Abstract
We study a quantization via fractional derivative of a nonminimal derivative coupling cosmological theory, namely, the Fab Four John theory. Its Hamiltonian version presents the issue of fractional powers in the momenta. That problem is solved here by the application of the so-called conformable fractional derivative. This leads to a Wheeler–DeWitt equation of second order, showing that a Bohm–de Broglie interpretation can be constructed. That combination of fractional quantization and Bohmian interpretation provides us a new quantization method, in which the quantum potential is the criterion to say if a quantum solution is acceptable or not to be further studied. We show that a wide range of solutions for the scale factor is possible. Among all of those, a bouncing solution analogous to the perfect fluid cosmology seems to deserve special attention.
Highlights
The expansion of the universe is one of the greatest discoveries of the last century
The scalar fields have shown to be an insightful way to modify gravity [2,3], by the introduction of an extra degree of freedom, that seems to be substantial for the early universe, since the canonical scalar field is the basis for inflationary models [4,5]
The aim of this paper is to study the second alternative: to apply a fractional calculus technique to investigate the quantum cosmology of (9) at a deeper level
Summary
The expansion of the universe is one of the greatest discoveries of the last century. It is shown how the contribution from the G5 family of Horndeski theories provides an inflationary mechanism in some different scenarios without a fine-tuned potential V (φ) This fact indicates that the G5 family strongly affects the cosmology of the very early universe. We shall not apply this method here because our goal in this paper is to study the simplest alternative to canonical transformations, and the bounded functional calculus would require a deeper mathematical analysis to guarantee that f ( p) satisfies all the required conditions for such a formalism This may be explored in a future work, with the insight given by this first approach.
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