Constraints on non-minimal coupling from quantum cosmology

Quantum cosmology is investigated in a de Sitter minisuperspace model with a quantized scalar field non-minimally coupled to curvature. Quantum states of the scalar field must satisfy the regularity condition, which requires that the probability of field fluctuations should not increase with their amplitude. We show that this condition imposes constraints on the allowed values of the curvature coupling parameter $\xi$. This is a surprising result, since the field dynamics depends only on the combination $m^2+\xi R$, where $m$ is the field mass and $R=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ is the curvature, and does not depend on $\xi$ separately.