The Trouble with De Sitter Space

In this paper we assume the de Sitter space version of black hole Complementarity which states that a single causal patch of de Sitter space is described as an isolated finite temperature cavity bounded by a horizon which allows no loss of information. We discuss the how the symmetries of de Sitter space should be implemented. Then we prove a no go theorem for implementing the symmetries if the entropy is finite. Thus we must either give up the finiteness of de Sitter space entropy or the exact symmetry of the classical space. Each has interesting implications for the very long time behavior. We argue that the lifetime of a de Sitter phase can not exceed the Poincare recurrence time. This is supported by recent results of Kachru, Kallosh, Linde and Trivedi. 1 Thermofield Dynamics and Horizons An exact formulation of the quantum theory of an accelerating universe appears to be very elusive [1]. Thus far the holographic principle [2, 3, 4] has not produced a dual description of de Sitter space analogous to the holographic duality between anti–de Sitter space and Super Yang Mills theory [5, 6, 7] A number of authors [10, 11, 12, 13, 14, 15, 16, 17, 18] have argued for a de Sitter complementarity principle similar to the black hole version [8, 9]. While differing in the details, all versions agree that physics in a single causal patch of de Sitter space should be described as an isolated quantum system at at finite temperature, and that the thermal entropy of the system should be finite. From previous experience, especially with Matrix theory and the AdS/CFT duality, we can expect that the holographic dual of de Sitter space will not look much like classical relativity. Most likely its degrees of freedom will be very nonlocally connected to to the local semiclassical description of the space. That raises an interesting question: Suppose we were presented with the de Sitter space holographic dual. How would we recognize it as such? The answer for Matrix theory and AdS/CFT was initially through the symmetries, which exactly matched. Supersymmetry, Galilean symmetry, and conformal invariance were especially important. Thus we might try to recognize the de Sitter dual by finding that its symmetries include the SO(d, 1) symmetry of d-dimensional de Sitter space. However, most of the group (and the most interesting part of it) involve transformations which take the causal patch to another patch. The manifest symmetries of the causal patch are the SO(d − 1) rotations which preserve the horizon, and the time translations. There are another d− 1 compact generators and d− 1 noncompact generators which displace the observer to a new patch. These generators do not act on the Hilbert space of a single observer. To understand how they act we have to introduce a bigger space called the thermofield double. Thermofield theory was invented [22] in the context of many body theory for the purpose of simplifying the calculation of real time correlation functions in finite temperature systems. Its connection with black holes was realized by Israel [21] and elaborated in the holographic context by Maldacena [20]. Begin with a thermal system characterized by a Hilbert space H, a hamiltonian H and a temperature T = 1/β. The thermal ensemble is described by the Bolzmann density matrix ρ = 1 Z exp (−βH). (1.1) Thermal expectation values of real time correlation functions (response functions) contain