De Sitter Holography with a Finite Number of States

We investigate the possibility that, in a combined theory of quantummechanics and gravity, de Sitter space is described by finitely many states.The notion of observer complementarity, which states that each observer hascomplete but complementary information, implies that, for a single observer,the complete Hilbert space describes one side of the horizon. Observercomplementarity is implemented by identifying antipodal states with outgoingstates. The de Sitter group acts on S-matrix elements. Despite the fact thatthe de Sitter group has no nontrivial finite-dimensional unitaryrepresentations, we show that it is possible to construct an S-matrix that isfinite-dimensional, unitary, and de Sitter-invariant. We present a class ofexamples that realize this idea holographically in terms of spinor fields onthe boundary sphere. The finite dimensionality is due to Fermi statistics andan `exclusion principle' that truncates the orthonormal basis in which thespinor fields can be expanded.Comment: 23 pages, 1 eps figure, LaTe