: (i) as the exponential of the creation operator on the ground state, and (ii) as eigenstates of the annhilation operator. In the $k\to\infty$ limit, all the known properties of ordinary coherent states are recovered. States (i) and (ii) (which are equivalent in the context of ordinary quantum mechanics and string theory in flat spacetime) are not equivalent in the context of WZWN models. The set (i) was constructed by these authors in a previous article. In this paper we provide the construction of states (ii), we compare the two sets and discuss their properties. We analyze the uncertainty relation, and show that states (ii) satisfy automatically the {\it minimal uncertainty} condition for any $k$; they are thus {\it quasiclassical}, in some sense more classical than states (i) which only satisfy it in the $k\to\infty$ limit. Modification to the Heisenberg relation is given by $2 {\cal H}/k$, where ${\cal H}$ is connected to the string energy.Comment: More discussion on relation to previous work. More references added. 14 pages, Latex, no figure

From the WZWN Model to the Liouville Equation: Exact String Dynamics in Conformally Invariant AdS Background

It has been known for some time that the SL(2,R) WZWN model reduces to Liouville theory. Here we give a direct and physical derivation of this result based on the classical string equations of motion and the proper string size. This allows us to extract precisely the physical effects of the metric and antisymmetric tensor, respectively, on the {\it exact} string dynamics in the SL(2,R) background. The general solution to the proper string size is also found. We show that the antisymmetric tensor (corresponding to conformal invariance) generally gives rise to repulsion, and it precisely cancels the dominant attractive term arising from the metric. Both the sinh-Gordon and the cosh-Gordon sectors of the string dynamics in non-conformally invariant AdS spacetime reduce here to the Liouville equation (with different signs of the potential), while the original Liouville sector reduces to the free wave equation. Only the very large classical string size is affected by the torsion. Medium and small size string behaviours are unchanged. We also find illustrative classes of string solutions in the SL(2,R) background: dynamical closed as well as stationary open spiralling strings, for which the effect of torsion is somewhat like the effect of rotation in the metric. Similarly, the string solutions in the 2+1 BH-AdS background with torsion and angular momentum are fully analyzed.Comment: 24 pages including 4 postscript figures. Enlarged version including a section on string solutions in 2+1 black hole background. To be published in Phys. Rev. D., December 199

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 $P=(\gamma-1)E,$ with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient $\gamma$ depending on the elliptic modulus. We semi-classically quantize the oscillating circular strings. The string mass is $m=\sqrt{C}/(\pi H\alpha'),\;C$ being the Casimir operator, $C=-L_{\mu\nu}L^{\mu\nu},$ of the $O(3,1)$-dS [$O(2,2)$-AdS] group, and $H$ is the Hubble constant. We find \alpha'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^2\alpha') 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.Comment: Latex, 26 pages + 2 tables and 5 figures that can be obtained from the authors on request. DEMIRM-Obs de Paris-9404

Quantum Coherent String States in AdS_3 and SL(2,R) WZWN Model

In this paper we make the connection between semi-classical string quantization and exact conformal field theory quantization of strings in 2+1 Anti de Sitter spacetime. More precisely, considering the WZWN model corresponding to SL(2,R) and its covering group, we construct quantum {\it coherent} string states, which generalize the ordinary coherent states of quantum mechanics, and show that in the classical limit they correspond to oscillating circular strings. After quantization, the spectrum is found to consist of two parts: A continuous spectrum of low mass states (partly tachyonic) fulfilling the standard spin-level condition necessary for unitarity |j|< k/2, and a discrete spectrum of high mass states with asymptotic behaviour m^2\alpha'\propto N^2 (N positive integer). The quantization condition for the high mass states arises from the condition of finite positive norm of the coherent string states, and the result agrees with our previous results obtained using semi-classical quantization. In the k\to\infty limit, all the usual properties of coherent or {\it quasi-classical} states are recovered. It should be stressed that we consider the circular strings only for simplicity and clarity, and that our construction can easily be used for other string configurations too. We also compare our results with those obtained in the recent preprint hep-th/0001053 by Maldacena and Ooguri.Comment: Misprints corrected. Final version to appear in Phys. Rev.

Circular String-Instabilities in Curved Spacetime

We investigate the connection between curved spacetime and the emergence of string-instabilities, following the approach developed by Loust\'{o} and S\'{a}nchez for de Sitter and black hole spacetimes. We analyse the linearised equations determining the comoving physical (transverse) perturbations on circular strings embedded in Schwarzschild, Reissner-Nordstr\"{o}m and de Sitter backgrounds. In all 3 cases we find that the "radial" perturbations grow infinitely for $r\rightarrow 0$ (ring-collapse), while the "angular" perturbations are bounded in this limit. For $r\rightarrow\infty$ we find that the perturbations in both physical directions (perpendicular to the string world-sheet in 4 dimensions) blow up in the case of de Sitter space. This confirms results recently obtained by Loust\'{o} and S\'{a}nchez who considered perturbations around the string center of mass.Comment: 24 pages Latex + 2 figures (not included). Observatoire de Paris, Meudon No. 9305