Quantum Belinski–Khalatnikov–Lifshitz scenario
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Abstract
We present the quantum model of the asymptotic dynamics underlying the Belinski–Khalatnikov–Lifshitz (BKL) scenario. The symmetry of the physical phase space enables making use of the affine coherent states quantization. Our results show that quantum dynamics is regular in the sense that during quantum evolution the expectation values of all considered observables are finite. The classical singularity of the BKL scenario is replaced by the quantum bounce that presents a unitary evolution of considered gravitational system. Our results suggest that quantum general relativity has a good chance to be free from singularities.
1 Introduction
It is believed that the cosmological and astrophysical singularities predicted by general relativity (GR) can be resolved at the quantum level. That has been shown to be the case in the quantization of the simplest singular GR solutions like FRWtype spacetimes (commonly used in observational cosmology). However, it is an open problem in the general case. Our paper addresses the issue of possible resolution of a generic singularity problem in GR due to quantum effects.
The Belinski, Khalatnikov and Lifshitz (BKL) conjecture is thought to describe a generic solution to the Einstein equations near spacelike singularity (see, [1, 2, 3] and references therein). Later, it was extended to deal with generic timelike singularity of general relativity [4, 5, 6]. According to the BKL scenario [1, 2], in the approach to a spacelike singularity neighbouring points decouple and spatial derivatives become negligible in comparison to temporal derivatives. The conjecture is based on the examination of the dynamics toward the singularity of a Bianchi spacetime, typically Bianchi IX (BIX). The BKL scenario presents the oscillatory evolution (towards the singularity) entering the phase of chaotic dynamics (see, e.g., [7, 8]), followed by approaching the spacelike manifold with diverging curvature and matter field invariants.
The most general scenario is the dynamics of the nondiagonal BIX model. However, this dynamics is difficult to exact treatment. Qualitative analytical considerations [9, 10, 11] and numerical analysis [12] strongly suggest that in the asymptotic regime near the singularity the exact dynamics can be well approximated by much simpler dynamics (presented in the next section).
The BKL scenario based on a diagonal BIX reduces to the dynamics described in terms of the three directional scale factors dependent on an evolution parameter (time). This dynamics towards the singularity has the following properties: (i) is symmetric with respect to the permutation of the scale factors, (ii) the scale factors are oscillatory functions of time, (iii) the product of the three scale factors is proportional to the volume density decreasing monotonically to zero, and (iv) the scale factors may intersect each other during the evolution of the system. The diagonal BIX is suitable to address the vacuum case and the cases with simple matter fields. More general cases, including perfect fluid with nonzero time dependent velocity, require taking nondiagonal space metric. However, the general dynamics simplifies near the singularity and can be described by three effective scale factors, which include contribution from matter field. This effective dynamics does not have the properties (i) and (iv) of the diagonal case. More details can be found in the paper [13].
Roughly speaking, the main advantage of the nondiagonal BIX scenario is that it can be used to derive the BKL conjecture in a much simpler way than when starting from the diagonal case. Namely, considering inhomogeneous perturbations of the nondiagonal BIX metric is sufficient to derive the BKL conjecture, whereas the diagonal case needs additionally considering inhomogeneous perturbation of the matter field that would correspond, e.g., to time dependent velocity of the perfect fluid [14].
The present paper concerns the quantum fate of the asymptotic dynamics of the nondiagonal BIX model. The quantum dynamics, described by the Schrödinger equation, is regular (no divergencies of physical observables) and the evolution is unitary. The classical singularity is replaced by quantum bounce due to the continuity of the probability density.
Our paper is organized as follows: In Sect. 2 we recall the Hamiltonian formulation of our gravitational system and identify the topology of physical phase space. Section 3 is devoted to the construction of the quantum formalism. It is based on using the affine coherent states ascribed to the physical phase space, and the resolution of the unity in the carrier space of the unitary representation of the affine group. The quantum dynamics is presented in Sect. 4. Finding an explicit solution to the Schrödinger equation enables addressing the singularity problem. We conclude in the last section. Appendix A presents an alternative affine coherent states. The basis of the carrier space is defined in Appendix B.
2 Classical dynamics
For selfconsistency of the present paper, we first recall some results of Ref. [15], followed by the analysis of the topology of the physical phase space.
2.1 Asymptotic regime of general Bianchi IX dynamics
The dynamics (1)–(2) defines the asymptotic regime of the BKL scenario, satisfying (3), that lasts quite a long time before the system approaches the singularity, marked by the condition \(a\, b\, c \rightarrow 0\) (see [9, 10] for more details).
2.2 Hamiltonian formulation with dynamical constraint
Equations (12) and (3) imply that the space of singular points includes the critical points surface \(S_B\). It is so because \((q_1 \rightarrow \infty ,~ q_2q_1 \rightarrow \infty ,~ q_3q_2 \rightarrow \infty )\) implies (3) (which means \(a\, b\, c \rightarrow 0\)) for any \((p_1, p_2, p_3) \in {\bar{{\mathbb {R}}}}^3\).
2.3 Hamiltonian formulation devoid of dynamical constraint
The reduced system (15)–(18) has been obtained in the procedure of mapping the system with the Hamiltonian constraint, defined by (5)–(11), into the Hamiltonian system devoid of the constraint. In the former, the Hamiltonian \(H_c\) is a dynamical constraint, in the latter the Hamiltonian H in a generator of dynamics without the constraint. As it is known, this procedure is a sort of “onetomany” mapping (for more details, see e.g. [16, 17] and references therein). Roughly speaking, it consists in resolving the dynamical constraint with respect to one phase space variable that is chosen to be a Hamiltonian. This procedure leads to the choice of an evolution parameter (time) as well so that the Hamiltonian and time emerge in a single step. In general, this is a highly nonunique procedure if there are no hints to this process. Here the choice of the reduction was motivated by the two circumstances: (i) the resolution of the constraint only with respect to one variable \(q_3\) is unique, and (ii) the resulting reduced phase space is isomorphic to the Cartesian product of two affine groups. The latter has unitary irreducible representation enabling the affine coherent states quantization of the underlying gravitational system (presented in the next section).
2.4 Numerical simulations of dynamics
The solution visible in Fig. 1 presents a wiggled curve in the physical phase space. This classical dynamics cannot be further extended towards the singularity (for \(t < t_0\)) due to the physical and mathematical reasons (and implied numerical difficulties).
2.5 Topology of phase space
Equations (15)–(18) define a coupled system of nonlinear ordinary differential equations. The solution defines the phase space of our gravitational system.
3 Quantization
Suppose we have reduced phase space Hamiltonian formulation of classical dynamics of a gravitational system. It means dynamical constraints have been resolved and the Hamiltonian is a generator of the dynamics. By quantization we mean (roughly speaking) a mapping of such Hamiltonian formulation into a quantum system described in terms of quantum observables (including Hamiltonian) represented by an algebra of operators acting in a Hilbert space. The construction of the Hilbert space may make use some mathematical properties of phase space like, e.g., symplectic structure, geometry or topology. The quantum Hamiltonian is used to define the Schrödinger equation. In what follows we make specific the above procedure by using the affine coherent states approach.
3.1 Affine coherent states
The Hilbert space \({\mathcal {H}}\) of the entire system consists of the Hilbert spaces \({{\mathcal {H}}}_1\) and \({{\mathcal {H}}}_2\) corresponding to the phase spaces \(\Pi _1\) and \(\Pi _2\), respectively. In the sequel the construction of \({\mathcal {H}}_1\) is followed by merging of \({\mathcal {H}}_1\) and \({\mathcal {H}}_2\).
As both halfplanes \(\Pi _1\) and \(\Pi _2\) have the same mathematical structure, the corresponding Hilbert spaces \({\mathcal {H}}_1\) and \({\mathcal {H}}_2\) are identical so we first consider only one of them. In what follows we present the formalism for \(\Pi _1\) and \({\mathcal {H}}_1\) to be extended later to the entire system.
3.1.1 Affine coherent states for halfplane
3.1.2 Structure of the fiducial vector
How to construct the fiducial vector to have \(\mathrm {G}_\Phi =\{e_G\}\), where \(e_G\) is the unit element in this group? It is seen that Eq. (50) cannot be fulfilled for \({\tilde{q}} \ne 0\), independently of chosen fiducial vector. This suggests that the generalized stationary group \(\mathrm {G}_\Phi \) is parameterized only by the momenta \((0,{\tilde{p}})\), i.e. it has to be a subgroup of the multiplicative group of positive real numbers, \(\mathrm {G}_\Phi \subseteq ({\mathbb {R}}_+,\cdot )\).
On the other hand, Eq. (50) implies that \(\Phi ({\tilde{p}} x)= \Phi (x)\) for all \((0,{\tilde{p}}) \in \mathrm {G}_\Phi \). In addition, for the fiducial vectors \(\Phi (x)=\Phi (x)e^{i\gamma (x)}\) the phases of these complex functions are bounded by \(0\le \gamma (x) < 2\pi \). Due to Eq. (50) the phases \(\gamma (x)\) and \(\beta (0,{\tilde{p}})\) have to fulfil the following condition \(\gamma ({\tilde{p}}x)\gamma (x)=\beta (0,{\tilde{p}})\). One of the solutions to this equation is the logarithmic function \(\gamma (x)= \ln (x)\).
In what follows, to have the unique representation of the phase space as a group manifold of the affine group, we require the generalized stationary group to be the group consisted only of the unit element. This can be achieved by the appropriate choice of the fiducial vector.
3.1.3 Phase space and quantum state spaces

The phase space \(\Pi \), which consists of two halfplanes \(\Pi _1\) and \(\Pi _2\) defined by (29). It is the background for the classical dynamics.^{6}
 The carrier spaces \( {\mathcal {H}}_1 :=L^2({\mathbb {R}}_+, d\nu (x))\) of the unitary representation U(q, p), with the scalar product defined as$$\begin{aligned} \langle \psi _2 \vert \psi _1 \rangle = \int _0^\infty \frac{dx}{x}\, \psi _2^\star (x) \psi _1(x). \end{aligned}$$(56)
 The space of square integrable functions on the affine group \({\mathcal {K}}_G = L^2(\mathrm {Aff}({\mathbb {R}}), d\mu _L(q,p))\). The scalar product is defined as followswhere \(\psi _G(q,p):=\langle q,p \vert \psi \rangle =\langle \Phi \vert U(q,p)^\dagger \vert \psi \rangle \) with \(\vert \psi \rangle \in {\mathcal {H}}_1\). The Hilbert space \({\mathcal {K}}_G \) is defined to be the completion in the norm induced by (57) of the span of the \(\psi _G\) functions.$$\begin{aligned} \langle \psi _{G2} \vert \psi _{G1} \rangle _{G} = \frac{1}{A_\phi }\int _{\mathrm {Aff}} d\mu _L(q,p) \psi _{G2}^\star (q,p) \psi _{G1}(q,p)\, ,\nonumber \\ \end{aligned}$$(57)
3.1.4 Affine coherent states for the entire system
3.2 Quantum observables
It is important to indicate that in the case the classical observable f is defined only on a subspace of the full phase space, the corresponding quantum operator \({\hat{f}}\) obtained via (67) acts in the entire Hilbert space corresponding to the full phase space. This is the peculiarity of the coherent states quantization [28]. Roughly speaking, it results from the nonzero overlap, \(\langle q,pq^\prime ,p^\prime \rangle \ne 0\), of any two coherent states of the entire Hilbert space. In particular, the classical Hamiltonian (31) is defined on the subspace \({\tilde{\Pi }}\) of \(\Pi \). Hovever, the corresponding quantum Hamiltonian (71) acts in the Hilbert space \(L^2({\mathbb {R}}_+ \times {\mathbb {R}}_+,d\nu (x_1,x_2))\) corresponding to \(\Pi \).
4 Quantum dynamics
In general, the parameters t and s are different. To get the consistency between the classical and quantum levels we postulate that \(t = s\), which defines the time variable at both levels. It is worth to mention that so defined time changes monotonically due to the special choice of the parameter t at the classical level (see the paragraph below (19)). This way we support the interpretation that Hamiltonian is the generator of classical and corresponding quantum dynamics.
In what follows, we extend the range of the time variable to include \(t_0 = 0\) (we quantize the sector \(t>0\) of classical dynamics).
The condition (86), which restricts \(x_2 \sim 1/p_2\), see (88) and the text below it, implies that the probability of finding the system in the region with \(x_2 < t_H\) vanishes.
4.1 Elementary quantum observables
4.2 Singularity of dynamics
4.3 Resolution of the singularity problem
In what follows, we first define an example of a regular state at \(t = t_s >0\). Next, we make generalization. Afterwards, we map the general regular state to the initial state at \(t = 0 \), by inverting the general form of the solution defined by Eqs. (84) (with \(t_0 = 0\)) and (86). Finally, we argue that the initial state is regular at \(t = 0\) due to the unitarity of the quantum evolution. This way we get the resolution of the initial singularity problem of the underlying classical dynamics.
4.3.1 Regular state at fixed time
Therefore, the state (97) is regular at \(t_s\).
4.3.2 Initial state obtained in backward evolution
4.3.3 Regularity of the initial state
Since the integrand defining Eq. (108) is positive definite, the equation is satisfied, which completes the proof.
Thus, the initial state is regular, i.e., does not satisfy Eqs. (92) and (93). This implies that whenever we have a regular state far away from the singularity (which is generic case), the initial quantum state at \(t = 0\) is regular so that the quantum evolution is well defined for any \(t \ge 0\). This is a direct consequence of the unitarity of considered quantum evolution.
4.4 Quantum bounce
Let us examine the issue of possible time reversal invariance of our quantum model. In what follows, we examine the time reversal invariance of our Schrödinger equation and its solution.
Comparing Eqs. (83) and (110) we can see that the dynamical equation fails to be time reversal invariant because the Hamiltonian \({\hat{H}}_0\) does not have this symmetry. However, the solutions to these equations have only different phases. Thus, the probability density is continuous at \(t = 0\) (that marks the classical singularity) due to Eqs. (84) and (111) (with \(t_0 = 0\)), which we call the quantum bounce.
5 Conclusions
Near the classical singularity the dynamics of the general Bianchi IX model simplifies. Due to the symmetry of the physical phase space of this model, we can apply the affine coherent states quantization method. The quantum dynamics, described by the Schrödinger equation, is devoid of singularities in the sense that the expectation values of basic operators are finite during the quantum evolution of the system. The evolution is unitary and the probability density of our system is continuous at \(t = 0\), which marks the classical singularity.
We name the state defined by Eqs. (102)–(103) the rescue state. It is chosen to be regular because one expects that the quantum state far away from the singularity is a proper quantum state. The Schrödinger evolution does not lead outside the space of such states.
The nondiagonal BIX underlies the BKL conjecture which concerns the generic singularity of general relativity. Therefore, our results suggest that quantum general relativity is free from singularities. Classical singularity is replaced by quantum bounce, which presents a unitary evolution of the quantum gravity system.
Since general relativity successfully describes almost all available gravitational data, it makes sense its quantization to get the extension to the quantum regime. The latter could be used to describe quantum gravity effects expected to occur near the beginning of the Universe and in the interior of black holes.
Our fully quantum results show that the preliminary results obtained for the diagonal BIX within the semiclassical affine coherent states approximation [29, 30] are correct. Appendix B presents the affine coherent states applied in these papers, which define another parametrization of our coherent states.
As far as we are aware, we are pioneers in addressing the issue of resolving the generic singularity problem of general relativity via quantization. Therefore, trying to confirm our preliminary results within different quantization scheme would be interesting. In particular, the choice of different time and corresponding Hamiltonian in the reduced phase space approach (see, Eq. (13)) would be valuable.
Footnotes
 1.
 2.
 3.
We stay with \(p_3 = t\) as we wish to discuss the possible sign of the time variable.
 4.The general notion of invariant measure dm(x) on the set X in respect to the transformation \(h: X \rightarrow X\) can be approximately defined as follows: for every function \(f:X \rightarrow {\mathbb {C}}\) the integral defined by this measure fulfils the invariance condition:This property is often written as: \(dm(h(x))=dm(x)\).$$\begin{aligned} \int _X dm(x) f(h(x)) = \int _X dm(x) f(x). \end{aligned}$$
 5.
We use Dirac’s notation whenever we wish to deal with abstract vector, instead of functional representation of the vector.
 6.
For simplicity we consider here only one halfplane, but the results can be easily extended to \(\Pi \).
 7.
The precise meaning of Eq. (93) will become clear in the next subsection.
Notes
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