Can Quantum de Sitter Space Have Finite Entropy?

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.