Can Quantum de Sitter Space Have Finite Entropy?

Abstract

If one tries to view de Sitter as a true (as opposed to a meta-stable) vacuum, there is a tension between the finiteness of its entropy and the infinite-dimensionality of its Hilbert space. We invetsigate the viability of one proposal to reconcile this tension using $q$-deformation. After defining a differential geometry on the quantum de Sitter space, we try to constrain the value of the deformation parameter by imposing the condition that in the undeformed limit, we want the real form of the (inherently complex) quantum group to reduce to the usual SO(4,1) of de Sitter. We find that this forces $q$ to be a real number. Since it is known that quantum groups have finite-dimensional representations only for $q=$ root of unity, this suggests that standard $q$-deformations cannot give rise to finite dimensional Hilbert spaces, ruling out finite entropy for q-deformed de Sitter.

10 pages, v2: references added, v3: minor corrections, abstract and title made more in-line with the result, v4: published version

References in corpus (8)

• The Acceleration of the Universe, a Challenge for String Theory
• Recurrent Nightmares?: Measurement Theory in de Sitter Space
• De Sitter Holography with a Finite Number of States
• q-Deformed de Sitter/Conformal Field Theory Correspondence
• Finite Number of States, de Sitter Space and Quantum Groups at Roots of Unity
• Finite dimensional unitary representations of quantum Anti-de Sitter groups at roots of unity
• Real forms of quantum orthogonal groups, q-Lorentz groups in any dimension
• Strings on Plane Waves, Super-Yang Mills in Four Dimensions, Quantum Groups at Roots of One

Cited by in corpus (3)

• Higher Spin Cosmology
• Vacuum Transitions in Two-Dimensions and their Holographic Interpretation
• Multiparametric Quantum Algebras and the Cosmological Constant