  We investigate the possibility that, in a combined theory of quantum
mechanics and gravity, de Sitter space is described by finitely many states.
The notion of observer complementarity, which states that each observer has
complete but complementary information, implies that, for a single observer,
the complete Hilbert space describes one side of the horizon. Observer
complementarity is implemented by identifying antipodal states with outgoing
states. The de Sitter group acts on S-matrix elements. Despite the fact that
the de Sitter group has no nontrivial finite-dimensional unitary
representations, we show that it is possible to construct an S-matrix that is
finite-dimensional, unitary, and de Sitter-invariant. We present a class of
examples that realize this idea holographically in terms of spinor fields on
the boundary sphere. The finite dimensionality is due to Fermi statistics and
an `exclusion principle' that truncates the orthonormal basis in which the
spinor fields can be expanded.
