De Sitter Space With Finitely Many States: A Toy Story

TL;DR

Parik and Verlinde address the clash between finite entropy in de Sitter space and the noncompact de Sitter symmetry by relaxing Hamiltonian Hermiticity and allowing de Sitter generators to mix in and out states. They construct a Dirac spinor based toy model with finite dimensional Fock spaces for antipodal observers and identify de Sitter invariant tensor products that serve as the building blocks of a unitary, finite dimensional $S$-matrix. Through an elliptic de Sitter space identification, they demonstrate how in to out processes can preserve de Sitter invariance, with explicit phase based S-matrix realizations. The work suggests that symmetry and finite state counting can coexist in a controlled setting and points toward a boundary like fuzzy sphere structure in larger representations, offering a route toward holographic descriptions of de Sitter space.

Abstract

The finite entropy of de Sitter space suggests that in a theory of quantum gravity there are only finitely many states. It has been argued that in this case there is no action of the de Sitter group consistent with unitarity. In this note we propose a way out of this if we give up the requirement of having a hermitian Hamiltonian. We argue that some of the generators of the de Sitter group act in a novel way, namely by mixing in- and out-states. In this way it is possible to have a unitary S-matrix that is finite-dimensional and, moreover, de Sitter-invariant. Using Dirac spinors, we construct a simple toy model that exhibits these features.