# Quantum Kasner transition in a locally rotationally symmetric Bianchi II
universe

Ana Alonso-Serrano<EMAIL_ADDRESS>Max-Planck-Institut für
Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, 14476 Potsdam,
Germany  David Brizuela<EMAIL_ADDRESS>Fisika Saila, Universidad del
País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Barrio Sarriena s/n, 48940
Leioa, Spain  Sara F. Uria<EMAIL_ADDRESS>Fisika Saila, Universidad
del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Barrio Sarriena s/n,
48940 Leioa, Spain

###### Abstract

The Belinski-Khalatnikov-Lifshitz (BKL) conjecture predicts a chaotic
alternation of Kasner epochs in the evolution of generic classical spacetimes
towards a spacelike singularity. As a first step towards understanding the
full quantum BKL scenario, we analyze a vacuum Bianchi II model with local
rotational symmetry, which presents just one Kasner transition. During the
Kasner epochs, the quantum state is coherent and it is thus characterized by
constant values of the different quantum fluctuations, correlations and
higher-order moments. By computing the constants of motion of the system we
provide, for any peaked semiclassical state, the explicit analytical
transition rules that relate the parametrization of the asymptotic coherent
state before and after the transition. In particular, we obtain the
modification of the transition rules for the classical variables due to
quantum back-reaction effects. This analysis is performed by considering a
high-order truncation in moments (the full computations are performed up to
fifth-order, which corresponds to neglecting terms of an order $\hbar^{3}$),
providing a solid estimate about the quantum modifications to the classical
model. Finally, in order to understand the dynamics of the state during the
transition, we perform some numerical simulations for an initial Gaussian
state, that show that the initial and final equilibrium values of the quantum
variables are connected by strong and rapid oscillations.

##  I Introduction

It has been conjectured that, close to a spacelike singularity, the classical
dynamics of any universe follows a chaotic behavior given by the Belinski-
Khalatnikov-Lifshitz (BKL) scenario BKL . According to this, different points
decouple and the dynamics of each point can be described by a Bianchi IX
universe, which is characterized by an alternation of different Kasner epochs
(corresponding to a Bianchi I vacuum or Kasner solution) connected by quick
transitions (as compared with the duration of the epochs). The alternation
among epochs implies that different (anisotropic) spatial directions are
squeezed and stretched during the evolution towards the singularity, resulting
in the mentioned chaotic behavior. One relevant characteristic of this
conjecture is that this behavior is dominated by the vacuum solution, and the
introduction of matter (besides particular cases) does not change the
qualitative behavior.

The analysis of Bianchi IX models can be very complicated and it is often
convenient to describe the system in terms of the Misner variables Misner ;
Misner2 , in the context of the so-called Mixmaster Universe; that is, instead
of using the scale factors, the system is characterized in terms of the
spatial volume and two anisotropic shape-parameters. The dynamics of these
models is then described as a free particle with a potential that drives the
transitions between Kasner regimes via exponential walls Misner ; Maccallum ;
Bojowald:2006da , in the so-called “cosmological billiard” Damour:2002et ;
Heinzle:2007kv . A simpler scenario corresponds to a Bianchi II universe. This
model presents only one transition between two different Kasner epochs on its
evolution towards the singularity. Nonetheless, any Bianchi IX spacetime can
be understood as a succession of Bianchi II models providing transitions among
different Kasner epochs Heinzle:2009eh .

This picture of the classical dynamics remains being a conjecture, although it
is widely accepted and numerous numerical studies confirm the described
behavior Berger:2002st ; Garfinkle:2003bb ; Heinzle:2012um . But let us remark
that the BKL scenario takes place very close to the singularity. Therefore, at
this stage, one would expect quantum effects to become relevant in the
analysis. However, it is still unclear how this scenario can be modified when
quantum effects are taken into account. In the literature, there are several
approaches to quantum models of Bianchi IX, focused on different features and
questions (as the avoidance of the singularity or the survival of the chaotic
behavior) Berger:1989jm ; Bojowald:2004ra ; Kheyfets:2005nz ; Benini:2006xu ;
Wilson2010 ; Ashtekar:2011ck ; Bergeron:2015ppa ; Czuchry:2016rlo ; Wilson-
Ewing:2018lyx ; Kiefer:2018uyv ; Gozdz:2018aai ; Giovannetti:2019ewe , as well
as simpler approaches to Bianchi II quantum models Ashtekar:2009um ;
Corichi:2012hy ; Bergeron:2014kea ; Saini:2017ipg .

In particular, one question concerns the survival of the structure of Kasner
epochs connected by quick transitions, and whether they follow the same
transition rules as in the classical picture. In this paper, we analyze this
question within the context of a simple locally rotationally symmetric (LRS)
Bianchi II model. We first analyze the canonical structure and exact solutions
of the classical model, identifying the transition rules of the canonical
variables of the system. Then we develop an exact quantization of the model
and focus on analyzing the survival of the Kasner transition and the quantum
modification of the transition rules. For such a purpose, we will perform a
decomposition of the wavefunction into its infinite set of moments following
the framework first presented in Bojowald:2005cw . This formalism is very
useful to understand the dynamics of peaked semiclassical states and it has
already been applied to the analysis of a variety of cosmological models; see,
e.g., Bojowald:2020emy –Bojowald:2010qm .

The rest of the paper is organized as follows. In Sec. II the canonical
structure of the classical model is described in terms of Misner variables and
the classical transition rules are obtained. The full quantum analysis is then
performed in Sec. III, which is divided into four subsections. In Subsec.
III.1 the quantum moments and their equations of motion are derived, whereas
in Subsec. III.2 the constants of motion of the system are constructed. Making
use of these constants of motion, in Subsec. III.3 the quantum transition
rules for any semiclassical peaked state are obtained. Subsec. III.4 completes
the study with a numerical simulation that describes in detail the dynamics
during the transition. The discussion of the results and possible future
developments are finally presented in Sec. IV.

##  II Canonical analysis of the classical system

The metric of a general homogeneous but anisotropic universe can be expressed
as follows

$ds^{2}=-N^{2}dt^{2}+\sum_{i,j,k=1}^{3}a_{k}^{2}l^{k}_{i}l^{k}_{j}dx_{i}dx_{j},$
(1)

where $N$ is the lapse function, $a_{k}$ is the scale factor in the
$\vec{l}^{k}=(l_{1}^{k},l_{2}^{k},l_{3}^{k})$ spatial direction, and the
$l^{k}_{i}$ vectors (or Kasner axes) form an orthonormal triad that determines
the spatial direction of the anisotropic expansion or contraction of the
universe. For the Bianchi II model, these vectors verify the following
conditions,

$\displaystyle\sum_{i,j=1}^{3}(l_{i,j}^{k}-l_{j,i}^{k})l^{i}_{n}l^{j}_{m}=\delta_{k,3}\epsilon_{knm},$

with $\epsilon_{knm}$ being the complete antisymmetric Levi-Civita symbol.

A very convenient way to describe this spacetime is given by the variables
introduced by Misner Misner ; Misner2 . The three different scale factors are
written as

$a_{k}=e^{\alpha+\beta_{k}},$ (2)

where $\alpha$ encodes the spatial volume,
$e^{\alpha}=(a_{1}a_{2}a_{3})^{1/3}$, and the variables $\beta_{k}$ satisfy
the constraint $\beta_{1}+\beta_{2}+\beta_{3}=0$. Therefore, the following two
independent shape-parameters can be defined

$\displaystyle\beta_{+}$ $\displaystyle:=$
$\displaystyle-\frac{1}{2}\beta_{3}=-\frac{1}{2}\ln\Bigg{[}\frac{a_{3}}{(a_{1}a_{2}a_{3})^{1/3}}\Bigg{]},$
$\displaystyle\beta_{-}$ $\displaystyle:=$
$\displaystyle\frac{1}{2\sqrt{3}}(\beta_{1}-\beta_{2})=\frac{1}{2\sqrt{3}}\ln\bigg{(}\frac{a_{1}}{a_{2}}\bigg{)}.$
(3)

As one can directly check, the range of $\alpha$ and $\beta_{\pm}$ corresponds
to the whole real line.

In this paper we will focus on the analysis of the vacuum case. For this
Bianchi II vacuum model, the Hamiltonian constraint reads

${\cal
C}=\frac{1}{2}e^{-3\alpha}\left(-p_{\alpha}^{2}+p_{-}^{2}+p_{+}^{2}\right)+e^{\alpha}U(\beta_{+},\beta_{-})=0,$
(4)

where $p_{\alpha}=-\frac{e^{3\alpha}}{N}\frac{d\alpha}{dt}$ and
$p_{\pm}=\frac{e^{3\alpha}}{N}\frac{d\beta_{\pm}}{dt}$ are the conjugate
momenta of $\alpha$ and $\beta_{\pm}$, respectively, and the potential term
takes the value $U(\beta_{+},\beta_{-})=e^{-8\beta_{+}}$. Other Bianchi models
are described by this very same Hamiltonian, but with a different form of the
potential. In particular, the vacuum Bianchi I –or Kasner– model, which will
be relevant in the subsequent discussion, is given by
$U(\beta_{+},\beta_{-})=0$.

Since it is a monotonic function, in these models it is usual to choose
$\alpha$ as the internal time. The above constraint is then deparametrized to
define the physical Hamiltonian

$\displaystyle H:=$ $\displaystyle-
p_{\alpha}=\big{[}p_{+}^{2}+p_{-}^{2}+2e^{4\alpha}U(\beta_{+},\beta_{-})\big{]}^{1/2}$
$\displaystyle=$
$\displaystyle\big{(}p_{+}^{2}+p_{-}^{2}+2e^{4\alpha-8\beta_{+}}\big{)}^{1/2}.$
(5)

In order to understand and study this system analytically in detail, we will
introduce a further simplifying assumption, and focus on models with a
preferred spatial direction, which are known as locally rotationally symmetric
(LRS) spacetimes111These spacetimes have been largely studied in cosmology;
check, e.g., Ellis1967 ; vanElst:1995eg ; Clarkson:2002jz ; Betschart:2004uu ;
Clarkson:2007yp ; Singh:2016qmr ; Khan:2016dqj ; Bergh:2017sps ; Singh:2017qxi
; Bojowald:2003md .. In this case, by choosing the third direction as the
preferred one, this corresponds to assuming $a_{1}=a_{2}$ and thus the shape-
parameter $\beta_{-}$, as well as its conjugate momentum $p_{-}$, are
vanishing: $\beta_{-}=0=p_{-}$. Therefore, defining for compactness
$\beta:=\beta_{+}$ and $p:=p_{+}$, in this LRS model the Hamiltonian (II) is
reduced to

$H=\big{(}p^{2}+2e^{4\alpha}U\big{)}^{1/2}=\big{(}p^{2}+2e^{4\alpha-8\beta}\big{)}^{1/2}.$
(6)

This is a system with just one degree of freedom $(\beta,p)$, and the
equations of motion for the canonical pair can readily be obtained,

$\displaystyle\dot{\beta}$ $\displaystyle=$ $\displaystyle\frac{p}{H},$ (7)
$\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle\frac{8}{H}e^{4\alpha}U=\frac{8}{H}e^{4\alpha-8\beta},$ (8)

where the dot represents the derivative with respect to the internal time
$\alpha$.

The regions where the potential term $e^{4\alpha}U$ has a negligible
contribution to the Hamiltonian $H$, that is, where the dimensionless ratio
$r=2e^{4\alpha}U/p^{2}$ is very small $r\ll 1$, are the so-called Kasner
regimes or epochs. (Note that, as commented above, in the Kasner model the
potential is exactly vanishing $U=0$ and thus also the corresponding
dimensionless ratio $r$.) In these regimes, the equations of motion can easily
be solved. On the one hand, $p$ is a constant of motion and, in this sense,
the system follows a free dynamics. On the other hand, the shape-parameter
$\beta$ is a linear function of $\alpha$

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\text{sign}(p)\alpha+c,$
(9)

with the integration constant $c$. Note that $\beta$ either grows or decreases
according to the sign of the variable $p$. Therefore, each Kasner epoch is
completely characterized by the values of the constants of motion $p$ and $c$.

In fact, the dynamics of the Bianchi II model can be understood as two
(asymptotic) Kasner epochs and a transition between them. Let us describe this
in more detail. We are interested in a model corresponding to an expanding
universe with a singularity in the past $\alpha\to-\infty$. Our analysis will
begin on a macroscopic universe far away from the singularity, that is, with a
large initial value of $\alpha$. If the initial values of $\beta$ and $p$ are
chosen in such a way that the ratio $r$ is small, the system will be on a
Kasner epoch. Depending on the sign of $p$, $\beta$ will be an increasing or
decreasing function (9). For definiteness, and without loss of generality, let
us choose the initial $p$ as positive.222If the initial $p$ is chosen as
negative, the system will follow the same qualitative behavior as described,
but with the direction of time inverted. In this case, towards larger values
of $\alpha$, the shape parameter $\beta$ will be growing, making the ratio $r$
more and more negligible. Therefore, in this direction, the Kasner condition
holds and the dynamics is given by (9) up to $\alpha\rightarrow\infty$. On the
contrary, towards lower values of $\alpha$, the shape-parameter $\beta$ will
be decreasing and thus the ratio $r$ increasing. Therefore, it will reach a
point where $r$ will be no longer negligible and the Kasner approximation will
break down. From this point on, $p$ is no longer constant and the dynamics of
the shape-parameter can not be approximated by (9) anymore.

Nonetheless, this period is very short. A very quick transition happens, which
changes the sign of $p$ and produces a bounce in $\beta$, and then the system
enters into another Kasner regime with corresponding parameters
$\widetilde{c}$ and $\widetilde{p}<0$. This new Kasner epoch evolves towards
lower values of $\alpha$ until reaching the singularity at
$\alpha\rightarrow-\infty$. During this evolution, the shape parameter $\beta$
increases linearly, following (9), and tending to infinity at the singularity.
For the sake of clarity, the evolution of $\beta$ and $p$ is illustrated in
Figs. 1 and 2 respectively.

Figure 1: Classical evolution of the variable $\beta$ with respect to
$\alpha$, where we have chosen $p>0$ for large values of $\alpha$ and $\beta$.
Figure 2: Classical evolution of the variable $p$ with respect to $\alpha$,
where we have chosen $p>0$ for large values of $\alpha$ and $\beta$.

From now on, we will focus on the study of this transition, specifically on
the rules that relate the parameters that characterize the first Kasner epoch
$(\overline{c},\overline{p})$, which will be denoted with a bar, with those
that describe the second one $(\widetilde{c},\widetilde{p})$, and will be
denoted with a tilde. The seminal idea to perform this analysis Misner  is to
find the constants of motion of the system. Since they are conserved all along
the evolution, these quantities shall be the same for both Kasner epochs.
Therefore, one can write them in terms of the corresponding parameters in each
Kasner epoch, before ($\alpha\to+\infty$) and after ($\alpha\to-\infty$) the
transition, and obtain the relations
$\widetilde{c}=\widetilde{c}(\overline{c},\overline{p})$ and
$\widetilde{p}=\widetilde{p}(\overline{c},\overline{p})$. These relations will
be known as the transition rules.

The first nontrivial task is thus to find the constants of motion. Any
conserved quantity $R=R(\alpha,\beta,p)$ must obey the following equation

$\displaystyle 0=\frac{dR}{d\alpha}$ $\displaystyle=$
$\displaystyle\frac{\partial R}{\partial\alpha}+\frac{\partial
R}{\partial\beta}\frac{d\beta}{d\alpha}+\frac{\partial R}{\partial
p}\frac{dp}{d\alpha}$ (10) $\displaystyle=$ $\displaystyle\frac{\partial
R}{\partial\alpha}+\frac{\partial R}{\partial\beta}\frac{\partial H}{\partial
p}-\frac{\partial R}{\partial p}\frac{\partial H}{\partial\beta},$

where we have made use of definitions (7)-(8). In order to find the solution,
it is convenient to perform the following transformation $\alpha\rightarrow
H$, so that $R=R(H,\beta,p)$, and the above equation reads

$0=\frac{\partial R}{\partial H}\frac{\partial
H}{\partial\alpha}+\frac{\partial R}{\partial\beta}\frac{\partial H}{\partial
p}-\frac{\partial R}{\partial p}\frac{\partial H}{\partial\beta}.$ (11)

Replacing in this expression the form of the Hamiltonian (6) one obtains,

$0=2(H^{2}-p^{2})\frac{\partial R}{\partial H}+\frac{\partial
R}{\partial\beta}p+4(H^{2}-p^{2})\frac{\partial R}{\partial p}.$ (12)

The general solution to this equation yields

$\displaystyle R$
$\displaystyle=F\Big{[}2H-p,e^{2(\alpha+\beta)}(H-p)\Big{]},$ (13)

for any generic function $F$. Consequently, we will consider the following two
independent conserved quantities to analyze the transition

$\displaystyle R_{1}$ $\displaystyle:=$ $\displaystyle 2H-p,$ (14)
$\displaystyle R_{2}$ $\displaystyle:=$ $\displaystyle
e^{2(\alpha+\beta)}(H-p).$ (15)

In the Kasner regimes, where the ratio is very small $r\ll 1$, the conserved
quantities are approximately given by

$\displaystyle R_{1}$ $\displaystyle\approx 2|p|-p,$ (16) $\displaystyle
R_{2}$ $\displaystyle\approx
e^{2(\alpha+\beta)}\Bigg{[}|p|\bigg{(}1+\frac{e^{4\alpha-8\beta}}{p^{2}}\bigg{)}-p\Bigg{]}.$
(17)

By considering the functional form for $\beta$ (9) in both asymptotic regions,
and taking into account that the initial value of $p$ is chosen as positive,
$\overline{p}>0$, the requirement of conservation of the above quantities
provides the following two equations,

$\displaystyle\overline{p}=[2\text{sign}(\widetilde{p})-1]\widetilde{p},$ (18)
$\displaystyle\dfrac{e^{-6\overline{c}}}{\overline{p}}=\widetilde{p}e^{2[1+\text{sign}(\widetilde{p})]\alpha+2\widetilde{c}}$
(19) $\displaystyle\hskip
34.14322pt\times\Bigg{[}\text{sign}(\widetilde{p})\bigg{(}1+\dfrac{e^{4[1-2\text{sign}(\widetilde{p})]\alpha-8\widetilde{c}}}{\widetilde{p}^{2}}\bigg{)}-1\Bigg{]},$

which relate the parameters of the first Kasner epoch
$(\overline{c},\overline{p})$ with those of the second Kasner regime
$(\widetilde{c},\widetilde{p})$. It is then straightforward to solve this
system and obtain the classical transition rules

$\displaystyle\widetilde{p}=-\frac{1}{3}\overline{p},$ (20)
$\displaystyle\widetilde{c}=-3\overline{c}-\frac{1}{2}\ln\bigg{(}\frac{2}{3}\overline{p}^{3}\bigg{)}.$
(21)

##  III Quantum analysis

Once we have provided a canonical description of the classical system and the
classical results are under control, we are ready to develop the quantum
analysis. The present section is divided into four subsections. In Subsec.
III.1 we will define the quantum moments, which will be the basic variables to
describe the quantum states. Their equations of motion are obtained and solved
in the asymptotic Kasner regimes. In particular, it is shown that, for
semiclassical states, the qualitative dynamical picture of the system is kept
as in the classical case: namely, there will be two different Kasner epochs
with a transition connecting them. Following the classical analysis, in
Subsec. III.2, the conserved quantities for the quantum system will be
derived. These constants of motion will be then used to obtain analytically
the quantum transition rules between the two Kasner epochs in Subsec. III.3.
This will provide the asymptotic behavior of all the quantum moments for any
initial state. Nonetheless, in order to understand better the dynamics during
the transition, in Subsec. III.4 a numerical implementation of the system of
equations is performed for the particular case of a Gaussian initial state.

###  III.1 Set up

The quantum dynamics of this model is governed by a Hamiltonian $\hat{H}$,
which is obtained by promoting the classical Hamiltonian (6) to an operator.
We will analyze the dynamics by using a formalism based on a moment
decomposition of the wave function Bojowald:2005cw . More precisely, we define
the quantum moments as

$\Delta(\beta^{i}p^{j}):=\Big{\langle}(\hat{\beta}-\langle\hat{\beta}\rangle)^{i}(\hat{p}-\langle\hat{p}\rangle)^{j}\Big{\rangle}_{\text{Weyl}},$
(22)

where $i,j\in\mathbb{N}$, $\langle\,\rangle$ indicates the expectation value
and the Weyl subscript refers to a totally symmetric ordering of the
operators. This infinite set of moments, together with
$\langle\hat{\beta}\rangle$ and $\langle\hat{p}\rangle$, fully characterizes
the quantum state. The sum of the indices $i+j$ will be referred as the order
of the corresponding moment. This order will be later used to introduce a cut-
off on this infinite system.

The dynamics of the moments (22) is driven by a quantum effective Hamiltonian,
that is defined as the expectation value of the operator $\hat{H}$. By
performing a Taylor expansion around the expectation values of $\hat{\beta}$
and $\hat{p}$, this effective Hamiltonian can be written as follows,

$\displaystyle H_{Q}$
$\displaystyle=\big{\langle}\hat{H}(\hat{p},\hat{\beta})\big{\rangle}$ (23)
$\displaystyle=H+\sum_{i+j=2}^{\infty}\frac{1}{i!j!}\dfrac{\partial^{i+j}H(\beta,p)}{\partial\beta^{i}\partial
p^{j}}\Delta(\beta^{i}p^{j}),$

where $\beta:=\langle\hat{\beta}\rangle$ and $p:=\langle\hat{p}\rangle$ have
been defined, and $H(\beta,p)$ is the classical Hamiltonian (6). As it can be
seen, the Hamiltonian operator has been chosen as Weyl-ordered.

The equations of motion for the variables of the system ($\beta$, $p$ and
$\Delta(\beta^{i}p^{j})$) can then be obtained by computing their Poisson
brackets with the effective Hamiltonian $H_{Q}$:

$\displaystyle\frac{d\beta}{d\alpha}=\\{\beta,H_{Q}\\}=\frac{\partial
H_{Q}}{\partial p},$ (24)
$\displaystyle\frac{dp}{d\alpha}=\\{p,H_{Q}\\}=-\frac{\partial
H_{Q}}{\partial\beta},$ (25)
$\displaystyle\frac{d\Delta(\beta^{i}p^{j})}{d\alpha}=\\{\Delta(\beta^{i}p^{j}),H_{Q}\\}$
(26) $\displaystyle\hskip
30.0pt=\sum_{m,n=0}^{+\infty}\frac{1}{m!n!}\dfrac{\partial^{m+n}H(\beta,p)}{\partial\beta^{m}\partial
p^{n}}\\{\Delta(\beta^{i}p^{j}),\Delta(\beta^{m}p^{n})\\},$

where $\\{\,,\,\\}$ are the Poisson brackets between expectation values
defined as
$\\{\langle{\hat{f}}\rangle,\langle{\hat{g}}\rangle\\}=\frac{1}{i\hbar}\langle[{\hat{f},\hat{g}}]\rangle$,
$\hat{f}$ and $\hat{g}$ being arbitrary operators. Moreover, the general
expression for the brackets between any two moments is known Bojowald:2005cw ;
Bojowald:2010qm  to be

$\displaystyle\\{\Delta(\beta^{k}p^{l}),\Delta(\beta^{m}p^{n})\\}=ml\Delta(\beta^{k}p^{l-1})\Delta(\beta^{m-1}p^{n})-kn\Delta(\beta^{k-1}p^{l})\Delta(\beta^{m}p^{n-1})+\sum_{\begin{subarray}{c}r=1\\\\[2.0pt]
\text{odd}\end{subarray}}^{S}\bigg{(}\frac{i\hbar}{2}\bigg{)}^{r-1}K_{klmn}^{r}\Delta(\beta^{k+m-r}p^{l+n-r}),$

where $S:=\min(k+l,m+n,k+m,l+n)$ and the coefficients $K^{r}_{klmn}$ are
defined as

$\displaystyle
K^{r}_{klmn}=\sum_{s=0}^{r}(-1)^{s}s!(r-s)!\binom{k}{r-s}\binom{l}{s}\binom{m}{s}\binom{n}{r-s}.$

Note that, in general, (24)–(26) form an infinite system of coupled equations.
Therefore, in order to analyze the system, one usually introduces a truncation
by considering negligible all the moments $\Delta(\beta^{n}p^{m})$ with an
order $n+m$ higher than a certain cut-off $N$. A moment
$\Delta(\beta^{n}p^{m})$ has the dimensions of $\hbar^{(n+m)/2}$ and, in this
sense, this kind of truncation is valid during semiclassical regimes, that is,
as long as the state remains peaked on a classical trajectory.

Nonetheless, if the Hamiltonian is at most quadratic on the basic variables,
the different orders decouple and one usually is able to solve the complete
infinite set of equations. In fact, this is the case in the Kasner epochs
analyzed above (when the ratio $r=2e^{4\alpha-8\beta}/p^{2}$ is negligible),
as then the classical Hamiltonian is linear in $p$. In this approximation, the
equations of motion take the form,

$\displaystyle\frac{d\beta}{d\alpha}\approx\text{sign}(p),\hskip
8.5359pt\frac{dp}{d\alpha}\approx 0,\hskip
8.5359pt\frac{d\Delta(\beta^{i}p^{j})}{d\alpha}\approx 0,$

which are immediate to solve

$\displaystyle\beta\approx\text{sign}(p)\alpha+c,$ (27) $\displaystyle
p\approx const.,$ (28) $\displaystyle\Delta(\beta^{i}p^{j})\approx const.,$
(29)

with the integration constant $c$. As can be seen, the dynamics during the
Kasner regime is that of a coherent state. There is no quantum back-reaction,
as the moments are constants of motion and evolve decoupled from the
expectation values, and the behavior of the expectation values remains as in
the classical case. In summary, the parameters that characterize each quantum
Kasner epoch will be, as in the classical case, the constants $c$ and $p$,
along with the infinite constant set of moments $\Delta(\beta^{i}p^{j})$.

Therefore, our goal is to obtain the quantum transition rules, that relate the
set of parameters that describe the initial Kasner epoch
$(\overline{c},\overline{p},\overline{\Delta(\beta^{i}p^{j})})$, which will be
denoted with a bar, with the parametrization of the second Kasner regime
$(\widetilde{c},\widetilde{p},\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{m}p^{n})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}})$, denoted with a tilde. In this way,
we will be able to see how the classical transition rules, described by
equations (20)-(21), are modified by quantum effects. For such a purpose, let
us obtain the constants of motion for the quantum system.

###  III.2 Conserved quantities

In order to obtain the constants of motion, we first note that, since we are
dealing with an infinite system of equations, we will need an infinite amount
of them. Nonetheless, if two independent conserved operators $\hat{R}_{1}$ and
$\hat{R}_{2}$ are known, their expectation values $\langle\hat{R}_{1}\rangle$
and $\langle\hat{R}_{2}\rangle$ would be trivially conserved. In fact, one
would be able to generate an infinite set of conserved quantities, just by
taking the expectation values of any product between them, that is,

$\langle\hat{R}_{i}^{n}\hat{R}_{j}^{m}\rangle=const.,\hskip
5.69046pt\text{for}\hskip
5.69046ptm,n\in\mathbb{N}\;\text{and}\;i,j\in\\{1,2\\}.$ (30)

Obtaining the operators $\hat{R}_{1}$ and $\hat{R}_{2}$ might be a very
complicate task, so we will proceed order by order. Up to third-order in
moments, one can construct these conserved operators just by directly
promoting the conserved classical quantities (14) to operators:

$\displaystyle\hat{R}_{1}=2\hat{H}-\hat{p},$ (31)
$\displaystyle\hat{R}_{2}={e^{2(\alpha+\hat{\beta})}(\hat{H}-\hat{p})}.$ (32)

Up to the mentioned order, the expectation values of these operators can then
be written in terms of moments by performing an expansion around the
expectation values of the basic variables

$\displaystyle\langle\hat{R}_{1}\rangle=R_{1}+\sum_{i+j=2}^{3}\frac{\partial^{i+j}R_{1}}{\partial\beta^{i}\partial
p^{j}}\Delta(\beta^{i}p^{j}),$ (33)
$\displaystyle\langle\hat{R}_{2}\rangle=R_{2}+\sum_{i+j=2}^{3}\frac{\partial^{i+j}R_{2}}{\partial\beta^{i}\partial
p^{j}}\Delta(\beta^{i}p^{j}).$ (34)

Note that these expressions are valid for any ordering of the basic operators
in the definitions (31)–(32). Any reordering of the operators would give a
factor proportional to $\hbar^{2}$ in the above expansions, which is of
fourth-order and thus negligible at this level of approximation. Therefore, up
to third order, one can iteratively construct all the necessary constants of
motion.

For instance, at second-order, the system is described by the two expectation
values ($\beta$, $p$) and the three moments $\Delta(\beta^{i}p^{j})$ with
$i+j\leq 2$, namely the correlation $\Delta(\beta p)$ and the two fluctuations
$\Delta(\beta^{2})$ and $\Delta(p^{2})$. Therefore, in order to completely
solve the system, one needs to construct five constants of motion. The
expectation values (33)–(34) are two of them, and another three can be defined
as,

$\displaystyle\langle(\hat{R}_{1}-R_{1})^{2}\rangle,$ (35)
$\displaystyle\langle(\hat{R}_{1}-R_{1})(\hat{R}_{2}-R_{2})+(\hat{R}_{2}-R_{2})(\hat{R}_{1}-R_{1})\rangle,$
(36) $\displaystyle\langle(\hat{R}_{2}-R_{2})^{2}\rangle.$ (37)

Writing these expressions in terms of moments, as performed in (33)–(34), and
truncating the series at second-order, one ends up with five constants of
motion for five variables and the system is thus completely determined. Note
that, for convenience, we have defined the conserved quantities as expectation
values of powers of differences, like $(\hat{R}_{i}-R_{i})^{2}$, instead of
expectation values of powers, like $\hat{R}_{i}^{2}$, and with a symmetric
ordering of $R_{1}$ and $R_{2}$. In this way one automatically gets pure
second-order real expressions (there is no zeroth order contributions when
expanding these expectation values).

The rationale within a third-order truncation scheme is exactly equivalent to
the one applied at second order, but with some more variables. In this case
the number of variables is nine: two expectation values ($\beta,p$), three
second-order moments, and four third-order moments (the two pure fluctuations
$\Delta(\beta^{3})$ and $\Delta(p^{3})$, in combination with the two high-
order correlations $\Delta(\beta p^{2})$, $\Delta(\beta^{2}p)$). In addition
to the conserved quantities (33)–(37), one can construct another four by
computing the following expectation values

$\displaystyle\langle(\hat{R}_{1}-R_{1})^{3}\rangle,$ (38)
$\displaystyle\langle(\hat{R}_{1}-R_{1})^{2}(\hat{R}_{2}-R_{2})+(\hat{R}_{2}-R_{2})(\hat{R}_{1}-R_{1})^{2}\rangle,$
(39)
$\displaystyle\langle(\hat{R}_{1}-R_{1})(\hat{R}_{2}-R_{2})^{2}+(\hat{R}_{2}-R_{2})^{2}(\hat{R}_{1}-R_{1})\rangle,$
(40) $\displaystyle\langle(\hat{R}_{2}-R_{2})^{3}\rangle.$ (41)

From fourth order on, the situation gets more involved, as the ordering of the
basic operators in the definitions (31)–(32) comes into play. We find out that
the expectation value of the operator $\hat{R}_{1}$, with a completely
symmetric ordering of basic operators,

$\hat{R}_{1}=2\hat{H}-\hat{p}\big{|}_{\text{Weyl}}$ (42)

is conserved up to seventh-order in moments. Nonetheless, this is not the case
for $\hat{R}_{2}$: by considering a Weyl-ordered form for $\hat{R}_{2}$, its
time derivative turns out not to be vanishing, due to some terms proportional
to $\hbar^{2}$. Still, we have been able to explicitly integrate these non-
vanishing terms and, by subtracting the result from the expectation value of
the Weyl-ordered operator version of $R_{2}$, construct the corresponding
conserved quantity. This leads to the following form for the second conserved
operator,

$\displaystyle\hat{R}_{2}$
$\displaystyle={e^{2(\alpha+\hat{\beta})}(\hat{H}-\hat{p})}+\hbar^{2}{e^{2(\alpha+\hat{\beta})}(\hat{H}-\hat{p})^{2}\frac{15\hat{p}\hat{H}^{2}+11\hat{H}^{3}+3\hat{H}\hat{p}^{2}-4\hat{p}^{3}}{2\hat{H}^{4}\hat{R}_{1}^{2}}}\bigg{|}_{\text{Weyl}}.$
(43)

In summary, up to seventh-order in moments, the two operators (42)–(43) are
conserved. Therefore, one can use these expressions to generate, at each
order, all the necessary constants of motion by computing expectation values
of their products, as explained above for second- and third-order truncations.

###  III.3 Quantum Kasner transition

At this point, in order to obtain the quantum transition rules, one can
proceed systematically. One just needs to compute the expectation values of
products of the conserved operators (42)–(43), and write them in terms of
moments by performing an expansion around the expectation values $\beta$ and
$p$. These constants of motion are then evaluated on each asymptotic Kasner
regions, by considering the behavior of different variables (27)–(29). The
requirement of conservation of these quantities leads to a system of equations
that relates the parameters of the first Kasner epoch
$(\overline{c},\overline{p},\overline{\Delta(\beta^{i}p^{j})})$ with those of
the second Kasner regime
$(\widetilde{c},\widetilde{p},\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{i}p^{j})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}})$. This system of equations must then
be solved to obtain the quantum transition rules.

For definiteness, we will complete this procedure with a fifth-order
truncation, by assuming that all moments $\Delta(\beta^{i}p^{j})$ with
$i+j>5=:N$ are negligible. Up to this order, there are 20 variables and one
needs to construct the same amount of constants of motion.

Regarding the classical variables we find out that, on the one hand, making
use of the conserved quantity $\langle\hat{R}_{1}\rangle$, which takes the
simple form $\langle\hat{R}_{1}\rangle=2|p|-p$ at every order in the Kasner
regimes, one can immediately obtain the transition rule for the momentum $p$,

$\displaystyle\widetilde{p}=-\frac{1}{3}\overline{p}.$ (44)

This expression keeps the same form as its classical counterpart (20) and it
is not affected by the moments. On the other hand, the quantum back-reaction
does affect the transition rule for the parameter $c$, which encodes the
asymptotic behavior of the shape-parameter $\beta$, and it takes the form

$\displaystyle\widetilde{c}=-3\overline{c}-\frac{1}{2}\ln\bigg{(}{\frac{2\overline{p}^{2}}{3}}\bigg{)}+\sum_{n=2}^{N}\frac{(-1)^{n}}{n\overline{p}^{n}}\overline{\Delta({p^{n}})}.$
(45)

However, note that only the initial relative pure fluctuations of the momentum
$\overline{\Delta({p^{n}})}/{\overline{p}}^{n}$ appear in this relation.

Concerning the moments, their transition rules are more involved, and they are
explicitly displayed in App. A. Nevertheless, the general transition rule can
be written as follows,333There is just one exception to this general rule. As
can be seen in App. A, apart from the terms given in (III.3), the transition
rule (54) for the moment $\Delta(\beta^{3})$ contains an extra term
proportional to $\hbar^{2}$.

$\displaystyle\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{n}p^{m})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}=(-3)^{n-m}\overline{\Delta(\beta^{n}p^{m})}+\sum_{r=0}^{r_{max}}\sum_{k=0}^{n-1}\sum_{l=l_{min}}^{l_{max}}\frac{a_{nmklr}}{\overline{p}^{n+l+r-k}}\overline{\Delta(p^{r})}\,\,\,\,\overline{\Delta(\beta^{k}p^{n+m+l-k})}$

with certain numerical coefficients $a_{nmklr}$. In fact, all explicit factors
$p$ that appear in this expression can be absorbed by considering relative
moments with respect to $p$ (that is, $\Delta(\beta^{n}p^{m})/p^{m})$, and by
taking into account the transition rule (44),

$\displaystyle\frac{\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{n}p^{m})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}}{\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle
p$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}^{m}}=(-3)^{n}\frac{\overline{\Delta(\beta^{n}p^{m})}}{\overline{p}^{m}}+\sum_{r=0}^{r_{max}}\sum_{k=0}^{n-1}\sum_{l=l_{min}}^{l_{max}}b_{nmklr}\frac{\overline{\Delta(p^{r})}}{\overline{p}^{r}}\frac{\overline{\Delta(\beta^{k}p^{n+m+l-k})}}{\overline{p}^{n+m+l-k}}$

with $b_{nmklr}=(-3)^{m}a_{nmklr}$. In this way, the transition rule for the
relative moments does not explicitly depend on the values of the asymptotic
classical parameters $p$ or $c$.

In the above expressions, the lower limit for $l$ is defined as
$l_{min}=\max\\{-r,k-(n+m)\\}$. This ensures that the right-hand sides of
(III.3) and (III.3) are exclusively given by contributions of order higher or
equal to $n+m$. Moreover, the upper limits of two of the sums, $r_{max}$ and
$l_{max}$, are an artifact of the truncation we are considering, and they just
impose that the product of two moments inside the sum to be, at most, of order
$N$, that is, $n+m+l+r\leq N$. However, the upper limit in the sum in $k$ is
independent of the truncation, and thus physically meaningful. More precisely,
the largest possible value for $k$ is $n-1$, that is, lower that the index in
$\beta$ of the final moment under consideration. Therefore, this means that
the asymptotic form of a moment
$\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{n}p^{m})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}$ after the Kasner transition, depends
on the initial value of the very same moment
$\overline{\Delta(\beta^{n}p^{m})}$ and all moments
$\overline{\Delta(\beta^{k}p^{q})}$, with any $q\geq 0$ but with $0\leq k\leq
n-1$. This translates into the fact that for a given moment, the larger the
index of the shape parameter $n$, the more complicate is the corresponding
transition rule, as more initial moments enter into play.

In particular, the relative pure fluctuations of the momentum $p$, that is
moments with $n=0$, have a very simple transition rule. In fact, they have the
same asymptotic value after and before the transition

$\frac{\ThisStyle{\stackengine{-.1\LMpt}{\SavedStyle\Delta(p^{m})}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}}{\ThisStyle{\stackengine{-.1\LMpt}{\SavedStyle
p}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}^{m}}=\frac{\overline{\Delta(p^{m})}}{\overline{p}^{m}}.$
(46)

This is due to the fact that, at any order, the conserved quantity
$\langle(\hat{R}_{1}-\langle{\hat{R}_{1}}\rangle)^{m}\rangle$ takes the simple
form
$\langle(\hat{R}_{1}-\langle{\hat{R}_{1}}\rangle)^{m}\rangle=(2\text{sign}(p)-1)^{m}\Delta(p^{m})$
in the Kasner regime.

For $n=1$ the transition rule complicates a bit,

$\displaystyle=-3\frac{\overline{\Delta(\beta
p^{m})}}{\overline{p}^{m}}+\sum_{k=m+1}^{N}\frac{(-1)^{k-m}}{k-m}\frac{\overline{\Delta(p^{k})}}{\overline{p}^{k}}$
$\displaystyle\qquad+\sum_{k=2}^{N-m}\frac{(-1)^{k-1}}{k}\frac{\overline{\Delta(p^{m})}}{\overline{p}^{m}}\frac{\overline{\Delta(p^{k})}}{\overline{p}^{k}},$
(47)

but, as can be seen, only moments $\overline{\Delta(\beta p^{m})}$ and
$\overline{\Delta(p^{n})}$ appear on the right-hand side. In order to check
the transition rules for higher values of $n$, we refer the reader to App. A.

Another important property of the transition rule (III.3) is that it is
composed by linear ($r=0$) as well as quadratic ($r\neq 0$) terms on initial
moments. Nonetheless, concerning these quadratic combinations, note that one
of the components is always given by pure fluctuations of the momentum
$\overline{\Delta(p^{r})}/\overline{p}^{r}$, which are conserved through the
Kasner transition (46).

In expressions (III.3)–(III.3), we have chosen to explicitly write the
contribution of the initial moment $\Delta(\beta^{n}p^{m})$ on the right-hand
side, instead of including it in the sum. In this way, one can see that the
transition produces two distinct effects on the moments.

On the one hand, the moment $\Delta(\beta^{n}p^{m})$ is modified by a
multiplicative factor $(-3)^{n-m}$. Therefore, for moments with more weight on
$\beta$ than on $p$, that is with $n>m$, the absolute value of the moment is
increased by this factor. On the contrary, for moments with a dominant
contribution of $p$ ($n<m$), this effect produces a decrease in their absolute
value. Finally, moments with $n=m$ are not affected by this global factor. All
in all, this effect produces a squeezing of the state on the phase space by
compressing it in the $p$ direction, and stretching it in the $\beta$
direction. The compression in the $p$ direction, given by the factor
$(-3)^{m}$, is clearly inherited from the transition rule for the momentum
(44). This is why the relative moments $\Delta(\beta^{n}p^{m})/p^{m}$ do not
suffer such compression in $p$. The effect in the $\beta$ direction is also
related to the factor $-3$ that appears in the transition rule (45) for the
parameter $c$ that defines the asymptotic behavior of $\beta$. But since, in
this case, there are other moments involved, it is not straightforward to
absorb the factor $(-3)^{n}$ in the definition of certain relative moments.

On the other hand, the moment $\Delta(\beta^{n}p^{m})$ is modified by the
presence of other moments $\Delta(\beta^{k}p^{q})$, with $0\leq k\leq n-1$.
This effect, which couples the final value of a given moment with the initial
value of other moments, has not a definite sign, since the numerical factors
$a_{nmklr}$ can be either positive or negative. Therefore, the interpretation
of this effect is not so clear as the previous one, and one has to check
individual moments (and particular states) to see how each moment is affected.

For instance, if we consider an initial Gaussian state in the momentum $p$,
these effects can easily been observed. Its corresponding moments, i.e., the
asymptotic moments before the transition, are given as follows

$\overline{\Delta(\beta^{2n}p^{2m})}=2^{-2(n+m)}\hbar^{2n}\sigma^{2(m-n)}\frac{(2n)!(2m)!}{n!m!},$
(48)

for $\forall n,m\in\mathbb{N}$, and vanishing otherwise. These moments will
evolve and after the transition, they will transform according to (III.3). For
the sake of clarity, let us focus on how the second-order moments are modified

$\overline{p}$ being the value of $p$ in the initial Kasner regime. First of
all, from (III.3) we observe that
$\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta
p)$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}>0$, which means that the transition
generates a positive correlation between $\beta$ and $p$. Thus, since this
correlation is vanishing for any Gaussian state, it is clear that the final
state does not belong to this class. Moreover, from relation (III.3) one can
see that the initial state is compressed in the $p$ direction. However, from
the transition rule (III.3) it is not straightforward to deduce whether the
initial state is stretched or compressed in the $\beta$ direction.
Nevertheless, by considering the explicit form (48) for the initial moments,
it can be shown that,

$\displaystyle\frac{\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\Delta(\beta^{2})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}}{\overline{\Delta(\beta^{2})}}=\frac{\frac{5\sigma^{6}}{4\overline{p}^{4}}+\frac{\sigma^{4}}{\overline{p}^{2}}}{\hbar^{2}}+9$

which is clearly larger than one. Therefore, an initial Gaussian state is
certainly stretched in the $\beta$ direction.

Finally, it is also interesting to evaluate the Heisenberg uncertainty
principle which, for the final state takes the form, In order to be satisfied,
the relative initial fluctuation of the momentum should be small,
$\sigma/p\leq 1$, which is consistent with our assumption of peaked
semiclassical states.

###  III.4 Analysis of the transition dynamics for a Gaussian initial state

In this section we are interested in examining in detail the evolution of the
state and, more specifically, the transition dynamics. In order to develop
this study, we will need to perform a numerical analysis. For this purpose,
let us consider an initial unsquezeed Gaussian state in the initial Kasner
epoch, characterized by a Gaussian width $\sigma=\sqrt{\hbar}$. The moments
for such a state take the form given in equation (48). In addition, for the
rest of the variables, we choose as initial conditions $p=1$, $\beta=100$ and
$\alpha=100$. In this way we ensure that the potential of the Hamiltonian is
negligible and thus the system begins in a Kasner regime.

As expected, the numerical study shows that, evolving the system towards lower
values of $\alpha$, the system follows the Kasner dynamics until it undergoes
a transition, which happens near $\alpha=0$. This transition takes place in a
very small period of time and then the system reaches its final equilibrium
value –the final Kasner regime– very quickly.

On the one hand, concerning the evolution of the expectation values $p$ and
$\beta$, we conclude that they remain almost identical as in the classical
case, that is, as depicted in Figs. 1 and 2. In fact, only in a very close
zoom into the transition (near $\alpha=0$) we can appreciate a slight
modification in their evolution, which is shown in Figs. 3 and 4. In
particular, from these figures one can see that quantum effects slightly
increase the value of $p$ and $\beta$ around the transition.

Figure 3: Comparison of classical and quantum evolution of the momentum $p$
during the transition. The orange line represents the classical evolution and
the blue one the quantum evolution. Figure 4: Comparison of classical and
quantum evolution of the shape-parameter $\beta$ during the transition. The
orange line represents the classical evolution and the blue one the quantum
evolution.

On the other hand, in Figs. 5–8 the evolution of the moments has been plotted.
As it can be seen, during the initial Kasner regime they hold a constant
value, at certain point they begin to perform strong oscillations but then
they quickly relax to another constant value, which characterizes the coherent
state during the final Kasner regime. It is particularly interesting to
observe that the moments that were vanishing in the initial Kasner epoch –due
to the choice of the Gaussian state– are not vanishing in the final one. This
is indeed an expected outcome according to the transition laws that we have
discussed in the previous section. In particular, this means that even when we
choose an initial state where there are no correlations between $\beta$ and
$p$, the transition generates correlations for the final state.

In fact, all the initial vanishing moments are activated in a similar way: as
they approach the transition they experience an excitation and start to grow
exponentially, until they begin to oscillate. This behavior is shown in Fig.
9, where one can notice that all of them grow with the same slope in a
logarithmic plot. This slope is related to the potential of the Hamiltonian
and is found to be equal to $-4$. That is, between the exact Kasner regime and
the transition, the moments begin to feel the presence of a nonvanishing
potential and their evolution is driven by $\Delta(\beta^{n}p^{m})\propto
e^{-4\alpha}$. Furthermore, we see that the lower the order of a given moment,
the sooner it is activated, which is in agreement with the semiclassical
hierarchy of moments we are assuming. The same applies to the beginning of the
oscillation period: lower-order moments begin their oscillation regime before
higher-order ones.

Figure 5: Evolution of the second-order moments. $\Delta(\beta p)$ is depicted
in blue, $\Delta(p^{2})$ in purple and $\Delta(\beta^{2})$ in orange. During
the transition a small positive correlation is generated, the fluctuation of
the shape-parameter is amplified, and the fluctuation of the moment is
diminished.

Figure 6: Evolution of the third-order moments, with the whole range of the
amplitude in the first plot and with a zoom in the second.
$\Delta(p^{2}\beta)$ is depicted in blue, $\Delta(\beta^{2}p)$ in purple,
$\Delta(p^{3})$ in orange and $\Delta(\beta^{3})$ in green.

Figure 7: Evolution of the fourth-order moments, with the whole range of the
amplitude in the first plot and with a zoom in the second.
$\Delta(\beta^{3}p)$ is depicted in blue, $\Delta(\beta p^{3})$ in purple,
$\Delta(\beta^{4})$ in orange, $\Delta(p^{4})$ in green and
$\Delta(\beta^{2}p^{2})$ in black.

Figure 8: Evolution of the fifth-order moments, with the whole range of the
amplitude in the first plot and with a zoom in the second. $\Delta(\beta^{5})$
is depicted in blue, $\Delta(p^{5})$ in purple, $\Delta(\beta^{3}p^{2})$ in
orange, $\Delta(\beta^{2}p^{3})$ in green, $\Delta(\beta^{4}p)$ in black and
$\Delta(\beta p^{4})$ in red. Figure 9: Logarithmic evolution of the initial
vanishing moments with respect to $\alpha$ as they approach the transition.

The precise dynamics of the moments is quite complicate but, if we look into
the transition in more detail, there are some general features that one can
see. For instance, we observe that lower-order moments tend to oscillate with
bigger amplitudes as higher-order ones, as can be seen in Figs. 5–8. Moreover,
at each given order, the pure fluctuations of $p$ have the biggest amplitudes.
And the higher the index of $p$ the earlier the moments reach their final
equilibrium value.

Furthermore, in general, it can be seen that moments with at least one odd
index, i.e., the initially vanishing ones, present more oscillations. In fact,
as shown in Fig. 10, pure fluctuations of the shape-parameter
$\Delta(\beta^{n})$ with an even index $n$ do not oscillate per se; they just
have a local minimum in the transition and then grow up to their final value.

Figure 10: Evolution of $\Delta(\beta^{2})$ and $\Delta(\beta^{4})$ during the
transition, in the first and second plot respectively.

##  IV Discussion

In this paper, we have considered a quantum locally rotationally symmetric
Bianchi II model in order to analyze the transition between Kasner epochs. The
behavior of the classical variables through this transition is well-known, but
there is an open discussion about how this behavior might be modified when
considering quantum effects that are expected to be relevant close to the
singularity.

In order to describe the dynamics of this system, we have considered Misner
variables, which are given by the (logarithm of the) spatial volume and two
anisotropic shape-parameters. In the locally rotationally symmetric case under
consideration, one of the shape-parameters is vanishing. In addition, the
Hamiltonian constraint is deparametrized by choosing the (logarithm of the)
spatial volume as the internal time, which leaves the system with one physical
degree of freedom $(\beta,p)$, $\beta$ being the nonvanishing shape-parameter
and $p$ its conjugate momentum.

Once performed the quantization of the system, the whole information of the
quantum state is encoded in the infinite set of moments (22). Our analysis
then shows that, for peaked semiclassical states, the classical qualitative
picture is maintained in the quantum regime: there are two asymptotic Kasner
regions, connected by a quick transition. In particular, during the Kasner
epochs, the state evolves coherently, as $\langle\hat{\beta}\rangle$ and
$\langle\hat{p}\rangle$ follow a classical trajectory on the phase space,
whereas all the fluctuations and higher-order moments are kept constant
throughout evolution.

By considering a fifth-order truncation in the moments, which corresponds to
neglecting terms of an order $\hbar^{3}$ and higher, we have been able to
obtain the explicit analytic transition rules, that relate the parametrization
of the state before and after the transition. These rules, schematically given
by equations (44)–(III.3) and explicitly displayed in App. A, comprise the
most relevant result of the present paper.

The transition rule for the momentum $p$ (44) remains unchanged with respect
to its classical counterpart. Nonetheless the transition rule for the
parameter $c$ (45), which characterizes the asymptotic behavior of the shape-
parameter $\beta$, is slightly modified by quantum back-reaction effects.
Interestingly, this back-reaction is completely encoded in the relative pure
fluctuations of the momentum $\Delta(p^{n})/p^{n}$.

Concerning the moments, their transition rule (III.3) is much more involved,
but one can notice certain general properties. In particular, the asymptotic
form of a given moment $\Delta(\beta^{n}p^{m})$ after the transition is given
by certain combination of the initial value of all moments
$\Delta(\beta^{k}p^{q})$ with $k\leq n$ and any $q$. Therefore, the complexity
of the transition rule is largely governed by the index of the shape-parameter
$n$. Furthermore, one can see that the transition introduces two distinct
deformations on the state. On the one hand, the initial value of the moment is
multiplied by certain coefficient, which produces a squeezing of the state on
the phase space by compressing it in the $p$ direction and stretching it in
the $\beta$ direction. On the other hand, the back-reaction of other moments
enters as an additive deformation in the transition rule. The physical
interpretation of this second effect is though not so clear.

Finally, in order to understand more precisely the dynamics during the
transition, in the last section, we have performed some numerical simulations
for an initial Gaussian state. The state is deformed throughout the
transition, as expected from the transition rules, and the initial vanishing
moments are excited. In particular the transition generates a positive
correlation of the system. In addition, we observe that the constant
asymptotic values of the moments are connected through quick and strong
oscillations.

Let us remark that this analysis has been carried out for a particular Kasner
transition. We have developed then a first approach to understand the
prevalence and changes of Kasner transitions, which play a key role in the
chaotic oscillatory BKL behavior, under quantum effects. A further study to a
general case should be analyzed in the future to have a full understanding of
the quantum survival of the BKL transitions.

## Acknowledgments

AA-S is supported by the ERC Advanced Grant No. 740209. SFU acknowledges
financial support from an FPU fellowship of the Spanish Ministry of
Universities. This work is funded by Projects FIS2017-85076-P and
FIS2017-86497-C2-2-P (MINECO/AEI/FEDER, UE), and by Basque Government Grant
No. IT956-16.

##  Appendix A Quantum transition rules

In this appendix we display the complete explicit transition rules for the
$20$ parameters that describe the Kasner regime of the system up to fifth-
order in moments, that is, $p$, $c$, and all moments $\Delta(\beta^{n}p^{m})$
with $n+m\leq 5$. In order to remove all explicit dependence on $p$ from the
expressions, we provide the transition rules for the relative moments
$Q^{nm}:=\Delta(\beta^{n}p^{m})/p^{m}$. As in the main text, the objects
before the transition are denoted with a bar
$(\overline{p},\overline{c},\overline{Q}^{nm})$, whereas after the transition
they are denoted with a tilde
$(\widetilde{p},\widetilde{c},\widetilde{Q}^{nm})$:

$\displaystyle\widetilde{p}$ $\displaystyle=-\frac{1}{3}p,$ (49)
$\displaystyle\widetilde{c}$
$\displaystyle=-3c-\frac{1}{2}\ln\bigg{(}{\frac{2p^{2}}{3}}\bigg{)}+\frac{1}{2}\overline{Q}^{02}-\frac{1}{3}\overline{Q}^{03}+\frac{1}{4}\overline{Q}^{04}-\frac{1}{5}\overline{Q}^{05},$
(50) $\displaystyle\widetilde{Q}^{11}$
$\displaystyle=-3\overline{Q}^{11}-\overline{Q}^{02}+\frac{1}{2}\overline{Q}^{03}-\frac{1}{3}\overline{Q}^{04}+\frac{1}{4}\overline{Q}^{05},$
(51) $\displaystyle\widetilde{Q}^{20}$
$\displaystyle=\overline{Q}^{02}+6\overline{Q}^{11}+9\overline{Q}^{20}-\overline{Q}^{03}-3\overline{Q}^{12}+2\overline{Q}^{13}-\frac{1}{4}\big{(}\overline{Q}^{02}\big{)}^{2}+\frac{11}{12}\overline{Q}^{04}+\frac{1}{3}\overline{Q}^{02}\overline{Q}^{03}$
$\displaystyle\quad-\frac{5}{6}\overline{Q}^{05}-\frac{3}{2}\overline{Q}^{14},$
(52) $\displaystyle\widetilde{Q}^{02}$ $\displaystyle=\overline{Q}^{02},$ (53)
$\displaystyle\widetilde{Q}^{30}$
$\displaystyle=-\frac{9}{4p^{2}}\hbar^{2}-27\overline{Q}^{30}-27\overline{Q}^{21}-9\overline{Q}^{12}-\overline{Q}^{03}-\frac{3}{2}\big{(}\overline{Q}^{02}\big{)}^{2}+\frac{3}{2}\overline{Q}^{04}+9\overline{Q}^{13}$
$\displaystyle\quad+\frac{27}{2}\overline{Q}^{22}-9\overline{Q}^{02}\overline{Q}^{11}-\frac{27}{2}\overline{Q}^{02}\overline{Q}^{20}-\frac{7}{4}\overline{Q}^{05}-\frac{33}{4}\overline{Q}^{14}-9\overline{Q}^{23}$
$\displaystyle\quad+6\overline{Q}^{03}\overline{Q}^{11}+9\overline{Q}^{03}\overline{Q}^{20}+\frac{5}{2}\overline{Q}^{02}\overline{Q}^{03}+\frac{9}{2}\overline{Q}^{02}\overline{Q}^{12},$
(54) $\displaystyle\widetilde{Q}^{03}$ $\displaystyle=\overline{Q}^{03},$ (55)
$\displaystyle\widetilde{Q}^{21}$
$\displaystyle=9\overline{Q}^{21}+6\overline{Q}^{12}+\overline{Q}^{03}-\overline{Q}^{04}-3\overline{Q}^{13}+\big{(}\overline{Q}^{02}\big{)}^{2}+3\overline{Q}^{02}\overline{Q}^{11}$
$\displaystyle\quad+2\overline{Q}^{14}+\frac{11}{12}\overline{Q}^{05}-2\overline{Q}^{03}\overline{Q}^{11}-\frac{7}{6}\overline{Q}^{02}\overline{Q}^{03},$
(56) $\displaystyle\widetilde{Q}^{12}$
$\displaystyle=-3\overline{Q}^{12}-\overline{Q}^{03}+\frac{1}{2}\overline{Q}^{04}-\frac{1}{2}\big{(}\overline{Q}^{02}\big{)}^{2}-\frac{1}{3}\overline{Q}^{05}+\frac{1}{3}\overline{Q}^{02}\overline{Q}^{03},$
(57) $\displaystyle\widetilde{Q}^{40}$
$\displaystyle=81\overline{Q}^{40}+\overline{Q}^{04}+12\overline{Q}^{13}+54\overline{Q}^{22}+108\overline{Q}^{31}-2\overline{Q}^{05}-18\overline{Q}^{14}-54\overline{Q}^{23}$
$\displaystyle\quad-54\overline{Q}^{32}+2\overline{Q}^{02}\overline{Q}^{03}+18\overline{Q}^{02}\overline{Q}^{12}+54\overline{Q}^{02}\overline{Q}^{21}+54\overline{Q}^{02}\overline{Q}^{30},$
(58) $\displaystyle\widetilde{Q}^{04}$ $\displaystyle=\overline{Q}^{04},$ (59)
$\displaystyle\widetilde{Q}^{13}$
$\displaystyle=-3\overline{Q}^{13}-\overline{Q}^{04}+\frac{1}{2}\overline{Q}^{05}-\frac{1}{2}\overline{Q}^{02}\overline{Q}^{03},$
(60) $\displaystyle\widetilde{Q}^{31}$
$\displaystyle=-27\overline{Q}^{31}-9\overline{Q}^{13}-27\overline{Q}^{22}-\overline{Q}^{04}+\frac{3}{2}\overline{Q}^{05}+9\overline{Q}^{14}+\frac{27}{2}\overline{Q}^{23}-\frac{3}{2}\overline{Q}^{02}\overline{Q}^{03}$
$\displaystyle\quad-9\overline{Q}^{02}\overline{Q}^{12}-\frac{27}{2}\overline{Q}^{02}\overline{Q}^{21},$
(61) $\displaystyle\widetilde{Q}^{22}$
$\displaystyle=9\overline{Q}^{22}+\overline{Q}^{04}+6\overline{Q}^{13}-3\overline{Q}^{14}-\overline{Q}^{05}+\overline{Q}^{02}\overline{Q}^{03}+3\overline{Q}^{02}\overline{Q}^{12},$
(62) $\displaystyle\widetilde{Q}^{50}$
$\displaystyle=-243\overline{Q}^{50}-\overline{Q}^{05}-15\overline{Q}^{14}-90\overline{Q}^{23}-270\overline{Q}^{32}-405\overline{Q}^{41},$
(63) $\displaystyle\widetilde{Q}^{05}$ $\displaystyle=\overline{Q}^{05},$ (64)
$\displaystyle\widetilde{Q}^{14}$
$\displaystyle=-3\overline{Q}^{14}-\overline{Q}^{05},$ (65)
$\displaystyle\widetilde{Q}^{41}$
$\displaystyle=81\overline{Q}^{41}+\overline{Q}^{05}+12\overline{Q}^{14}+54\overline{Q}^{23}+108\overline{Q}^{32},$
(66) $\displaystyle\widetilde{Q}^{23}$
$\displaystyle=9\overline{Q}^{23}+6\overline{Q}^{14}+\overline{Q}^{05},$ (67)
$\displaystyle\widetilde{Q}^{32}$
$\displaystyle=-27\overline{Q}^{32}-\overline{Q}^{05}-9\overline{Q}^{14}-27\overline{Q}^{23}.$
(68)

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