De Sitter Space With Finitely Many States: A Toy Story

TL;DR

This paper proposes a model for de Sitter space with finite states that maintains de Sitter invariance and unitarity by relaxing the hermiticity of the Hamiltonian, using a toy model with Dirac spinors.

Contribution

It introduces a novel approach to reconcile finite entropy, unitarity, and de Sitter invariance by allowing non-hermitian generators that mix in- and out-states.

Findings

Finite-dimensional, de Sitter-invariant S-matrix constructed.

A toy model with Dirac spinors demonstrating the proposed features.

Shows a way to preserve unitarity in de Sitter space with finite states.

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.