De Sitter Holography with a Finite Number of States

TL;DR

This paper explores a finite-dimensional, de Sitter-invariant S-matrix in quantum gravity, proposing a holographic model with Fermi statistics that limits the state space, aligning with observer complementarity.

Contribution

It introduces a novel finite-dimensional, unitary, de Sitter-invariant S-matrix constructed via holography and Fermi statistics, addressing the challenge of finite states in de Sitter space.

Findings

Constructed a finite-dimensional, unitary S-matrix for de Sitter space.

Demonstrated holographic realization using spinor fields on the boundary.

Showed Fermi statistics and an exclusion principle truncate the state basis.

Abstract

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.