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

In this work, we present a complete analysis of the quantisation of the classical Brans-Dicke Theory using the method of affine quantisation in the Hamiltonian description of the theory. The affine quantisation method is based on the symmetry of the phase-space of the system, in this case the (positive) half-plane, that is identified with the affine group. We consider a FLRW-type spacetime, and since the scale factor is always positive, the affine method seems to be more suited than the canonincal quantisation for our Quantum Cosmology. We find the wave function of the Brans-Dicke universe, and its energy spectrum. A smooth bounce is expected at the semi-classical level in the quantum phase-space portrait. We also adress the problem of equivalence between the Jordan and Einstein frames.