Robustness of the quantum BKL scenario

The quantum Belinski–Khalatnikov–Lifshitz scenario presents an unitary evolution of the system. However, the affine coherent states quantization applied to the quantization of the underlying classical scenario depends on the choice of the group parametrization. Using the two simplest parameterizations of the affine group, we show that qualitative features of our quantum system do not depend on the choice. It means that the quantum bounce replacing a singular classical scenario is expected to be a generic feature of the considered system. This paper complements our recent article (Góźdź et al. in Eur Phys J C 79:45, 2019).


I. INTRODUCTION
Recently, we have found that the affine coherent states, ACS, quantization depends on the parametrization of the affine group [2].Since our paper [1] concerning the quantization of the Belinski-Khalatnikov-Lifshitz (BKL) scenario is based on the ACS quantization, it is reasonable to examine the dependence of the results on the group parametrization.This is the main motivation of the present paper.To have analytical results, as in previous paper [1], we consider the second, the most popular parametrization of the affine group.
It is worth to recall that the BKL scenario concerns the generic singularity of general relativity (see [3][4][5] and [6,7]).The resolution of the singularity at the quantum level is of the primary importance for the quantum gravity programme.Recently, we have found [1] that the quantum BKL scenario presents an unitary process so that singular classical BKL evolution is replaced by regular quantum bounce.However, the quantization method we have applied is not unique (as any quantization scheme).Thus, the examination of the robustness of obtained results is an important issue that cannot be omitted.

II. CLASSICAL DYNAMICS
For self-consistency of the present paper, we first recall the main results of Ref. [1].
The two form Ω defining the Hamiltonian formulation, devoid of the dynamical constraints, is given by where the variables {q 1 , q 2 , p 1 , p 2 } parameterise the phase space, H is the Hamiltonian generating the dynamics, and where t is an evolution parameter (time) corresponding to the specific choice of H.The Hamiltonian reads where F (t, q 1 , q 2 , p 1 , p 2 ) > 0.
The examination of the topology of the phase space and well definiteness of the logarithmic function in (2) requires [1] + , where R + := {p ∈ R | p > 0}, and (ii) p 1 → 0 and p 2 → 0 implies t → 0 + .Thus, considered gravitational system evolves away from the singularity at t = 0.The range of the variables q 1 and q 2 results from the physical interpretation ascribed to them [8] so that (q 1 , q 2 ) ∈ R 2 .Thus, the physical phase space Π consists of the two half planes: It is important to notice that only the subspace is available to the dynamics.It is due to the logarithmic function in the expression defining the Hamiltonian.To make this restriction explicit, we rewrite the Hamiltonian (2) in the form with lim F →0 − H = 0 and lim F →0 + H = +∞.

III. HILBERT SPACE AND QUANTUM OBSERVABLES
Each Π k (k = 1, 2) can be identified with the manifold of the affine group G := Aff(R) acting on R, which is sometimes denoted as "px + q".In the case considered in [1] the actions of this group on R + is defined to be and the corresponding multiplication law of the group G reads (q ′ , p′ ) • (q, p) = (p ′ q + q′ , p′ p) .
In the present paper we apply another simple parametrization, considered in [2,9], with the action of the group G on R + defined as The corresponding multiplication law of the group is defined to be The affine group G = Aff(R) has two (nontrivial) inequivalent irreducible unitary representations, [10] and [11,12], defined in the Hilbert space L 2 (R + , dν(x)), where dν(x) := dx/x.In what follows, we choose the one defined by where Integration over the affine group is defined as where the measure in ( 11) is left invariant.Any coherent state can be obtained as where L 2 (R + , dν(x)) ∋ Φ(x) = x|Φ , with Φ|Φ = 1, is the so called fiducial vector.
The resolution of the identity in the Hilbert space where A. Affine coherent states for the entire system Here, we again recall some essentials of the formalism of [1], and insert suitable modifications resulting from the different parametrization (8) of the affine group.
In the Cartesian product Π = Π 1 × Π 2 , the partial phase spaces Π 1 or Π 2 are identified with the corresponding affine groups The product of both affine groups G Π = G 1 × G 2 can be identified with the whole phase space and its action reads where , and where the entire Hilbert space is the tensor product of two Hilbert spaces )) with the measure dν(x 1 , x 2 ) = dν(x 1 )dν(x 2 ).The scalar product in H is defined as The fiducial vector See [1] for some subtleties concerning the choice of the vector Φ.

B. Quantum observables
Making use of the resolution of identity in the Hilbert space H, we define the quantization of a classical observable f defined in the phase space Π as follows [1]: where dµ(ξ 1 , ξ 2 ) := dµ(q 1 , p 1 )dµ(q 2 , p 2 ).
We recommend [1] for a discussion of the properties of the mapping that leads to (18).If the operator f : H → H is unbounded, its possible self-adjoint extensions require further examination [13][14][15].

IV. QUANTUM DYNAMICS
The mapping (18) applied to the classical Hamiltonian reads where t is an evolution parameter of the classical level and where In our article [1] we apply the reduced phase space quantization.It means, we quantize the classical system with already resolved dynamical constraint.Its Hamilton's dynamics, corresponding to (1), includes the generator of evolution in the physical phase space, i.e. the Hamiltonian H, and corresponding evolution parameter t.As this Hamiltonian system has no dynamical constraint, no quantum constraint occurs.This is quite different from the Dirac quantization where the classical constraint1 is kept unsolved and is promoted to the quantum level so that it leads to an operator type equation.The latter serves as the quantum transformation that sometimes can be used to define a kind of quantum evolution, but in most cases stays timeless (see [16] for more details).
As H and t is a classical canonical pair in (1), it is reasonable to assume that the quantum operator Ĥ corresponding to H is a generator of the evolution of the system in the Hilbert space H.More precisely, the operator Ĥ is the generator of translations of the wave function of our quantum system with the corresponding shift parameter τ .It is natural to identify the classical shift parameter t and the quantum shift parameter τ , i.e. we assume τ = t.This is a reasonable assumption as t changes monotonically [1], and it introduces the consistency between the classical and quantum levels.Assuming the above identification of the evolution parameters, the translation of the system from t 0 to t is represented by the unitary operator U(t, t 0 ) generated by Ĥ(t).The standard properties of the unitary evolution operators in the Hilbert space H: and continuity imply Ψ(t) = U(t, t 0 )Ψ(t 0 ).It further means that the quantum evolution of our gravitational system can be equivalently defined by the Schrödinger type equation: The classical time t occurs in (22) because it enters the integrand of (19).We do not quantize the classical time t.In the case t were a quantum observable, it would be mapped into a quantum operator [17], but we do not consider here such a case.

A. Classical dynamics near the singularity
Near the gravitational singularity, the terms exp(2q 1 ) and exp(q 2 − q 1 ) in the function F can be neglected (see [1] for more details) so that we have This form of F leads to the simplified form of the Hamiltonian ( 5) which now reads with lim F 0 →0 − H 0 = 0 and lim F 0 →0 + H 0 = +∞.In fact, the condition defines the available part of the physical phase space Π for the classical dynamics, defined by (3), which corresponds to the approximation (23).Eqs. ( 23)-( 24) define the approximation to our original Hamiltonian system describing the dynamics in the close vicinity of the singularity.

B. Quantum dynamics near the singularity
Calculations similar to the ones carried out in our paper [1], applied to the Hamiltonian (24), lead to the Schrödinger equation (22) in the form: where Ψ(t, x 1 , x 2 ) := x 1 , x 2 |Ψ(t) .The function K reads The fiducial function Φ 2 (x) ∈ R should satisfy the conditions: and the solution to (26) is expected to have the properties: The mathematical structure of Eq. ( 26) is similar to the corresponding one of the paper [1].The difference concerns just one part of these equations, namely the actual function K(t, x 1 , x 2 ) and the function K(t, x 1 , x 2 ) in [1].Therefore, the general solution to (26) reads where t ≥ t 0 > 0, and where η(x 1 , x 2 ) := Ψ(t 0 , x 1 , x 2 ) is the initial state satisfying the condition with t H > 0 being the parameter of our model.This condition is consistent with (30) and for t < t H we get (see [1]) so that the inner product is time independent, which implies that the quantum evolution is unitary.Due to (32), the probability of finding the system in the region with x 2 < t H vanishes so that this region does not contribute to the expectation values of observables.These results are consistent with the results of [1].
The unitarity of the evolution (with t 0 = 0) can be obtained again if which corresponds to the condition (32).Since the solutions (31) and (35) differ only by the corresponding phases, the probability density is continuous at t = 0, which means that we are dealing with quantum bounce at t = 0 (that marks the classical singularity).

V. CONCLUSIONS
The quantum dynamics we have obtained does not depend essentially on the applied parametrization of the affine group.Two different parametrizations give qualitatively the same results, which differ only slightly quantitatively.The latter is meaningless if we only insist on the main result which is the resolution of the classical singularity.
We have applied the simplest two group parametrizations.The general one can be presented in the form of the action of the group on R + as follows [2]: We expect that in the case ξ(p, q) = ξ(q) and η(p, q) = η(p), the result of quantization will be qualitatively the same as the one obtained in the present paper.The effect of using quite general parametrization considered in [2], applied to the quantization of our gravitational system, would need separate examination and is beyond the scope of the present paper.We may stay with the simplest parametrizations if we do not test the quantization method as such, but intend to get the result with satisfactory physics.After experimental or observational data on quantum gravity become available, the way of choosing the most suitable group parametrization will obtain sound guideline.