Quantum fractionary cosmology: K-essence theory

Using a particular form of the quantum K-essence scalar field, we show that in the quantum formalism, a fractional differential equation in the scalar field variable, for some epochs in the Friedmann-Lema\^itre-Robertson-Walker (FLRW) model (radiation and inflation-like epochs, for example), appears naturally. In the classical analysis, the kinetic energy of scalar fields can falsify the standard matter in the sense that we obtain the time behavior for the scale factor in all scenarios of our Universe by using the Hamiltonian formalism, where the results are analogous to those obtained by an algebraic procedure in the Einstein field equations with standard matter. In the case of the quantum Wheeler-DeWitt (WDW) equation for the scalar field $\phi$, a fractional differential equation of order $\beta=\frac{2\alpha}{2\alpha-1}$ is obtained. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; that is to say, when $\omega_X \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $\omega_X \in [-1,0)$, the order belongs to the interval $0< \beta \leq 1$. The corresponding quantum solutions are also given.