# Ontologising Trustworthy in the Telecommunications Domain

Ian Oliver  Pekka Kuure  Wiktor Sedkowski  Thore Sommer  Nokia Bell-Labs,
Espoo, Finland
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-8319-2612 Nokia Mobile Networks, Espoo, Finland
Nokia Strategy and Technology, Wrocław, Poland  Christian-Abrecht-Universität
zu Kiel, Germany

###### Abstract

Based upon trusted and confidential computing platforms, telecommunications
systems must provide guaranteed security for the processes and data running
atop them. This in turn requires us to provide trustworthy systems. The term
trustworthy is poorly defined with corresponding misunderstanding and
misapplication. We present a definition of this term, as well as others,
demonstrate its application against certain telecommunications use cases and
address how the learnings from ontologising these structures contribute to
standardisation and the necessity for FAIR ontologies across
telecommunications standards and hosting organisations.

###### keywords:

Trustworthy, Trusted Computing, Telecommunications, O-RAN, 5G, 6G,
Confidential Computing

,

## Introduction

The telecommunications domain is build on an extensive set of interoperable
standards mediated by ETSI, GSMA, 3GPP, IETF etc, covering almost every aspect
from radio physics to cloud and service architectures. This process, described
by some [1] as Machiavellian or as a prisoner’s dilemma with multiple,
sometimes irrational opponents, generates specifications that have enabled to
the present day, five generations of telecommunications systems spanning over
40 years to the current day work on sixth generation (6G) standards.

Standards adhere to the ‘FRAND’ or fair, reasonable, and non-discriminatory
terms of usage [2] and terminological agreement must be made, not just at a
technical level but also at the legal and patenting levels. The use of
terminologies and concepts adhering to findable, accessible, interoperable and
reusable (FAIR) principles in therefore critical; and more so when standards
cross multiple standardisation bodies and are integrated and referenced to.
Such terminologies or ontologies are found inside the standards documents and
rarely are presented as their own stand-alone terminological or ontological
documents. This causes problems when terminology is then inconsistently
applied across standards, even in the same series.

The use of the term trustworthy is becoming endemic within standardisation and
is especially used in telecommunications because of the critical nature of the
systems and the critical nature of data being processed [3]. Increasing
applied to almost everything, for example trustworthy artificial intelligence
in telecoms systems [4] is and will be a focus of 6G. The term trustworthy is
not defined and nor is it rooted or linked in any other definition of the
term, and associated terms in any other standard, especially those
specifically dealing with trust.

The most interesting specifications are the IETF Remote Attestation Procedures
(RATS) and IETF Trusted Execution Environment Provisioning (TEEP) which both
define a number of terms related to trusted and confidential computing [5].
These are directed at the supporting architectures (and may be included in the
later architectural models here) but do not explicitly address the definition
of trustworthy.

The term trustworthy originates from the concepts of trusted computing and the
later confidential computing - mechanisms for establishing the identity,
integrity of systems, and then ensuring that any workload (e.g.: AI processes,
telecoms databases etc) that requires protection (privacy, secure processing)
can run in a secure environment: a trusted and attested machine [6, 7, 8], a
CPU enclave etc. [9].

A trustworthy system is the minimal basis for providing any form of
confidential computing [10, 11]. As this is the chosen technology for
providing ‘trustworthy AI’ and securing data processing, ensuring a good,
common understanding has wide reaching effects.

This introduces a further problem of how trust is established [12], how that
trust is transferred and limited across the system - the chain of trust - and
how that trust is maintained over time.

In this paper we present the outline ontology for trusted and confidential
computing, the underlying semantics and its use in selected telecoms domains
and how the terms related to trust can be utilised. From this we can properly
frame the discussion about what makes something trustworthy and if presented
as its own standard or supporting documentation can be utilised as a common
references. We conclude with our learnings and how this work proceeds with
regards to standardisation and their implementations.

## Semantics of Trustworthy

The definition of trustworthy according to the Cambridge English and Merriam
Webster dictionaries:

> trustworthy, adjective. Deserving of trust, or able to be trusted,
> dependable or worthy of confidence

Trusted Computing based on some root of trust with necessary infrastructure
such as Hardware Security Module (HSM) or Trusted Platform Module (TPM),
provides a mechanism for establishing and endorsing trust. Remote attestation
[13] as part of the management, operations, and supply-chain provides
verification point.111The Nokia Attestation Engine is one implementation of
this: https://github.com/nokia/AttestationEngine.

Remote attestation is based around the idea of measurement, for example,
cryptographic hashes calculated during the boot-time of a device, or as part
of file system integrity monitoring. This further involves collecting those
measurements - known as claims - and verifying them against known good values.
Once this is complete, a decision is made whether to trust the device. This
gives four actions: measure, attest, verify and decide as expressed in figure
1 utilising the formalisms here [14, 15].

${Element}$${Measure}$${Claim}$${Result}$${Decision}$$\scriptstyle{measure}$$\scriptstyle{attestable}$$\scriptstyle{trustworthy}$$\scriptstyle{attest}$$\scriptstyle{verify}$$\scriptstyle{decide}$
Figure 1: Definitions of Trust

If measurements - cryptographic hashes, boot logs, identity structures,
TPM/SGX quote structures etc. - can be taken and the element has some
interface providing access to those then the element is attestable and a claim
produced. Good examples of claims are the TPM 2.0 Quote and SGX tcbblock
structures, while the IEFT RATS Entity Attestation Token (EAT) [16] is a
structure for generalising this.

If the correctness of the obtained claim can be verified then the element is
trustworthy \- note the dictionary definition above. That this does not mean
we trust the element, but rather that we can obtain information that we can
utilise to make a confident decision whether it is trusted or not.

Figure 2 shows how the notion of trust222As a categorical sub-object
classifier is formed. Trusted elements are those elements (a subset of in this
formalism) those elements that map to positive decisions. The statement
$\phi_{Decision}$ represents the whole measurement, attestation, verification
and decision process. Decisions are not necessary binary true/false in nature
and from this we could333and in the Nokia Attestation Engine we do have
intermediate values, to handle explicitly network failure, incorrect requests
etc, which in turn provides the possibility of degrees or levels of trust and
their [partial] ordering. This is out of scope for this discussion.

${TrustedElement}$${1}$${Element}$${Decision}$$\scriptstyle{true}$$\scriptstyle{\phi_{Decision}}$
Figure 2: Trusted Elements

The above diagrams (figs.1 and 2) are simplified - we do not show the
mechanisms for internal structures and their composition444Objects, such as
elements, measures etc, are composed utilising push-outs which very naturally
gives us structures such as the TPM2.0 quote. \- but forms the core principles
for these terms, how they are formed and what they mean.

This gives our semantic framework for trust and trustworthiness and relates
the objects in our system: elements, measure, claims, results and decisions,
with the actions over these and the descriptions such as trusted, attestable,
trustworthy [17]. In the following sections we apply this to the concepts and
models found in telecommunications systems.

## NFV, O-RAN and 5G

The ETSI Network Function Virtualisation Architecture [18] and Open Radio-
Access Network Architecture [19] are two of the main components of modern
telecommunication systems [20]. These each have their own architecture models
which serve to define the major functional components and their interfaces.
These are used to define terminology and plan the standardisation processes,
especially in terms of the functionality and data required by the interfaces.
The elements in these models should not be taken to be component definitions
but rather than broad roles the implementing components take; in the highly
simplified (and incomplete!) figures 3, 4 and 5 we utilise the relationships
between components rather than their interface names as from the specification
[21].

Figure 3: Network Function Virtualisation Elements Figure 4: Open Radio Access
Network Elements Figure 5: 5G System Elements

  * •

O-Cloud is constructed from [physical] infrastructures nodes. This may conform
to the NFV architecture principles, and some may reside within a 5G
deployment.

  * •

O-eNB is an eNB (4G) or gNB(5G) that supports the E2 (SMO-*NB) interface

  * •

A 5G base station (gNB) comprises of from an O-RU, O-DU and O-CU and potential
additional services.

  * •

O-CU, O-DU and the 5G services (AMF, SMF, UPF etc) can be realised and
deployed as bare-metal processes or as virtualised functions, typically
containers but also as virtual machines.

  * •

O-RU are physical network function with one or more antennas. O-DU may be co-
located with the O-RU for real-time requirements and constraints

  * •

An O-RU, being a physical computing (albeit specialised) unit may managed as
part of the NFVI

  * •

O-DU and some 5G services may be distributed towards the network edge.

  * •

Customer services provided OSS/BSS may be implemented as bare-metal,
containers etc., and distributed accordingly.

  * •

O-RAN SMO and NFV MANO will be implemented similarly to other services.

Figure 6: 5G, O-RAN, NFV Relationships

We can further describe these relationships by relating the implementation to
the network function virtualisation infrastructure (NFVI in fig.3) which
defines a collection of computing elements - in effect, things with CPUs and
in our case trusted elements such as TPMs, CPU enclaves etc. It is important
to note that that radio units, despite their internal complexity including
ASICs and other specialised hardware for processing radio signals are in
effect just computers as any other.

Figure 7: Implementation via NFVI Elements

## Providing Trust

As discussed earlier trust in a system is provided from a core root-of-trust
using hardware root-of-trust modules such as the TPM or some other mechanism
such as that found in CPU enclaving systems such as Intel SGX, AMD SEV-SNP and
TrustZone. Figure 8 describes these elements.

Figure 8: Trust Elements

A device consists of a number of components which are capable of running
network applications. Device power-on from immutable code [22]; the
configuration will vary between ASICs, FPGAs, PCs, microcontrollers up to
mainframes and supercomputers.

The measured boot process [23] in forms such as the UEFI measured boot on x86
devices, Dynamtic Root-of-Trust found in Intel TXT equipped systems, similar
in some Power based systems and [very] occasionally in IoT devices [24],
starts with an explicitly trusted piece of code [25], which then
cryptographically measures (hashes) the next piece of code to run - the
firmware proper - and write this to a TPM. This process continues during boot
until run-time forming a chain-of-trust through a Merkel tree of hashes
describing the system’s integrity [26].

Secure boot is a complimentary process providing verification of a component
or software through digital signatures and prevents locally the execution of
unverified components. Secure boot only provides evidence about the provenance
and not the validity of the contents. If the signing keys are lost - an
unfortunately regular occurrence - then anyone can potentially sign
components.

Figure 9 shows some of the more important quote structures which provide
cryptographically signed evidence or claims about the machine’s identity and
integrity.

We have abstracted away a number of details, for example, a CPU enclave
generating an enclave key is a very specific and complex process. However
these vendor specific details are not required at this level.

Figure 9: Quotes and Their Derivations and Relationships

Important to note is the amount of cross-referencing, such as the TPM’s
Endorsement Certificate being verifiable against a manufacturer’s certificate
authority, or the UEFI eventlog’s contents used to calcualte PCRs values etc.
The TPM’s quote (TPMS_ATTEST) structure itself is signed by an attestation key
(AK) derived from the endorsement key (EK) and contains a hash of the hashed
public parts of the AK and the EK.

Referring back to figure 1 these are the measurements from any given element.
Figure 10 shows how these (with composition) link together in a specific
example: here a PC’s firmware and firmware configuration form measurements and
the composition of these a quote which becomes a claim through the process of
attestation.

${t:TPM}$${ak:Key}$${q:TPMQuote}$${e:PC}$${f:FW}$${pcr0:Hash}$${pcr0||pcr1:Hash}$${c:Config}$${pcr1:Hash}$$\scriptstyle{measure}$$\scriptstyle{measure}$$\scriptstyle{measure}$
Figure 10: Elements and Measures

This shows the structure of a PC composed of a TPM, firmware and
configuration, each of these with its own measurement(s). A TPM quote is a
measurement formed by the composition of these measures. Composition of PCRs
means an ordered concatenation and hashing (Merkel tree) and composition of a
Key means signing by that key.

It is worth noting the fields of the TPM’s quote or TPMS_ATTEST structure as
described in table 1 \- the quote structure for SGX can be found in [27].

Field | Description
---|---
pcrDigest | hash of the selected PCR entries
pcrSelect | list of the selected PCR fields
magic, type | Fixed values denoting the type of the structure (TCG defined)
firmwareVersion | Firwmare version of the TPM 2.0 device
clockInfo::clock | Value of the TPM’s internal clock
clockInfo::resetCount | Counters showing the number of power cycles
clockInfo::restartCount | Counters showing the number of sleep (S3) cycles
clockInfo::safe | Flag showing whether the TPM was powered off explicitly
extraData | Any user supplied data, eg: a nonce for replay attack prevention
qualifiedSigner | a hash of the public parts of the signing key’s hierachy
Table 1: TPM 2.0 Quote (TPMS_ATTEST) Fields

It follows that the term ‘integrity of a system‘ is just the composition of
all possible measures for that system. This implies a lattice in which we can
delineate a bound above which gives a minimal set of sufficient measurements
that must be taken.

## Trusting a Telecommunications Deployment

Utilising the models in the earlier sections we apply these to some selected
use cases for prototypical telecommunications deployments. These are used to
both to validate our models and assist in deriving the properties a
‘trusted/trustworthy’ O-RAN/5G/NFV system requires. It is exactly these use
case that are used to drive the standardisation processes and the notions of
trustworthy.

### O-RU Trust in NFV

A simple case is an O-RAN Radio Unit is an NFVI element containing in this
case a TPM - this is a typical deployment. The O-RU provides an attestation
interface towards an attestation server within the O-RAN SMO as shown in in
figure 11.

Figure 11: O-RU Relationships

The following questions deriving from this must be answered in order to
establish trust:

  * •

Which measurements and combinations are are required?

  * •

What is the decision mechanism by which O-RAN SMO denotes the O-RU as trusted?

  * •

What is the decision mechanism by which NFV MANO denotes the O-RU as trusted?

  * •

What is the correlation between the O-RAN SMO view of trust and NFV MANO view
of trust?

We may also establish failure scenarios such as if the NFV MANO detects a loss
of trust but the SMO not, is the O-RU still trusted? Is there a trust ordering
such that $x:RU\;|\;trusted_{SMO}(d)\Rightarrow trusted_{NFV}(d)$?

### O-DU Trust

The question of O-DU Trust is more complex; the DU is often deployed as a
container, which in turn runs on some NFVI element. The DU uses one or more
RUs as part of its operation. Figure 12 shows a scenario - NB: the RU here is
the same RU (with measurements) as in fig.11.

Figure 12: O-DU Relationships

Again, we infer similar questions about the NFVI Element and how MANO decides
on its trust; the O-RU is as earlier. For the O-DU we have the following
questions inferred:

  * •

Can an O-DU only run on trusted NFVI elements?

  * •

Does a similar trust ordering exist as before:
$d:DU\;|\;trusted_{SMO}(d)\Rightarrow trusted_{NFV}(d)$??

  * •

Can an O-DU manage untrusted O-RUs and vice versa, can a trusted O-RU be
utilised by an untrusted O-DU?

### O-DU Confidentiality

The final case depicted in figure 13 involves confidential computing and that
some functionality (maybe some AI function that needs to be trustworthy?)
within a O-DU must run within a CPU enclave.

Figure 13: O-DU Confidentiality Relationships

The questions raised earlier again are all valid here, but now with the
following additions:

  * •

Does a relationship of trust exist between the enclave and the NFVI element on
which that enclave is implemented (via the CPU)

  * •

Does the enclave need to be on the same NFVI element? Does this imply ‘enclave
mobility’ or is there a ‘locality’ of processing constraint?

  * •

Can a function requiring an enclave run on an untrusted machine, even though
the enclave may be trustworthy (and trusted)?

  * •

How is trust communicated across the composition (part-of) relationships?

In current trustworthy systems there is no direct mechanism to relate the
quote generated by an enclave with the quote generated by the the host
machine’s TPM. If a link does exist then it is very ephemeral and hidden in
firmware measurements during boot (if at all, and even then it is most likely
proprietary and unavailable, which fails our attestatble definition from
earlier).

From this model we are forced to address the nature of composition
relationships and the establishment and maintenance chain-of-trust over these
relationships. We are also forced to confront the fact that a function may be
trusted, but is part of an component and running on trusted elements. Putting
it in this manner it becomes obvious that trustworthy of individual components
is not the same as trustworthy when applied to larger systems.

## Conclusions

We have presented the domain, and set out to define the underlying semantics
for the terms trust, trustworthy, attestable and outlined how these terms fit
together. We have then shown an example applying this to the O-RAN, 5G and NFV
concepts which underpin modern telecommunications systems. Obviously for
brevity much has been simplified in how these systems are composed but
satisfies the definition of ontology itself, in that we provide a mechanism
for the kinds of things and their nature in these systems.

Through the use of some simple examples drawn from the telecoms domain we have
shown that we can easily generate a large number of questions about how
trustworthiness is establish, how a trust decision might be made, as well as
identifying where attestation must take place. Larger questions then arise
from how the chains-of-trust and system composition interact.

From a FAIR point of view it has been obvious to many in this domain that such
ontologies must be made to provide a common terminological grounding. In day-
to-day work we interact with over ten international standards organisations
specifically for telecommunications - this does not include governmental
regulations and internal specifications and requirements.

Any such ontology in this area is most likely an evolving document rather than
a fixed standard. It is also necessary that the terms here do not conflict but
are mappable to existing terms. For example, the use of EAT in IETF RATS is a
‘claim’ in our ontology, and its contents ‘measurements.

Implementation of the ontologies here has been made in Protege and will be
released as open source, possibly via one or more standards organisations. At
minimum these ontologies will remain open, freely accessible and become part
of any input to standards relating to cybersecurity across the standardisation
organisations.

A final note on the complexity of the ontologies: in the current model we have
approximately 100 types, 200 relationships and the description logic is
$\mathcal{ALCHIF(\circ,*)}$.

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# Quantum dynamics corresponding
to chaotic BKL scenario

Andrzej Góźdź<EMAIL_ADDRESS>Institute of Physics, Maria Curie-
Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland
Aleksandra Pȩdrak<EMAIL_ADDRESS>Department of Fundamental
Research, National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa,
Poland  Włodzimierz Piechocki<EMAIL_ADDRESS>Department of
Fundamental Research, National Centre for Nuclear Research, Pasteura 7, 02-093
Warszawa, Poland

(August 28, 2024)

###### Abstract

We quantize the solution to the Belinski-Khalatnikov-Lifshitz (BKL) scenario
using the integral quantization method. Quantization smears the gravitational
singularity avoiding its localization in the configuration space. The latter
is defined in terms of spatial and temporal coordinates, which are treated on
the same footing that enables respecting covariance of general relativity. The
relative quantum perturbations grow as the system evolves towards the
gravitational singularity. The quantum randomness amplifies the deterministic
classical chaos of the BKL scenario. Additionally, our results suggest that
the generic singularity of general relativity can be avoided at quantum level
giving support to the expectation that quantum gravity has good chance to be a
regular theory.

###### Contents

  1. I Introduction
  2. II Solution to the BKL scenario
  3. III Affine coherent states quantization
  4. IV Quantization of the solution to BKL scenario
    1. IV.1 Evolving wave packets
      1. IV.1.1 Coherent states and expectation values
      2. IV.1.2 The modified exponential packet and expectation values
    2. IV.2 Variances in the Hilbert space $\mathcal{H}$
      1. IV.2.1 Using coherent states
      2. IV.2.2 Using exponential wave packet
  5. V Stochastic aspects of quantum evolution
  6. VI Conclusions
  7. A Essence of integral quantization

##  I Introduction

The Belinski, Khalatnikov and Lifshitz (BKL) conjecture states that general
relativity includes the solution with generic gravitational singularity [1,
2]. The evolution towards the BKL singularity, the so-called BKL scenario,
consists of the deterministic dynamics turning into stochastic process near
the generic singularity. There are at least two fundamental questions to be
addressed in that context: What is the fate of the BKL chaos at quantum level?
Can the singularity be avoided in the corresponding quantum theory?

The evolution process presented in [1, 2] is complicated and difficult to map
into quantum evolution. There exists well defined and comparatively simple
model of the BKL scenario [3, 4, 5] that can be used in the derivation of the
BKL conjecture [6]. The model has been obtained from the general model of the
Bianchi IX spacetime for perfect fluid. The equation of state of that fluid
reads $p=k\varepsilon,\leavevmode\nobreak\ 0\leq k<1,$ where $p$ and
$\varepsilon$ denote the pressure and energy density of the fluid,
respectively. The case $k=1$ is excluded as it does not lead to the
oscillatory dynamics specific to the Bianchi IX model. The massive BKL
scenario model has been obtained from that general Bianchi IX model by making
the assumption (see Eq. (3)) that in the dynamics, near the singularity, the
anisotropy of space grows without bound so that each of the so-called
directional scale factors oscillate, but never crosses each other, and evolves
towards vanishing, i.e., singularity. The resulting dynamics, specified in the
next section, is different from the commonly known mixmaster dynamics [7, 8]
in which one can divide the oscillatory evolution of the system into eras,
each consisting of Kasner’s epochs, evolving towards the singularity. The
mixmaster model is the vacuum Bianchi IX model and can serve to derive the BKL
conjecture as well [6] so that we call it the vacuum model of the BKL
scenario. We have made the comparison of the dynamics of both massive and
vacuum models within dynamical systems method recently [9]. The dynamics of
the massive BKL model depends on the matter field implicitly via the
directional scale factors which are effective ones, see Eqs. (2.6), (2.7) and
(2.24) in [3]. Also, due to the dependence of the general Bianchi IX model on
matter components, it was possible to obtain the asymptotic form (near the
singularity) of the general dynamics [3]. That dynamics is mathematically
simple enough to be solved analytically [10]. The dynamics of the vacuum
Bianchi IX model is the same faraway and near the singularity. It is so
complex that the model is non-integrable [11]. Therefore, the massive BKL
model is both simpler for analysis than the vacuum BKL model, and better
suitable to describe the dynamics in the neighborhood of the cosmological
singularity.

Recently, we have verified numerically that the classical dynamics underlying
the present paper leads to the gravitational singularity [12, 13], and it is
generically unstable turning into chaotic process near the singularity [10].
These features are consistent with the original BKL scenario [1, 2, 5].

As far as we are aware, there are not available any results concerning the
issue of the construction of quantum theory corresponding directly to the
original BKL scenario [1, 2]. The existing results concern the Hamiltonian
framework, in terms of Ashtekar’s variables, that is supposed to be convenient
to address the BKL scenario problem [14, 15]. We have used the homogeneous
sector of that formalism to consider the possibility of existence of classical
end quantum spikes within that sector [16, 17]. Recent article on spacelike
singularities of general relativity makes promise of coming back to the issue
of addressing the original BKL singularity problem within loop quantum gravity
[18]. We do not exclude the case of joining that development in the future,
but presently we rather prefer to follow our programme.

Quantization of the dynamics presented in [3] can be used in the examination
of the fate of the corresponding quantum dynamics. In fact, we have already
quantized that model with the conclusion that quantization of the dynamics
leads to avoiding gravitational singularity [19, 20]. In these papers, we have
quantized Hamilton’s dynamics derived in [23]. However, since quantization is
known to be an ambiguous procedure, we have decided to examine the robustness
of these results by making use of a completely different quantization method,
which is one of the goals of the present paper. That method, applied recently
to the quantization of the Schwarzschild spacetime [24], includes quantization
of the temporal and spatial variables on the same footing. The rationale for
such dealing is that the distinction between time and space variables violates
the general covariance of arbitrary transformations of temporal and spatial
coordinates.

The results of the present paper, in the context of resolving the cosmological
singularity, are similar to the results of the paper [24] addressing the issue
of a singularity of an isolated object. In both cases quantization smears the
singularity avoiding its localization in configuration space. The issue of
resolving the singularity within our two quite different approaches will be
further discussed in the last section.

The new phenomenon we deal with, in the present paper, is the fate of the
classical chaos at quantum level. Our analysis shows that the quantum
randomness turns deterministic classical chaos into stochastic quantum chaos.

The paper is organized as follows: In Sec. II we recall the main results of
the paper [10] to have our paper self-contained. Sec. III presents the main
aspects of the coherent states quantization method adopted to our
gravitational system. In Sec. IV we quantize the solution to the BKL scenario.
Stochastic aspects of quantum evolution are presented in Sec. V. We conclude
in the last section. Appendix presents the essence of the coherent states
quantization.

In the following we choose $\;G=c=1=\hbar\;$ except where otherwise noted.

##  II Solution to the BKL scenario

To have the paper self-contained, we recall in this section the main results
of Ref. [10] to be used later.

The massive model of the BKL scenario is defined to be [3, 5]

$\frac{d^{2}\ln a}{dt^{2}}=\frac{b}{a}-a^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \frac{d^{2}\ln
b}{dt^{2}}=a^{2}-\frac{b}{a}+\frac{c}{b},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \frac{d^{2}\ln
c}{dt^{2}}=a^{2}-\frac{c}{b},$ (1)

subject to the constraint

$\frac{d\ln a}{dt}\;\frac{d\ln b}{dt}+\frac{d\ln a}{dt}\;\frac{d\ln
c}{dt}+\frac{d\ln b}{dt}\;\frac{d\ln c}{dt}=a^{2}+\frac{b}{a}+\frac{c}{b}\,,$
(2)

where $\,a=a(t)>0,\,b=b(t)>0$ and $\,c=c(t)>0$ are the so-called directional
scale factors, while $t\in\mathbb{R}$ is a monotonic function of proper time.

Eqs. (1)–(2) have been derived from the general dynamics of the Bianchi IX
model under the condition that near the singularity the following strong
inequalities are satisfied [3]

$a\gg b\gg c>0\,.$ (3)

It has been found in [10] that the analytical solutions to Eqs. (1)–(2), for
$t>t_{0}$, read

$a(t)=\frac{3}{t-t_{0}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ b(t)=\frac{30}{(t-t_{0})^{3}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ c(t)=\frac{120}{(t-t_{0})^{5}}\,,$
(4)

where $t-t_{0}\neq 0$ and $t_{0}$ is an arbitrary real number. Thus, the
solutions are parameterized by the number $t_{0}\in\mathbb{R}$.

The solution (4) corresponds, for instance in the case $t>t_{0}$ and
$t_{0}<0$, to the following choice of the initial data

$\displaystyle a(0)$ $\displaystyle=-3\;t_{0}^{-1},$ $\displaystyle\dot{a}(0)$
$\displaystyle=-3\;t_{0}^{-2}\,,$ $\displaystyle b(0)$
$\displaystyle=-30\;t_{0}^{-3},$ $\displaystyle\dot{b}(0)$
$\displaystyle=-90\;t_{0}^{-4}\,,$ (5) $\displaystyle c(0)$
$\displaystyle=-120\;t_{0}^{-5},$ $\displaystyle\dot{c}(0)$
$\displaystyle=-600\;t_{0}^{-6}\,.$

The stability analyses carried out in [10] have shown that the solution (4) is
unstable against small perturbation. More precisely, substituting the
following functions

$\displaystyle a(t)=3(t-t_{0})^{-1}+\epsilon\alpha(t),$ (6a) $\displaystyle
b(t)=30(t-t_{0})^{-3}+\epsilon\beta(t),$ (6b) $\displaystyle
c(t)=120(t-t_{0})^{-5}+\epsilon\gamma(t)\,,$ (6c)

into (1)–(2) leads, in the first order in the small parameter $\epsilon$, to
the following solution of the resulting equations

$\displaystyle\alpha(t)=$
$\displaystyle\exp(-\theta/2)\\!\left[K_{1}\cos(\omega_{1}\theta\\!+\\!\varphi_{1})\\!+\\!K_{2}\cos(\omega_{2}\theta\\!+\\!\varphi_{2})\right]\\!+\\!K_{3}\exp(-2\theta),$
(7a) $\displaystyle\beta(t)=$
$\displaystyle\exp(-5\theta/2)\left[\left(4+6\sqrt{6}\right)K_{1}\cos(\omega_{1}\theta+\varphi_{1})\right.$
$\displaystyle\left.+\left(4-6\sqrt{6}\right)K_{2}\cos(\omega_{2}\theta+\varphi_{2})\right]+30K_{3}\exp(-4\theta),$
(7b) $\displaystyle\gamma(t)=$
$\displaystyle-4\exp(-9\theta/2)\left[\left(26+9\sqrt{6}\right)K_{1}\cos(\omega_{1}\theta+\varphi_{1})\right.$
$\displaystyle\left.+\left(26-9\sqrt{6}\right)K_{2}\cos(\omega_{2}\theta+\varphi_{2})\right]+200K_{3}\exp(-6\theta)\,,$
(7c)

where $\theta=\ln(t-t_{0})$. The two frequencies read

$\omega_{1}=\frac{1}{2}\sqrt{95-24\sqrt{6}},\qquad\omega_{2}=\frac{1}{2}\sqrt{95+24\sqrt{6}}\,,$
(8)

where $K_{1},K_{2},K_{3},\varphi_{1},$ and $\varphi_{2}$ are constants.

The manifold $\mathcal{M}$ defined by
$\\{K_{1},K_{2},K_{3},\varphi_{1},\varphi_{2}\\}$ is a submanifold of
$\mathbb{R}^{5}$. The solution defined by (6) and (7) corresponds to the
choice of the set of the initial data $\mathcal{N}$ that is a small
neighborhood of the initial data (II). $\mathcal{N}$ is a submanifold of
$\mathbb{R}^{5}$ as (II) defines five independent constants due to the
constraint (2). Therefore, it is clear that (7) presents a generic solution as
the measures of both $\mathcal{M}$ and $\mathcal{N}$ are nonzero. The exact
solution (4) alone is of zero-measure in the space of all possible solutions
to Eqs. (1)–(2).

The relative perturbations $\alpha/a,\beta/b,$ and $\gamma/c$ grow
proportionally as $\exp(\frac{1}{2}\theta)$. The multiplier $1/2$ plays the
role of a Lyapunov exponent, describing the rate of their divergence. Since it
is positive, the evolution of the system towards the gravitational singularity
($\theta\rightarrow+\infty$) is chaotic. The transition into the chaos occurs
if the evolution begins with the initial data which belong, for instance, to
the neighbourhood of the conditions (II).

The original BKL scenario [1, 2] is known to enter the chaotic phase near the
singularity. Its vacuum model, the mixmaster universe, has been proved to
include the chaotic dynamics [25, 26, 27, 28, 29]. Its massive model [10],
underlying the present paper, has never been examined earlier in the context
of stochasticity. Finding that its dynamics is chaotic, opens the door for
studies of that issue at quantum level, which is the main subject of our
article. The main difference between both BKL models (in terms of physics) is
that the latter is more realistic near the singularity as it includes
effectively some contribution from matter field.

##  III Affine coherent states quantization

We propose to quantize the classical BKL scenario by using the integral
quantization called the affine coherent states quantization, see Appendix. We
have recently applied this approach in the context of cosmology [19, 20] and
astrophysics [24, 22].

In general relativity time and position in space are treated on the same
level, however, in quantum mechanics time is not considered to be a quantum
observable, but a parameter enumerating events. In this paper, we treat time
and position on the same footing in the quantum description. They are related
to operators obtained by the affine coherent states quantization. This idea
requires introducing the notion of an extended classical configuration space
by including time as an additional coordinate. The correspondence between the
classical time and position is done by comparing their classical values with
expectation values of their quantum counterparts.

In the following, we extend the method of quantization used in [19, 20, 24].
In the present description of the BKL scenario, the Hilbert space has to be
extended to the carrier space of infinite dimensional unitary irreducible
representation of the direct product of three affine groups with additional
constraints determining a model of physical time. In this approach there is no
distinction between kinematical and dynamical Hilbert spaces; we construct the
state space of the system with time treated on the same footing as other
observables. In addition, we obtain quantum states evolving similarly to
classical solutions. It is achieved by using the idea of the correspondence
principle between quantum mechanics and its classical approximation.

To begin with, we introduce two configuration spaces defined as follows: the
classical gravitational configuration space $T_{BKL}$

$T_{BKL}:=\\{(t,a,b,c):(t,a,b,c)\in\mathbb{R}\times\mathbb{R}_{+}^{3}\\}\,,$
(9)

where $\mathbb{R}_{+}=(0,+\infty)$, and the affine configuration space $T$ is
defined as

$T=\\{\xi=(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6}):\xi\in(\mathbb{R}\times\mathbb{R}_{+})\times(\mathbb{R}\times\mathbb{R}_{+})\times(\mathbb{R}\times\mathbb{R}_{+})\\}\,,$
(10)

where every pair $(\xi_{k},\xi_{k+1})$, ($k=1,3,5$), parameterizes the affine
group ${\textrm{Aff}(\mathbb{R})}$.

The variables with even indices correspond to the scale factors
$\xi_{2}=a,\,\xi_{4}=b,\,\xi_{6}=c$. Because $a,b,c>0$ and
$\xi_{1},\xi_{3},\xi_{5}\in\mathbb{R}$, the configuration space $T$
parametrizes the simple product of 3 affine groups
${\textrm{Aff}(\mathbb{R})}\times{\textrm{Aff}(\mathbb{R})}\times{\textrm{Aff}(\mathbb{R})}=:G$
to be used in the affine quantization.

As the observational data are parameterized by a single time parameter, the
variables $\xi_{1},\,\xi_{3},\,\xi_{5}$ should be mapped onto a single
variable representing time.

The affine group ${\textrm{Aff}(\mathbb{R})}$ is known to have two nontrivial
unitary irreducible representations in the Hilbert space
$\mathcal{H}_{x}:=L^{2}(\mathbb{R}_{+},d\nu(x))$, where $d\nu(x):=dx/x$. We
choose the one defined as follows (the second representation would give
exactly the same results):

$U(\xi_{k},\xi_{k+1})\Psi(x)=e^{i\xi_{k}x}\Psi(\xi_{k+1}x),$ (11)

where, $k=1,3,5$ and $\langle x|\Psi\rangle=:\Psi(x)\in\mathcal{H}_{x}$. The
action (11) corresponds to the standard parametrization of the affine group
${\textrm{Aff}(\mathbb{R})}$ defined by the multiplication law

$(\xi_{k},\xi_{k+1})\cdot(\xi_{k}^{\prime},\xi_{k+1}^{\prime}):=(\xi_{k}+\xi_{k+1}\xi_{k}^{\prime},\xi_{k+1}\xi_{k+1}^{\prime})\in{\textrm{Aff}(\mathbb{R})}\,.$
(12)

The left invariant measure on the group ${\textrm{Aff}(\mathbb{R})}$ reads

$d\mu(\xi_{k},\xi_{k+1}):=d\xi_{k}\;\frac{d\xi_{k+1}}{\xi_{k+1}^{2}},$ (13)

and the corresponding invariant integration over the affine group is defined
to be

$\int_{{\textrm{Aff}(\mathbb{R})}}d\mu(\xi_{k},\xi_{k+1}):=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\xi_{k}\int_{0}^{\infty}d\xi_{k+1}/\xi_{k+1}^{2}\,.$
(14)

It is clear that the direct product of three affine groups $G$ has the unitary
irreducible representation in the following Hilbert space
$\mathcal{H}=\mathcal{H}_{x_{1}}\otimes\mathcal{H}_{x_{2}}\otimes\mathcal{H}_{x_{3}}=L^{2}(\mathbb{R}_{+}^{3},d\nu(x_{1},x_{2},x_{3}))$,
where $d\nu(x_{1},x_{2},x_{3})=d\nu(x_{1})d\nu(x_{2})d\nu(x_{3})$. It enables
defining in $\mathcal{H}$ the continuous family of affine coherent states
$\langle
x_{1},x_{2},x_{3}|\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6}\rangle:=\langle
x_{1}|\xi_{1},\xi_{2}\rangle\langle x_{2}|\xi_{3},\xi_{4}\rangle\langle
x_{3}|\xi_{5},\xi_{6}\rangle$, as follows

$\mathcal{H}\ni\langle
x_{1},x_{2},x_{3}|\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6}\rangle:=U(\xi)\Phi_{0}(x_{1},x_{2},x_{3})\,,$
(15)

where $U(\xi):=U(\xi_{1},\xi_{2})U(\xi_{3},\xi_{4})U(\xi_{5},\xi_{6})$, and
$|\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6}\rangle:=|\xi_{1},\xi_{2}\rangle|\xi_{3},\xi_{4}\rangle|\xi_{5},\xi_{6}\rangle$
and where

$\mathcal{H}\ni\Phi_{0}(x_{1},x_{2},x_{3})=\Phi_{1}(x_{1})\Phi_{2}(x_{2})\Phi_{3}(x_{3})\,.$
(16)

In (16) the vectors $\Phi_{k}(x_{k})\in L^{2}(\mathbb{R}_{+},d\nu(x_{k}))$,
$k=1,2,3$ are the so-called fiducial vectors. They are required to satisfy the
two conditions

$\int_{0}^{\infty}\frac{dx}{x}|\Phi_{k}(x)|^{2}=1\,,$ (17)

and

$A_{\phi_{l}}:=\int_{0}^{\infty}\frac{dx}{x^{2}}|\Phi_{k}(x)|^{2}<\infty\,.$
(18)

where $l=1$ for $k=1$, $l=3$ for $k=2$ and $l=5$ for $k=3$. The fiducial
vectors are the free “parameters” of this quantization scheme.

Finally, we have

$\displaystyle
U(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\Phi_{0}(x_{1},x_{2},x_{3})$
$\displaystyle=$ $\displaystyle
U(\xi_{1},\xi_{2})U(\xi_{3},\xi_{4})U(\xi_{5},\xi_{6})\Phi_{1}(x_{1})\Phi_{2}(x_{2})\Phi_{3}(x_{3})$
$\displaystyle=$ $\displaystyle
e^{i(\xi_{1}x_{1}+\xi_{3}x_{2}+\xi_{5}x_{3})}\Phi_{1}(\xi_{2}x_{1})\Phi_{2}(\xi_{4}x_{2})\Phi_{3}(\xi_{6}x_{3})\,.$
(19)

The irreducibility of the representation leads to the resolution of the unity
in $\mathcal{H}$ as follows

$\displaystyle\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(\xi)|\xi\rangle\langle\xi|:=\bigotimes_{k=1,3,5}\frac{1}{A_{\phi_{k}}}\int_{{\textrm{Aff}(\mathbb{R})}}d\mu(\xi_{k},\xi_{k+1})|\xi_{k},\xi_{k+1}\rangle\langle\xi_{k},\xi_{k+1}|$
$\displaystyle=\frac{1}{A_{\Phi_{1}}A_{\Phi_{3}}A_{\Phi_{5}}}\int_{\textrm{Aff}(\mathbb{R})}d\mu(\xi_{1},\xi_{2})|\xi_{1},\xi_{2}\rangle\langle\xi_{1},\xi_{2}|\otimes\int_{{\textrm{Aff}(\mathbb{R})}}d\mu(\xi_{3},\xi_{4})|\xi_{3},\xi_{4}\rangle\langle\xi_{3},\xi_{4}|$
$\displaystyle\otimes\int_{{\textrm{Aff}(\mathbb{R})}}d\mu(\xi_{5},\xi_{6})|\xi_{5},\xi_{6}\rangle\langle\xi_{5},\xi_{6}|=\hat{1\kern-4.75pt1}_{1}\otimes\hat{1\kern-4.75pt1}_{2}\otimes\hat{1\kern-4.75pt1}_{3}=\hat{1\kern-4.75pt1}\,,$
(20)

where $d\mu(\xi)=\Pi_{k=1,3,5}d\mu(\xi_{k},\xi_{k+1})$.

The resolution (III) can be used for mapping a classical observable
$f:T\rightarrow\mathbb{R}$ into an operator
$\hat{f}:\mathcal{H}\rightarrow\mathcal{H}$ as follows [19, 20, 21, 24]

$\displaystyle\hat{f}:=\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(\xi)|\xi\rangle
f(\xi)\langle\xi|$
$\displaystyle=\frac{1}{A_{\Phi_{1}}A_{\Phi_{3}}A_{\Phi_{5}}}\int_{\textrm{Aff}(\mathbb{R})}d\mu(\xi_{1},\xi_{2})\int_{\textrm{Aff}(\mathbb{R})}d\mu(\xi_{3},\xi_{4})\int_{\textrm{Aff}(\mathbb{R})}d\mu(\xi_{5},\xi_{6})$
$\displaystyle|\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6}\rangle
f(\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6})\langle\xi_{1},\xi_{2};\xi_{3},\xi_{4};\xi_{5},\xi_{6}|\,,$
(21)

where $A_{\Phi}:=A_{\Phi_{1}}A_{\Phi_{3}}A_{\Phi_{5}}$.

Here, we recall two standard characteristics of quantum observables: (i)
expectation values and variances of quantum observables are the most important
characteristics which allow to compare quantum and classical worlds, and (ii)
expectation values of quantum observables correspond to classical values of
measured quantities and their variances describe quantum smearing of these
observables.

A general form of expectation value of the observable $\hat{f}$ obtained from
the classical function $f$, while the quantum system is in the state
$|\Psi\rangle$, reads

$\langle\hat{f};\Psi\rangle:=\langle\Psi|\hat{f}|\Psi\rangle=\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(\xi)f(\xi)|\langle\xi|\Psi\rangle|^{2}\,.$
(22)

The variance of an observable $\hat{f}$ defined as
$\mathrm{var}(\hat{f};\Psi):=\langle(\hat{f}-\langle\hat{f};\Psi\rangle)^{2},\Psi\rangle$
is more difficult for calculations because it requires
$\langle\hat{f}^{2};\Psi\rangle$ which involves an overlap between the
coherent states, and usually depends on the fiducial vector explicitly:

$\langle(\hat{f})^{2};\Psi\rangle:=\langle\Psi|(\hat{f})^{2}|\Psi\rangle=\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(\xi)\frac{1}{A_{\phi}}\int_{\textrm{G}}d\mu(\xi^{\prime})\langle\Psi|\xi\rangle
f(\xi)\langle\xi|\xi^{\prime}\rangle
f(\xi^{\prime})\langle\xi^{\prime}|\Psi\rangle\,.$ (23)

The variance $\mathrm{var}(\hat{f};\Psi)$ can be rewritten as

$\mathrm{var}(\hat{f};\Psi)=\langle\hat{f}^{2};\Psi\rangle-\langle\hat{f};\Psi\rangle^{2}\,.$
(24)

The important quantum observables correspond to the variables of the
configuration space (10). These elementary variables $\xi_{k}$
($k=1,2,\dots,6$), can be mapped into the quantum operators as follows

$\hat{\xi}_{k}=\frac{1}{A_{\Phi}}\int_{\textrm{G}}d\mu(\xi)|\xi\rangle\xi_{k}\langle\xi|\,.$
(25)

For every $k$ the above equality (25) reduces to integration over a single
affine group. The other integrations give the unit operators in two remaining
spaces $\mathcal{H}_{x_{l}}$, $l\not=k$. For example,

$\hat{\xi}_{2}=\frac{1}{A_{\Phi_{1}}}\int_{{\textrm{Aff}(\mathbb{R})}}d\mu(\xi_{1},\xi_{2})|\xi_{1},\xi_{2}\rangle\xi_{2}\langle\xi_{1},\xi_{2}|\otimes\hat{1\kern-4.75pt1}_{x_{1}}\otimes\hat{1\kern-4.75pt1}_{x_{3}}\
.$ (26)

To deal with a single time variable at quantum level, one needs to choose a
model of time in the configuration space $T$, defined by (10). In general, it
can be introduced either as a real function or distribution,
$\mathcal{T}:T\to\mathbb{R}$. Its quantization leads to the time operator
$\hat{\mathcal{T}}$. However, we can impose the appropriate constraints to
have the common time variable for all three operators
$\hat{\xi}_{1},\hat{\xi}_{3}$, and $\hat{\xi}_{5}$. In this paper we realise
that option assuming that the only allowed quantum states $\Psi$ of our BKL
system are the states which satisfy the condition

$\langle\Psi|\hat{\xi}_{1}|\Psi\rangle=\langle\Psi|\hat{\xi}_{3}|\Psi\rangle=\langle\Psi|\hat{\xi}_{5}|\Psi\rangle\
,$ (27)

which is consistent with the choice of the configuration space in the form
(9). It means, we require the same expectation values for all three operators
which represent three “times” related to appropriate quantum observables
$\hat{a}=\hat{\xi}_{2},\hat{b}=\hat{\xi}_{4}$, and $\hat{c}=\hat{\xi}_{6}$.

##  IV Quantization of the solution to BKL scenario

The above quantization scheme can be now applied to the solutions (7) of the
BKL scenario111In what follows we denote by $\tilde{a},\tilde{b}$ and
$\tilde{c}$ the solution (4).

$a(t)=\tilde{a}(t)+\epsilon\alpha(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
b(t)=\tilde{b}(t)+\epsilon\beta(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c(t)=\tilde{c}(t)+\epsilon\gamma(t)\,,$ (28)

ascribing to them appropriate quantum states and the corresponding operators.

In quantum mechanics, contrary to classical one, one needs two kinds of
objects to describe physical world. These are quantum observables represented
by either appropriate operators or operator valued measures, and quantum
states being the vectors in a Hilbert space or the so called density
operators. In the classical mechanics the functions on either configuration or
phase space are at the same time states and observables.

To quantize solutions of the BKL scenario, we already have the elementary
observables $\hat{\xi}_{k}$. However, we also have to find the appropriate
family of states related to the solutions (28). This family of states has to
reproduce the classical solutions (28) by comparing them with expectation
values of the corresponding observables.

The classical solutions are represented by three time dependent functions. In
the configuration space $T$ we have 6 variable, where 3 of them
$\xi_{1},\xi_{3},\xi_{5}$ represent the time in the state space which
satisfies the condition (27). As is it was mentioned above, the classical
observables should be related to their quantum counterparts by the
corresponding expectation values. This idea leads directly to the conditions
for a family of states $\\{\Psi_{\eta}(x_{1},x_{2},x_{3})=\langle
x_{1},x_{2},x_{3}|\Psi_{\eta}\rangle,\eta\in\mathbb{R}^{s}\\}$ parameterized
by a set of evolution parameters $\eta=(\eta_{1},\eta_{2},\dots\eta_{s})$
enumerating the set of trial functions.

We require the states $|\Psi_{\eta}\rangle$ to satisfy the following
conditions [24]:

$\displaystyle\langle\Psi_{\eta}|\hat{\xi}_{k}|\Psi_{\eta}\rangle=t,\quad
k=1,3,5$ (29)
$\displaystyle\langle\Psi_{\eta}|\hat{\xi}_{2}|\Psi_{\eta}\rangle=a(t)\ ,$
(30) $\displaystyle\langle\Psi_{\eta}|\hat{\xi}_{4}|\Psi_{\eta}\rangle=b(t)\
,$ (31)
$\displaystyle\langle\Psi_{\eta}|\hat{\xi}_{6}|\Psi_{\eta}\rangle=c(t)\ .$
(32)

The equations (29) represent the single time constraint (27). The parameter
$\eta$ labels the family of states to be found, and it should be a function of
$t$ as the r.h.s. of (29)–(32) depends on $t$. The solution of Eqs. (29)–(32)
allows to construct the vector state dependent on classical time,
$|\Psi_{\eta(t)}\rangle\in\mathcal{H}$, in our Hilbert space.

In this way we relate the quantum dynamics to the classical one. Obviously,
there may exist more than one family of states satisfying the above equations
of motion.

In what follows, we determine the states $|\Psi_{\eta(t)}\rangle$ satisfying
the conditions (29)–(32). This will enable examination of the issue of the
fate of the gravitational singularity and chaos of the BKL scenario at the
quantum level.

###  IV.1 Evolving wave packets

In our paper we consider two kinds of wave packets satisfying the conditions
(29)–(32). The first kind are the affine coherent states themselves. The
second type is a set of modified “exponential” wave packets, which represent a
dense set of states in the Hilbert space $\mathcal{H}$.

####  IV.1.1 Coherent states and expectation values

One can verify that considered coherent states generated by a single affine
group satisfy, due to the results of the recent paper [24], the following
equations

$\langle\xi_{k},\xi_{k+1}|\hat{\xi}_{l}|\xi_{k},\xi_{k+1}\rangle=\xi_{l},\text{
where }l=k,k+1;\ k=1,3,5\,,$ (33)

where the operators $\hat{\xi}_{l}$ are defined to be

$\hat{\xi}_{l}:=\frac{1}{2\pi}\frac{1}{A_{\Phi_{k}}}\int_{\mathbb{R}}d\xi_{k}\int_{\mathbb{R}_{+}}\frac{d\xi_{k+1}}{\xi_{k+1}^{2}}\;|\xi_{k},\xi_{k+1}\rangle\xi_{l}\langle\xi_{k},\xi_{k+1}|\,,$
(34)

and where $l=k,k+1$, $k=1,3,5$.

The conditions (33) are the consistency conditions between the affine group
parametrization and the configuration space of the quantized physical system
[24].

This implies that the coherent states generated by the product of three affine
groups also satisfy the consistency condition

$\langle\xi|\hat{\xi}_{l}|\xi\rangle=\frac{1}{A_{\phi}}\int_{G}d\mu(\xi^{\prime})\langle\xi|\xi^{\prime}\rangle\xi_{l}^{\prime}\langle\xi^{\prime}|\xi\rangle=\xi_{l},\text{
where }l=1,2,\dots,6\,.$ (35)

The consistency conditions coincide with an idea that the coherent state
$|\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6}\rangle$ represents a state
localized at the point $\\{\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6}\\}$
of the affine configuration space and at the same time in the spacetime.

Therefore, the coherent states

$|CS_{\epsilon};t\rangle:=|t,\tilde{a}(t)+\epsilon\alpha(t);\,t,\tilde{b}(t)+\epsilon\beta(t);\,t,\tilde{c}(t)+\epsilon\gamma(t)\rangle$
(36)

satisfy the equations of motions (29)–(32). In such case, we propose to use a
one dimensional parameter $\eta$ and identify it with the classical time $t$,
i.e., the classical time is a label of the evolving family of quantum states.

Realization of (36) as a wave packet constructed in the space of square
integrable functions $L^{2}(\mathbb{R}_{+}^{3},d\nu(x_{1},x_{2},x_{3}))$ reads

$\displaystyle\Psi_{CS_{\epsilon}}(t,x_{1},x_{2},x_{3})=\langle
x_{1},x_{2},x_{3}|CS_{\epsilon};t\rangle$
$\displaystyle=e^{it(x_{1}+x_{2}+x_{3})}\Phi_{1}(a(t)x_{1})\Phi_{2}(b(t)x_{2})\Phi_{3}(c(t)x_{3})\,.$
(37)

Using equation (35) and the fact that the vector (36) factorises

$|CS_{\epsilon};t\rangle=|t,\tilde{a}(t)+\epsilon\alpha(t)\rangle|t,\tilde{b}(t)+\epsilon\beta(t)\rangle|t,\tilde{c}(t)+\epsilon\gamma(t)\rangle\,,$
(38)

we can compute the expectation value of the volume operator $\hat{V}$, where
$V:=\xi_{2}\xi_{4}\xi_{6}$, as follows

$\langle
CS_{\epsilon};t|\hat{V}|CS_{\epsilon};t\rangle=(\tilde{a}(t)+\epsilon\alpha(t))\,(\tilde{b}(t)+\epsilon\beta(t))\,(\tilde{c}(t)+\epsilon\gamma(t))\,.$
(39)

####  IV.1.2 The modified exponential packet and expectation values

Let us consider the set of Gaussian distribution wave packets (with modified
exponential part)

$\Psi_{n}(x;\tau,\gamma)=Nx^{n}\exp\left[i\tau
x-\frac{\gamma^{2}x^{2}}{2}\right]\,,\quad
N^{2}=\frac{2\gamma^{n}}{(n-1)!}\,,$ (40)

which according to [24] is dense in $L^{2}(\mathbb{R}_{+},d\nu(x))$.

The expectation values and variances of the operators $\hat{\xi}_{k}$ and
$\hat{\xi}_{k+1}$ have the following values

$\displaystyle\langle\Psi_{n}|\hat{\xi}_{k}|\Psi_{n}\rangle=\tau,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ k=1,3,5\,,$
(41)
$\displaystyle\langle\Psi_{n}|\hat{\xi}_{k+1}|\Psi_{n}\rangle=\frac{1}{A_{\Phi}}\frac{\Gamma\left(n-\frac{1}{2}\right)}{(n-1)!}\gamma\,,$
(42)
$\displaystyle\mathrm{var}(\hat{\xi}_{k};\Psi_{n})=\frac{4n-3}{4(n-1)}\gamma^{2}\,,$
(43)
$\displaystyle\mathrm{var}(\hat{\xi}_{k+1};\Psi_{n})=\frac{1}{A_{\Phi}^{2}}\left(\frac{1}{n-1}-\frac{\Gamma\left(n-\frac{1}{2}\right)^{2}}{(n-1)!^{2}}\right)\gamma^{2}\,.$
(44)

In this case, the evolution parameters $\eta$ consists of $\eta_{1}=\tau$ and
$\eta_{2}=\gamma$.

In the space $L^{2}(\mathbb{R}_{+}^{3},d\nu(x_{1},x_{2},x_{3}))$ we take the
corresponding wave packets in the form

$\Psi_{n_{1},n_{3},n_{5}}(x_{1},x_{2},x_{3};\,\tau_{1},\tau_{3},\tau_{5},\gamma_{1},\gamma_{3},\gamma_{5})=\Psi_{n_{1}}(x_{1};\tau_{1},\gamma_{1})\Psi_{n_{3}}(x_{2};\tau_{3},\gamma_{3})\Psi_{n_{5}}(x_{3};\tau_{5},\gamma_{5})\,.$
(45)

To meet the properties (29)–(32) for the wave packets
$\Psi_{n_{1},n_{3},n_{5}}$, we choose the parameters $\tau_{k}$ and
$\gamma_{k}$ as follows

$\displaystyle\tau_{1}=\tau_{3}=\tau_{5}=t\,,$ (46)
$\displaystyle\gamma_{k}=A_{\Phi_{k}}\frac{(n_{k}-1)!}{\Gamma\left(n_{k}-\frac{1}{2}\right)}\cdot
f_{k}(t)\,,\quad k=1,3,5\,,$ (47)

where

$f_{k}(t)=\left\\{\begin{array}[]{ll}\tilde{a}(t)+\epsilon\alpha(t)\,,&k=1\\\
\tilde{b}(t)+\epsilon\beta(t)\,,&k=3\\\
\tilde{c}(t)+\epsilon\gamma(t)\,,&k=5\end{array}\right.\,.$ (48)

Using similar technique as in case equation (39), one can calculate the
expectation value of the volume operator in the state (45) and one gets

$\langle\Psi_{n_{1},n_{3},n_{5}}|\hat{V}|\Psi_{n_{1},n_{3},n_{5}}\rangle=f_{1}(t)f_{3}(t)f_{5}(t)\,,$
(49)

which coincides with the result obtained within coherent states method (39).

It is clear that the expectation value of the volume operator converges very
fast to zero as $t\rightarrow\infty$.

###  IV.2 Variances in the Hilbert space $\mathcal{H}$

The wave packets obtained above follow the classical solutions. However, in
quantum mechanics the observables which do not commute with time and position
operators fluctuate at every spacetime point. This smearing of quantum
observables is determined by the Heisenberg uncertainty principle. The most
important ingredients of it are variances of the corresponding observables. In
the following we perform analysis of the variances of our elementary
variables.

####  IV.2.1 Using coherent states

The variance of the operators $\hat{\xi}_{k}$, $\hat{\xi}_{k+1}$, $k=1,3,5$ in
coherent states (36) read

$\displaystyle\mathrm{var}(\hat{\xi}_{k};|CS_{\epsilon};t\rangle)=\langle\hat{\xi}^{2}_{k}\rangle_{0}f_{k}(t)^{2}\,,$
(50)
$\displaystyle\mathrm{var}(\hat{\xi}_{k+1};|CS_{\epsilon};t\rangle)=\left(\langle\hat{\xi}^{2}_{k+1}\rangle_{0}-1\right)f_{k}(t)^{2}\,,$
(51)

where

$\displaystyle\langle\hat{\xi}^{2}_{i}\rangle_{0}=\langle
0,1|\hat{\xi}^{2}_{i}|0,1\rangle,\quad i=1,2,\dots 6\,,$ (52)

and where $\langle\hat{\xi}^{2}_{i}\rangle_{0}$ is a constant which depends on
the choice of the fiducial vector.

The variance of the volume operator is found to be

$\mathrm{var}(\hat{V};|CS_{\epsilon};t\rangle)=\left[\prod_{k=2,4,6}\langle\hat{\xi}^{2}_{k}\rangle_{0}-1\right]f_{1}(t)^{2}f_{3}(t)^{2}f_{5}(t)^{2}\,.$
(53)

For more details concerning the r.h.s. of (52) see App. D of [24].

####  IV.2.2 Using exponential wave packet

The corresponding results for the wave packets (45), under the conditions
(46)-(47), read

$\displaystyle\mathrm{var}(\hat{\xi}_{k};\Psi_{n_{1},n_{3},n_{5}})=\mathcal{A}_{k}f_{k}(t)^{2}\,,$
(54)
$\displaystyle\mathrm{var}(\hat{\xi}_{k+1};\Psi_{n_{1},n_{3},n_{5}})=\mathcal{B}_{k}f_{k}(t)^{2}\,,$
(55)

where

$\displaystyle\mathcal{A}_{k}=A^{2}_{\Phi_{k}}\frac{(4n_{k}-3)(n_{k}-1)!(n_{k}-2)!}{4\Gamma\left(n_{k}-\frac{1}{2}\right)^{2}}\,,$
(56)
$\displaystyle\mathcal{B}_{k}=\frac{(n_{k}-1)!(n_{k}-2)!}{\Gamma\left(n_{k}-\frac{1}{2}\right)^{2}}-1\,.$
(57)

The variance of the volume operator has the form

$\mathrm{var}(\hat{V};\Psi_{n_{1},n_{3},n_{5}})=\left[\prod_{k=2,4,6}\frac{(n_{k}-1)!(n_{k}-2)!}{\Gamma\left(n_{k}-\frac{1}{2}\right)}-1\right]f_{1}(t)^{2}f_{3}(t)^{2}f_{5}(t)^{2}\,.$
(58)

These results show that all positions of our system in time and space are
smeared owing to nonzero variances. It is an important fact about possibility
of avoiding singularities in this dynamics.

##  V Stochastic aspects of quantum evolution

The results of recent paper [10] give strong support to the expectation that
near the generic gravitational singularity the evolution becomes chaotic. In
what follows we examine that fundamental property of the BKL scenario at
quantum level.

To make direct comparison with the results of [10], we split the variances
(55) into the contributions from unperturbed and perturbed states. We have:

$\displaystyle f_{2}(t)^{2}$ $\displaystyle=$
$\displaystyle(\tilde{a}(t)+\epsilon\alpha(t))^{2}=\tilde{a}(t)^{2}+2\epsilon\tilde{a}(t)\alpha(t)+\epsilon^{2}\alpha(t)^{2}\simeq\tilde{a}(t)^{2}+2\epsilon\tilde{a}(t)\alpha(t)\,,$
(59) $\displaystyle f_{4}(t)^{2}$ $\displaystyle=$
$\displaystyle(\tilde{b}(t)+\epsilon\beta(t))^{2}=\tilde{b}(t)^{2}+2\epsilon\tilde{b}(t)\beta(t)+\epsilon^{2}\beta(t)^{2}\simeq\tilde{b}(t)^{2}+2\epsilon\tilde{b}(t)\beta(t)\,,$
(60) $\displaystyle f_{6}(t)^{2}$ $\displaystyle=$
$\displaystyle(\tilde{c}(t)+\epsilon\gamma(t))^{2}=\tilde{c}(t)^{2}+2\epsilon\tilde{c}(t)\gamma(t)+\epsilon^{2}\gamma(t)^{2}\simeq\tilde{c}(t)^{2}+2\epsilon\tilde{c}(t)\gamma(t)\,.$
(61)

The corresponding dimensionless functions describing relative quantum
perturbations are defined to be

$\kappa_{k}:=\frac{\mathrm{var}(\hat{\xi}_{k};\Psi_{pert})-\mathrm{var}(\hat{\xi}_{k};\Psi_{unpert})}{\mathrm{var}(\hat{\xi}_{k};\Psi_{unpert})},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ k=2,4,6\,,$ (62)

where $\Psi_{pert}$ and $\Psi_{unpert}$ denote perturbed and unperturbed wave
packets, respectively.

The explicit form of (62), up to the 1-st order in $\epsilon$, reads:

$\displaystyle\kappa_{a}(t):=\kappa_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{2\epsilon\tilde{a}(t)\alpha(t)}{\tilde{a}(t)^{2}}=2\epsilon\frac{\alpha(t)}{\tilde{a}(t)}\,,$
(63) $\displaystyle\kappa_{b}(t):=\kappa_{4}(t)$ $\displaystyle=$
$\displaystyle\frac{2\epsilon\tilde{b}(t)\beta(t)}{\tilde{b}(t)^{2}}=2\epsilon\frac{\beta(t)}{\tilde{b}(t)}\,,$
(64) $\displaystyle\kappa_{c}(t):=\kappa_{6}(t)$ $\displaystyle=$
$\displaystyle\frac{2\epsilon\tilde{c}(t)\gamma(t)}{\tilde{c}(t)^{2}}=2\epsilon\frac{\gamma(t)}{\tilde{c}(t)}\,.$
(65)

It is clear that the relative perturbations (63)–(65) are the same for the
coherent states and the exponential wave packets.

Figure 1 presents the parametric curve visualizing the relative quantum
perturbations. The time dependence of the expectation values of
$\hat{\xi}_{2}$ operator and corresponding variances of unperturbed and
perturbed solutions are presented in Fig. 2. The plots for $\hat{\xi}_{4}$
and $\hat{\xi}_{6}$ operators would look similarly so that we do not present
them.

Figure 1:  The $t$ dependence of quantum perturbation defined by (63)-(65) for
$K_{1}=K_{2}=0.01$, $K_{3}=0$, $\phi_{1}=\phi_{2}=0$, $t_{0}<0$,
$\epsilon=0.01$. The plot presents the parametric curve
$\\{\kappa_{a}(t),\kappa_{b}(t),\kappa_{c}(t)\\}$, where
$t-t_{0}\in(0.01,35)$.

Figure 2:  The $t$ dependence of the expectation value of the operator
$\hat{\xi}_{2}$ defined by (42),(47) for $K_{1}=K_{2}=0.01$, $K_{3}=0$,
$\phi_{1}=\phi_{2}=0$, any $t_{0}<0$, $n_{1}=3$. Axis of $t$ is in logarithmic
scale. The left panel corresponds to unperturbed solution ($\epsilon=0$), the
right panel corresponds to perturbed solution ($\epsilon=0.01$). The blue area
defines the points for which distance from expected value is smaller than
$\sqrt{\mathrm{var}(\xi_{2};\Psi_{n})}$ defined by (55) (the distance is
counted along fixed $t$ line).

##  VI Conclusions

It has been shown [10] that the perturbed classical solution to the dynamics
of the massive model of the BKL scenario exhibits the chaotic behaviour. Our
results present the quantum dynamics corresponding to that dynamics. Fig. 1
shows that the relative quantum perturbations grow as the system evolves
towards the singularity, which is consistent with the corresponding classical
evolution, see Fig. 2 of [10]. Since our quantum and classical perturbations
have quite similar time evolutions, we conclude that quantization does not
destroy classical chaos. In fact, the quantum chaos corresponds to the
classical chaos in the lowest order approximation. Nonlinearity of classical
dynamics creates deterministic chaos. Non-vanishing variances of observables
of the corresponding quantum dynamics leads to stochastic chaos.

To show that behaviour, we have constructed the wave packets for which
expectation values of elementary observables follow the corresponding
classical ones. This choice of quantum states leads to the scenario in which
calculated expectation values of quantum directional scale factors evolve to
vanishing similarly as their classical counterparts. However, quantum states
do not represent sharp properties of a physical system. Expectation values are
smeared quantities. That smearing is represented by quantum variance.

More exactly, the variance is a measure of the stochastic deviation from an
expectation value of given operator. A quantum system is in an eigenstate of
an operator iff the variance of this operator in that state equals zero (see,
[24] for more details). The calculated variances depicted in Fig. 2 are
always non-zero, which mean that the probability of hitting the gravitational
singularity is equal to zero. The non-zero variance removes the singularity
from considered quantum evolution.

The quantum randomness amplifies the deterministic classical chaos. This
supports the hypothesis that in the region corresponding to the neighbourhood
of the classical singularity the dynamics, both classical and quantum, enters
the stochastic phase. The oscillatory behaviour of the expectation value of
the quantum scale factor increases as $t\rightarrow\infty$, which is
consistent with the classical BKL scenario [2, 5].

The results of the present paper support our previous results [19, 20]
concerning the fate of the BKL singularity at the quantum level. One of the
main differences between the results of the papers [19, 20] and the present
article is that in the former case the evolution parameter (time) used at the
quantum level was purely mathematical and was taken being equal to the
classical time by an assumption. In the present paper, the quantum time is
established by the requirement that the temporal and spatial variables should
be treated on the same footing at quantum level, which supports the covariance
of arbitrary transformations of these variables in general relativity. Another
important difference is quite different implementation of the dynamics both at
classical and quantum levels. In the case [19, 20] it was based on Hamilton’s
dynamics and corresponding Schrödinger’s equation at quantum level. Here, we
quantize the solution to the classical dynamics ascribing to it corresponding
quantum system. Surprisingly, both approaches give physically similar result:
avoiding of classical singularity at quantum level. Additionally, the approach
of the present paper addresses the issue of the fate of the classical chaos at
quantum level, which was beyond the scope of the approach used in [19, 20]. We
have examined that issue by calculating the variances of expectation values of
quantum observables. The variances are measures of stochasticity of considered
observables at quantum level. In fact, the calculations of variances have been
ignored in the recent quantizations of the vacuum BKL models [30, 31, 32].

The BKL conjecture states that general relativity includes generic
gravitational singularity. Our results strongly suggest that generic
singularity can be avoided at quantum level so that one can expect that a
theory of quantum gravity (to be constructed) has good chance to be regular.

###### Acknowledgements.

We would like to thank Vladimir Belinski and Piotr Goldstein for helpful
discussions.

##  Appendix A Essence of integral quantization

If the configuration space $\Pi$ is a half-plane,

$\Pi:=\\{(p,q)\in\mathbb{R}\times\mathbb{R}_{+}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}_{+}:=\\{x\in\mathbb{R}\leavevmode\nobreak\ |\leavevmode\nobreak\
x>0\\},$

it can be identified with the affine group Aff$(\mathbb{R})$ =: G.

The multiplication law can be defined as

$(p_{1},q_{1})\cdot(p_{2},q_{2}):=(p_{1}+q_{1}p_{2},q_{1}q_{2})\,.$ (66)

The unity of the group is $(0,1)$ and the inverse reads
$(p,q)^{-1}=(-p/q,1/q)$.

This group has two nontrivial unitary irreducible representations realized in
the Hilbert space $L^{2}(\mathbb{R}_{+},d\nu(x))=:\mathcal{H}$, where
$d\nu(x)=dx/x$. We choose the one defined as follows (the second
representation would lead to exactly the same results):

$U(p,q)\psi(x)=e^{ipx}\psi(qx),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\psi(x)\in\mathcal{H}\leavevmode\nobreak\ \;.$ (67)

Eq. (67) enables defining the continuous family of affine coherent states
(ACS), denoted $\langle x|p,q\rangle\in\mathcal{H}$, as follows

$\langle x|p,q\rangle=U(p,q)\langle x|\phi\rangle\;,$ (68)

where $\langle x|\phi\rangle=:\phi(x)\in\mathcal{H}$ is the so-called fiducial
vector, which is a free parameter (to some extent) of ACS quantization scheme.

Eq. (68) can be interpreted as the correspondence

$(p,q)\longrightarrow|p,q\rangle\langle p,q|$ (69)

between the point of the configuration space $\Pi$ and the quantum projection
operator acting in $\mathcal{H}$.

The irreducibility of the representation leads (due to Schur’s lemma) to the
resolution of the unity in $L^{2}(\mathbb{R}_{+},d\nu(x))$:

$\frac{1}{A_{\phi}}\int_{G}d\mu(p,q)|p,q\rangle\langle
p,q|=\hat{1\kern-4.75pt1}\;,$ (70)

where $d\mu(p,q):=dp\;dq/q^{2}$ is the left invariant measure on $G$, and
where $A_{\phi}:=\int_{0}^{\infty}|\phi(x)|^{2}\,\frac{dx}{x^{2}}<\infty$ is a
constant.

Using (70) enables quantization of almost any observable
$f:\Pi\rightarrow\mathbb{R}$

$f\longrightarrow\hat{f}=\frac{1}{A_{\phi}}\int_{G}d\mu(p,q)|p,q\rangle
f(p,q)\langle p,q|\,.$ (71)

The operator $\hat{f}:\mathcal{H}\rightarrow\mathcal{H}\leavevmode\nobreak\ $
is symmetric by construction. No ordering ambiguity occurs (notorious problem
of canonical quantization). That operator is self-adjoint if it is bounded.

For more details concerning the integral quantization, see [21] and references
therein.

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