Is There Really a de Sitter/CFT Duality

In this paper a de Sitter Space version of Black Hole Complementarity is formulated which states that an observer in de Sitter Space describes the surrounding space as a sealed finite temperature cavity bounded by a horizon which allows no loss of information. We then discuss the implications of this for the existence of boundary correlators in the hypothesized dS/cft correspondence. We find that dS complementarity precludes the existence of the appropriate limits. We find that the limits exist only in approximations in which the entropy of the de Sitter Space is infinite. The reason that the correlators exist in quantum field theory in the de Sitter Space background is traced to the fact that horizon entropy is infinite in QFT. ∗ Permanent address, Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 † Permanent address, Computational Physics Laboratory, Howard University, Washington, DC 20059 ‡ Permanent address, Department of Physics, Stanford University, Stanford, CA 94305-4060 1 The Complementarity Principle Recent evidence suggests that we may live in a space-time that will asymptotically tend to de Sitter Space. If this is so, it is important to understand how quantum gravity should be formulated in such a geometry. Since de Sitter Space has an event horizon many of the questions that confused theorists about the quantum theory of black holes become relevant to cosmology. Perhaps the most important lesson we have learned from black hole quantum mechanics concerns the complementary way that different observers describe events in the black hole environment. That together with the Holographic principle, the UV/IR connection and the counting of black hole microstates is providing a new paradigm for the quantum mechanics of horizons. It is therefore natural to try to apply them to de Sitter Space. According to the Principle of Black Hole Complementarity [1] the horizon of a black hole may be regarded by an external observer as an impenetrable thermal membrane which can absorb, thermalize and reemit all information. The principle also says that a freely falling observer encounters nothing special at the horizon. The principle has recieved strong support from the study of gravity in AdS space and its equivalence to the boundary conformal field theory. The horizon of a de Sitter Space is structurally very similar to that of a black hole. A static patch of de Sitter Space is described by the metric ds = R [ (1 − r)dt − (1− r)dr − rdΩ ] . (1.1) In this case the horizon surrounds the observer. It is well known that the static patch has an entropy given by S = Area 4G = 4πR 4G . (1.2) The proper temperature at r = 0 is given by T = 1 2πR . More generally the local proper temperature is given by T (r) = 1 2πR √ 1− r2 . (1.3) Note that as in the black hole case the local proper temperature formally diverges near the horizon. The analog of the Black Hole Complementarity Principle for de Sitter space can be expressed as follows: An observer in de Sitter Space sees the surrounding spacetime as a finite closed cavity bounded by a horizon. The cavity is described by a thermal ensemble at coordinate temperature 1/2π. As for any closed system information can never be lost but can be scrambled or thermalized at the hot horizon. Also as in the black hole case, a freely falling observer will experience nothing out of the ordinary when passing through the horizon. These complementary descriptions of the horizon comprise the Complementarity Principle for de Sitter Space. Next let us consider the attempt by Strominger [2] to formulate a holographic description of de Sitter Space. The proposal is inspired by the AdS/CFT duality and is based on the fact that the symmetries of