Classical and quantum cosmology of Fab Four John theories

We present here a quantum cosmological model with Bohm-de Broglie interpretation of the theory described by a combination of two terms of the Fab Four cosmological theory. The first term is the John Lagrangian and the second is a potential representing matter content to avoid classical trivial solutions. This model has two free functions that provide an adjustment mechanism known classically as self-tuning. The self-tuning is a way to address the cosmological constant problem by allowing a partial break of symmetry in the scalar field sector. The Fab Four is the most general set of self-tuning scalar-tensor gravitational theories in four dimensions. The minisuperspace Hamiltonian thus obtained from this combination of Fab Four terms has fractional powers in the momenta, leading to a problem in applying canonical quantization. We have solved this problem by generalizing the canonical quantization rule using the so-called conformable fractional derivative. We show that this analysis leads to both singular and bouncing (non-singular) solutions, depending on the initial conditions over the scale factor and the homogeneous scalar field, and also depending on the free functions mentioned. This provides an adjustment mechanism in analogy with the classical self-tuning of the Fab Four, but with another interpretation.