Quantum Cosmological Scenarios of Brans-Dicke Gravity in Einstein and Jordan Frames

Abstract

The Hamiltonian of any quantum cosmological model must obey some conditions in order to be self-adjoint, or to admit self-adjoint extensions. We will consider in this paper the self-adjoint character of the Hamiltonian resulting from the Brans-Dicke theory, both in Einstein and Jordan frames. In both cases we find the wave-function solutions in order to obtain concrete predictions for the evolution of the Universe. We show that the specific cosmological scenarios obtained depend very weakly on the conditions of having a self-adjoint operator. The problem of equivalence between Einstein and Jordan frames is considered, and it is shown that this equivalence implies a specific ordering parameter, as well as a particular choice of the physical variables.

Affine quantization of the Brans-Dicke theory: Smooth bouncing and the equivalence between the Einstein and Jordan frames

“…The equivalence between these frames is still debatable (see e.g. [33][34][35][36][37]), and our results may contribute to this debate. In the Einstein frame, the Schrödingerlike equation is separable, and becomes easier to deal with.…”

“…These formulations, related by a conformal transformation, are the target of a long debate on which of these frames is physically relevant. Some authors claim they are equivalent classically but should be different at the quantum level [33,34], while others claim that both are equivalent at classical and quantum level [35][36][37][38]. Some also claim that the equivalence is broken by off-shell one-loop quantum corrections, but recovered on-shell [39].…”

Quantum and Classical Cosmology in the Brans–Dicke Theory

“…Here, again, we have thatp andq are ordering factors. Notice that Equations ( 37) and (38) are Schrödinger-like, that is,…”

“…It is known [37] that the Hamiltonian operator of this quantized BD model with a radiative fluid can be self-adjoint only if q =q = 1. On the other hand, we have the quantum equivalence between Jordan's and Einstein's frames [38] if, and only if, p =p = 1, and thus we can use Einstein's frame. Now, in Einstein's frame, we choose the coordinate φ = e…”

Quantum and classical cosmology in the Brans-Dicke theory

“…It is known [22] that the Hamiltonian operator of this quantized Brans-Dicke model with a radiative fluid can be self-adjoint only if q = q = 1. On the other hand, we have the quantum equivalence between Jordan's and Einstein's frames [23] if, and only if, p = p = 1, and thus we can use Einstein's frame. Now, in Einstein's frame, we choose the coordinate…”