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

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Context: The question of frame equivalence in modified gravity is still open. I introduced with Carla Almeida (São Paulo University, Brazil) the affine quantisation of the Brans-Dicke theory (BDT), which is a quantisation procedure based on continuous wavelet analysis, widely used to recover data from singularities in a signal. A mathematical model describing a physical system is scale-dependent, and the arbitrariness in the choice of the wavelet ψ allows to probe the system at different scales.

Method: I applied the affine quantisation to the Hamiltonian constraint of the BDT in both Jordan and Einstein frames, and analysed the differences between the two Wheeler-deWitt (WdW) equations. After, I used the separability of the WdW equation in the Einstein frame to find the wavefunction of the Universe and its energy spectrum. Finally, I calculated the expectation value of the scalar field and the scale factor to depict the semi-classical behaviour of the system using Mathematica.

Results:

• I proved that unitarity in both frames depends on the arbitrary wavelet only, i.e. an infinite number of unitarily equivalent models exist, hinting towards the equivalence between frames.
• In the Einstein frame, I was able to find a discrete energy spectrum.
• At the semi-classical level, the self-adjointness of the kinetic operator gives a smooth bounce…
• …and surprisingly, there must be an upper limit on the velocity of the scalar field, otherwise the semi-classical solutions diverge in both frames.