#  n-dimensional non-commutative GUP quantization and application to the
Bianchi I model

Sebastiano Segreto<EMAIL_ADDRESS>Physics Department,
’Sapienza’ University of Rome, P.le Aldo Moro 5, Rome, 00185, Italy  INFN,
Sezione di Roma 1, P.le A. Moro 2, 00185 Rome, Italy  Giovannni Montani
<EMAIL_ADDRESS>ENEA, Fusion and Nuclear Safety Department, C. R.
Frascati, Via E. Fermi 45, 00044 Frascati (RM), Italy  Physics Department,
’Sapienza’ University of Rome, P.le Aldo Moro 5, Rome, 00185, Italy

###### Abstract

We analyse a $n$-dimensional Generalized Uncertainty Principle (GUP)
quantization framework, characterized by a non-commutative nature of the
configurational variables. First, we identify a set of states which are
maximally localized only along a single direction, at the expense of being
less localized in all the other ones. Subsequently, in order to recover
information about localization on the whole configuration space, we use the
only state of the theory which exhibits maximal localization simultaneously in
every direction to construct a satisfactory quasi-position representation, by
virtue of a suitable translational operator.

The resultant quantum framework is then applied to model the dynamics of the
Bianchi I cosmology. The corresponding Wheeler-DeWitt equation is reduced to
Schrödinger dynamics for the two anisotropy degrees of freedom, using a WKB
representation for the volume-like variable of the Universe, in accordance
with the Vilenkin scenario. The main result of our cosmological implementation
of the constructed quantum theory demonstrates how the dynamics of a wave
packet peaked at some point in the configuration space represented in the
quasi-position variables, favours as the most probable configuration exactly
the initial one for a relatively long time, if compared with the ordinary
quantum theory. This preference arises from the distinct behavioral dynamics
exhibited by wave packets in the two quantum theories.

##  I Introduction

As widely acknowledged, any quantum theory of gravity lays down its
foundations in the attempt to reconcile General Relativity with Quantum
Mechanics. Despite the lack of a comprehensive theory and the absence of
consensus regarding the predominant approach, the idea that addressing this
issue demands a profound reconsideration of the fundamental structure of
space-time is broadly spread and accepted.

Consequently, several models and theories based on a space-time structure
fundamentally different with respect to the one arising in General Relativity
have been constructed so far [1].

While certain frameworks aim to construct entirely new foundational paradigms,
there exist different formulations that can be considered effective models.
These models are capable of reproducing and making manifest some essential
aspects derived from a potentially unspecified ultimate theory that, in
principle, accounts for the altered structure of space-time.

Generalized uncertainty principle (GUP) theories are quantum non-relativistic
theories based on a deformation of the ordinary Heisenberg’s uncertainty
principle (HUP) and they do belong to this last category.

There exist several arguments, stemming from String and Gedanken experiments
[2], which suggest that a modification of the HUP is indeed necessary at some
level.

The first examples in which the subject has been comprehensively investigated
and analysed can be found in [3, 4], where considerations are specifically
grounded in the low energy limits of string dynamics [5, 6].

From these works clearly emerges that the most rigorous way to proceed in this
direction is to deform the usual Heisenberg algebra between the quantum
conjugate operators, hence modifying the canonical structure of ordinary
quantum mechanics. According to the specific algebra, this operation results
in two noteworthy consequences: firstly, the possible appearance of a nonzero
absolute minimal uncertainty in the coordinate operators, which equips the
theory with some kind of ”minimal structure”; secondly, the possible emergence
of a non-commutative ”geometry”, manifesting itself in the non-commutativity
of the operators associated to the configurational variables.

Despite it can help to gain useful insights, when this scheme is applied to
the quantum dynamics of a particle, not surprisingly, the corresponding
observables turn out to have negligible corrections. This is essentially due
to the fact that the scale at which the effect generated by the deformations
of the algebra becomes relevant is usually Planckian.

Nonetheless, the implementation of such generalized approaches in the Mini-
superspace [7] of cosmological models [8, 9] is potentially able to offer an
interesting perspective to introduce cut-off physics effects on the Universe
dynamics during the Planck era of its evolution.

Driven by these motivations, significant interest has emerged in the
literature (see in particular [10, 11] or more recently [12, 13, 14] for
general aspects as well as [15, 16, 17] for applications and relation with
different formalisms) for the Generalized Uncertainty Principle (GUP) physics,
in particular toward possible extensions of the original paradigm proposed in
[4].

Among these, a formulation that possesses a certain degree of generality and
hence significance is the one proposed in [10, 18]. More recently in [19], it
has been argued that this GUP theory, distinguished for the presence of a
square root in the position-momentum commutator, is a viable extension of the
theory, still preserving the existence of a non-zero minimal uncertainty for
the coordinate operator. This statement has been amended in [12], where it has
been demonstrated the necessity to deal with a compact momentum space in order
to recover the emergence of a minimal structure in this deformed algebra.

The effort to generalize these modified algebra theories also involved
constructing consistent multi-dimensional extensions. In this context, the key
consideration is that, as previously stated, modified Heisenberg algebras in
the $n$-dimensional scenario can determine a non-commutativity in the
coordinate operators. While there are studies exploring $n$-dimensional
commutative GUP theories [20], there is an evident absence of analysis
regarding their non-commutative counterparts.

it is evident that, in a mini-superspace formulation of the GUP framework, the
possibility for a higher-dimensional (and, in particular for a two-
dimensional) case gains in relevance for the simple fact that the Universe
dynamics can be in general described by three scale factors [21].
Specifically, referring to the Bianchi models class, these degrees of freedom
are essentially a volume-like variable and two physical anisotropy variables
[22].

The novelty of the present study is the construction of a quantum non-
commutative GUP theory in the case of a generic number of space coordinates,
able to offer a consistent dynamical scenario in which it is possible to
investigate the dynamics of the Bianchi I Universe in a WKB reduction [23] of
its quantum dynamics à la Vilenkin [24] (see also [25, 26, 27, 28]).

As a first and fundamental step of our analysis, we define a family of states,
within the $n$-dimensional GUP framework, able to reach the maximal
localization with respect to one generic coordinate, while the remaining one
unavoidably exhibits greater uncertainties. The states we construct from this
procedure have a general structure, wherein the minimal uncertainty can apply
to any of the $n$ coordinates. However, this consistently results in an
anisotropic effect, leaving the other coordinates non-uniformly localized.
This inherent anisotropy is a natural manifestation of the non-commutativity
of our GUP theory. While these derived states prove useful in determining
localization along a single direction, they fall short in mapping the entire
configuration space. Consequently, to achieve a suitable generalization of the
”quasi-position” representation of the quantum particle dynamics [4, 29, 30],
we leverage the only isotropic with respect to the maximal localization
properties state allowed within our proposed scenario. This state describes a
particle located at the origin of the coordinate system, having simultaneously
the same (minimal) uncertainty in all the space directions.

By employing a properly defined translational operator, we can shift the
expectation position value of this isotropic state to a generic point in the
configuration space, finally obtaining a family of states which can serve as a
generalization of the quasi-position basis. This way, we are able to properly
describe the localization properties and the evolution of a wave packet in the
quasi-position space. This method permits us to extract some phenomenological
considerations and insights from our study when applied to the behaviour of
the Bianchi I model.

We describe the Bianchi I cosmology via standard Misner variables [31], as
mentioned above. In order to deal with a Schrödinger equation for the quantum
dynamics - which is, in principle, fixed by a Wheeler-DeWitt formulation [32,
33] \- we adopt a WKB scenario à la Vilenkin, in which the isotropic
coordinate approaches a quasi-classical limit, while the two anisotropy
coordinates explore a small quantum phase-space (for previous applications of
the same type see [34, 35, 36]).

Our formulation of the two-dimensional GUP paradigm is then implemented to
describe the two anisotropy degrees of freedom as non-commutative coordinates
and, since the quantization has to refer to the momentum representation, we
adopt the construction proposed in [37].

The most relevant characteristic of our cosmological model lies in the
tendency of dynamics to favour the initial configurations as the most probable
states of our Universe, at least for a certain time. Specifically, the initial
states are probabilistically favoured for longer time intervals compared to
what happens in the ordinary theory. The greater significance of this fact
becomes clear when considering an isotropic state as the initial condition.
This behaviour is essentially due to the fact that the peak of our wave
packets is moving ”slowly” in time, exhibiting a certain degree of ”inertia”
in leaving the initial point of the quasi-anisotropies plane in which we have
placed it, i.e. the quasi-position plane in the case of the anisotropy
variables. It is clear that this signals a departure from ordinary quantum
mechanics, highlighting how non-commutative GUP algebras can introduce novel
and relevant effects in the dynamics of the early Universe and provide
insights - perhaps in more comprehensive models - on some of the open problems
in quantum cosmology.

The paper is organized as follows: in Section II we introduce the
$n$-dimensional non-commutative GUP Heisenberg algebra and we outline a
functional analysis of the framework. In Section III we face and address the
problem of the maximally localized states and their definition in a non-
commutative setting. In Section IV, exploiting the results of the previous
section, we proceed to define and construct a suitable generalization of the
quasi-position basis and its relation with the momentum representation. In
Section V, using Vilenkin’s scheme for quantum cosmology, we apply the
developed GUP quantum theory to the dynamics of the Bianchi I model and
explore its behaviour, comparing it to what happens in the ordinary quantum
scenario. Finally, in Section VI we give our conclusions, summarizing our
results and their relevant aspects.

Additionally, two appendices, Appendix A and Appendix B, containing explicit
and detailed calculations concerning relevant parts of the article are
included.

##  II Non-commutative GUP framework

The associative modified Heisenberg algebra we are going to analyze in this
section is the $n$-dimensional generalization of the Kempf-Mangano-Mann
algebra, extensively presented in [4]. In the one-dimensional scenario, the
deformation of the canonical commutation relation is written as:

$[\hat{\mathbf{x}},\hat{\mathbf{p}}]=i\hbar(1+\beta\hat{\mathbf{p}}^{2}),$ (1)

where $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$ are conjugate operators and
$\beta$ is a positive deformation parameter, with proper dimensions.

On momentum space wave functions $\psi(p)=\innerproduct{p}{\psi}$, the
operators can be represented as follows:

$\displaystyle\hat{\mathbf{p}}\ket{\psi}\rightarrow p\psi(p),$ (2)
$\displaystyle\hat{\mathbf{x}}\ket{\psi}\rightarrow i\hbar(1+\beta
p^{2})\partial_{p}\psi(p).$ (3)

It is straightforward to verify that this representation satisfies (1).

As exhaustively discussed in [3, 4], we can choose as domain of definition of
these operators the Schwartz space $\mathcal{S}$, which is a dense domain on
the Hilbert space of the theory
$\mathcal{H}=\mathcal{L}^{2}(\mathbb{R},dp/(1+\beta p^{2})$. The modified
Lebesgue measure assures the symmetry of the conjugate operators on the chosen
domain $\mathcal{S}$. Nevertheless, a careful functional analysis shows that,
while the $\hat{\mathbf{p}}$ operator is essentially self-adjoint on
$\mathcal{S}$, the $\hat{\mathbf{x}}$ operator is not. Rather, the coordinate
operator admits a one-parameter family of self-adjoint extensions. This
feature is the mathematical sign of a departure from ordinary quantum
mechanics.

On a more physical ground, all of this is translated in the emergence of an
absolute minimal uncertainty in the coordinate operator different from zero.

This, in principle, should be true on the physical domain of the theory, which
can be defined as the intersection of the domains of some fundamental
operators 111Here $\mathbf{\hat{x}}$ and $\hat{\mathbf{x}}^{2}$ refer to
proper self-adjoint extensions of these operators:

$\mathcal{D}_{phys}:=\mathcal{D}_{\mathbf{\hat{x}}}\cap\mathcal{D}_{\hat{\mathbf{x}}^{2}}\cap\mathcal{D}_{\mathbf{\hat{p}}}\cap\mathcal{D}_{\mathbf{\hat{p}^{2}}}\cap\mathcal{D}_{\commutator{\mathbf{\hat{x}}_{\lambda}}{\mathbf{\hat{p}}}}.$
(4)

Nevertheless, this condition can be weakened, imposing that the minimization
procedure holds for all the states belonging to the domain :

$\mathcal{D}_{min}:=\mathcal{D}_{\mathbf{\hat{x}}_{\lambda}}\cap\mathcal{D}_{\hat{\mathbf{p}}}.$
(5)

Indeed, it can be shown that this condition is enough to define the
uncertainties relations for a wave function (see [38, 12].

These states cannot be considered physical states in a strong way, as those
which belong to $\mathcal{D}_{phys}$, but they can be considered physical
states in a more ”kinematic” sense.

Throughout the paper, we will make reference to this last definition.

The presence of this absolute minimum in the physical quantity
$\Delta\hat{\mathbf{x}}$ is easily interpreted as a limit in the possibility
of arbitrarily localizing objects in the configuration space of the theory. In
light of this, the physical meaning of the coordinate representation, which is
a point-wise concept, is lost. The eigenstates of the $\hat{\mathbf{x}}$
operator do exist but they have only a formal value as well as the basis they
form.

In this scenario, in order to recover information on localization in the
configuration space, those states which realize the absolute minimal
uncertainty in the coordinate operator can be used as a new basis. These
states are called ”maximally localized states” and, in the case in which the
$\hat{\mathbf{x}}$ operator is the position operator, the new representation
they give birth to is known as ”quasi-position representation”.

The generalisation of this algebra to the $n$-dimensional case can be achieved
by assuming the commutation relation (1) for every couple of conjugate
variables:

$[\hat{\mathbf{x}}_{i},\hat{\mathbf{p}}_{j}]=i\hbar\delta_{ij}(1+\beta\hat{\vec{\mathbf{p}}}^{2}),$
(6)

where now the momentum on the right-hand side is the total momentum.

We require that:

$[\hat{\mathbf{p}}_{i},\hat{\mathbf{p}}_{j}]=0.$ (7)

In this way, momentum representation is well-defined and we can generalize the
representations (2)-(3) of the conjugate operators:

$\displaystyle\hat{\mathbf{p}}_{i}\ket{\psi}\rightarrow p_{i}\psi(p),$ (8)
$\displaystyle\hat{\mathbf{x}}_{i}\ket{\psi}\rightarrow
i\hbar(1+\beta\vec{p}^{\;2})\partial_{p_{i}}\psi(p).$ (9)

By exploiting these representations it is possible to derive explicitly the
commutation relations between coordinate operators:

$[\hat{\mathbf{x}}_{i},\hat{\mathbf{x}}_{j}]=2i\hbar\beta(\hat{\mathbf{p}}_{i}\hat{\mathbf{x}}_{j}-\hat{\mathbf{p}}_{j}\hat{\mathbf{x}}_{i}).$
(10)

From (10) it is clear that in this framework the $\hat{\mathbf{x}}$ operators
do not commute between each other. This relevant feature of the theory can be
interpreted as the emergence of a ”non-commutative geometry” in the
configuration space, fixed by the non-trivial non-commutativity of the
coordinate operators.

By introducing the operator:

$\hat{\mathbf{L}}_{ij}=\frac{1}{1+\beta\hat{\vec{\mathbf{p}}}^{2}}(\hat{\mathbf{x}}_{i}\hat{\mathbf{p}}_{j}-\hat{\mathbf{x}}_{j}\hat{\mathbf{p}}_{i}),$
(11)

we can rewrite (10) as follows:

$[\hat{\mathbf{x}}_{i},\hat{\mathbf{x}}_{j}]=-2i\hbar\beta\frac{1}{\left(1+\beta\hat{\vec{\mathbf{p}}}^{2}\right)}\hat{\mathbf{L}}_{ij}.$
(12)

The operator $\hat{\mathbf{L}}_{ij}$ is the generator of n-dimensional
rotations in this quantum theory or, in other words, is the generalisation of
ordinary orbital angular momentum operator, as it is clear from its momentum
representation.

Given that, it appears evident how the $n$-dimensional modified Heisenberg
algebra we are dealing with preserves rotational symmetry, while the
translation invariance is lost. This entails that the translation group is not
defined in the usual sense in this quantum framework.

The functional analysis of the operators in the $n$-dimensional case, even if
more involved, is basically the same as the one-dimensional case.

The Hilbert space of the theory is the space
$\mathcal{H}=\mathcal{L}^{2}\left(\mathbb{R}^{n},dp/(1+\beta\vec{p}^{\;2})\right)$.
The operators $\hat{\mathbf{x}}_{i}$ and $\hat{\mathbf{p}}_{i}$ can be defined
on the Schwartz space $\mathcal{S}$, which is dense in $\mathcal{H}$. Here the
$\hat{\mathbf{p}}_{i}$ operators results to be symmetric and essentially self-
adjoint. The formal eigenstates of the total momentum operator are
$n$-dimensional Dirac deltas and for consistency we define their inner product
as follows:

$\innerproduct{p}{p^{\prime}}=(1+\beta\vec{p}^{\;2})\delta^{n}(p-p^{\prime}).$
(13)

On the other hand, the coordinate operators in every direction are symmetric
operators on $\mathcal{S}$ but they are not essentially self-adjoint. Once
again, each of them admits a one-parameter family of self-adjoint extensions.

This fact suggests, by analogy with the one-dimensional case, that a limit in
localization in configuration space arises in every direction. Nevertheless,
in order to properly understand the structure of this quantum theory, it is
important to recognize the relevant role played by the non-commutativity in
the configuration space, which adds another challenge concerning objects
localization.

##  III Maximally localized states

We can use the squeezed states method [4, 11] to identify the maximally
localized states of theory and the corresponding uncertainty in the
configuration variables, for every direction. In the case of these
uncertainties being different from zero, we can state that there is a limit in
localization in the configuration space, i.e. minimal structures emerge within
this framework.

In this context by ”squeezed states” we are referring to those states which
saturate the uncertainty relation between conjugate operators:

$\Delta\hat{\mathbf{x}}_{i}\Delta\hat{\mathbf{p}}_{j}\geq\frac{\hbar}{2}\absolutevalue{\expectationvalue{[\hat{\mathbf{x}}_{i},\hat{\mathbf{p}}_{j}]}}\rightarrow\Delta\hat{\mathbf{x}}_{i}\Delta\hat{\mathbf{p}}_{j}=\frac{\hbar}{2}\absolutevalue{\expectationvalue{[\hat{\mathbf{x}}_{i},\hat{\mathbf{p}}_{j}]}}.$
(14)

Following [4, 11], from (14) it is possible to obtain a first order
differential equation:

$\displaystyle\left[\hat{\mathbf{x}}_{i}-\xi_{i}+i\hbar\delta_{ij}\Lambda(\hat{\mathbf{p}}_{j}-\eta_{j})\right]\ket{\Psi_{\Lambda}^{i}}=0\Rightarrow$
$\displaystyle\left[i\hbar(1+\beta\vec{p}^{\;2})\partial_{p_{i}}-\xi_{i}+i\hbar\delta_{ij}\Lambda(p_{j}-\eta_{j})\right]\Psi_{\Lambda}^{i}(p)=0,$
(15)

where we have set the following identities:

$\displaystyle\xi_{i}:=\expectationvalue{\hat{\mathbf{x}}_{i}},$ (16)
$\displaystyle\eta_{j}:=\expectationvalue{\hat{\mathbf{p}}_{j}},$ (17)
$\displaystyle\Lambda:=\frac{\expectationvalue{[\hat{\mathbf{x}}_{i},\hat{\mathbf{p}}_{j}]}}{2(\Delta\hat{\mathbf{p}}_{j})^{2}}.$
(18)

The solutions of (III) are exactly the squeezed states of theory if and only
if they satisfy some specific conditions which are necessary to guarantee the
connection between the differential equation (III) and relation (14) and the
validity of the previous identities. An extensive discussion can be found in
[11] and in [12].

In the $n$-dimensional case we are analyzing we can write $n$ differential
equations (III), one for every couple of conjugate variables. Let us call
$\hat{\mathcal{A}}^{i}_{\Lambda}$ the operator which annihilates the state
$\Psi_{\Lambda}^{i}$ in (III).

The first thing to notice is that:

$[\hat{\mathcal{A}}^{i}_{\Lambda},\hat{\mathcal{A}}^{j}_{\Lambda}]=[\hat{\mathbf{x}}_{i},\hat{\mathbf{x}}_{j}]\neq
0\quad\text{if \;\;}i\neq j.$ (19)

From this simple commutation relation we can affirm that in this theory is not
possible in general to find states that are maximally localized in different
directions at the same time. This aspect of the theory is, as expected, the
first consequence of the presence of a non-commutativity in configuration
space: we can maximally localize an object in any direction, at the expense of
de-localize it of a certain amount in all the other ones.

If we solve (III) for the generic $i$-th direction we obtain the following
wave function 222Here we solved the equation for $\eta_{i}=0$ since numerical
computations show that the uncertainty in the coordinate operator is minimized
by this choice. This can be further supported by the analysis of the one-
dimensional case in [4]:

$\displaystyle\Psi^{i}_{\Lambda}(p)=\mathcal{C}\;g\left(p_{\\{j\neq
i\\}}\right)(1+\beta p_{r}p_{s}\delta^{rs})^{-\frac{\Lambda}{2\beta}}$ (20)
$\displaystyle\times\exp{-i\xi_{i}\frac{\tan[-1](\frac{\sqrt{\beta}p_{i}}{\sqrt{1+\beta
p_{r}p_{s}\delta^{rs}|_{r,s\neq i}}})}{\hbar\sqrt{\beta}\sqrt{1+\beta
p_{r}p_{s}\delta^{rs}|_{r,s\neq i}}}},$ (21)

where $g\left(p_{\\{j\neq i\\}}\right)$ is a not-specified function of momenta
different from $p_{i}$, $\mathcal{C}$ is the normalization constant and the
indices $r,s$ run over all the momenta, including $p_{i}$, except when
differently specified.

From expression (20) it is clear that the final form of the maximally
localized states and the respective uncertainty in $\hat{\mathbf{x}}_{i}$,
will depend not only on the parameter $\Lambda$ but also on the function
$g\left(p_{\\{j\neq i\\}}\right)$.

Since we need to deal whit wave functions for which all the uncertainties
relations hold and are well-defined, it is clear that this $g\left(p_{\\{j\neq
i\\}}\right)$ function cannot be completely arbitrary.

In order to fulfill our requests, the state (20) have to be a normalizable
state belonging to the domain of all the $\hat{\mathbf{p}}_{k}$ and
$\hat{\mathbf{x}}_{k}$ operators (see above). Imposing these conditions
entails some integrability requirements and accordingly some constraints over
the possible values of $\Lambda$. More specifically, the bounds on $\Lambda$
are produced by the explicit integration over the $p_{i}$ variable, that we
can perform in any case since the function $g\left(p_{\\{j\neq i\\}}\right)$
does not depend on $p_{i}$.

The conditions the state has to meet are the following:

  * •

normalizability:

$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!\\!\\!d^{n-1}p\,(1+p_{l}p_{m}\delta^{lm})^{-\frac{1}{2}-\Lambda}\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}<+\infty$
$\displaystyle\text{with}\quad\Lambda>-\frac{1}{2}$ (22)

  * •

belonging to the domain of $\hat{\mathbf{p}}_{i}$:

$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!\\!\\!d^{n-1}p\,(1+p_{l}p_{m}\delta^{lm})^{\frac{1}{2}-\Lambda}\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}<+\infty$
$\displaystyle\text{with}\quad\Lambda>\frac{1}{2}$ (23)

  * •

belonging to the domain of $\hat{\mathbf{p}}_{j\neq i}$:

$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!\\!\\!d^{n-1}p\,p_{j}^{2}(1+p_{l}p_{m}\delta^{lm})^{-\frac{1}{2}-\Lambda}\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}<+\infty$
$\displaystyle\text{with}\quad\Lambda>-\frac{1}{2}$ (24)

  * •

belonging to the domain of $\hat{\mathbf{x}}_{i}$:

$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!\\!\\!d^{n-1}p\,(1+p_{l}p_{m}\delta^{lm})^{-\frac{1}{2}}\left[(1+p_{l}p_{m}\delta^{lm})\Lambda^{2}+\right.$
(25)
$\displaystyle\left.(2\Lambda-1)\left(\xi_{i}\right)^{2}\right]\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}<+\infty$
$\displaystyle\text{with}\quad\Lambda>\frac{1}{2}$ (26)

  * •

belonging to the domain of $\hat{\mathbf{x}}_{j\neq i}$ 333This condition was
verified numerically:

$\displaystyle\int_{\mathbb{R}^{n}}\\!\\!\\!\\!\\!\\!d^{n}p\,\absolutevalue{i\hbar(1+p_{r}p_{s}\delta^{rs}\partial_{p_{j}}\Psi^{i}_{\Lambda}(p)}^{2}<+\infty$
$\displaystyle\text{with}\quad\Lambda>\frac{3}{2}$ (27)

As it should be clear, in the previous expressions the indices $j,l,m\neq i$
and we have recast everything in the ”natural” units of the theory, that is
$[\Lambda]=\beta$ and $[p]=1/\sqrt{\beta}$.

We pause a moment to clarify the choice of domains with respect to which we
determined the previous conditions. For the operators $\hat{\mathbf{p}}_{i}$
and $\hat{\mathbf{p}}_{j\neq i}$ we considered the only extension of
$\mathcal{S}$ in which they result to be self-adjoint (see e.g. [12]).
Concerning the position operators - which we remind are not essentially self-
adjoint - we considered the domain of the adjoint operators. This is indeed a
suitable choice since all the self-adjoint extensions of the starting domain
$\mathcal{S}$ are contained in the domain of the $\hat{\mathbf{x}}^{\dagger}$
operators. Therefore, even though we are not particularly concerned with
ensuring that the maximally localized states belong to the domain of adjoint
operators, due to the aforementioned inclusion relation, imposing belonging in
these domains establishes a minimal condition that these states must still
satisfy. The specific belonging to ”smaller” self-adjoint domains may at most
impose more stringent conditions.

Now, in order to understand the role played by this function in determining
the minimum of the uncertainty of the $\hat{\mathbf{x}}_{i}$ operator, we can
explicitly derive the expression of $\Delta\hat{\mathbf{x}}^{i}_{\Lambda}$ for
a generic function $g\left(p_{\\{j\neq i\\}}\right)$, which results to be:

$\displaystyle\Delta\hat{\mathbf{x}}^{i}_{\Lambda}=$
$\displaystyle\left[\frac{\bigintsss_{\mathbb{R}^{n-1}}\\!d^{n-1}p\,\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}(1+p_{l}p_{m}\delta^{lm})^{\frac{1}{2}-\Lambda}}{\bigintsss_{\mathbb{R}^{n-1}}\\!d^{n-1}p\,\absolutevalue{g\left(p_{\\{j\\}}\right)}^{2}(1+p_{l}p_{m}\delta^{lm})^{-\frac{1}{2}-\Lambda}}\right.$
$\displaystyle\left.\times\left(\frac{\Lambda^{2}}{2\Lambda-1}\right)\right]^{\frac{1}{2}}\hbar\sqrt{\beta}$
(28)

where again the indices $j,l,m\neq i$ and we have carried out the integration
in the $p_{i}$ variable.

From basic properties of the Lebesgue integral, we can affirm that the ratio
of the two integrals in (III) is always greater than one. Indeed the first
integrand is always greater than the second one on the whole
$\mathbb{R}^{n-1}$ space, except at the origin, where the two are equals.

On this ground we can write:

$\Delta\hat{\mathbf{x}}^{i}_{\Lambda}>\frac{\Lambda^{2}}{2\Lambda-1}\hbar\sqrt{\beta}$
(29)

Since the right-hand side of the previous expression reaches its minimum for
$\Lambda=1$ (in $\beta$ units), finally we deduce that
$\Delta\hat{\mathbf{x}}^{i}_{\Lambda}>1$ (in $\hbar\sqrt{\beta}$ units). We
notice that the obtained lower bound is exactly the minimal uncertainty found
in [4] for the one-dimensional case.

The existence of the bound itself for $\Delta\hat{\mathbf{x}}^{i}_{\Lambda}$
suggests that, in principle, it should be possible to model the function
$g(p_{\\{j\\}})$ in order to make the uncertainty arbitrarily close to one.

Thinking of the $g-$functions as weights for integral, it is not hard to
understand that this is exactly what happens when we select functions centered
in the origin, with a gradually narrower profile.

Indeed, by choosing this kind of functions, the main contribution for the two
integrals comes from the region around the origin, where the two integrands
are essentially the same.

Therefore, in a similar scenario, while the ratio of the two integrals tends
towards one, accordingly we will have:

$\Lambda_{min}\to 1\quad\text{and}\quad\Delta\hat{\mathbf{x}}_{i}\to 1.$ (30)

Nevertheless, all those states which tend to realize this condition are not
acceptable, since it is not possible to consistently define for them the
uncertainties in all the other directions. Indeed, all the states for which
$\Delta\hat{\mathbf{x}}_{i}\to 1$, realize this condition for
$\Lambda_{min}\to 1$ and hence they violate the previous fundamental
conditions that we have derived on $\Lambda$ in order to deal with a proper
wave function. In particular, it is the condition of belonging to the domain
of the $\hat{\mathbf{x}}_{j\neq i}$ operators which is not fulfilled in the
case of $\Lambda\to 1$.

From a physical point of view, by analogy with ordinary quantum mechanics, we
could say that these states, while almost maximally localized in one
direction, tend to be completely delocalized in all the other ones, as it
could be reasonably expected due to the presence of non-commutativity.

From all these considerations it should be now clear that - if an absolute
minimum in the uncertainty of the $i$-th coordinate operator exists - the
$g$-function realizing this condition must be identified without contravening
the constraints we have established.

This could be done by means of a constrained variational principle (see [11,
12]). Nevertheless, we will not carry out this task further, here. We simply
observe, for example, that a constant $g$-function, while not guaranteed to be
the one leading to the minimum configuration, satisfies all the requirements
and, as it will be clearer from what follows, it plays somehow a special role.

The point is that, through this procedure, we could be able to obtain states
maximally localized only along one direction, containing only information
about that specific direction. This information is carried by the $\xi_{i}$
parameter, which is the expectation value of the $i$-th coordinate operator
for the considered wave function.

These states alone cannot help us to map the whole configuration space and
hence to recover in a satisfactory way information on position.

To pursue this task we start from the following consideration: even if it is
true that in general the $\hat{\mathcal{A}}^{i}_{\Lambda}$ operators do not
commute each other, there is a particular case that represents an exception.
Indeed, if the $\xi_{i}$ parameters are zero the
$\hat{\mathcal{A}}^{i}_{\Lambda}$ operators do commute and this means that
they have a common eigenstates basis.

From the wave function in (20) it appears evident that this family of states
is represented by:

$\Phi_{\Lambda}(p)=\mathcal{N}(1+\beta
p_{s}p_{r}\delta^{sr})^{-\frac{\Lambda}{2\beta}}.$ (31)

This object is a rotational invariant state in momentum space and its
expectation value of position in configuration space is zero.

We can easily compute the normalization constant $\mathcal{N}$ using
hyperspheric coordinates:

$\Phi_{\Lambda}(p)=\sqrt{\left(\frac{\beta}{\pi}\right)^{n/2}\frac{\Gamma\left[1+\frac{\Lambda}{\beta}\right]}{\Gamma\left[1-\frac{n}{2}+\frac{\Lambda}{\beta}\right]}}(1+\beta
p_{r}p_{s}\delta^{rs})^{-\frac{\Lambda}{2\beta}}.$ (32)

With the same technique, we can also obtain the uncertainties in the
coordinate operators, which are the same in every direction:

$\displaystyle(\Delta\hat{\mathbf{x}}_{i})^{2}_{\Lambda}\\!=\\!\\!-\hbar^{2}\\!\\!\int_{\mathbb{R}^{n}}\\!\\!\\!\\!d^{n}p\;\Phi^{*}_{\Lambda}(p)\partial_{p_{i}}\left[\left(1+\beta
p_{r}p_{s}\delta^{rs}\right)\partial_{p_{i}}\Phi_{\Lambda}(p)\right]\\!=$
$\displaystyle\hbar^{2}\left(\frac{\beta}{\pi}\right)^{n/2}\frac{\Gamma\left[1+\frac{\Lambda}{\beta}\right]}{\Gamma\left[1-\frac{n}{2}+\frac{\Lambda}{\beta}\right]}\int
d\Omega_{n-1}\int_{\mathbb{R}^{+}}dr\;r^{n-2}$
$\displaystyle\times\int_{\mathbb{R}}\\!\\!dp_{i}\\!\left(1+\beta
r^{2}\\!\\!+p^{2}_{i}\right)^{-\frac{\Lambda}{\beta}-1}(1+\beta
r^{2}\\!-\\!p^{2}_{i}(-\beta+\Lambda))\Lambda=$
$\displaystyle\frac{\Lambda^{2}}{(2\Lambda-n)}\;\hbar^{2}\beta,\quad\quad\Lambda>\frac{n}{2},\;n\in\mathbb{N},$
(33)

where $r=\sum_{j\neq i}p^{2}_{j}$, $\Omega_{n-1}$ is the $(n-1)$-dimensional
solid angle and, in the result, we are using again the natural units suggested
by theory, i.e. $[x_{i}]=\hbar\sqrt{\beta}$ and $[\Lambda]=\beta$.

In order to reach the absolute minimum value of the quantity
$(\Delta\hat{\mathbf{x}}_{i})^{2}_{\Lambda}$ for the wave function (32), we
need to minimize (III) with respect to $\Lambda$. This operation leads to the
following result:

$\Lambda_{min}=n\Rightarrow\Delta\hat{\mathbf{x}}_{i}^{min}=\sqrt{n}\;\hbar\sqrt{\beta}.$
(34)

In the end we can write the desired expression for the wave function (32):

$\Phi^{ml}(p)=\sqrt{\left(\frac{\beta}{\pi}\right)^{n/2}\frac{\Gamma\left[1+n\right]}{\Gamma\left[1+\frac{n}{2}\right]}}(1+\beta\;p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}}.$
(35)

We stress the fact that, even if the second line integral in (III) is defined
only for $n>1$, the final result is valid $\forall n\in\mathbb{N}$. This is
confirmed by expressions (34) and (35), which, for $n=1$, are exactly the
maximally localized state (with $\xi=0$) and the relative uncertainty found in
[4] for the one-dimensional case.

The state (35) appears then to be a special state for the theory, since it is
the only state - at least according to the squeezed state method - which can
be maximally localized by the same amount, simultaneously in every direction.

##  IV Quasi-position basis

As stated in the previous section, maximally localized states along one
direction are not enough to recover information on position in configuration
space. This holds also for the special state (35), which cannot be used as a
basis to construct a new representation, since there is no parameter that can
promoted to a variable. Nevertheless, the obtained wave function preserves its
privileged role as a function able to minimize by the same amount at the same
time the uncertainties in the coordinate operators. Hence, to reintroduce the
parameter $\xi_{i}$ and construct a new basis to map the configuration space,
following the path suggested in [4] for the quasi-position representation (for
a discussion of this representation see [29, 30], we propose to ”translate” in
every direction the state (35), so that we can obtain a new state for which
holds the following:

$\frac{\bra{\Phi^{ml}_{T}}\hat{\mathbf{x}}_{k}\ket{\Phi^{ml}_{T}}}{\innerproduct{\Phi^{ml}_{T}}{\Phi^{ml}_{T}}}=\xi_{k},\quad\forall
k.$ (36)

The latter expression has to be taken as a definition for what we mean in this
scenario for a translation, since, strictly speaking, the translation group in
the usual sense is not defined in this kind of theories.

The new wave function which allows us to achieve the validity of the
expression (36) is suggested from the form of the state (20):

$\displaystyle\Phi^{ml}_{T}(p)=\sqrt{\left(\frac{\beta}{\pi}\right)^{n/2}\frac{\Gamma\left[1+n\right]}{\Gamma\left[1+\frac{n}{2}\right]}}(1+\beta\;p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}}$
$\displaystyle\times\prod_{k=1}^{n}\exp{-i\xi_{k}\frac{\arctan(\frac{\sqrt{\beta}p_{k}}{\sqrt{1+\beta
p_{r}p_{s}\delta?{rs}|_{r,s\neq k}}})}{\hbar\sqrt{\beta}\sqrt{1+\beta
p_{r}p_{s}\delta^{rs}|_{r,s\neq k}}}}.$ (37)

The normalization constant is unchanged since we have only added complex
exponential factors to the state (35) and it is not difficult to show that the
conditions (36) are satisfied. The uncertainties in the coordinate operators
can be found after performing several $n$-dimensional integrals by means of
hyperspheric coordinates and the obtained expression is the same for any
direction. The details of this calculation can be found in the appendix. In
the end we can write:

$\displaystyle(\Delta{\hat{\mathbf{x}}_{i}})^{2}=\frac{1}{2}$
$\displaystyle\left\\{2n+\frac{2+4n}{(n-1)(4n^{2}-1)}\sum_{k\neq
i}\xi^{2}_{k}+\right.$
$\displaystyle\left.\\!\\!\frac{n!}{\Gamma[3/2+n]}\sqrt{\pi}\biggl{(}\alpha(n)+\theta(n)\pi^{2}\biggr{)}\sum_{k\neq
i}\xi^{2}_{k}\right\\}\hbar^{2}\beta,$ (38)

where $\alpha(n)$ and $\theta(n)$ are respectively a real negative function
and a real positive function depending on the dimension $n$, for which there
is no closed form. A table of their values can be found in the appendix A.

The most relevant fact to notice is that the uncertainties in the coordinate
operators are now dependent on the $\xi_{i}$ parameters, i.e. the expectation
values on the configuration variables $x_{i}$. Therefore, by using states (IV)
as a basis, the possibility of localizing an object in the configuration space
and the limit to this procedure will depend on where we want to localize. This
means that with respect to the localizing properties, the configuration space
of the theory appears to be somehow ”inhomogeneous”. In particular, from (IV),
it is clear that, once we have fixed an origin in our configuration space,
away from this point the localization process becomes increasingly worse.
Accordingly, we can strictly talk about localized objects only around the
fixed origin.

Now that we have constructed our basis we can define a map from momentum space
to quasi-position space:

$\displaystyle\tilde{\psi}(\xi):=$
$\displaystyle\int_{\mathbb{R}^{n}}\frac{d^{n}p}{(1+\beta
p_{r}p_{s}\delta^{rs})}\innerproduct{\Phi_{T}^{ml}}{p}\innerproduct{p}{\psi}=$
$\displaystyle\int_{\mathbb{R}^{n}}\frac{d^{n}p}{(1+\beta
p_{r}p_{s}\delta^{rs})}\overline{\Phi_{T}^{ml}(p)}\psi(p).$ (39)

This map can be properly inverted, obtaining a transformation from quasi-
position space to momentum space:

$\displaystyle\psi{(p)}:=$ $\displaystyle\frac{(1+\beta
p_{r}p_{s}\delta^{rs})^{\frac{n}{2}+1}}{\sqrt{2\pi}^{n}}\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}\int_{\mathbb{R}^{n}}d^{n}\xi\;\psi(\xi)$
(40)
$\displaystyle\times\prod_{k=1}^{n}\exp{-i\xi_{k}\frac{\arctan(\frac{\sqrt{\beta}p_{k}}{\sqrt{1+\beta
p_{r}p_{s}\delta^{rs}|_{r,s\neq k}}})}{\hbar\sqrt{\beta}\sqrt{1+\beta
p_{r}p_{s}\delta^{rs}|_{r,s\neq k}}}},$ (41)

where $V(p)$ is a vector field properly defined and $\mathcal{J}$ is the
associated Jacobian matrix.

How we have constructed the inverse transform guarantees the unitarity of the
transform itself.

The details of the derivation of the inverse transform and all the necessary
definitions can be found in the appendix B.

##  V Non-commutative GUP quantization of Bianchi I model

In this section, we are going to discuss the quantization procedure of a
cosmological model, namely the Bianchi I model, within the framework we have
developed so far. As it is widely known, one-particle quantization schemes are
well-suited for the cosmological context. This is true since space-times
considered in cosmology are usually homogeneous space-times, therefore the
gravitational field we are dealing with has a finite number of degrees of
freedom. In this scenario, a GUP quantization procedure is potentially able to
provide the theory with some new relevant features, such as the emergence of
minimal structures and different localization properties due to the presence
of non-commutativity in configuration space, which can modify in a meaningful
way the dynamics of the Universe and its behavior.

In order to apply rigorously the machinery we have developed to our
cosmological model, we need to work in a quantum formulation where the
probabilistic interpretation of the wave function of the system is well-posed
and meaningful and where the dynamics is dictated by a true Schrödinger
equation.

This is a delicate issue in quantum cosmology, which can be addressed in
different ways. In this paper we will adopt the Vilenkin approach [24]. In
Vilenkin’s idea, in order to achieve the above configuration, we need to
separate the system, by means of its degrees of freedom, in a semi-classical
part and a full quantum part. As we will see, it is the presence of a part of
the system treated semi-classically that allows us to obtain the desired
dynamics for the quantum part.

The conditions that must be satisfied in order to have a consistent and well-
defined procedure are extensively discussed in [24].

Here we stress the fact that we need to assume that the quantum subsystem has
a negligible effect on the semi-classical part. In this sense we can talk
about a ”small” quantum part of the system.

Now, we are going to re-derive explicitly every step for the particular chosen
case, working in momentum representation. As it should be clear from the
previous sections, this is the natural choice within a GUP theory.

The configurational variables and the respective conjugate momenta we are
adopting are the so-called Misner variables $(\alpha,\gamma_{+},\gamma_{-})$
and $(p_{\alpha},p_{+},p_{-})$ [31]. These variables are able to diagonalize
the kinetic part of the Hamiltonian of our system - and this is true for
several Bianchi models - and it can be shown that they have a clear physical
meaning. Indeed, the variable $\alpha$ is related to the isotropic volume of
the Universe, while the variables $\gamma_{+}$ and $\gamma_{-}$ are known as
anisotropies, since they ”measure” the deviation from the completely isotropic
Universe.

The Hamiltonian constraint describing a Bianchi I universe is the following:

$\mathcal{H}=N(t)e^{-3\alpha}\left(-p_{\alpha}^{2}+p_{+}^{2}+p_{-}^{2}\right)\approx
0,$ (42)

where $N(t)$ is the lapse function, coming out from the $3+1$ reduction
procedure of the metric [39], where we have set the speed of light in vacuum
$c$ and the Einstein constant $\kappa$ is equal to one.

For a consistency matter, as it will be clear later, we need to consider
additionally a positive cosmological constant term in our Universe, hence
rewriting the Hamiltonian constraint as:

$\mathcal{H}=N(t)e^{-3\alpha}\left(-p_{\alpha}^{2}+p_{+}^{2}+p_{-}^{2}+\Lambda
e^{6\alpha}\right)\approx 0.$ (43)

By inspecting (43) it is evident how the cosmological constant part acts as a
potential for the system.

The constraint (43) has to be promoted to be a quantum operator:

$\hat{\mathbf{\mathcal{H}}}=\left(-\hat{\mathbf{p}}_{\alpha}^{2}+\Lambda
e^{6\hat{\mathbf{\alpha}}}+\hat{\mathbf{p}}_{+}^{2}+\hat{\mathbf{p}}_{-}^{2}\right)\ket{\Psi}=0,$
(44)

where we can identify two parts:

  * •

the semi-classical part
$\hat{\mathcal{H}}_{0}=-\hat{\mathbf{p}}_{\alpha}^{2}+\Lambda
e^{6\hat{\mathbf{\alpha}}}$, which concerns essentially the volume of the
Universe;

  * •

the full quantum part
$\hat{\mathcal{H}}_{q}=\hat{\mathbf{p}}_{+}^{2}+\hat{\mathbf{p}}_{-}^{2}$,
which is completely described by the anisotropies.

By considering a parameter $\lambda\propto\hbar$, we can express the
”smallness condition” of the quantum subsystem with respect to the semi-
classic part in a more precise way:

$\hat{\mathcal{H}}_{q}/\hat{\mathcal{H}}_{0}=o(\lambda).$ (45)

By following the reasoning line of [24], we can write the ansatz for the
solution of (44) in momentum space:

$\Psi(p_{\alpha},p_{\pm})\\!=\\!A(p_{\alpha})e^{iS(p_{\alpha})/\hbar}\chi(p_{\alpha},p_{\pm})\\!:=\\!\Psi_{0}(p_{\alpha})\chi(p_{\alpha},p_{\pm}).$
(46)

We can distinguish the $\Psi_{0}$ term, which is a WKB expansion of the semi-
classical part with respect to the $\hbar$ parameter and the $\chi$ term
containing information about the quantum behavior of the system.

We will now set two different equations for the two different terms of the
wave function of the Universe and we will solve them in order to obtain a
complete description of the system, according to the method we are using.

The wave function $\Psi_{0}$ will satisfy the equation:

$\bra{p}\hat{\mathcal{H}}_{0}\ket{\Psi_{0}}=\left(-p^{2}_{\alpha}+\Lambda
e^{i6\hbar\partial_{p_{\alpha}}}\right)A(p_{\alpha})e^{iS(p_{\alpha})/\hbar}=0.$
(47)

In the spirit of the WKB expansion, being $\hbar$ the expansion parameter, we
can solve (47) at different orders [23]. Specifically, we will solve (47) at
the zeroth and first order in $\hbar$, obtaining in this way respectively a
Hamilton-Jacobi equation, which will allow us to determine $S(p_{\alpha})$ and
a continuity equation from which will be possible to derive $A(p_{\alpha}).$

We will refer to the formal series expansion of the pseudo-differential
operator in (47) to identify the different orders in $\hbar$:

$e^{i6\hbar\partial_{p_{\alpha}}}=\sum_{n=0}^{\infty}\frac{\left(i6\hbar\partial_{p_{\alpha}}\right)^{n}}{n!}.$
(48)

By letting (48) acting on $\Psi_{0}$, at the zeroth order in $\hbar$, the
equation (47) will read as:

$\displaystyle(-p_{\alpha}^{2}+\Lambda\sum_{n}^{\infty}\frac{1}{n!}(6\;\partial_{p_{\alpha}}S(p_{\alpha}))^{n})A(p_{\alpha})e^{iS(p_{\alpha})/\hbar}=0$
$\displaystyle-p_{\alpha}^{2}+\Lambda
e^{6\partial_{p_{\alpha}}S(p_{\alpha})}=0,$ (49)

from which it is straightforward to obtain:

$S(p_{\alpha})=\frac{1}{6}p_{\alpha}\left(\ln(\frac{p_{\alpha}^{2}}{\Lambda})-2\right)+c_{1}.$
(50)

The function that we have just determined, by working at the zeroth order in
$\hbar$, is the classical action of the system only concerning the variables
$(\alpha,p_{\alpha})$.

On the other hand, at the first order in $\hbar$, considering the validity of
(V), we can write the following differential equation for $A(p_{\alpha})$:

$(6S^{\prime})^{2}A^{\prime}e^{6S^{\prime}}+6AS^{\prime\prime}(e^{6S^{\prime}}-6S^{\prime}-1)=0,$
(51)

where the symbol ′ refers to the differentiation with respect to $p_{\alpha}$.
The solution of (51) is:

$A(p_{\alpha})=c_{2}\;e^{-\frac{\Lambda-p^{2}_{\alpha}}{p^{2}_{\alpha}\ln(\frac{p^{2}_{\alpha}}{\Lambda})}}.$
(52)

Now that we have completely described the semi-classical part of the wave
function of the Universe, we can turn our attention to the quantum part. The
quantum piece of the wave function $\chi$ depends not only on the quantum
variables $p_{\pm}$ but also on the semi-classical one $p_{\alpha}$. This
means that, in order to obtain a correct description of the dynamics, we need
to consider both the action of $\hat{\mathcal{H}}_{q}$ and
$\hat{\mathcal{H}}_{0}$ on $\chi$. Working at the first order in $\hbar$ \-
which is the first order containing quantum information - and keeping in mind
the validity of the equations (V) and (51) for the semi-classical part, we
arrive at the following equation for the quantum subsystem:

$6i\hbar\Lambda
e^{6S^{\prime}(p_{\alpha})}\frac{\partial\chi(p_{\alpha},p_{\pm})}{\partial
p_{\alpha}}=-\mathcal{H}_{q}\chi(p_{\alpha},p_{\pm}).$ (53)

Too obtain, in the end, a real Schrödinger equation from (53), we need to
exploit a series of relations which holds for the classical action
$S(p_{\alpha})$ and the classical quantities related to this object. First of
all, from the Hamilton-Jacobi construction, we know that
$\partial_{p_{\alpha}}S=\alpha$ and we can rewrite (53) as:

$6i\hbar\Lambda e^{6\alpha}\frac{\partial\chi(p_{\alpha},p_{\pm})}{\partial
p_{\alpha}}=-\mathcal{H}_{q}\chi(p_{\alpha},p_{\pm}).$ (54)

Secondly, we need to consider the classical Hamiltonian described by the
variables $(\alpha,p_{\alpha})$:

$\mathcal{H}_{cl}=c^{3}N(t)e^{-3\alpha}\left(\ p_{\alpha}^{2}+\Lambda
e^{6\alpha}\right)\approx 0.$ (55)

By taking into account the respective Hamilton equations we can arrive at the
following expression:

$6\Lambda e^{6\alpha}=-\frac{\dot{p_{\alpha}}}{N(t)}e^{3\alpha}$ (56)

where the dot refers to the differentiation with respect to the coordinate
time.

Finally, putting together (54)-(56), we are able to write (53) as:

$\displaystyle
i\hbar\dot{p_{\alpha}}\frac{\partial\chi(p_{\alpha},p_{\pm})}{\partial
p_{\alpha}}=N(t)e^{-3\alpha}\mathcal{H}_{q}\chi(p_{\alpha},p_{\pm})\Rightarrow$
$\displaystyle i\hbar\frac{\partial\chi(p_{\alpha},p_{\pm})}{\partial
t}=N(t)e^{-3\alpha}\mathcal{H}_{q}\chi(p_{\alpha},p_{\pm}),$ (57)

which is the Schrödinger equation for the quantum dynamics we were looking
for.

We notice that in (V) the lapse function $N(t)$ is present: this is a
reflection of the fact that our theory is invariant under time-diffeomorphism,
as it should be. Indeed, we can choose different times to follow the evolution
of the system and consequently we will deal with different Hamiltonians.

In our case, we want the Hamiltonian of the quantum subsystem of the Universe
to be the one of a free particle. This implies to set $N(t)$ to a precise
form, specifically:

$N(t)=e^{3\alpha(t)}.$ (58)

Consequently, we are able to define our physical time $\tau$, which is
essentially a WKB time, from the relation:

$d\tau=e^{3\alpha(t)}dt.$ (59)

In the end:

$i\hbar\frac{\partial\chi(p_{\alpha},p_{\pm})}{\partial\tau}=\left(p^{2}_{+}+p^{2}_{-}\right)\chi(p_{\alpha},p_{\pm})$
(60)

The general solution of (60) is given by a superposition of free particle
time-dependent wave functions. In the non-commutative GUP quantization scheme
- which is the one we are adopting for the description of the quantum degrees
of freedom of our Bianchi I Universe - these wave functions can be obtained by
the map (IV), transforming into quasi-position representation - or in this
case, quasi-anisotropies - two dimensional Dirac deltas, defined as in (13).
This allows us to write the general solution of (60) as:

$\displaystyle\tilde{\chi}(\tilde{\gamma}_{+},\tilde{\gamma}_{-})=\int_{\mathbb{R}^{2}}\frac{dp_{+}dp_{-}}{\left(1+\beta\;(p_{+}^{2}+p_{-}^{2})\right)^{2}}$
(61)
$\displaystyle\exp{i\tilde{\gamma}_{+}\frac{\arctan(\frac{\sqrt{\beta}p_{+}}{\sqrt{1+\beta
p_{-}^{2}}})}{\hbar\sqrt{\beta}\sqrt{1+\beta p_{-}^{2}}}}$
$\displaystyle\times\exp{i\tilde{\gamma}_{-}\frac{\arctan(\frac{\sqrt{\beta}p_{-}}{\sqrt{1+\beta
p_{+}^{2}}})}{\hbar\sqrt{\beta}\sqrt{1+\beta p_{+}^{2}}}}$
$\displaystyle\times\exp{-i(p_{+}^{2}+p_{-}^{2})\frac{(\tau-\tau_{0})}{\hbar}}\phi_{0}(p_{+},p_{-}).$
(62)

To explore explicitly the behavior of the system we need to choose a precise
profile $\phi_{0}(p_{+},p_{-})$, which represents also the wave function of
the system in momentum space at some initial time $\tau_{0}$.

As a first thing, we choose a Gaussian profile with a zero position
expectation value in the non-commutative variables $(\gamma_{+},\gamma_{-})$,
at the initial time $\tau_{0}=0$:

$\phi_{0}(p_{+},p_{-})=\exp{-\left[\frac{(p_{+}-\nu_{+})^{2}}{2\sigma_{+}^{2}}+\frac{(p_{-}-\nu_{-})^{2}}{2\sigma_{-}^{2}}\right]},$
(63)

where $\nu_{\pm},\sigma_{\pm}$ are real parameters related respectively to the
momentum expectation value and uncertainty of the state itself.

Figure 1: Non-commutative case I: dynamics evolution in the time $\tau$
($\hbar\beta$ units, whit [c]=[$\kappa$]=1) of the Bianchi I wave packet
probability density. The values set for the parameters of the Gaussian profile
are $\nu_{\pm}=3$, $\sigma_{\pm}=3.5$, in units of $1/\sqrt{\beta}$. It is
possible to appreciate the flow of probability mostly along the direction
identified by the classical vector of the initial expectation values of
momenta and the consequent trajectory of the peak.

Figure 2: Ordinary case I: dynamics evolution in the time $\tau$ ($\hbar\beta$
units, whit [c]=[$\kappa$]=1) measured in units of $\hbar\beta$, of the
Bianchi I wave packet probability density. The values of the parameters of the
Gaussian profile are fixed in order to match the initial conditions of the
corresponding non-commutative wave packet. By comparison with the non
commutative scenario it is possible to observe the different dynamical
behavior of the wave packets, concerning both the spreading geometry and the
motion’s velocity of the peaks. Be aware of the different range of the quasi-
anisotropy variables represented in this figure and in Fig (1).

It is important to keep in mind that, in a non-commutative space, we cannot
really talk about points, therefore when we refer to a point in the
$(\tilde{\gamma}_{+},\tilde{\gamma}_{-})$-space we are actually talking about
a region around that point in the not-accessible space
$(\gamma_{+},\gamma_{-})$, in the spirit of the quasi-position representation
we have constructed. In particular, from our results in Sec. IV, in this
theory the origin is the point in which the localization properties are
maximized, hence the extension of this region of uncertainty is the minimal
one we can reach. From a physical point of view, this means that the origin in
the configuration space $(\tilde{\gamma}_{+},\tilde{\gamma}_{-})$ represents
the best realization of an isotropic Universe.

Once these parameters are properly fixed, numerical integration of (61) with
the profile (63) allows us to follow the evolution of the quantum Universe at
different times $\tau$. Two different choices for set of the involved
parameters has been done, in order to distinguish between two general cases,
as shown in Fig (1) and Fig (3). In the first scenario, in the profile (63),
we have fixed $\nu\\_{\pm}>\sigma_{\pm}$, while in the second we have set
$\nu\\_{\pm}<\sigma_{\pm}$.

By visual inspection, we can infer features of our non-commutative quantum
model and appreciate the difference between the two general cases. In both
scenarios, while time elapses, the system is spreading, signaling that the
probability of finding the particle Universe in another region of the
configuration space $(\tilde{\gamma}_{+},\tilde{\gamma}_{-})$ away from the
origin increases in time. Nevertheless, the geometry of the two spreading
processes is quite different. While in the first case the probability density
of the wave packet is flowing mostly along a precise direction, which is the
one roughly identified by the classical vector of the initial expectation
values of momenta, in the second case the wave packet is spreading almost
symmetrically with respect to the origin, in the direction identified by the
diagonals of the plane.

The peak of the wave packets will move somehow accordingly: in the scenario
depicted in Fig (1), it follows the flow of probability density and it is
possible to appreciate its trajectory; in the scenario presented in Fig (3),
due to the different spreading progress, the peak’s movement is much slower
and, consequently, it is harder to observe a trajectory.

Finally we note the different initial localization properties of the two wave
packets in the quasi-anisotropies space, namely in the second example the wave
packet appears to be more localized with respect to the first one.

Figure 3: Non-commutative case II: dynamics evolution in the time $\tau$
($\hbar\beta$ units, whit [c]=[$\kappa$]=1) of the Bianchi I wave packet
probability density. The values set for the parameters of the Gaussian profile
are $\nu_{+}=3.5$, $\nu_{-}=2.5$, $\sigma_{+}=5.5$, $\sigma_{-}=4.5$, in units
of $1/\sqrt{\beta}$. It is possible to appreciate the almost symmetric spread
of the wave packet with respect to the origin, in the direction identified by
the diagonals of the plane, while it is harder to observe a trajectory of the
peak.

Figure 4:  Ordinary case II: dynamics evolution in the time $\tau$
($\hbar\beta$ units, whit [c]=[$\kappa$]=1), of the Bianchi I wave packet
probability density. The values of the parameters of the Gaussian profile are
fixed in order to match the initial conditions of the corresponding non-
commutative wave packet. By comparison with the non commutative case it is
possible to notice the different the dynamical behavior of the wave packets,
regarding both the spreading geometry and the motion’s velocity of the peaks.
Be aware of the different range of the quasi-anisotropy variables represented
in this figure and in Fig (3).

This behavior of the Universe wave packet in this non-commutative setting is
remarkably different with respect to the behavior of a properly defined wave
packet describing the same Universe, namely Bianchi I, in the ordinary quantum
theory.

As it is clear from Fig (2) and Fig (4) and the visual comparison with the
respective non-commutative counterparts, the ordinary Bianchi I wave packet
does not display any different spreading geometries in the two illustrations
and it moves from the initial point towards regions with higher absolute
values of the anisotropies, with a greater velocity with respect to the non-
commutative cases.

Due to this tendency, the initial configuration loses ”rapidly” and in a
definitive way the status of favored one, differently from what happens in the
non-commutative framework, where this point remains the one in which we can
find the particle Universe with a higher probability for a longer time if
compared to the ordinary quantum dynamics.

To make sense of this qualitative statement, considering as initial
configuration the isotropic one, we can compare how the relative probability
density at the origin of the wave packets Fig (1) and Fig.(2) changes over
time.

The situation for the wave packets of the second scenario is identical.

In Fig (5), it can be appreciated how the point $(0,0)$, denoting the
isotropic state in both theories, experiences a faster relative decline in
probability within the ordinary theory compared to the non-commutative one.
This suggests that in the non-commutative theory, the isotropic configuration
(or any initial configuration, actually) maintains a greater probabilistic
advantage with respect to the ordinary theory.

Figure 5: Plot of the relative probability density
$p(\tau)=\absolutevalue{\Psi(0,0,\tau)}^{2}/\absolutevalue{\Psi(0,0,0)}^{2}$
at the origin concerning the non-commutative wave packet of Fig (1) and the
ordinary wave packet of Fig (2), where $\Psi$ stands for the generic wave
function. It can be noticed how the relative probability in the non-
commutative case prevails over the relative probability of the ordinary theory
at any displayed time $\tau$.

The same scenario is verified also if we initially peak our wave packet in an
anisotropic configuration: the particular point chosen for the initial
condition will be the favored one for a relatively long time, again if
compared to what happens to the ordinary wave packet.

Here it is important to stress that in our theory it only makes sense to
explore ”small” regions around the origin. This is true for two orders of
reasons:

  * •

the Vilenkin approximation we have used constrains us to remain in a ”small”
quantum phase space (see above);

  * •

in our non-commutative theory we can talk about localization only in proximity
of the origin.

These insights emphasize how the non-commutative GUP quantum theory modifies
some aspects of the dynamics of the Bianchi I Universe, establishing new
different relationships between the initial probability densities and their
evolution, concerning both isotropic and anisotropic configurations of the
Universe.

##  VI Conclusions

In our analysis, we faced a subtle and challenging problem, concerning the
construction of a consistent quantum theory for a $n$-dimensional non-
commutative GUP formulation of the Heisenberg algebra. Our main concern
specifically regarded the possibility of recovering a way to describe the
localization properties of the theory. After a brief extension of the
necessary functional analysis of the fundamental operators, as a first result,
we were able to identify those states that minimize the uncertainty in the
coordinate operator along a specific direction. The price to pay in order to
achieve this maximal localization in one of the space coordinates is to
delocalize the state itself of a certain amount in all other directions. This
means that, in general, it is not possible to achieve at the same time a
maximal localization with respect to all the $n$ variables. This fact is a
natural manifestation of the non-commutative nature of the algebra we are
considering.

Since these states alone cannot be used to describe the position on the whole
configuration space, we defined a procedure to construct a consistent
generalization of the quasi-position representation introduced in [4]. We
successfully identified the only state of the theory that exhibits isotropic
properties of localization, that is that state which admits simultaneously the
same maximal localization in all the $n$ variables. This state is centered
around the origin of the system of coordinates and it is a spherically
symmetric state in momentum space. The translation - properly defined with
respect to the expectation position value - of this state in all directions of
a certain arbitrary amount, allowed us to generate a family of states suitable
to be used as a quasi-position basis.

Once reached the task of setting up a consistent quantum theory for the non-
commutative $n$-dimensional GUP, we provided a two-dimensional implementation
of our quantum scenario to the evolution of the Bianchi I cosmological model
in the WKB approximation of a quasi-classical volume, while dealing with pure
quantum anisotropy degrees of freedom.

More precisely, by adopting the standard Misner variables, we identified the
so-called isotropic variable $\alpha$ as the one approaching a quasi-classical
limit and the two variables $\gamma_{\pm}$ with the ones belonging to a small
quantum subsystem. In cosmology, the two anisotropy degrees of freedom play
the role of the two physical degrees of the gravitational field, while
$\alpha$ is often called an ”embedding” variable and, here, it restores, via
its classical limit, the dependence of the anisotropy variables on the label
time $t$. Accordingly, in a scenario à la Vilenkin, we were able to arrive at
a real two-dimensional Schrödinger equation in momentum space for the
description of the dynamics of the quantum part of the Universe, now dictated
by our GUP-modified Heisenberg algebra.

By analyzing wave packets describing our Bianchi I Universe, we were able to
show how differently these wave functions behave with respect to the ordinary
quantum case.

We showed how these wave packets exhibit more complex spreading properties,
with a ”geometry” depending on the their initialization in momentum space, and
we observed how their overall motion in the quasi-anisotropies plane -
identified with the motion of the peak - is slower in time when compared to
the ordinary quantum theory.

This particular tendency makes the initial configurations more ”stable”, in
the sense that, from a probabilistic point of view, they are the favored ones
for a time which is longer if compared with the same situation in the ordinary
theory.

This holds for any initial configuration in the permitted region restricted by
the Vilenkin approximation (see [40]) and the localization properties of the
GUP framework and clearly also for an isotropic one, that is a configuration
in which the wave packet describing our Universe is peaked and sufficiently
localized around the origin of the plane
$(\tilde{\gamma}_{+},\tilde{\gamma}_{-})$.

The observed behavior suggests the idea that non-trivial Heisenberg algebras,
thought in the Mini-superspace as the way for accounting of cut-off physics,
can potentially provide new physical insights on some open questions of
quantum cosmology, such as the classicalization of the Universe [41], the
emergence of an isotropic Universe from more general configuration, which
appear to be very natural in the proximity of the initial singularity [42, 43]
or the possibility to generate Big Bounce scenarios, as in the case of Polymer
quantum mechanics [9] and Loop quantum cosmology [44, 45, 46].

More complex Bianchi models with richer structures will be for sure the best
candidates to test the validity of these ideas and hence the effectiveness of
this non-commutative GUP quantization framework.

##  Appendix A Computation of the coordinate operator uncertainty in the i-th
direction

The computation of the uncertainty in the $i$-th coordinate for the state (IV)
essentially relies on the computation of the integral quantity:

$\displaystyle\bra{\Phi_{T}^{ml}}{\hat{\mathbf{x}}_{i}^{2}}\ket{\Phi_{T}^{ml}}=-\\!\\!\\!\int_{\mathbb{R}^{n}}$
$\displaystyle
d^{n}p\left(\Phi^{ml}_{T}(p)\right)^{*}\left[2p_{i}\partial{p_{i}}\Phi^{ml}_{T}(p)\right.$
$\displaystyle\left.+(1+p_{r}p_{s}\delta^{rs})\partial^{2}_{p_{i}}\Phi^{ml}_{T}(p)\right]\frac{\hbar^{2}\beta}{\beta^{n/2}},$
(64)

where $i,j$ run over all the $n$ indices. Notice that we have factored out the
dimensional quantities in order to work, inside the integral, with
adimensional objects.

Every resulting integrals can be resolved by using hyperspherical coordinates.

The first integral, i.e. the one involving the first partial derivative, once
made explicit, has only one contributing term:

$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!d^{n-1}p\int_{\mathbb{R}}dp_{i}\,2p_{i}(1+p_{r}p_{s}\delta^{rs})^{-n}\left[\frac{-np_{i}+I\xi_{i}}{1+p_{r}p_{s}\delta^{rs}}\right]=$
$\displaystyle\int
d\Omega_{n-1}\int_{\mathbb{R}^{+}}dr\;r^{n-2}(1+r^{2})^{1/2-n}\sqrt{\pi}\frac{\Gamma[n-\frac{1}{2}]}{\Gamma[n]}=$
$\displaystyle\frac{2^{1-n}\pi^{\frac{n+1}{2}}}{\Gamma[\frac{n+1}{2}]},$ (65)

where $r:=\sum_{l\neq i}p_{l}^{2}$ and $\Omega_{n-1}$ is the solid angle in
$(n-1)$ dimensions.

The second integral, i.e. the one involving the second partial derivative, can
be decomposed in different contributing terms.

The first term is given by:

$\displaystyle\int_{\mathbb{R}^{n}}d^{n}p\left[-n(2+n)p^{2}_{i}(1+p_{r}p_{s}\delta^{rs})^{-1-n}\right]=$
$\displaystyle\int_{\mathbb{R}^{n-1}}\\!\\!\\!\\!\\!d^{n-1}p\int_{\mathbb{R}}dp_{i}\left[-n(2+n)p^{2}_{i}(1+p_{r}p_{s}\delta^{rs})^{-1-n}\right]=$
$\displaystyle\\!\\!\int\\!\\!\\!\,{d\Omega_{n-1}}\\!\\!\int_{\mathbb{R}^{+}}\\!\\!\\!\\!\\!\\!dr\,r^{n-2}\\!\\!\left(\\!\\!\frac{-n(2+n)\sqrt{\pi}(1+r^{2})^{\frac{1}{2}-n}\Gamma\left[n-\frac{1}{2}\right]}{2\Gamma\left[n+1\right]}\\!\right)\\!=$
$\displaystyle-\frac{2^{-n}(2+n)\pi^{\frac{1+n}{2}}}{\Gamma\left[\frac{n+1}{2}\right]},\qquad\text{with}\quad
r:=\sum_{l\neq i}p_{l}^{2}$ (66)

The second term is:

$\displaystyle\int_{\mathbb{R}^{n}}\\!\\!d^{n}p(1+p_{r}p_{s}\delta^{rs})^{-n}=\\!\int\\!\\!d{\Omega_{n}}\int_{\mathbb{R}^{+}}\\!\\!\\!\\!\\!dr\;r^{n-1}n(1+r^{2})^{-n}\\!=$
$\displaystyle\frac{2^{1-n}n\pi{\frac{1+n}{2}}}{\Gamma\left[\frac{n+1}{2}\right]},\qquad\text{with}\quad
r:=\sum_{l\neq i}p_{l}^{2}$ (67)

Finally, the third and last term, which is the most complex one, is the
following:

$\displaystyle\int_{\mathbb{R}^{n}}\\!\\!\\!\\!\\!\\!d^{n}p(1+p_{r}p_{s}\delta^{rs})^{1-n}\\!\\!\left\\{\\!\\!\left[\sum_{\begin{subarray}{c}k=1\\\
k\neq
i\end{subarray}}^{n}\left(\\!\frac{\\!p_{i}p_{k}\xi_{k}\\!}{\left(1+p_{r}p_{s}\delta^{rs}|_{r,s\neq
k}\right)}\\!\right)\\!-\\!\xi_{i}\\!\right]\right.$
$\displaystyle\left.\times\frac{1}{\left(1+p_{r}p_{s}\delta^{rs}\right)}\\!+p_{i}\\!\\!\left[\sum_{\begin{subarray}{c}k=1\\\
k\neq
i\end{subarray}}^{n}\frac{\xi_{k}\tan[-1](\\!\frac{p_{k}}{\sqrt{1+p_{r}p_{s}\delta^{rs}|_{r,s\neq
k}}}\\!)}{(1+p_{r}p_{s}\delta^{rs}|_{r,s\neq
k})^{3/2}}\right]\\!\\!\right\\}^{\\!2}\\!\\!\\!.$ (68)

We can further identify three pieces in this last integral, one for every term
of the square.

The sum of first two integrals, that is the square of the first term and the
mixed term, gives the following results:

$\displaystyle\sum_{\begin{subarray}{c}k=1\\\ k\neq i\end{subarray}}^{n}$
$\displaystyle\left[-\frac{\pi^{n/2}\xi_{k}^{2}\Gamma\left[1+\frac{n}{2}\right]}{n\Gamma[n]-4n^{3}\Gamma[n]}+\frac{n\pi^{n/2}\xi_{k}^{2}\Gamma[\frac{n}{2}]}{4n^{3}-4n^{2}-n+1)\Gamma[n]}\right]$
$\displaystyle\times\frac{2^{-n}\pi^{\frac{n+1}{2}}\xi^{2}_{i}}{\Gamma[\frac{n+1}{2}]}.$
(69)

The technique to resolve these integrals is once again the same as the
previous used above: first we integrate in those $p$-variables which do not
enter the integral in a rotational invariant way, then, once the integral is
reduced to only a $n$-dimensional spherical symmetric term we employ
hyperspherical coordinates to compute it.

The resolution of the last integral, that is the one containing the square of
the arctangent term, is more involved.

The integral we need to solve is:

$\displaystyle\sum_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n}\int_{\mathbb{R}^{n}}\\!\\!\\!d^{n}p(1+p_{k}^{2}+p_{l}p_{m}\delta^{lm})^{(-n+1)}p_{i}^{2}\xi_{k}^{2}$
$\displaystyle\times\left[\tan[-1](\frac{p_{k}}{\sqrt{1+p_{l}p_{m}\delta^{lm}}})\right]^{2}\frac{1}{\left(1+p_{k}^{2}+p_{l}p_{m}\delta^{lm}\right)^{3}},$
(70)

where we remember that $l,m\neq k$.

Considering just a generic term of the sum, we make a change of variable,
namely $q:=\arctan[\frac{p_{k}}{\sqrt{1+p_{i}^{2}+r^{2}}}]$, where we have set
$r:=p_{l}p_{m}\delta^{lm}|_{l,m\neq k,i}$. The integral (A) then become:

$\xi_{s}^{2}\\!\int_{\mathbb{R}^{n-2}}\\!\\!\\!\\!\\!\\!d^{n-2}p\int_{\mathbb{R}}dp_{i}\,p_{i}^{2}(1+p_{i}^{2}+r^{2})^{-\frac{3}{2}-n}\\!\\!\int_{-\pi/2}^{\pi/2}\\!\\!dq\frac{q^{2}}{\cos[4-2n](q)}$
(71)

The $q-$integral can be solved for every $n>1$ but it does not admit a closed
form.

By induction we can show that the general result can be written as:

$\int_{-\pi/2}^{\pi/2}dq\frac{q^{2}}{\cos[4-2n](q)}=\alpha(n)\pi+\theta(n)\pi^{3},$
(72)

where $\alpha(n)$ is a real negative function and $\theta(n)$ is a real
positive function, both defined for $n>1$. A table with the value of the
integral for some values of $n>1$ can be find below (1).

$n$ | $\alpha(n)$ | $\theta(n)$
---|---|---
2 | 0 | $\frac{1}{12}$
3 | $-\frac{1}{4}$ | $\frac{1}{24}$
4 | $-\frac{15}{64}$ | $\frac{1}{32}$
5 | $-\frac{245}{1152}$ | $\frac{5}{192}$
Table 1: Table with the values of the real functions $\alpha(n)$ and
$\theta(n)$ for some $n$, where $n\in\mathbb{N}$ is the number of dimensions
in which we are working.

Considering this, by performing the integration in $p_{k}$ and then using
again hyperspherical coordinates for the last integration, in the end we
obtain:

$\left(\alpha(n)\pi+\theta(n)\pi^{3}\right)\pi^{\frac{n-1}{2}}\frac{\Gamma[1+\frac{n}{2}]}{2\Gamma[\frac{3}{2}+n]}\left[\sum_{s\neq
k}\xi_{s}^{2}\right]$ (73)

where we have re-introduced the initial sum over all $s\neq k$.

Finally - summing all the contributes properly re-scaled with the norm of the
state shown in (IV) and considering the dimensional quantities factored out at
the beginning - after some algebra involving properties of the Euler Gamma
function, we are able to obtain the expression (IV), concerning the
uncertainty of the coordinate operator in a generic direction $k$, for the
state (IV):

$\displaystyle(\Delta{\hat{\mathbf{x}}_{i}})^{2}=\frac{1}{2}$
$\displaystyle\left\\{2n+\frac{2+4n}{(n-1)(4n^{2}-1)}\sum_{k\neq
i}\xi^{2}_{k}+\right.$
$\displaystyle\left.\\!\\!\\!\\!\frac{n!}{\Gamma[3/2+n]}\sqrt{\pi}\biggl{(}\alpha(n)+\theta(n)\pi^{2}\biggr{)}\sum_{k\neq
i}\xi^{2}_{k}\right\\}\hbar^{2}\beta.$ (74)

##  Appendix B Inverse transform

In Section IV we have defined the integral transform (IV), which maps wave
functions from momentum space to quasi-position space:

$\displaystyle\psi(\xi):=$
$\displaystyle\int_{\mathbb{R}^{n}}\frac{d^{n}p}{(1+\beta
p_{r}p_{s}\delta^{rs})}\prod_{k=1}^{n}e^{i\xi_{k}g\left(p_{k},p_{\\{l\\}\neq
k}\right)}$ $\displaystyle\times(1+\beta
p_{r}p_{s}\delta^{ij})^{-n/2}\psi(p),$ (75)

where we have defined:

$g\left(p_{k},p_{\\{l\\}\neq
k}\right):=\frac{\tan[-1](\frac{\sqrt{\beta}p_{k}}{1+\beta\sum_{l}p^{2}_{l}})}{\sqrt{1+\beta\sum_{l}p^{2}_{l}}},\qquad\forall
k$ (76)

and $i,j$ run over all the $n$ indices.

To construct the inverse of this transform we can rewrite the previous
relation as:

$\displaystyle\int_{\mathbb{R}^{n}}d^{n}\xi\prod_{k=1}^{n}e^{-i\xi_{k}g\left(p^{\prime}_{k},p^{\prime}_{\\{l\\}\neq
k}\right)}\psi(\xi)=$
$\displaystyle\int_{\mathbb{R}^{n}}\prod_{k=1}^{n}e^{-i\xi_{k}g\left(p^{\prime}_{k},p^{\prime}_{\\{l\\}\neq
k}\right)}\left(\int_{\mathbb{R}^{n}}\frac{d^{n}p}{(1+\beta
p_{r}p_{s}\delta^{rs})}\right.$
$\displaystyle\left.\times\prod_{k=1}^{n}e^{i\xi_{k}g\left(p_{k},p_{\\{l\\}\neq
k}\right)}(1+\beta p_{i}p_{j}\delta^{ij})^{-n/2}\psi(p)\right)d^{n}\xi.$ (77)

By exchanging the order of integration and performing the $\xi$-integral, we
can write for the right-hand side of (B):

$\displaystyle(\sqrt{2\pi})^{n}\int_{\mathbb{R}^{n}}$ $\displaystyle
d^{n}p\,(1+\beta p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p)$ (78)
$\displaystyle\times\prod_{k=1}^{n}\delta\left(g\left(p_{k},p_{\\{l\\}\neq
k}\right)-g\left(p^{\prime}_{k},p^{\prime}_{\\{l\\}\neq k}\right)\right).$
(79)

The presence of the deltas inside the integral define a specific region of
$\mathbb{R}^{n}$ on which the integral itself has to be evaluated. This region
is given by the intersection of $n$ $(n-1)$-dimensional hypersurfaces in a
$n$-dimensional space, hence corresponds to a point or a set of points.

To explicitly perform the integration we need to use the so-called coarea
formula [47].

Let us consider the integral over an open set $A$ of $\mathbb{R}^{n}$ of an
integrable function $f(x)$.

Introducing the vector function $u(x)$:

$u:A\subseteq\mathbb{R}^{n}\to\mathbb{R}^{k},\qquad k\leq n,$ (80)

the coarea formula states that:

$\int_{A}f(x)d^{n}x:=\int_{\mathbb{R}^{k}}\left(\int_{u^{-1}(t)}\frac{f(x)}{\absolutevalue{\mathcal{J}_{k}u(x)}}dH_{n-k}(x)\right)d^{k}t$
(81)

where $\absolutevalue{\mathcal{J}_{k}u(x)}$ is the determinant of the Jacobin
matrix of $u(x)$, which can be defined also in the rectangular case, and the
measure in the second integral is the Hausdorff measure in $(n-k)$ dimensions.

We want to apply this formula to our integral (78).

Rigorously speaking, due to the Dirac deltas, our function is not an
integrable one over $\mathbb{R}^{n}$. Nevertheless we can define the delta as
the limit of a suitable integrable function in order to deal with a proper
overall integrable function in every steps and, in the end, consider the limit
itself to reintroduce the deltas. In this way we consider our integral
containing delta distributions well-defined with respect to the possibility of
applying the theorem.

In order to proceed, first, we define two vector fields, namely:

$\displaystyle\qquad\qquad\qquad G:\mathbb{R}^{n}\to\mathbb{R}^{n}$
$\displaystyle G(p):=\left(g\left(p_{1},p_{\\{l\\}\neq
1}\right),...,g\left(p_{n},p_{\\{l\\}\neq n}\right)\right)$ (82)

and

$\displaystyle\qquad\qquad\qquad V:\mathbb{R}^{n}\to\mathbb{R}^{n}$
$\displaystyle V(p):=\left(g\left(p_{1},p_{\\{l\\}\neq
1}\right)-g\left(p^{\prime}_{1},p^{\prime}_{\\{l\\}\neq 1}\right),...,\right.$
$\displaystyle\left.\qquad\quad\quad g\left(p_{n},p_{\\{l\\}\neq
n}\right)-g\left(p^{\prime}_{n},p^{\prime}_{\\{l\\}\neq n}\right)\right),$
(83)

where we can consider the prime quantities as fixed.

Therefore, we can rewrite the integral (78) as:

$\displaystyle(\sqrt{2\pi})^{n}\int_{\mathbb{R}^{n}}$ $\displaystyle
d^{n}p(1+\beta p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p)$
$\displaystyle\times\prod_{k=1}^{n}\delta\left(V_{k}(p)\right).$ (84)

where $V_{k}(p)$ are the components of the vector field $V(p)$.

By applying the coarea formula (81), the previous integral becomes:

$\displaystyle\int_{\mathbb{R}^{n}}\left(\int_{V(p)=t}\\!\\!\\!\\!\\!\\!dH_{0}(p)\frac{(1+\beta
p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p)}{\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}}\right.$
$\displaystyle\left.\qquad\quad\times\prod_{k=1}^{n}\delta\left(V_{k}(p)\right)\right)d^{n}t=$
$\displaystyle\int_{\mathbb{R}^{n}}\\!\\!\,\prod_{k=1}^{n}\delta(t_{k})\\!\left(\int_{V(p)=t}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dH_{0}(p)\frac{(1+\beta
p_{i}p_{j}\delta^{ij})^{-\frac{n}{2}-1}\psi(p)}{\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}}\right)d^{n}t.$
(85)

We notice that the Jacobian is well-defined and it does not have any singular
points in any dimension, as can be verified by a direct calculation.

The integral inside the parenthesis can be considered as function $F(t)$ to be
evaluated in $t=0$ because of the presence of the deltas:

$\displaystyle\int_{V(p)=0}\\!\\!\\!\\!\\!dH_{0}(p)\frac{(1+\beta
p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p)}{\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}}=$
$\displaystyle\int_{G(p)=G(p^{\prime})}\\!\\!\\!\\!\\!dH_{0}(p)\frac{(1\\!+\\!\beta
p_{r}p_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p)}{\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}}$
(86)

The zero-dimensional Hausdorff measure is a discrete measure which prescribes
the evaluation of the integral on the points of the domain. Since
$G(p)=G(p^{\prime})$ if and only if $p=p^{\prime}$, the integral (B) in the
end is just the value of the integrand in $p=p^{\prime}$.

Therefore, coming back to the relation (B), we can write:

$\displaystyle\int_{\mathbb{R}^{n}}\\!\\!\\!\\!\\!\\!d^{n}\xi\\!\prod_{k=1}^{n}\\!e^{-i\xi_{k}g\left(p^{\prime}_{k},p^{\prime}_{\\{l\\}\neq
k}\right)}\psi(\xi)\\!=\\!\frac{(1\\!+\\!\beta
p^{\prime}_{r}p^{\prime}_{s}\delta^{rs})^{-\frac{n}{2}-1}\psi(p^{\prime})}{\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}|_{p=p^{\prime}}}$
(87)

and from this we can obtain the final expression of the inverse transform,
which maps wave function from quasi-position space to momentum space:

$\displaystyle\psi(p)=$ $\displaystyle(1+\beta
p_{r}p_{s}\delta^{rs})^{\frac{n}{2}+1}\absolutevalue{\mathcal{J}_{n}\left[V(p)\right]}$
$\displaystyle\times\int_{\mathbb{R}^{n}}d^{n}\xi\prod_{k=1}^{n}e^{-i\xi_{k}g\left(p_{k},p_{\\{l\\}\neq
k}\right)}\psi(\xi)$ (88)

where we have dropped the prime.

###### Acknowledgements.

S.S. would like to thank Matteo Bruno for useful mathematical insights and
discussions.

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