Equivalence of the wave function of the universe in the Einstein and Jordan frames

Abstract

We investigate the question of equivalence between the Einstein and Jordan frames for generic scalar-tensor and $f(R)$ gravity theories in the context of minisuperspace quantum cosmology. We consider, particularly, the minisuperspace of a homogeneous and isotropic universe with the Friedmann-Lemaître-Robertson-Walker metric. We show that the equivalence between the frames depends on the operator ordering scheme of the quantum Hamiltonian: if the quantum theory is covariant under point canonical transformations in the minisuperspace, then the equivalence between the two frames holds. We find that for a large class of operator orderings, including the Laplace-Beltrami operator in the minisuperspace, the wave functions of the universe obtained in the two frames are related by the same transformations that connect the two frames, \textit{i.e.}, the frames are quantum mechanically equivalent. Further, we show that adding a fluid clock with an arbitrary equation of state parameter $w$ to the system breaks the equivalence, except for the case of a radiation fluid ($w=1/3$). All our arguments are presented at a formal level in both canonical and path integral approaches, and for any scalar-tensor, $f(R)$ theories without assuming any particular model.