# Non commutative classical and Quantum fractionary Cosmology: Anisotropic
Bianchi Type I case.

J. Socorro1<EMAIL_ADDRESS>J. Juan Rosales 2<EMAIL_ADDRESS>Leonel
Toledo-Sesma3<EMAIL_ADDRESS>1 Department of Physics, Division of Science and
Engineering, University of Guanajuato, Campus León, León 37150, Mexico
2 Department of Electrical Engineering, Engineering Division Campus Irapuato-
Salamanca, University of Guanajuato, Salamanca 36885, Mexico<EMAIL_ADDRESS>
3 National Polytechnic Institute, UPIIH, Carretera Pachuca—Actopan Kilometer
1+500, San Agustín Tlaxiaca 42162, Hidalgo, México<EMAIL_ADDRESS>
###### Abstract

In this work, we will explore the effects of non-commutativity in fractional
classical and quantum schemes using the anisotropicc Bianchi Type I
cosmological model coupled to a scalar field in the K-essence formalism. We
introduce non-commutative variables considering that all minisuperspace
variables $q^{i}_{nc}$ do not commute, so the symplectic structure was
modified, resulting in some changes with respect to the traditional formalism.
In the quantum regime, the probability density presents a new structure in the
scalar field corresponding to the value of the non-commutative parameter.

Keywords: Fractional derivative, Fractional non-commutative classical and
quantum cosmology, K-essence formalism.

##  I Introduction

Many works investigate the Fractional Calculus (FC) and its applications r-s ,
being a powerful mathematical tool for describing complex processes, such as
the tautochrone problem Abel , models based on memory mechanism Caputo ,
anomalous diffusion Wyss , linear capacitor theory Westerlund , non-local
description of quantum mechanics Hermann , processing of medical images Jorge
; Leo , among others. Recently, CF has been applied to the general theory of
relativity co1 ; co2 ; co3 ; co4 ; co5 ; co6 ; co7 ; co8 ; co9 ; co10 ; co11 ;
co12 ; paulo3 ; paulo4 ; paulo5 , quantum mechanics and quantum cosmology
paulo1 ; paulo2 ; with interesting results in cosmology, such as that the age
of the Universe is greater than or of the order of 13.8196 Gyr, Jalalzadeh2023
. More recently, a new fractal cosmological model $\Lambda$CDM has been
created, where it is shown that the fractal spatial dimension is two and the
age of the Universe is 13.91 Gyr, Jalalzadeh2024 . The study and applications
of fractional calculus to cosmology is a new line of research and we have
recently worked along this line in the theory of K-essence where we find
several relevant indicators for its study. When this theory is analyzed as a
perfect fluid and in particular the barotropic parameter is constant, it is
generally demonstrated that this theory is equivalent to general relativity
coupled to ordinary matter with a barotropic equation of state sasaki2010 ,
which has been verified in particular with the standard FRW model universe
and an anisotropic model, the Bianchi type I. The second indicator relevant to
our case is that this is the only theory that when quantized under the ADM
formalism, a fractional Wheeler-DeWitt (WDW) equation, in the scalar field
component, is obtained naturally at different stages of the universe universe
. We have recently worked on non-commutativity (NC) in the space phase in this
formalism universe2024-nc . It is well known that there are various ways to
introduce non-commutativity in the phase space and that they produce different
dynamical systems from the same Lagrangian Abreu-2006 , as can be shown for
example in reference De-andrade  and references cited therein. Therefore,
distinct choices for the NC algebra among the brackets render distinct dynamic
systems. We will use non-commutativity in the coordinate space, leaving the
application of moments space for the future, Guzman-2011 , where other
quantities such as angular momentum appear between coordinates and momenta
sabido2018 ; sabido2024 . In this work, the known mathematical structure on
non-commutativity undergoes relevant changes since a prefactor appears that
was previously absorbed by the choice of the gauge. At the classical level
this prefactor does not cause a problem, since several of the moments turn out
to be constant, but at the quantum level we do have to make an adequate
separation to have something reasonable physically, and will depend strongly
on how it is used to reproduce plausible results that allow us to recover the
commutative quantum world, however, the combination of fractional differential
equations and non-commutativity in anisotropic models seems to be not a good
combination, since when solving the resulting quantum equations, this
combination causes the domain of definition of the non-commutative parameter
to change to another domain of definition that does not allow us to viably
recover the commutative quantum world. In this way, we discarded the other
choices of the prefactor given that due to the way in which the non-
commutative parameter is worked, it seems that we are facing an ambiguity or
separation of definition space.

Let us consider the Misner parameterization misner  in the line element of
Bianchi-type I cosmological models

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N^{2}dt^{2}+A^{2}dx^{2}+B^{2}dy^{2}+C^{2}dz^{2},$ (1)
$\displaystyle=$
$\displaystyle-N^{2}dt^{2}+e^{2\Omega+2\beta_{+}+2\sqrt{3}\beta_{-}}dx^{2}+e^{2\Omega+2\beta_{+}-2\sqrt{3}\beta_{-}}dy^{2}+e^{2\Omega-4\beta_{+}}dz^{2},$
(2)

where $N(t)$ is the lapse function, $\Omega(t)$ is a scalar function. We use
the Misner’s parametrization, then the radii are given by

$A=e^{\Omega+\beta_{+}+\sqrt{3}\beta_{-}},\qquad=e^{\Omega+\beta_{+}-\sqrt{3}\beta_{-}},\qquad
C=e^{\Omega-2\beta_{+}}.$

and the K-essence Lagrangian density of the form 1 ; roland ; chiba ; bose ;
arroja ; tejeiro

$S=\int d^{4}x\,\sqrt{-g}\,\left[f(\phi)\,{\cal G}(X)\right],$ (3)

where the canonical kinetic energy is given by
$\mathcal{G}(X)=X=-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi$, $R$ is the
scalar of curvature, $f(\phi)$ is an arbitrary function of the scalar field
$\phi$. In this work, we choose $f(\phi)=1$, and $g$ is the determinant of the
metric tensor. The K-essence model has been proposed as an inflationary model,
also as a model for dark energy along with explorations to unify energy and
dark matter roland ; bilic ; bento . Another motivation to consider this type
of Lagrangian originates from string theory string . For more details on
K-essence applied to dark energy, see copeland  and references therein.

The simplest K-essence Lagrangian density is

$\mathcal{L}_{geo}=\sqrt{-g}\left(\frac{R}{2}+{\cal G}(X)\right),$ (4)

hence, the field equations are given by

$\displaystyle G_{\alpha\beta}+\left[{\cal
G}_{X}\phi_{,\alpha}\phi_{,\beta}+{\cal G}g_{\alpha\beta}\right]$
$\displaystyle=$ $\displaystyle T_{\alpha\beta},$ (5a)
$\displaystyle\left[{\cal G}_{X}\phi^{,\beta}_{\,;\beta}+{\cal
G}_{XX}X_{;\beta}\phi^{,\beta}\right]$ $\displaystyle=$ $\displaystyle 0,$
(5b)

where, we have assumed the units with $8\pi G=1$, the semicolon means a
covariant derivative and the subscript $X$ denotes differentiation with
respect to $X$.

The same set of equations (5a) and (5b) are obtained if we consider the scalar
field $X(\phi)$ as part of the matter content, i.e.
$\mathcal{L}_{X,\phi}=\mathcal{G}(X)$, with the corresponding energy-momentum
tensor

$\mathcal{T}_{\alpha\beta}=\left[\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}(X)g_{\alpha\beta}\right],$
(6)

also, considering the energy-momentum tensor of a barotropic perfect fluid,

$T_{\alpha\beta}=(\rho+P)u_{\alpha}u_{\beta}+Pg_{\alpha\beta},$ (7)

with $u_{\alpha}$ being the four-velocity satisfying the relation
$u_{\mu}u^{\mu}=-1$, $\rho$ the energy density and $P$ the pressure of the
fluid. For simplicity, we consider a comoving perfect fluid. The pressure and
energy density corresponding to the energy momentum tensor of the field $X$,
are

$P(X)={\cal G},\qquad\rho(X)=\left[2X{\cal G}_{X}-{\cal G}\right],$ (8)

thus, the barotropic parameter $\omega_{X}=\frac{P(X)}{\rho(X)}$ for the
equivalent fluid is given by

$\omega_{X}=\frac{\mathcal{G}}{2X\mathcal{G}_{X}-\mathcal{G}}.$ (9)

The case of a constant barotropic parameter $\omega_{X}$, (with the exception
$\omega_{X}=0$) can be obtained by the solution to equation (9), $\mathcal{G}$
function

$\mathcal{G}=X^{\frac{1+\omega_{X}}{2\omega_{X}}},$ (10)

choosing the barotropic parameter as

$\omega_{X}=\frac{2\kappa-1}{2\kappa+1},\qquad\to\qquad\mathcal{G}=X^{\alpha},$

where the $\alpha$ parameter

$\alpha=\frac{2\kappa}{2\kappa-1},$ (11)

will be relevant in our study. Thus, we can write the barotropic parameter in
terms of $\alpha$ as $\omega_{X}=\frac{1}{2\alpha-1}$, being
$\kappa=\frac{\alpha}{2(\alpha-1)}$. Taking into account the above, we can
write the states in the evolution of our universe in the table 1, where we
have introduced the $\beta$ parameter that has to do with the fractional order
of the Wheeler-DeWitt equation in the scalar field.

States | $\kappa$ | $\alpha$ | $\beta=\frac{2\alpha}{2\alpha-1}$ | $\omega_{X}=\frac{1}{2\alpha-1}$ | Kinetic energy $(\mathcal{G}(X))$
---|---|---|---|---|---
Stiff matter | $\to\infty$ | 1 | 2 | 1 | $X$
Radiation | 1 | 2 | $\frac{4}{3}$ | $\frac{1}{3}$ | $X^{2}$
Dust-like | $\to\frac{1}{2}$ | $\to\infty$ | 1 | $\to 0$ | $X^{m},\qquad m\to\infty$
Inflation | 0 | 0 | 0 | -1 | $1,\qquad\Lambda=cte.$
Inflation-like | $\frac{1}{4}$ | -1 | $\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{X}$
| $\frac{1}{10}$ | $-\frac{1}{4}$ | $\frac{1}{3}$ | $-\frac{2}{3}$ | $\frac{1}{\sqrt[4]{X}}$
Table 1: States of universe’s evolution.

##  II Lagrange and Hamilton formalism

Introducing the line element (2) into the Lagrangian (4), we get

$\mathcal{L}_{I}=e^{3\Omega}\left[6\frac{\dot{\Omega}^{2}}{N}-6\frac{{\dot{\beta}_{+}}^{2}}{N}-6\frac{{\dot{\beta}_{-}}^{2}}{N}-\,\left(\frac{1}{2}\right)^{\alpha}\left(\dot{\phi}\right)^{2\alpha}\,N^{-2\alpha+1}\right].$
(12)

Using the standard definition of the momenta
$\Pi_{q^{\mu}}=\frac{\partial\mathcal{L}}{\partial\dot{q}^{\mu}}$, where
$q^{\mu}$ are the coordinate fields
$q^{\mu}=(\Omega,\beta_{+},\beta_{-},\phi)$, we obtain the momenta associated
to each field;

$\displaystyle\Pi_{\Omega}$
$\displaystyle=12e^{3\Omega}\frac{\dot{\Omega}}{N},$
$\displaystyle\dot{\Omega}$
$\displaystyle=Ne^{-3\Omega}\frac{\Pi_{\Omega}}{12},$ $\displaystyle\Pi_{+}$
$\displaystyle=-12e^{3\Omega}\frac{\dot{\beta}_{+}}{N},$
$\displaystyle\dot{\beta_{+}}$
$\displaystyle=-Ne^{-3\Omega}\frac{\Pi_{+}}{12},$ $\displaystyle\Pi_{-}$
$\displaystyle=-12e^{3\Omega}\frac{\dot{\beta}_{-}}{N},$
$\displaystyle\dot{\beta_{-}}$
$\displaystyle=-Ne^{-3\Omega}\frac{\Pi_{-}}{12},$ (13)
$\displaystyle\Pi_{\phi}$
$\displaystyle=-\,\left(\frac{1}{2}\right)^{\alpha}\frac{2\alpha}{N^{2\alpha-1}}\,e^{3\Omega}{\dot{\phi}}^{2\alpha-1},$
$\displaystyle\dot{\phi}$ $\displaystyle=-N\left[\frac{2^{\alpha-1}}{\alpha
e^{3\Omega}}\Pi_{\phi}\right]^{\frac{1}{2\alpha-1}},$

and introducing them into the Lagrangian density, we obtain the canonical
Lagrangian as
$\mathcal{L}_{canonical}=\Pi_{q^{\mu}}\dot{q}^{\mu}-N\mathcal{H}$. When we
perform the variation of this canonical Lagrangian with respect to $N$,
$\frac{\delta\mathcal{L}_{canonical}}{\delta N}=0$, we obtain the constraint
$\mathcal{H}\approx 0$.

Then, we obtain the corresponding Hamiltonian density

$\mathcal{H}_{I}=N\frac{e^{-\frac{3}{2\alpha-1}\Omega}}{24}\left\\{e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{12(2\alpha-1)}{\alpha}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}\right\\}.$
(14)

Next we take the gauge $N=24e^{\frac{3}{2\alpha 1}\Omega}$ and we will
continue the analysis with the reduced Hamiltonian

$\mathcal{H}=e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{12(2\alpha-1)}{\alpha}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}.$
(15)

The corresponding Hamilton equations become

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2\Pi_{\Omega}e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega},$ (16)
$\displaystyle\dot{\beta}_{\pm}$ $\displaystyle=$
$\displaystyle-2\Pi_{\pm}e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega},$ (17)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\Pi_{\phi}^{\frac{1}{2\alpha-1}},$
(18) $\displaystyle\dot{\Pi}_{\Omega}$ $\displaystyle=$
$\displaystyle\frac{6(\alpha-1)}{2\alpha-1}\,e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]$
(19) $\displaystyle\dot{\Pi}_{+}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{+}=p_{+}=constant,$ (20) $\displaystyle\dot{\Pi}_{-}$
$\displaystyle=$ $\displaystyle 0,\qquad\Pi_{-}=p_{-}=constant,$ (21)
$\displaystyle\dot{\Pi}_{\phi}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{\phi}=p_{\phi}=constant,$ (22)

substituting (22) into (18) we have the time solution for the scalar field as

$\phi(t)=\phi_{i}-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{1}{2\alpha-1}}\left(t-t_{\alpha_{i}}\right)$
(23)

where $\phi_{i}$ and $t_{\alpha_{i}}$ are the initial value for the scalar
field and the initial time in the $\alpha$ scenario, respectively.

On the other hand, using the Hamiltonian constraint (15), we found

$e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]=\frac{12(2\alpha-1)}{\alpha}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{2\alpha}{2\alpha-1}}=constant,$
(24)

and substituting into equations (19) we have the time solution

$\Pi_{\Omega}(t)=p_{0}+\frac{72(\alpha-1)}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{2\alpha}{2\alpha-1}}\left(t-t_{\alpha_{0}}\right),$
(25)

where $p_{0}$ is an integration constant.

The equation (16) can be rewritten as

$\left(\frac{2\alpha-1}{6(\alpha-1)}e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}\right)^{\bullet}=2\Pi_{\Omega},$
(26)

substituting the solution for $\Pi_{\Omega}(t)$, we get

$e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}=a_{0}+\frac{12(\alpha-1)}{2\alpha-1}p_{0}(t-t_{\alpha_{0}})+\frac{432(\alpha-1)^{2}}{\alpha(2\alpha-1)}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{2\alpha}{2\alpha-1}}\left(t-t_{\alpha_{0}}\right)^{2}.$
(27)

Hence, the $\Omega(t)$ becomes

$\Omega(t)=\frac{2\alpha-1}{6(\alpha-1)}\,Ln\left[a_{0}+\frac{12(\alpha-1)}{2\alpha-1}p_{0}(t-t_{\alpha_{0}})+\frac{432(\alpha-1)^{2}}{\alpha(2\alpha-1)}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{2\alpha}{2\alpha-1}}\left(t-t_{\alpha_{0}}\right)^{2}\right]$
(28)

and the corresponding solutions for the anisotropic parameter $\beta_{\pm}$
are

$\beta_{\pm}(t)=\beta_{{\pm}_{0}}-2p_{\pm}\,\int\frac{dz}{a_{0}+b_{\alpha}\,z+c_{\alpha}z^{2}},\qquad
z=t-t_{\alpha_{0}}.$ (29)

with
$b_{\alpha}=\frac{12(\alpha-1)}{2\alpha-1}p_{0},c_{\alpha}=\frac{432(\alpha-1)^{2}}{\alpha(2\alpha-1)}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}p_{\phi}^{\frac{2\alpha}{2\alpha-1}}$,
and ${\beta_{0}}_{\pm}$ are integration constants, so, the solution becomes

$\int\frac{dz}{a_{0}+b_{\alpha}\,z+c_{\alpha}z^{2}}=\frac{1}{\sqrt{b_{\alpha}^{2}-4a_{0}c_{\alpha}}}\,\ln{\left[\frac{2c_{\alpha}(t-t_{\alpha_{0}})+b_{\alpha}-\sqrt{b_{\alpha}^{2}-4a_{0}c_{\alpha}}}{2c_{\alpha}(t-t_{\alpha_{0}})+b_{\alpha}+\sqrt{b_{\alpha}^{2}-4a_{0}c_{\alpha}}}\right]},$

with the condition over the constants $b_{\alpha}>2\sqrt{a_{0}c_{\alpha}}$.

As we have mentioned, the Hamiltonian (15) has parts that are fractional
derivatives and parts that are integers. The case of integer derivatives must
be resolved in a particular way.

###  II.1 Stiff matter scenario, $\alpha=1$, $\beta=2$.

When we analyze the stiff matter scenario for the universe follows from the
Hamiltonian constriction (15)

$\mathcal{H}=\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}-12\Pi_{\phi}^{2},$ (30)

we have the following Hamilton equations

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle 2\Pi_{\Omega},$
(31) $\displaystyle\dot{\beta}_{\pm}$ $\displaystyle=$
$\displaystyle-2\Pi_{\pm},$ (32) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-24\Pi_{\phi},$ (33) $\displaystyle\dot{\Pi}_{\Omega}$
$\displaystyle=$ $\displaystyle 0,\qquad\Pi_{\Omega}=p_{0}=constant,$ (34)
$\displaystyle\dot{\Pi}_{\pm}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{\pm}=p_{\pm}=constants,$ (35) $\displaystyle\dot{\Pi}_{\phi}$
$\displaystyle=$ $\displaystyle 0,\quad\Pi_{\phi}=p_{\phi}=constant,$ (36)

Then, the $\Omega$ solution becomes

$\Omega(t)=\Omega_{0}+2p_{0}(t-t_{s}).$ (37)

and the anisotropic variables have the solutions
$\beta_{\pm}(t)=\beta_{\pm_{0}}-2p_{\pm}(t-t_{s})$. The volume of the universe
for this scenario is, $V(t)=e^{3\Omega(t)}$, given by

$V(t)=V_{0}\,e^{6p_{0}(t-t_{s})}.$ (38)

And the solution for the scalar field $\phi$ is

$\phi(t)=\phi_{s}-24p_{\phi}(t-t_{s}).$ (39)

where $\phi_{s}$ is the initial value in the stiff matter scenario.

###  II.2 Dust scenario, $\alpha\to\infty$, $\beta=1$.

For this scenario, the Hamiltonian constraint (15) is

$\mathcal{H}=e^{-3\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-12\sqrt{2}\Pi_{\phi},$
(40)

with the Hamilton equations

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2e^{-3\Omega}\Pi_{\Omega},$ (41) $\displaystyle\dot{\beta}_{\pm}$
$\displaystyle=$ $\displaystyle-2e^{-3\Omega}\Pi_{\pm},$ (42)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-12\sqrt{2},\qquad\phi(t)=\phi_{d}-12\sqrt{2}(t-t_{d}),$ (43)
$\displaystyle\dot{\Pi}_{\Omega}$ $\displaystyle=$ $\displaystyle
3e^{-3\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$ (44)
$\displaystyle\dot{\Pi}_{\pm}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{\pm}=p_{\pm}=constant,$ (45) $\displaystyle\dot{\Pi}_{\phi}$
$\displaystyle=$ $\displaystyle 0,\qquad\Pi_{\phi}=p_{\phi}=constant.$ (46)

where the constant $\phi_{d}$ must be very huge in such a way that this scalar
field survives actually.

Now, using the Hamiltonian constraint, we found that
$\Pi_{\Omega}(t)=p_{d}+36\sqrt{2}(t-t_{d})$, and substituting into (41), we
have that the volume of this universe becomes

$V(t)=e^{3\Omega}=a_{0}+6p_{d}(t-t_{d})+108\sqrt{2}p_{\phi}(t-t_{d})^{2},$
(47)

with $a_{0}$ an integration constant, and the other solutions are

$\beta_{\pm}={\beta_{0}}_{\pm}-2p_{\pm}\int\frac{dz}{a_{0}+b_{0}z+c_{0}z^{2}},\qquad
z=t-t_{d}.$ (48)

where $b_{0}=6p_{d}$ and $c_{0}=108\sqrt{2}p_{\phi}$, and the integral

$\int\frac{dz}{a_{0}+b_{0}z+c_{0}z^{2}}=\frac{1}{\sqrt{b_{0}^{2}-4a_{0}c_{0}}}\ln{\left[\frac{2c_{0}(t-t_{d})+b_{0}-\sqrt{b_{0}^{2}-4a_{0}c_{0}}}{2c_{0}(t-t_{d})+b_{0}+\sqrt{b_{0}^{2}-4a_{0}c_{0}}}\right]}$

with the condition over the constants
$36p_{d}^{2}-432a_{0}\sqrt{2}p_{\phi}>0$, and
$p_{d}>2\sqrt{3\sqrt{2}a_{0}p_{\phi}}$.

##  III Non-commutative fractional classical exact solution

Rewriting the classical Hamiltonian (15) in terms of the fractional parameter
$\beta=\frac{2\alpha}{2\alpha-1}$, we find the commutative equation of motion
in the classical phase space variables
$q^{\mu}=(\Omega,\beta_{+},\beta_{-},\phi)$, with the Poisson algebra:

$\left\\{q^{\mu},q^{\nu}\right\\}=0\qquad\left\\{\Pi_{q^{\mu}},\Pi_{q^{\nu}}\right\\}=0,\qquad\left\\{q^{\mu},\Pi_{q^{\nu}}\right\\}=\delta^{\mu}_{\nu},$
(49)

$\mathcal{H}=e^{-3(2-\beta)\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha\,f(\phi)}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(50)

Now, the natural extension is to consider the non-commutative version of our
model, with the idea of non-commutative between all variables
$(\Omega_{nc},{\beta_{\pm}}_{nc},\phi_{nc})$, so we apply a deformation of the
Poisson algebra. For this, we start with the usual Hamiltonian (50), but the
symplectic structure is modified as follows

$\displaystyle\left\\{\Pi_{\Omega},\Pi_{\phi}\right\\}_{\star}$
$\displaystyle=0,$
$\displaystyle\left\\{q^{\mu},\Pi_{q^{\mu}}\right\\}_{\star}$
$\displaystyle=1,$ $\displaystyle\left\\{\Omega,\beta_{+}\right\\}_{\star}$
$\displaystyle=\theta_{1},$
$\displaystyle\left\\{\Omega,\beta_{-}\right\\}_{\star}$
$\displaystyle=\theta_{2},$
$\displaystyle\left\\{\Omega,\phi\right\\}_{\star}$
$\displaystyle=\theta_{3},$
$\displaystyle\left\\{\beta_{+},\beta_{-}\right\\}_{\star}$
$\displaystyle=\theta_{4},$
$\displaystyle\left\\{\beta_{+},\phi\right\\}_{\star}$
$\displaystyle=\theta_{5},$
$\displaystyle\left\\{\beta_{-},\phi\right\\}_{\star}$
$\displaystyle=\theta_{6},$ (51)

where the $\star$ is the Moyal product Szabo2  and the resulting Hamiltonian
density is

$\mathcal{H}_{nc}=e^{-3(2-\beta)\Omega_{nc}}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha\,}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(52)

When we used the sifted variables (Bopp shift approach) but with the original
(commutative) symplectic structure

$\dot{q}^{\mu}_{nc}=\left\\{q^{\mu}_{nc},\mathcal{H}_{nc}\right\\},\qquad\dot{\Pi_{\mu}}_{nc}=\left\\{{\Pi_{\mu}}_{nc},\mathcal{H}_{nc}\right\\},$
(53)

the commutation relations (51) can be implemented in terms of the commutation
coordinates and it results in a modification of the potential-like terms in
the Hamiltonian density. In other words, this same result can be found
employing the Bob Shift between the commutative coordinates
$q^{\mu}=(\Omega,\beta_{+},\beta_{-},\phi)$ and the non-commutative
coordinates
$q^{\mu}_{nc}=(\Omega_{nc},\beta_{+}{{}_{nc}},\beta_{-}{{}_{nc}},\phi_{n})$ by
the relation

$q^{\mu}=q^{\mu}_{nc}+\frac{1}{2}\Theta^{\mu\nu}\Pi_{\nu},$ (54)

where we choose the matrix $\Theta^{\mu\nu}$ as

$\Theta^{\mu\nu}=\left(\begin{tabular}[]{cccc}0&$\theta_{1}$&$\theta_{2}$&$\theta_{3}$\\\
$-\theta_{1}$&0&$\theta_{4}$&$\theta_{5}$\\\
$-\theta_{2}$&$-\theta_{4}$&0&$\theta_{6}$\\\
$-\theta_{3}$&$-\theta_{5}$&$-\theta_{6}$&0\end{tabular}\right)$ (55)

we have, then

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\Omega_{nc}+\frac{\theta_{1}}{2}\Pi_{+}+\frac{\theta_{2}}{2}\Pi_{-}+\frac{\theta_{3}}{2}\Pi_{\phi},$
(56) $\displaystyle\beta_{+}$ $\displaystyle=$
$\displaystyle\beta_{+}{{}_{nc}}-\frac{\theta_{1}}{2}\Pi_{\Omega}+\frac{\theta_{4}}{2}\Pi_{-}+\frac{\theta_{5}}{2}\Pi_{\phi},$
(57) $\displaystyle\beta_{-}$ $\displaystyle=$
$\displaystyle\beta_{-}{{}_{nc}}-\frac{\theta_{2}}{2}\Pi_{\Omega}-\frac{\theta_{4}}{2}\Pi_{+}+\frac{\theta_{6}}{2}\Pi_{\phi},$
(58) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi_{nc}-\frac{\theta_{3}}{2}\Pi_{\Omega}-\frac{\theta_{5}}{2}\Pi_{+}-\frac{\theta_{6}}{2}\Pi_{-}.$
(59)

Therefore

$\displaystyle\Omega_{nc}$ $\displaystyle=$
$\displaystyle\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi},$
(60) $\displaystyle\beta_{+}{{}_{nc}}$ $\displaystyle=$
$\displaystyle\beta_{+}+\frac{\theta_{1}}{2}\Pi_{\Omega}-\frac{\theta_{4}}{2}\Pi_{-}-\frac{\theta_{5}}{2}\Pi_{\phi},$
(61) $\displaystyle\beta_{-}{{}_{nc}}$ $\displaystyle=$
$\displaystyle\beta_{-}+\frac{\theta_{2}}{2}\Pi_{\Omega}+\frac{\theta_{4}}{2}\Pi_{+}-\frac{\theta_{6}}{2}\Pi_{\phi},$
(62) $\displaystyle\phi_{nc}$ $\displaystyle=$
$\displaystyle\phi+\frac{\theta_{3}}{2}\Pi_{\Omega}+\frac{\theta_{5}}{2}\Pi_{+}+\frac{\theta_{6}}{2}\Pi_{-}.$
(63)

Therefore, our Hamiltonian density has the form

$\mathcal{H}_{nc}=e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta}.$
(64)

The Hamilton equations are

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2\Pi_{\Omega}e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]},$
(65) $\displaystyle\dot{\beta}_{+}$ $\displaystyle=$
$\displaystyle-2\Pi_{+}\,e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}$
(66)
$\displaystyle\qquad+\theta_{1}\,\frac{3(2-\beta)}{2}e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$
$\displaystyle\dot{\beta}_{-}$ $\displaystyle=$
$\displaystyle-2\Pi_{-}\,e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}$
(67)
$\displaystyle\quad+\theta_{2}\,\frac{3(2-\beta)}{2}e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-24\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta-1}$
(68)
$\displaystyle\quad+\theta_{3}\,\frac{3(2-\beta)}{2}e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$
$\displaystyle\dot{\Pi}_{\Omega}$ $\displaystyle=$ $\displaystyle
3(2-\beta)e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$
(69) $\displaystyle\dot{\Pi}_{\pm}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{\pm}=p_{\pm}=constants,$ (70) $\displaystyle\dot{\Pi}_{\phi}$
$\displaystyle=$ $\displaystyle 0,\qquad\Pi_{\phi}=p_{\phi}=constant.$ (71)

Similarly, using the Hamiltonian constraint and partial results in the
Hamilton equations, we find, that

$e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]=\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}=constant,$

then

$\dot{\Pi}_{\Omega}=\frac{72(2-\beta)}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta},\qquad\Pi_{\Omega}=p_{0}+\frac{72(2-\beta)}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}\left(t-t_{\alpha_{i}}\right).$
(72)

the $\Omega$ equation can be rewritten as

$\dot{\Omega}=2\Pi_{\Omega}e^{-3(2-\beta)\Omega}\,e^{\frac{3}{2}(2-\beta)F(\theta_{i}q^{i})}$

where $F(\theta_{i}q^{i})=\theta_{1}p_{+}+\theta_{2}p_{-}+\theta_{3}p_{\phi}$,
then we have

$\frac{1}{3(2-\beta)}\left(e^{3(2-\beta)\Omega}\right)^{\bullet}=e^{\frac{3}{2}(2-\beta)F(\theta_{i}q^{i})}\left[2p_{0}+\frac{144(2-\beta)}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}\left(t-t_{\alpha_{i}}\right)\right]$
(73)

thus,

$e^{3(2-\beta)\Omega}=a_{0}+e^{\frac{3}{2}(2-\beta)F(\theta_{i}q^{i})}\,P_{\alpha,\beta}(t)$

with
$P_{\alpha,\beta}(t)=\left[6(2-\beta)p_{0}\left(t-t_{\alpha_{i}}\right)+\frac{216(2-\beta)^{2}}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}\left(t-t_{\alpha_{i}}\right)^{2}\right]$,
having

$\Omega(t)=\frac{1}{3(2-\beta)}\ln\left|a_{0}+e^{\frac{3}{2}(2-\beta)F(\theta_{i}q^{i})}\,P_{\alpha,\beta}(t)\right|,$
(74)

with $a_{0}$ an integration constant. Then, the volume of the universe is

$V(t)=e^{3\Omega}=\left(a_{0}+e^{\frac{3}{2}(2-\beta)F(\theta_{i}q^{i})}P_{\alpha,\beta}(t)\right)^{\frac{1}{2-\beta}}.$
(75)

The anisotropic parameters $\beta_{\pm}(t)$ and scalar field $\phi(t)$ become

$\displaystyle\beta_{\pm}$ $\displaystyle=$
$\displaystyle{\beta_{0}}_{\pm}-2p_{\pm}\int\frac{dz}{a_{0}+b_{\alpha}z+c_{\alpha}z^{2}}+\frac{36(2-\beta)}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}\left(t-t_{\alpha_{i}}\right)\left\\{\begin{tabular}[]{l}$\theta_{1},\quad
for\,\beta_{+}$\\\ $\theta_{2},\quad for\,\beta_{-}$\end{tabular}\right.,$
(78) $\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\phi_{\alpha_{i}}+\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta-1}\left[-24+\theta_{3}\frac{36(2-\beta)}{\beta}p_{\phi}\right]\left(t-t_{\alpha_{i}}\right),$
(79)

where the integrals are the same as those that appear in the equation (48). As
a test that these non-commutative solutions are correct, when all
$\theta_{i}=0$, we recover the commutative solutions found in the previous
section.

##  IV Quantum commutativity

The WDW equation for these models is obtained by making the usual substitution
$\Pi_{q^{\mu}}=-i\hbar\partial_{q^{\mu}}$ in (15) and promoting the classical
Hamiltonian density in the differential operator applied to the wave function
$\Psi(\Omega,\beta_{\pm},\phi)$, $\hat{\cal H}\Psi=0$; we have

$\hat{\mathcal{H}}_{c}\Psi=\hbar^{2}e^{-3(2-\beta)\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}+\frac{\partial^{2}\Psi}{\partial\beta_{+}^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{-}^{2}}\right]-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}=0.$
(80)

a fractional differential equation of degree
$\beta=\frac{2\alpha}{2\alpha-1}$, and employing the following ansatz for the
wave function
$\Psi(\Omega,\beta_{+},\beta_{-},\phi)=\mathcal{B}_{+}(\beta_{+})\mathcal{B}_{-}(\beta_{-})\mathcal{A}(\Omega)\mathcal{F}(\phi)$
and next dividing between this one, we have the following expressions:

$\displaystyle\frac{d^{2}\mathcal{A}}{d\Omega^{2}}-Q\frac{d\mathcal{A}}{d\Omega}\pm\left[\ell_{a}^{2}+\frac{\ell^{2}}{\hbar^{2}}e^{3(2-\beta)}\Omega\right]\mathcal{A}$
$\displaystyle=$ $\displaystyle 0,$ (81)
$\displaystyle\frac{d^{\beta}\mathcal{F}}{d\phi^{\beta}}\mp\frac{\ell^{2}\beta}{24\hbar^{\beta}}\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}{\cal
F}$ $\displaystyle=$ $\displaystyle 0,$ (82)
$\displaystyle\frac{d^{2}\mathcal{B}_{+}}{d\beta_{+}^{2}}\pm\ell_{+}^{2}\mathcal{B}_{+}$
$\displaystyle=$ $\displaystyle 0,$ (83)
$\displaystyle\frac{d^{2}\mathcal{B}_{-}}{d\beta_{-}^{2}}\pm\ell_{-}^{2}{\cal
B}_{-}$ $\displaystyle=$ $\displaystyle 0,$ (84)

where $\ell^{2}$, $\ell_{+}^{2}$, $\ell_{-}^{2}$ are separations constants for
the scalar field equation, and the anisotropic function $\mathcal{B}_{\pm}$,
respectively, also $\ell_{a}^{2}=\ell_{+}^{2}+\ell_{-}^{2}$.

Following the book of Polyanin polyanin  (page 179.10), we discovered the
solution for the first equation for $\beta\not=2$, considering different
values in the factor ordering parameter and both signs in the constant
$\ell^{2}$, and the corresponding solutions become

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{0}e^{\frac{Q}{2}\Omega}\left\\{\begin{tabular}[]{ll}$J_{\rho_{-}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&\quad for\, $+\ell^{2}$\\\
$K_{\rho_{+}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&\quad for\,
$-\ell^{2}$\end{tabular}\right.$ (87) $\displaystyle\mathcal{B}_{\pm}$
$\displaystyle=$
$\displaystyle{\mathcal{B}_{0}}_{\pm}\left\\{\begin{tabular}[]{ll}$\cos{\left(\ell_{\pm}\beta_{\pm}-\alpha_{\pm}\right)}$,&\qquad
for \, $+\ell_{\pm}^{2}$\\\
$\cosh{\left(\ell_{\pm}\beta_{\pm}+\alpha_{\pm}\right)}$,&\qquad for \,
$-\ell_{\pm}^{2}$\end{tabular}\right.$ (90) $\displaystyle\mathcal{F}$
$\displaystyle=$
$\displaystyle\left\\{\begin{tabular}[]{ll}$f(0)\mathcal{E}_{\beta}\Big{(}\mp
A\phi^{\beta}\Big{)}$,&for \, $0<\beta\leq 1$\\\
$f(0)\mathcal{E}_{\beta}\Big{(}\mp
A\phi^{\beta}\Big{)}+f^{\prime}(0)\sum_{n=0}^{\infty}\frac{(-1)^{n}A^{n}}{\Gamma(n\beta+2)}\phi^{n\beta+1}$,&for
\,$1<\beta\leq 2$,\end{tabular}\right.$ (92)

with the order in the Bessel function $\rho_{\pm}=\frac{\sqrt{Q^{2}\pm
4\ell_{a}^{2}}}{3(2-\beta)}$,
$A=\mp\frac{\ell^{2}\beta}{24\hbar^{\beta}}\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}$
and $\mathcal{E}_{\beta}$ is the Mittag-Leffler function Haubold2011 .

Then, we have the probability density $|\Psi|^{2}$ by considering only
$\beta\not=2$, taking two sectors

###  IV.1 for $0<\beta\leq 1$

In the figure 1, we present the probability density of the wave function (95)
in the dust scenario ($\alpha\to\infty,\beta=1$), considering some values in
the factor ordering parameter $Q=-0.5,0,0.5,2$, where we are taking the
initial values;
$\psi_{0}=\frac{\sqrt{5}}{500},\frac{1}{100},\frac{\sqrt{5}}{5},\frac{\sqrt{3}}{3}$,
respectively. Employing the modified Bessel function, which has decreasing
behavior in the direction of evolution of the scale factor like ($\Omega$),
presenting a structure in the direction of the scalar field, as two parallel
universes, where for the the value Q=2, this behavior is more noticeable,
making the scalar field relevant in quantum evolution and remaining in the
classical evolution, here the anisotropic behavior is in the $\ell_{a}=0.03$
parameter, being very small in these plots.

As the factor ordering parameter increases its values (small values), the
universe has a more uniform structure, just as the normalization condition
increases, presenting the non-normalized behavior that characterizes quantum
cosmology.

Figure 1: Shows the behaviour of the probability density, equation (95), for
dust era considering the modified Bessel function. It is done by taking the
values of the factor ordering parameter $Q=-0.5,0,0.5,2$, with the initial
values
$\psi_{0}=\frac{\sqrt{5}}{500},\frac{1}{100},\frac{\sqrt{5}}{5},\frac{\sqrt{3}}{3}$,
respectively. The values to the constants are; $f(0)=f^{\prime}(0)=1$,
$\ell=0.5$, $\ell_{a}=0.03$, and the order in the modified Bessel function is
$\rho_{+}=0.167862;0.2;0.167862;0.666967,$ respectively.

$\displaystyle|\Psi|^{2}$ $\displaystyle=$
$\displaystyle\psi_{0}^{2}\,e^{Q\Omega}f^{2}(0)\mathcal{E}^{2}_{\beta}\Big{(}\mp
A\phi^{\beta}\Big{)}\times$ (95)
$\displaystyle\times\left\\{\begin{tabular}[]{ll}$\cos^{2}{\left(\ell_{+}\beta_{+}-\alpha_{+}\right)}\cos^{2}{\left(\ell_{-}\beta_{-}-\alpha_{-}\right)}\,J^{2}_{\rho_{-}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$+\ell^{2},+\ell_{+}^{2},+\ell_{-}^{2}$\\\
$\cosh^{2}{\left(\ell_{+}\beta_{+}+\alpha_{+}\right)}\cosh^{2}{\left(\ell_{-}\beta_{-}+\alpha_{-}\right)}K^{2}_{\rho_{+}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$-\ell^{2},-\ell_{+}^{2},-\ell_{-}^{2}$\end{tabular}\right.$

In the figure 2, we show the probability density in the radiation stage of the
universe, where it is perceived that in the evolution of the universe the
scalar field enters late as the factor ordering parameter increases, which
causes this field to remain longer in the quantum world and, therefore, remain
as a background in the transition to the classical world, just when the
amplitude of the probability density decreases. This behavior results in the
dust stage, where this probability density can be normalized manually (see
figure 1). Here the behavior of the probability density of the wave function
remains in the non-normalized quantum cosmology format, which is why we do not
give a specific value to the initial value of $\psi_{0}$.

###  IV.2 for $1<\beta\leq 2$,

Figure 2: Shows the behavior of the probability density, equation (98), for
radiation era, considering the modified Bessel function. It is done by taking
the values in the factor ordering parameter $Q=-2,-1,0,2$, with the initial
value
$\psi_{0}=\frac{\sqrt{5}}{500},\frac{1}{100},\frac{\sqrt{5}}{5},\frac{\sqrt{3}}{3}$,
respectively. The values to the constants $f(0)=f^{\prime}(0)=1$, $\ell=0.5$,
$\ell_{a}=0.03$, and the order in the modified Bessel function
$\rho_{+}=1.00045,0.500899,0.03,1.00045$, respectively.

$\displaystyle|\Psi|^{2}$ $\displaystyle=$
$\displaystyle\psi_{0}^{2}\,e^{Q\Omega}\left[f(0)\mathcal{E}_{\beta}\Big{(}\mp
A\phi^{\beta}\Big{)}+f^{\prime}(0)\sum_{n=0}^{\infty}\frac{(-1)^{n}A^{n}}{\Gamma(n\beta+2)}\phi^{n\beta+1}\right]^{2}\times$
(98)
$\displaystyle\times\left\\{\begin{tabular}[]{ll}$\cos^{2}{\left(\ell_{+}\beta_{+}-\alpha_{+}\right)}\cos^{2}{\left(\ell_{-}\beta_{-}-\alpha_{-}\right)}\,J^{2}_{\rho_{-}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$+\ell^{2},+\ell_{+}^{2},+\ell_{-}^{2}$\\\
$\cosh^{2}{\left(\ell_{+}\beta_{+}+\alpha_{+}\right)}\cosh^{2}{\left(\ell_{-}\beta_{-}+\alpha_{-}\right)}K^{2}_{\rho_{+}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$-\ell^{2},-\ell_{+}^{2},-\ell_{-}^{2}$\end{tabular}\right.$

##  V Quantum non-commutativity

The non-commutative version of fractional quantum cosmology presents several
difficulties due to the prefactor that remains associated with the canonical
moments that classically does not affect the mathematical analysis, but when
non-commutativity is used the moments appear in that prefactor. The question
that arises is: what is the classical way of stating this prefactor that gives
a better understanding of the quantum behavior of our problem? Since this
prefactor can be separated into several parts and applied to a different
sector of the Hamiltonian density. With this, the mathematical behavior is
presented differently in each case. We will use the following separation of
the classical non-commutative Hamiltonian density (64) and proceed to perform
the quantization using the correspondence principle.

###  V.1 Separating the prefactor term in two parts

The classical non commutative hamiltonian density

$\mathcal{H}_{nc}=e^{-3(2-\beta)\left[\Omega-\frac{\theta_{1}}{2}\Pi_{+}-\frac{\theta_{2}}{2}\Pi_{-}-\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(99)

can be written as

$\mathcal{H}_{nc}=e^{-3(2-\beta)\left[\Omega\right]}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,e^{-3(2-\beta)\left[\frac{\theta_{1}}{2}\Pi_{+}\frac{\theta_{2}}{2}\Pi_{-}+\frac{\theta_{3}}{2}\Pi_{\phi}\right]}\Pi_{\phi}^{\beta},$
(100)

then, the corresponding quantum version, (including the factor ordering
problem), becomes

$\displaystyle\hbar^{2}e^{-3(2-\beta)\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}+\frac{\partial^{2}\Psi}{\partial\beta_{+}^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{-}^{2}}\right]$
$\displaystyle-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}e^{3(2-\beta)\left[i\hbar\frac{\theta_{1}}{2}\frac{\partial}{\partial\beta_{+}}+i\hbar\frac{\theta_{2}}{2}\frac{\partial}{\partial\beta_{-}}+i\hbar\frac{\theta_{3}}{2}\frac{\partial}{\partial\phi}\right]}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}=0.$
(101)

and employing the following ansatz for the wave function
$\Psi(\Omega,\beta_{+},\beta_{-},\phi)={\cal B_{+}}(\beta_{+}){\cal
B_{-}}(\beta_{-}){\cal A}(\Omega){\cal F}(\phi)$ and next dividing between
this one, we have the following expression

$\displaystyle\hbar^{2}e^{-3(2-\beta)\Omega}\left[\frac{-\frac{d^{2}{\cal
A}}{d\Omega^{2}}+Q\frac{d{\cal A}}{d\Omega}}{{\cal A}}+\frac{\frac{d^{2}{\cal
B}_{+}}{d\beta_{+}^{2}}}{{\cal B}_{+}}+\frac{\frac{d^{2}{\cal
B}_{-}}{d\beta_{-}^{2}}}{{\cal B}_{-}}\right]$
$\displaystyle-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{1}}{2}\frac{d}{d\beta_{+}}\right]}{\cal
B}_{+}}{{\cal
B}_{+}}\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{2}}{2}\frac{d}{d\beta_{-}}\right]}{\cal
B}_{-}}{{\cal
B}_{-}}\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{3}}{2}\frac{d}{d\phi}\right]}\frac{d^{\beta}{\cal
F}}{d\phi^{\beta}}}{{\cal F}}=0.$ (102)

If we want to separate the differential equations, as a first approximation we
can consider that the terms that accompany the differential equation of ${\cal
F}$ are constrained to be constants, nearly at unity, because the $\theta_{i}$
parameter is closed to zero and subsequently use that information to solve the
corresponding differential equations as a system of equations.

$\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{1}}{2}\frac{d}{d\beta_{+}}\right]}{\cal
B}_{+}}{{\cal B}_{+}}=a_{+}^{2}\approx
1,\quad\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{2}}{2}\frac{d}{d\beta_{-}}\right]}{\cal
B}_{-}}{{\cal B}_{-}}=a_{-}^{2}\approx 1,$ (103)

(we choose for convenience $a_{\pm}^{2}$ and not $a_{\pm}$ in the last
equations) thus, equation (102) is separated as

$\displaystyle\hbar^{2}e^{-3(2-\beta)\Omega}\left[\frac{-\frac{d^{2}{\cal
A}}{d\Omega^{2}}+Q\frac{d{\cal A}}{d\Omega}}{{\cal A}}+\frac{\frac{d^{2}{\cal
B}_{+}}{d\beta_{+}^{2}}}{{\cal B}_{+}}+\frac{\frac{d^{2}{\cal
B}_{-}}{d\beta_{-}^{2}}}{{\cal B}_{-}}\right]$
$\displaystyle=\frac{24a_{+}^{2}a_{-}^{2}}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\frac{e^{3(2-\beta)\left[i\hbar\frac{\theta_{3}}{2}\frac{d}{d\phi}\right]}\frac{d^{\beta}{\cal
F}}{d\phi^{\beta}}}{{\cal F}}=\pm\ell^{2}=constant.$ (104)

with this, we can obtain the following separated equations

$\displaystyle\frac{d^{2}{\cal B}_{+}}{d\beta_{+}^{2}}\pm\ell_{+}^{2}{\cal
B}_{+}$ $\displaystyle=$ $\displaystyle 0,$ (105)
$\displaystyle\frac{d^{2}{\cal B}_{-}}{d\beta_{-}^{2}}\pm\ell_{-}^{2}{\cal
B}_{-}$ $\displaystyle=$ $\displaystyle 0,$ (106)
$\displaystyle\frac{d^{2}{\cal A}}{d\Omega^{2}}-Q\frac{d{\cal
A}}{d\Omega}\pm\left[\ell_{a}^{2}+\frac{\ell^{2}}{\hbar^{2}}e^{3(2-\beta)\Omega}\right]{\cal
A}$ $\displaystyle=$ $\displaystyle 0,$ (107) $\displaystyle
i\hbar\theta_{3}\frac{3(2-\beta)}{2}\frac{d^{\beta+1}{\cal
F}}{d\phi^{\beta+1}}+\frac{d^{\beta}{\cal
F}}{d\phi^{\beta}}\mp\frac{\ell^{2}\beta}{24\hbar^{\beta}a_{+}^{2}a_{-}^{2}}\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}{\cal
F}$ $\displaystyle=$ $\displaystyle 0.$ (108)

The solution for the function ${\cal A}$ is the same that in the commutative
case, equation (87).

To solve the fractional equation (108), we write it as follow:

$\frac{d^{\beta+1}F}{d\phi^{\beta+1}}+a\frac{d^{\beta}F}{d\phi^{\beta}}+bF=0,$
(109)

where

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{2}{3i\hbar\theta_{3}(2-\beta)},$ $\displaystyle b$
$\displaystyle=$
$\displaystyle\mp\frac{\ell^{2}\beta}{24\hbar^{\beta}a_{+}^{2}a_{-}^{2}}\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}\cdot\frac{2}{3i\hbar\theta_{3}(2-\beta)}=\pm\frac{i\ell^{2}\beta}{36\hbar^{\beta+1}\theta_{3}(2-\beta)a_{+}^{2}a_{-}^{2}}\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}.$

Applying the Laplace transform to (109), when $0<\beta\leq 2$, we have

$s^{\beta+1}F(s)-s^{\beta}f(0)-s^{\beta-1}f^{\prime}(0)-s^{\beta-2}f^{\prime\prime}(0)+as^{\beta}F(s)-as^{\beta-1}f(0)-as^{\beta-2}f^{\prime}(0)+bF(s)=0.$
(110)

Solving with respect to $F(s)$, we obtain

$\displaystyle F(s)$ $\displaystyle=$ $\displaystyle
f(0)\Big{[}\frac{s^{\beta}}{s^{\beta+1}+as^{\beta}+b}+\frac{as^{\beta-1}}{s^{\beta+1}+as^{\beta}+b}\Big{]}+f^{\prime}(0)\Big{[}\frac{s^{\beta-1}}{s^{\beta+1}+as^{\beta}+b}+\frac{as^{\beta-2}}{s^{\beta+1}+as^{\beta}+b}\Big{]}+$
(111) $\displaystyle+$ $\displaystyle
f^{\prime\prime}(0)\frac{s^{\beta-2}}{s^{\beta+1}+as^{\beta}+b}.$

To take the inverse Laplace transform, we use the following formula
universe2024-nc

$\mathcal{L}^{-1}\Big{[}\frac{s^{\gamma}}{s^{\alpha}+as^{\beta}+b}\Big{]}=t^{\alpha-\gamma-1}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[k(\alpha-\beta)+(n+1)\alpha-\gamma]}t^{k(\alpha-\beta)+n\alpha}$
(112)

Taking the inverse Laplace transform in (111), the solution of the fractional
differential equation (109), for the case $1<\beta\leq 2$, is given by

$\displaystyle{\cal F}_{3}(\phi)$ $\displaystyle=$ $\displaystyle
f(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k]}\Big{]}\phi^{n(\beta+1)+k}+$
(113) $\displaystyle+$ $\displaystyle
af(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+1]}\Big{]}\phi^{n(\beta+1)+k+1}+$
$\displaystyle+$ $\displaystyle
f^{\prime}(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+1]}\Big{]}\phi^{n(\beta+1)+k+1}+$
$\displaystyle+$ $\displaystyle
af^{\prime}(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+2]}\Big{]}\phi^{n(\beta+1)+k+2}+$
$\displaystyle+$ $\displaystyle
f^{\prime\prime}(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+2]}\Big{]}\phi^{n(\beta+1)+k+2}.$

So, from here, we obtain the case, when $0<\beta\leq 1$,

$\displaystyle{\cal F}_{1}(\phi)$ $\displaystyle=$ $\displaystyle
f(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k]}\Big{]}\phi^{n(\beta+1)+k}+$
(114) $\displaystyle+$ $\displaystyle
af(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+1]}\Big{]}\phi^{n(\beta+1)+k+1}+$
$\displaystyle+$ $\displaystyle
f^{\prime}(0)\Big{[}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{n+k}b^{n}a^{k}\binom{n+1+k}{k}}{\Gamma[(n+1)(\beta+1)-\beta+k+1]}\Big{]}\phi^{n(\beta+1)+k+1}.$

In the way in which the parameters of the differential equation for the
functional ${\cal F}$ are arranged, its solution for the $\theta=0$ case does
not reproduce the commutative case, since this parameter remains in the
denominator of the corresponding solution. It is in this sense that non-
commutativity makes the equations involved in anisotropic fractional cosmology
more complicated than in the non-fractional case.

On the other hand, employing the constraints equations (103) into equations
(105) and (106), we have

$\displaystyle\frac{d^{2}{\cal B}_{+}}{d\beta_{+}^{2}}\pm
i\theta_{1}\hbar\frac{3(2-\beta)\ell_{+}^{2}}{2a_{+}}\frac{d{\cal
B}_{+}}{d\beta_{+}}\pm\frac{\ell_{+}2}{a_{+}^{2}}{\cal B}_{+}$
$\displaystyle=$ $\displaystyle 0,$ (115) $\displaystyle\frac{d^{2}{\cal
B}_{-}}{d\beta_{-}^{2}}\pm
i\theta_{2}\hbar\frac{3(2-\beta)\ell_{-}^{2}}{2a_{-}}\frac{d{\cal
B}_{-}}{d\beta_{-}}\pm\frac{\ell_{-}2}{a_{-}^{2}}{\cal B}_{-}$
$\displaystyle=$ $\displaystyle 0,$ (116)

the auxiliary polynomial for the first equation becomes

$r^{2}\pm
i\theta_{1}\hbar\frac{3(2-\beta)\ell_{+}^{2}}{2a_{+}^{2}}r\pm\frac{\ell_{+}2}{a_{+}^{2}}=0,$

being the roots (in these roots the quadratic terms in $\theta_{j}$ have been
eliminated),

$r_{1}=\mp
i\theta_{1}\hbar\frac{3(2-\beta)\ell_{+}^{2}}{4a_{+}^{2}}+\sqrt{\mp\frac{\ell^{2}_{\pm}}{a_{+}^{2}}},\qquad
r_{2}=\mp
i\theta_{1}\hbar\frac{3(2-\beta)\ell_{+}^{2}}{4a_{+}^{2}}-\sqrt{\mp\frac{\ell^{2}_{\pm}}{a_{+}^{2}}}.$
(117)

The corresponding solutions are:

$\mathcal{B}_{\pm}={\mathcal{B}_{0}}_{\pm}Exp\left[\mp
i\theta_{j}\hbar\frac{\ell_{\pm}^{2}}{a_{\pm}^{2}}\frac{3(2-\beta)}{4}\beta_{\pm}\right]\,\,\left\\{\begin{tabular}[]{ll}$\cos{\left(\ell_{\pm}\beta_{\pm}-\alpha_{\pm}\right)}$,&\qquad
for \, $+\ell_{\pm}^{2}$\\\
$\cosh{\left(\ell_{\pm}\beta_{\pm}+\alpha_{\pm}\right)}$,&\qquad for \,
$-\ell_{\pm}^{2}$\end{tabular}\right.$ (118)

and taking into account the limit of $\theta_{j}=0$, this equation is reduced
to the commutative solutions (90).

Then, we have the probability density $|\Psi|^{2}$ by considering only
$\beta\not=2$, taking two sectors

###  V.2 for $0<\beta\leq 1$

Figure 3: In the following graphs, we show the behavior of the probability
density of the equation (121) for the dust era, considering the modified
Bessel function. Here, the values have been taken in the factor ordering
parameter $Q=-1,0,1$ and the non-commutative parameter in the direction
$\phi$, $\theta_{3}=0.1$ with $\psi_{0}=10^{-26}$ and $\theta_{3}=0.9$ with
$\psi_{0}=10^{-18}$. We have taken the initial value and the constant values
$f(0)=f^{\prime}(0)=f^{\prime\prime}(0)=1$, $\ell=0.5$, $\ell_{a}=0.03$,
$a_{+}=0.0,a_{-}=0.95$, as well as, the order in the modified Bessel function
$\rho_{+}=0.333933,0.02$.

Figure 4: In the following plots, we show the behavior of the probability
density, equation (121), for dust era, considering the modified Bessel
function. Here, the values have been taken in the factor ordering parameter
$Q=-1,0,1$ and the non commutative parameter in the direction with $\phi$ and
$\theta_{3}=9\times 10^{9}$. We have taken the initial value and the constant
values $f(0)=f^{\prime}(0)=f^{\prime\prime}(0)=1$, $\ell=0.5$,
$\ell_{a}=0.03$, $a_{+}=0.9,a_{-}=0.95$, $\psi_{0}=10^{-3}$ and the order in
the modified Bessel function $\rho_{+}=0.333933,0.02$.

$\displaystyle|\Psi|^{2}$ $\displaystyle=$
$\displaystyle\psi_{0}^{2}\,e^{Q\Omega}\,{\cal F}_{1}^{2}\times$ (121)
$\displaystyle\times\left\\{\begin{tabular}[]{ll}$e^{-\Theta_{i}}\,\cos^{2}{\left(\ell_{+}\beta_{+}-\alpha_{+}\right)}\cos^{2}{\left(\ell_{-}\beta_{-}-\alpha_{-}\right)}\,J^{2}_{\rho_{-}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\, $+\ell^{2}$\\\
$e^{\Theta_{i}}\,\,\cosh^{2}{\left(\ell_{+}\beta_{+}+\alpha_{+}\right)}\cosh^{2}{\left(\ell_{-}\beta_{-}+\alpha_{-}\right)}K^{2}_{\rho_{+}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$-\ell^{2}$\end{tabular}\right.$

with $e^{\pm\Theta_{i}}=Exp\left[\pm
i\hbar\frac{3(2-\beta)}{2}\left[\theta_{1}\frac{\ell_{+}^{2}}{a_{+}^{2}}\beta_{+}+\theta_{2}\frac{\ell_{-}^{2}}{a_{-}^{2}}\beta_{-}\right]\right]$

###  V.3 for $1<\beta\leq 2$

In figure 5, we show the probability density in the radiation stage of the non
commutative universe, where it is perceived that the evolution of the
universe, the global effect of the non-commutativity between the field
coordinates of the system in the fractional quantum cosmology scheme causes
the probability density to shift or shrink in the opposite direction in both
fields $(\Omega,\phi)$, causing the classical universe to emerge sooner, which
would mean that the current universe should have more time than is usually
mentioned, as in mentioned in the reference Jalalzadeh2023 , employing the
fractional framework. Here the behavior of the probability density of the wave
function remains in the format of quantum cosmology, not normalized, which is
why we do not give a specific value to the initial value of $\psi_{0}$.

Figure 5: In the following plots, we show the behavior of the probability
density, equation(124), for radiation era, considering the modified Bessel
function. It is done by taking the values of the factor ordering parameter
$Q=-1,0,1$, and the non commutative parameter in the direction $\phi$,
$\theta_{3}=0.1,0.9$. We have taken the initial value and the constant values
$f(0)=f^{\prime}(0)=f^{\prime\prime}(0)=0.1$, $\ell=0.5$, $\ell_{a}=0.03$,
$a_{+}=0.0,a_{-}=0.95$ and the order in the modified Bessel function
$\rho_{+}=0.500899,0.03$.

$\displaystyle|\Psi|^{2}$ $\displaystyle=$
$\displaystyle\psi_{0}^{2}\,e^{Q\Omega}\,{\cal F}_{3}^{2}\times$ (124)
$\displaystyle\times\left\\{\begin{tabular}[]{ll}$e^{-\Theta_{i}}\,\cos^{2}{\left(\ell_{+}\beta_{+}-\alpha_{+}\right)}\cos^{2}{\left(\ell_{-}\beta_{-}-\alpha_{-}\right)}\,J^{2}_{\rho_{-}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\, $+\ell^{2}$\\\
$e^{\Theta_{i}}\,\,\cosh^{2}{\left(\ell_{+}\beta_{+}+\alpha_{+}\right)}\cosh^{2}{\left(\ell_{-}\beta_{-}+\alpha_{-}\right)}K^{2}_{\rho_{+}}\left[2\frac{\ell}{\hbar
3(2-\beta)}e^{\frac{3}{2}(2-\beta)\Omega}\right]$,&for\,
$-\ell^{2}$\end{tabular}\right.$

Other ways of separating this prefactor do not lead us to anything plausible,
since when we make $\theta\to 0$, the values of the parameters on which the
roots of a cubic equation depend diverge, then the function ${\cal B}_{\pm}$
is indeterminate, so we decided not to put those cases.

##  VI Conclusions

From the quantum perspective of this work, some theories applied to fractional
cosmology in the K-essence formalism become very complicated, because a
prefactor survives that does not appear in non-fractional cosmology, since it
is absorbed in the choice of the appropriate gauge. Commutative behaviors
cannot be reproduced from these solutions, since the non-commutative parameter
enters the evolution of the functions in the denominator, and can diverge in
this limit. We only report the cases that we consider most plausible from a
physical point of view.

In this sense, the combination of fractional differential equations and non-
commutativity in anisotropic models seems to be not a good combination, since
when solving the resulting quantum equations, this combination causes the
domain of definition of the non-commutative parameter to change to another
domain of definition that does not allow us to viably recover the commutative
quantum world, as are shown in the figures 3,4 and 5. In this way, we
discarded the other choices of the prefactor given that due to the way in
which the non-commutative parameter is worked, it seems that we are facing an
ambiguity or separation of definition space.

For the very particular case treated in this work, the quantum evolution of
the probability density in the radiation stage is very similar to the standard
evolution, but with the novelty that the scalar field enters later in that
evolution according to the value of factor ordering parameter increases, and
the amplitude of this density decreases, which means that in the dust era of
the universe this evolution gives us the idea of two parallel universes, a
product of the fractional process. The fact that the scalar field enters later
means that it remains as a classical background for longer in the evolution of
the universe, possibly with large values, between stage and stage of the
universe.

In the classical fractional sector, the non-commutative parameter is appearing
in the scalar field relations, since that is where the fractional behavior
comes from, and not from the scale factor solutions (up to an exponential
prefactor of non-commutative parameters), as it usually appears in non-
fractional commutative cosmology, making The behavior of these functions is
different in both formalisms. However here, we can recover the classical
commutative world from the non-commutative sector.

###### Acknowledgements.

J.S. was partially supported by PROMEP grants UGTO-CA-3 and CIIC-118/2024. All
authors were partially supported by SNI-CONACyT. J.J.R appreciates the support
of DAIP, within the framework of the project Sobre la Dimensionalidad de los
Operadores Diferenciales Fraccionarios: grant 203/2024; We thank Itzayana
A.M.R for illuminating this work. This work is part of the collaboration
within the Instituto Avanzado de Cosmología. Many calculations were done by
Symbolic Program REDUCE 3.8.

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