Eur. Phys. J. C (2019) 79:45 https://doi.org/10.1140/epjc/s10052-019-6571-4

Regular Article - Theoretical Physics

Quantum Belinski–Khalatnikov–Lifshitz scenario

Andrzej Gó´zd´z1,a, Włodzimierz Piechocki2,b, Grzegorz Plewa2,c 1 Institute of Physics, Maria Curie-Skłodowska University, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland 2 Department of Fundamental Research, National Centre for Nuclear Research, Ho˙za 69, 00-681 Warsaw, Poland

Received: 17 November 2018 / Accepted: 2 January 2019 © The Author(s) 2019

Abstract We present the quantum model of the asymptotic 4.3.1 Regular state at fixed time ...... dynamics underlying the Belinski–Khalatnikov–Lifshitz 4.3.2 Initial state obtained in backward evolution (BKL) scenario. The symmetry of the physical phase space 4.3.3 Regularity of the initial state ...... enables making use of the affine coherent states quantization. 4.4 Quantum bounce ...... Our results show that quantum dynamics is regular in the 5 Conclusions ...... sense that during quantum evolution the expectation values Appendix A: Alternative affine coherent states for half-plane of all considered observables are finite. The classical singu- Appendix B: Orthonormal basis of the carrier space ... larity of the BKL scenario is replaced by the quantum bounce References ...... 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 singu- Contents larities predicted by general relativity (GR) can be resolved at the quantum level. That has been shown to be the case in 1 Introduction ...... the quantization of the simplest singular GR solutions like 2 Classical dynamics ...... FRW-type spacetimes (commonly used in observational cos- 2.1 Asymptotic regime of general Bianchi IX dynamics mology). However, it is an open problem in the general case. 2.2 Hamiltonian formulation with dynamical constraint Our paper addresses the issue of possible resolution of a 2.3 Hamiltonian formulation devoid of dynamical generic singularity problem in GR due to quantum effects. constraint ...... The Belinski, Khalatnikov and Lifshitz (BKL) conjecture 2.4 Numerical simulations of dynamics ...... is thought to describe a generic solution to the Einstein equa- 2.5 Topology of phase space ...... tions near spacelike singularity (see, [1–3] and references 3 Quantization ...... therein). Later, it was extended to deal with generic time- 3.1 Affine coherent states ...... like singularity of general relativity [4–6]. According to the 3.1.1 Affine coherent states for half-plane .... BKL scenario [1,2], in the approach to a space-like singu- 3.1.2 Structure of the fiducial vector ...... larity neighbouring points decouple and spatial derivatives 3.1.3 Phase space and quantum state spaces ... become negligible in comparison to temporal derivatives. 3.1.4 Affine coherent states for the entire system The conjecture is based on the examination of the dynam- 3.2 Quantum observables ...... ics toward the singularity of a Bianchi spacetime, typically 4 Quantum dynamics ...... Bianchi IX (BIX). The BKL scenario presents the oscilla- 4.1 Elementary quantum observables ...... tory evolution (towards the singularity) entering the phase of 4.2 Singularity of dynamics ...... chaotic dynamics (see, e.g., [7,8]), followed by approaching 4.3 Resolution of the singularity problem ...... the spacelike manifold with diverging curvature and matter field invariants. a e-mail: [email protected] The most general scenario is the dynamics of the non- b e-mail: [email protected] diagonal BIX model. However, this dynamics is difficult c e-mail: [email protected] to exact treatment. Qualitative analytical considerations [9–

0123456789().: V,-vol 123 45 Page 2 of 16 Eur. Phys. J. C (2019) 79:45

11] and numerical analysis [12] strongly suggest that in the 2 Classical dynamics asymptotic regime near the singularity the exact dynamics can be well approximated by much simpler dynamics (pre- For self-consistency of the present paper, we first recall some sented in the next section). results of Ref. [15], followed by the analysis of the topology The BKL scenario based on a diagonal BIX reduces to the of the physical phase space. dynamics described in terms of the three directional scale factors dependent on an evolution parameter (time). This 2.1 Asymptotic regime of general Bianchi IX dynamics dynamics towards the singularity has the following prop- erties: (i) is symmetric with respect to the permutation of The dynamical equations of the general (nondiagonal) the scale factors, (ii) the scale factors are oscillatory func- Bianchi IX model, in the evolution towards the singularity, tions of time, (iii) the product of the three scale factors is take the following asymptotic form [9,10] proportional to the volume density decreasing monotoni- d2 ln a d2 ln b cally to zero, and (iv) the scale factors may intersect each = b/a − a2, = a2 − b/a + c/b, other during the evolution of the system. The diagonal BIX dτ 2 dτ 2 is suitable to address the vacuum case and the cases with d2 ln c = a2 − c/b, (1) simple matter fields. More general cases, including perfect dτ 2 fluid with nonzero time dependent velocity, require taking where a, b, c are functions of an evolution parameter τ, and non-diagonal space metric. However, the general dynamics are interpreted as the effective directional scale factors of simplifies near the singularity and can be described by three considered anisotropic universe. The solution to (1) should effective scale factors, which include contribution from mat- satisfy the dynamical constraint ter field. This effective dynamics does not have the properties (i) and (iv) of the diagonal case. More details can be found d ln a d ln b + d ln a d ln c + d ln b d ln c in the paper [13]. dτ dτ dτ dτ dτ dτ Roughly speaking, the main advantage of the non- −a2 − b/a − c/b = 0. (2) diagonal BIX scenario is that it can be used to derive the BKL conjecture in a much simpler way than when starting Equations(1) and (2) define a coupled highly nonlinear sys- from the diagonal case. Namely, considering inhomogeneous tem of ordinary differential equations. perturbations of the non-diagonal BIX metric is sufficient to The derivation of this asymptotic dynamics (1)–(2)from derive the BKL conjecture, whereas the diagonal case needs the exact one is based, roughly speaking, on the assumption τ → additionally considering inhomogeneous perturbation of the that in the evolution towards the singularity ( 0) the matter field that would correspond, e.g., to time dependent following conditions are satisfied: velocity of the perfect fluid [14]. a → 0, b/a → 0, c/b → 0. (3) The present paper concerns the quantum fate of the asymp- totic dynamics of the non-diagonal BIX model. The quantum We recommend Sect.6 of Ref. [10] for the justification of tak- dynamics, described by the Schrödinger equation, is regular ing this assumption.1 The numerical simulations of the exact (no divergencies of physical observables) and the evolution dynamics presented in the recent paper [12] give support to is unitary. The classical singularity is replaced by quantum the assumption (3) as well. bounce due to the continuity of the probability density. The dynamics (1)–(2) defines the asymptotic regime of Our paper is organized as follows: In Sect. 2 we recall the BKL scenario, satisfying (3), that lasts quite a long time the Hamiltonian formulation of our gravitational system and before the system approaches the singularity, marked by the identify the topology of physical phase space. Section 3 is condition abc→ 0 (see [9,10] for more details). devoted to the construction of the quantum formalism. It is based on using the affine coherent states ascribed to the phys- 2.2 Hamiltonian formulation with dynamical constraint ical phase space, and the resolution of the unity in the car- rier space of the unitary representation of the affine group. Roughly speaking, using the canonical phase space variables The quantum dynamics is presented in Sect. 4. Finding an {qk, pl }=δkl introduced in [15]: explicit solution to the Schrödinger equation enables address- ing the singularity problem. We conclude in the last section. q1 := ln a, q2 := ln b, q3 := ln c, and p1 =˙q2 +˙q3,

Appendix A presents an alternative affine coherent states. p2 =˙q1 +˙q3, p3 =˙q1 +˙q2, (4) The basis of the carrier space is defined in Appendix B.

1 The directional scale factors {a, b, c} considered here and in [9], and the ones considered in [10] named {A, B, C}, are connected by the relations: a = A2, b = B2, c = C2. 123 Eur. Phys. J. C (2019) 79:45 Page 3 of 16 45

/ τ ( , , , , ) where “dot” denotes d d , turns the constraint (2) into the H t q1 q2 p1 p2 =   Hamiltonian constraint Hc 0 defined by − 1 =−q − ln −e2q1 − eq2 q1 − p2 + p2 + t2 2 4 1 2 1 1 2 2 2  Hc := (p p + p p + p p ) − (p + p + p ) 2 1 2 1 3 2 3 4 1 2 3 1 + (p1 p2 + p1t + p2t) , (14) − exp(2q1) − exp(q2 − q1) − exp(q3 − q2) = 0, 2 (5) and Hamilton’s equations are and the corresponding Hamilton’s equations read ∂ − − dq1 = H = p1 p2 t , 1 ∂ (15) q˙ = (−p + p + p ), (6) dt p1 2F 1 2 1 2 3 ∂ − + − dq2 = H = p1 p2 t , 1 ∂ (16) q˙2 = (p1 − p2 + p3), (7) dt p2 2F 2 − dp ∂ H −2e2q1 + eq2 q1 1 1 =− = , (17) q˙3 = (p1 + p2 − p3), (8) dt ∂q F 2 1 q −q p˙ = 2exp(2q ) − exp(q − q ), (9) dp2 ∂ H e 2 1 1 1 2 1 =− = 1 − , (18) dt ∂q F p˙2 = exp(q2 − q1) − exp(q3 − q2), (10) 2 p˙3 = exp(q3 − q2). (11) where   The dynamical systems analysis applied to the system (5)– − 1 F(t, q , q , p , p ) := −e2q1 − eq2 q1 − p2 + p2 +t2 (11) leads to the conclusion that there exists the set of the 1 2 1 2 4 1 2 nonhyperbolic type of critical points S , corresponding to 1 B + (p p + p t + p t)>0. (19) this dynamics, defined by [15] 2 1 2 1 2

SB := {(q1, q2, q3, p1, p2, p3) In what follows, we do not use explicitly the relationship ¯ 6 between the evolution parameter τ and t, but it does exist. ∈ R | (q1 →−∞, q2 − q1 →−∞, q3 − q2 →−∞) Namely, since t = p3, solving the dynamics (5)–(11) would ∧(p1 = 0 = p2 = p3} , (12) give p3 = p3(τ). Moreover, making use of Eq. (11) we can ˙ > where R¯ := R ∪{−∞, +∞}. get the essential information on the time variable: (i) p3 0 so that t is an increasing function of τ, and (ii) integrating Equations (12) and (3) imply that the space of singular  τ (11)givesp = 2 dτ exp(q − q )>0 as the integrand is points includes the critical points surface SB. It is so because 3 τ1 3 2 (q1 →−∞, q2 − q1 →−∞, q3 − q2 →−∞) implies (3) positive definite. ¯ 3 (which means abc→ 0) for any (p1, p2, p3) ∈ R . The reduced system (15)–(18) has been obtained in the procedure of mapping the system with the Hamiltonian con- 2.3 Hamiltonian formulation devoid of dynamical straint, defined by (5)–(11), into the Hamiltonian system constraint devoid of the constraint. In the former, the Hamiltonian Hc is a dynamical constraint, in the latter the Hamiltonian H There exists the reduced phase space formalism correspond- in a generator of dynamics without the constraint. As it is ing to the dynamics (5)–(11) presented in [15]. The two known, this procedure is a sort of “one-to-many” mapping form  defining the Hamiltonian formulation, devoid of the (for more details, see e.g. [16,17] and references therein). dynamical constraint (5), is given by Roughly speaking, it consists in resolving the dynamical con- straint with respect to one phase space variable that is chosen  = dq1 ∧ dp1 + dq2 ∧ dp2 + dt ∧ dH, (13) to be a Hamiltonian. This procedure leads to the choice of an evolution parameter (time) as well so that the Hamiltonian where t := p . The Hamiltonian H is defined to be H = 3 and time emerge in a single step. In general, this is a highly −q , where q is determined from the dynamical constraint 3 3 non-unique procedure if there are no hints to this process. (5). The variables {q , q , p , p } parameterise the physical 1 2 1 2 Here the choice of the reduction was motivated by the two phase space, H = H(t, q , q , p , p ) is the Hamiltonian 1 2 1 2 circumstances: (i) the resolution of the constraint only with generating the dynamics, and t is an evolution parameter respect to one variable q is unique, and (ii) the resulting corresponding to the specific choice of H. The Hamiltonian 3 reduced phase space is isomorphic to the Cartesian product reads2 of two affine groups. The latter has unitary irreducible rep- resentation enabling the affine coherent states quantization 2 In what follows, the choice of H differs from the one presented in [15] by a factor minus one to fit properly the third term of the r.h.s. of of the underlying gravitational system (presented in the next (13). section). 123 45 Page 4 of 16 Eur. Phys. J. C (2019) 79:45

2.4 Numerical simulations of dynamics Applying the simple algebraic identity (α + β + γ)2 = α2 + β2 + γ 2 + 2αβ + 2βγ + 2αγ to Eq. (19)gives: In what follows we present the numerical solutions of Eqs. 2q1 q2−q1 (15)–(18) near the singularity, which corresponds to the case: F(t, q1, q2, p1, p2) =−e − e →−∞ − →−∞ → + q1 , q2 q1 , F 0 . Accordingly, we 1 − (p − p + t)2 + p t, (22) choose the initial conditions in the form: 4 1 2 1   2q1 q2−q1 F(t, q1, q2, p1, p2) =−e − e 1 2 − q1(t0) =−, p1(t0) = t0 + t − 2e (4 + δ) , 2 0 1 2  − (−p1 + p2 + t) + p2t, (23)  4 1 2 − − q (t ) =−2, p (t ) = t − t − 2e (4 + δ) , ( , , , , ) =− 2q1 − q2 q1 2 0 2 0 2 0 0 F t q1 q2 p1 p2 e e 1 (20) − (p + p − t)2 + p p . (24) 4 1 2 1 2 where t is the initial “time”, whereas  and δ denote two 0 Combining (19) and (22) we get additional parameters. Inserting (20)into(19) one gets   1 − 1 > 2q1 + q2 q1 + ( − + )2 , ( , ( ), ( ), ( ), ( )) p1 e e p1 p2 t (25) F t0q1 t0 q2 t0 p1 t0 p2 t0 t 4 δ − −2 ˜ = 1 + e − e =: F0. (21) whereas (19) and (23)give 2   1 1 2q1 q2−q1 2 p2 > e + e + (−p1 + p2 + t) . (26) Thus, taking the limit  →∞, while keeping δ and t0 to be t 4 fixed, one gets F˜ → 0+, q →−∞, q −q →−∞. There- 0 1 2 1 Making use of (19) and (24) leads to fore, (20) can be regarded as an example that ensures that we are close to the singularity. In particular, taking   1 and − 1 p p > e2q1 + eq2 q1 + (p + p − t)2. (27) choosing δ>0 to be small (but not necessarily δ  1), for 1 2 4 1 2 afixedt0, we get the vicinity of the singularity. In the next It is clear that the signs of both r.h.s. of Eqs. (25) and (26) step we solve the system (15)–(18) with the boundary condi- depend only on the sign of t. Since t > 0, we get p1 > 0 and tions (20). To be more specific, starting with the point in the p2 > 0. phase space defined by (20), we can solve (15)–(18)forward To examine the issue of well definiteness of the logarith- or backward in time. Below, we consider the first possibility mic function in (14), we rewrite Eq.(14) in terms of (q , p ) ≥ k k assuming t t0. An example solution is presented in Fig. 1. variables3 as follows The evolution with t0 > t → 0, becomes quickly meaning-  − 1 less because numerical errors grow so much that the results − = − 2q1 − q2 q1 − ( 2 + 2 + 2) q3 q2 ln e e p1 p2 t cannot be trusted. This is due to rapidly increasing curva- 4  ture of the underlying spacetime and increasing nonlinearity 1 + (p p + p t + p t) . (28) effects near the surface of the nonhyperbolic critical points 2 1 2 1 2 (12). These lead to faster and faster changes of the phase At the critical surface S , defined by (12), we have q −q → space variables parameterizing the dynamics, and finally to B 3 2 −∞ so the r.h.s. of (28) should have this property as well. breaking off our numerics. We have tested that our choice + − It means that F(t, q , q , p , p ) → 0 on approaching S t = 10 20 enables performing reliable numerical simula- 1 2 1 2 B 0 so the problem of well definiteness reduces to solving the tions for t ≥ t . 0 equation t2 − 2(p + p )t + (p − p )2 < 0 with respect to The solution visible in Fig. 1 presents a wiggled curve in 1 2 1 2 √ √ the time variable. The solution reads ( p − p )2 < t < the physical phase space. This classical dynamics cannot be √ √ 1 2 ( p + p )2, which means that as p → 0 and p → 0, further extended towards the singularity (for t < t ) due to the 1 2 1 2 0 we have t → 0+. Therefore, our gravitational system evolves physical and mathematical reasons (and implied numerical away from the singularity at t = 0. difficulties). The range of the variables q1 and q2 results from the phys- ical interpretation ascribed to them [15]. Since 0 < a < +∞ 2 2.5 Topology of phase space and 0 < b < +∞,wehave(q1, q2) ∈ R . Thus, the physical phase space consists of the two half planes: Equations (15)–(18) define a coupled system of nonlin- 3 ear ordinary differential equations. The solution defines the We stay with p3 = t as we wish to discuss the possible sign of the phase space of our gravitational system. time variable. 123 Eur. Phys. J. C (2019) 79:45 Page 5 of 16 45

(a) (b)

(c) (d)

−20 Fig. 1 The solution to Eqs.(15)–(18) with the initial conditions:  = 100, δ = t0 = 10 , which corresponds to q1(t0) = q2(t0)−q1(t0) =−100, ˜ −44 F0 3.72 × 10

= 1 × 2 := {(q1, p1) ∈ R × R+} where defined by (29), is available to the dynamics. It is due to the logarithmic function in the expression (14) defining ×{(q2, p2) ∈ R × R+} , (29) the Hamiltonian. To make this restriction explicit, we rewrite the Hamiltonian (14)intheform where R+ := {p ∈ R | p > 0}. Each k (k = 1, 2) can be identified with the manifold of the affine group Aff(R) H(t, q , q , p , p ) acting on R, which is sometimes denoted as “px + q” 1 2 1 2 − − ( , , , , ), ( , , , , )> = q2 ln F t q1 q2 p1 p2 for F t q1 q2 p1 p2 0 , ( , , , , )< x = (q, p) · x = px + q, where p > 0 and q ∈ R. 0 for F t q1 q2 p1 p2 0 (31) This opens the possibility for quantization by affine coherent − = + =+∞ states. with limF→0 H 0 and limF→0 H . It is important to notice that only the subspace

˜ ={(q1, p1, q2, p2): F(t, q1, p1, q2, p2)>0}⊂ , (30) 123 45 Page 6 of 16 Eur. Phys. J. C (2019) 79:45

3 Quantization where5 |ψ∈L2(R+, dν(x)). Equation(34) defines the rep- resentation as we have Suppose we have reduced phase space Hamiltonian formula- ( , )[ ( , ,)ψ( )]= ( , )[ iqxψ( )] tion of classical dynamics of a gravitational system. It means U q p U q p x U q p e px ( + ) dynamical constraints have been resolved and the Hamilto- = ei p q q x ψ(p px), nian is a generator of the dynamics. By quantization we mean (roughly speaking) a mapping of such Hamiltonian formu- and on the other hand lation into a quantum system described in terms of quantum [U(q , p )U(q, p,)]ψ(x) = U(p q + q , p p)ψ(x) observables (including Hamiltonian) represented by an alge- = ei(p q+q )x ψ(p px). bra of operators acting in a Hilbert space. The construction of the Hilbert space may make use some mathematical proper- This action is unitary in respect to the scalar product in ties of phase space like, e.g., symplectic structure, geometry L2(R+, dν(x)): or topology. The quantum Hamiltonian is used to define the ∞ Schrödinger equation. In what follows we make specific the dν(x)[U(q, p) f2(x)] [U(q, p) f1(x)] above procedure by using the affine coherent states approach. 0 ∞ iqx iqx = dν(x)[e f2(px)] [e f1(px)] 3.1 Affine coherent states 0 ∞ ∞ = ν( ) ( ) ( ) = ν( ) ( ) ( ). H d x f2 px f1 px d x f2 x f1 x The Hilbert space of the entire system consists of the 0 0 H H Hilbert spaces 1 and 2 corresponding to the phase spaces (35) 1 and 2, respectively. In the sequel the construction of H1 is followed by merging of H1 and H2. The last equality results from the invariance of the measure ν( ) = ν( ) As both half-planes 1 and 2 have the same mathe- d px d x . matical structure, the corresponding Hilbert spaces H1 and The affine group is not the unimodular group. The left and H2 are identical so we first consider only one of them. In right invariant measures are given by H what follows we present the formalism for 1 and 1 to be dp dp extended later to the entire system. dμ (q, p) = dq and dμ (q, p) = dq , (36) L p2 R p 3.1.1 Affine coherent states for half-plane respectively. The left and right shifts of any group G are defined differ- The phase space 1 may be identified with the affine group ently by different authors. Here we adopt the definition from G1 ≡ Aff(R) by defining the multiplication law as follows [22]: ( , ) · ( , ) = ( + , ), LL ( ) = ( −1 ) LR ( ) = ( −1) q p q p p q q p p (32) h f g f h g and h f g f gh (37) with the unity (0, 1) and the inverse : → C ∈  for a function f G and all g G. For simplicity of notation, let us define integrals over the −1 q 1 (q , p ) = − , . (33) = (R) p p affine group G1 Aff as:

The affine group has two, nontrivial, inequivalent irreducible 1 +∞ ∞ dp dμ (q, p) = dq L 2 and unitary representations [18–20]. Both are realized in the G 2π −∞ 0 p 2 1 Hilbert space H1 = L (R+, dν(x)), where dν(x) = dx/x μ ( , ) is the invariant measure4 on the multiplicative group (R+, ·). d R q p G In what follows we choose one of it defined by the following 1 1 +∞ ∞ dp action: = dq . (38) 2π −∞ 0 p U(q, p)ψ(x) = eiqxψ(px), (34) In many formulae it is useful to use shorter notation for points ξ ≡ ( , ) 4 The general notion of invariant measure dm(x) on the set X in respect in the phase space q p and identify them with ele- to the transformation h : X → X can be approximately defined as ments of the affine group. In this case the product (32)is follows: for every function f : X → C the integral defined by this denoted as ξ · ξ. Depending on needs we will use both nota- measure fulfils the invariance condition: tions. dm(x) f (h(x)) = dm(x) f (x). X X 5 Weuse Dirac’s notation whenever we wish to deal with abstract vector, This property is often written as: dm(h(x)) = dm(x). instead of functional representation of the vector. 123 Eur. Phys. J. C (2019) 79:45 Page 7 of 16 45

∞ Fixing the normalized vector |∈L2(R+, dν(x)), A = dμL (q, p) × dν(x ) called the fiducial vector, one can define a continuous family G 1 0 2 ∞ of affine coherent states |q, p∈L (R+, dν(x)) as follows − × dν(x)( f (x ) eiqx (px ))(e iqx(px) f (x)) 0 |q, p=U(q, p)|. (39) ∞ ∞ ∞  +∞  dx dx dp 1 = dqeiq(x −x) 2 π As we have two invariant measures, one can define two oper- 0 x 0 x 0 p 2 −∞ ators which potentially can lead to the unity in the space × f (x ) f (x)(px )(px) ∞ ∞ L2(R+, dν(x)): dx dp = | f (x)|2 |(px)|2 x2 p2 0 0 = μ ( , )| ,  , | ∞ ∞ BL d L q p q p q p and dx 2 dp 2 G = | f (x)| |(p)| 1 x p2 0 0 B = dμ (q, p)|q, pq, p|. (40) ∞ dp R R = |(p)|2 G1 2 (48) 0 p Let us check which one is invariant under the action U(q, p) because  f | f =1. Thus, the normalization constant is of the affine group: dependent on the fiducial vector. Appendix A presents alter- ( , ) ( , )† native affine coherent states. U q p BLU q p

= dμ (q, p)|p q + q , p pp q + q , p p|. (41) L 3.1.2 Structure of the fiducial vector G1

One needs to replace the variables under integral: The problem which influences the structure of quantum state

q˜ = p q + q and p˜ = p p (42) space is a possible degeneration of the space due to specific structure of the fiducial vector. In the case of quantum states 1 p˜ q = (q˜ − q ) and p = . (43) the vectors which differ by a phase factor represent the same p p quantum state. Thus, let us consider the states satisfying the ∂(q,p) = 1 above condition for physically equivalent state vectors [24]: Calculating the Jacobian ∂(q˜,p˜) (p )2 one gets ( ˜, ˜)( ) = iβ(q˜,p˜)( ), β(˜, ˜) ∈ R. 1 1 1 U q p x e x where q p dμ (q, p) = dqd˜ p˜ = dqd˜ p˜ = dμ (q˜, p˜). L p2 (p )2 p˜2 L (49) (44) The phase space points ξ˜ = (q˜, p˜) treated as elements of the The last result proves that affine group Aff(R) forms its subgroup G. The left-hand side of Eq. (49) can be rewritten as: † U(q , p )B U(q, p) = dμ (q˜, p˜)|˜q, p˜q ˜, p˜|=B . ˜ β(˜, ˜) L L L eiqx(px˜ ) = ei q p (x). (50) G1 (45) If the generalized stationary group G of the fiducial This also means that B is not invariant under the action vector  is a nontrivial group, then the phase space points R U(q, p). (q , p ) and (q˜, p˜) · (q , p ) = (pq˜ , pp˜ ) are represented by

The irreducibility of the representation, used to define the same state vector U(q , p )(x), for all transformations the coherent states (39), enables making use of Schur’s (q˜, p˜) ∈ G. This is due to the equality lemma [23], which leads to the resolution of the unity in U(q , p )(x) = U((q˜, p˜) · (q , p ))(x). (51) L2(R+, dν(x)):

In this case, to have a unique relation between phase space μ ( , )| ,  , |= I, d L q p q p q p A (46) and the quantum states, the phase space has to be restricted to G1 the quotient structure Aff(R)/G. From the physical point where the constant A can be determined by using any arbi- of view, in most cases, this is an undesired property. | ∈ 2(R , ν( )) trary, normalized vector f L + d x : How to construct the fiducial vector to have G ={eG },

where eG is the unit element in this group? It is seen that A = dμL (q, p)  f |q, pq, p| f . (47) Eq. (50) cannot be fulfilled for q˜ = 0, independently of cho- G1 sen fiducial vector. This suggests that the generalized station- This formula can be calculated by making use of the invari- ary group G is parameterized only by the momenta (0, p˜), ance of the measure: 123 45 Page 8 of 16 Eur. Phys. J. C (2019) 79:45 i.e. it has to be a subgroup of the multiplicative group of 3.1.3 Phase space and quantum state spaces positive real numbers, G ⊆ (R+, ·). On the other hand, Eq. (50)impliesthat|(px˜ )|=|(x)| The quantization procedure requires understanding the rela- for all (0, p˜) ∈ G. In addition, for the fiducial vectors tions among the classical phase space and quantum states (x) =|(x)|eiγ(x) the phases of these complex func- space. We have three spaces to be considered: tions are bounded by 0 ≤ γ(x)<2π.DuetoEq.(50) the phases γ(x) and β(0, p˜) have to fulfil the following con- • The phase space , which consists of two half-planes dition γ(px˜ ) − γ(x) = β(0, p˜). One of the solutions to this 1 and 2 defined by (29). It is the background for the equation is the logarithmic function γ(x) = ln(x). classical dynamics.6 2 In what follows, to have the unique representation of the • The carrier spaces H1 := L (R+, dν(x)) of the unitary phase space as a group manifold of the affine group, we representation U(q, p), with the scalar product defined require the generalized stationary group to be the group con- as sisted only of the unit element. This can be achieved by the ∞ appropriate choice of the fiducial vector. ψ |ψ = dx ψ ( )ψ ( ). 2 1 2 x 1 x (56) The unit operator (46) depends explicitly on the fiducial 0 x vector • The space of square integrable functions on the affine 1 2 † group KG = L (Aff(R), dμL (q, p)). The scalar prod- I[]= dμL (ξ)U(ξ)||U(ξ) , (52) A G1 uct is defined as follows This suggests that the most natural transformation of vectors | ψ |ψ  = 1 μ ( , )ψ ( , )ψ ( , ), from the representation given by the fiducial vector to the G2 G1 G d L q p G2 q p G1 q p Aφ representation given by another fiducial vector |  can be Aff constructed as the product of two unit operators I[ ]I[]. (57) |∈ 2(R+, ν( )) † Let us consider an arbitrary vector L d x where ψG (q, p) := q, p|ψ=|U(q, p) |ψ with and its representation in the space spanned with a help of the |ψ∈H1. The Hilbert space KG is defined to be the fiducial vector |: completion in the norm induced by (57) of the span of

the ψG functions. 1 † |=I[]|= dμL (ξ)U(ξ)||U(ξ) | A G1 We show below that the spaces H and K are unitary (53) 1 G isomorphic. First, one needs to check that the functions ψ ∈ K The same vector can be represented in terms of another fidu- G G are square integrable function belonging to 2( (R), μ ( , )) cial vector | : L Aff d L q p . Using the decomposition of unity we get

1 † |=I[ ]|= dμL (ξ)U(ξ)|  |U(ξ) | 1 μ ( , )| , |ψ|2 < ψ|ψ < ∞. A G d L q p q p H (58) 0 A 1 (54) G1 The definition of the space KG shows that for every ψ1,ψ2 ∈ However, one can transform the vector (53) into the vector Hl we have the corresponding functions ψG1,ψG2 for which (54) using the product of two unit operators: the scalar products are equal (unitarity of the transformation

between both spaces): ψ2|ψ1H =ψG1|ψG2K . 1 1 1 G |=I[ ]I[]|= dμL (q, p) Let us now denote by |en the orthonormal basis in H1 (see A A G 1 App.B). The corresponding functions eGn(q, p) =q, p|en furnish the orthonormal set: dμL (q , p )U(ξ)|  G1 − eGn|eGmK  |U(ξ 1 · ξ )||U(ξ )†| (55) G = 1 μ ( , ) ( , ) ( , ) d L q p eGn q p eGm q p A Thus, the choice of the fiducial vector is formally irrelevant. G1 However, as we will see later a relation between the classical 1 = dμ (q, p)e |q, pq, p|e  model and its quantum realization depends on this choice. L n m A G1 The sets of affine coherent states generated from different =en|em=δnm. (59) fiducial vectors may be not unitarily equivalent, but lead in each case to acceptable affine representations of the Hilbert 6 For simplicity we consider here only one half-plane, but the results space [21]. can be easily extended to . 123 Eur. Phys. J. C (2019) 79:45 Page 9 of 16 45

It is obvious that the vectors |e  define the orthonormal ˆ 1 † Gn Ik = dμL (ξk)Uk(ξk)|kk|Uk(ξk) , k = 1, 2.  basis in the space KG . For every vector |ψ∈H1 A k Gk (64) |ψ= en|ψ|en. (60) (1) (2) n Let us consider the orthonormal basis {en (x1) ⊗ en (x2)} H ( , ) =  , | in the Hilbert space and an arbitrary vector x1 x2 Closing both sides of the above equation with q p gives the (1) (2) anmen (x1) ⊗ em (x2) belonging to this space (where unique decomposition of the vector ψG (q, p) ≡q, p|ψ∈ nm K |  the basis en(x) is defined in App. B). Acting on this vector G in the basis eGn G : ˆ with the operator I12 one gets: ψG (q, p) ≡q, p|ψ= en|ψq, p|en     Iˆ (x , x ) = a Iˆ e(1)(x ) ⊗ Iˆ e(2)(x ) n 12 1 2 nm 1 n 1 2 m 2 = e |ψ|e  . (61) nm n Gn G = (x , x ). (65) n 1 2 Iˆ Iˆ Note that the vector |ψ∈H1 and the vector |ψG ∈KG The operator 12 is identical with the unit operator on the have the same expansion coefficients in the corresponding space H. bases. This define the unitary isomorphism between both The explicit form of the action of the group G on the spaces. It means that we can work either with the quantum vector (x1, x2) reads: state space represented by the space H or K . 1 G U(q , p , q , p )(x , x ) 1 1 2 2 1 2    = ( , ) (1)( ) ⊗ ( , ) (2)( ) 3.1.4 Affine coherent states for the entire system anm U1 q1 p1 en x1 U q2 p2 em x2 nm     ( ) ( ) = iqx1 1 ( ) ⊗ iqx2 2 ( ) The phase space of our classical system has the structure of anm e en p1x1 e em p2x2 = × the Cartesian product of the two phase spaces: 1 2. nm ( ) ( ) The partial phase spaces l , where l = 1, 2, are identified = iq1x1 iq2x2 1 ( ) ⊗ 2 ( ) e e anmen p1x1 em p2x2 with the corresponding affine groups which we denote by nm Gl = Affl (R). The simple product of both affine groups iq1x1 iq2x2 = e e (p1x1, p2x2). (66) G = (G1 = Aff1(R)) × (G2 = Aff2(R)) can be identified with the whole phase space : 3.2 Quantum observables

(ξ ,ξ ) →|ξ ,ξ = (ξ ,ξ )|:= (ξ ) ⊗ (ξ )|, 1 2 1 2 U 1 2 U1 1 U2 2 Making use of the resolution of the identity (46), we define (62) the quantization of a classical observable f on a half-plane as follows [26] where ξ = (q , p ), l = 1, 2, the fiducial vector | l l l belongs to the simple product of two Hilbert spaces (H = 1 F  −→ ˆ := 1 μ ( , )| ,  ( , ) , |∈A, 2(R , ν( ))) × (H = 2(R , ν( ))) = 2(R × f f d L q p q p f q p q p L + d x1 2 L + d x2 L + A G R , ν( , )) ν( , ) = 1 + d x1 x2 , and where the measure d x1 x2 (67) 2 dν(x1)dν(x2). The scalar product in H = L (R+ × R+, dν(x1, x2)) reads where F is a vector space of real continuous functions ∞ ∞ on a phase space, and A is a vector space of operators

ψ2|ψ1= dν(x1) dν(x2)ψ1(x1, x2) ψ2(x1, x2). (quantum observables) acting in the Hilbert space H1 = 0 0 L2(R+, dν(x)). It is clear that (67) defines a linear mapping (63) and the observable fˆ is a symmetric (Hermitian) operator. ˆ The fiducial vector (x1, x2) is constructed as a product of Let us evaluate the norm of the operator f : ( , ) =  ( ) ( ) two fiducial vectors x1 x2 1 x1 2 x2 generating 1  fˆ≤ dμ (q, p)| f (q, p)||q, pq, p| the appropriate quantum partners for the phase spaces 1 and L A G . The fiducial vector of this type does not add any corre- 1 2 1 lations between both partial phase spaces. A nonseparable ≤ dμL (q, p)| f (q, p)|. (68) A G form of (x1, x2) might lead to reducible representation of 1 G in which case Schur’s lemma could not be applied to get This implies that, if the classical function f belongs to the 1 the resolution of unity in H. space of integrable functions L (Aff(R), dμL (q, p)),the ˆ ˆ Let us denote by I12 the linear extension of the tensor operator f is bounded so it is a self-adjoint operator. Oth- ˆ ˆ 2 product I1 ⊗I2, where the unit operators in Hk are expressed erwise, it is defined on a dense subspace of L (R+, dν(x)) in terms of the appropriate coherent states and its possible self-adjointness becomes an open problem as 123 45 Page 10 of 16 Eur. Phys. J. C (2019) 79:45 symmetricity does not assure self-adjointness so that further postulate that t = s, which defines the time variable at both examination is required [27]. The quantization (67) can be levels. It is worth to mention that so defined time changes applied to any type of observables including non-polynomial monotonically due to the special choice of the parameter t ones, which is of primary importance for us due to the func- at the classical level (see the paragraph below (19)). This tional form of the Hamiltonian (14). way we support the interpretation that Hamiltonian is the It is important to indicate that in the case the classical generator of classical and corresponding quantum dynamics. observable f is defined only on a subspace of the full phase Near the gravitational singularity, the terms exp(2q1) and ˆ space, the corresponding quantum operator f obtained via exp(q2 − q1) in the function F can be neglected, see Eqs. (67) acts in the entire Hilbert space corresponding to the full (19) and (12), so that we have phase space. This is the peculiarity of the coherent states ( , , , , ) −→ ( , , ) quantization [28]. Roughly speaking, it results from the non- F t q1 q2 p1 p2 F0 t p1 p2 zero overlap, q, p|q , p  = 0, of any two coherent states 1 2 := p1 p2 − (t − p1 − p2) . (73) of the entire Hilbert space. In particular, the classical Hamil- 4 ˜ tonian (31) is defined on the subspace of . Hovever, the This form of F leads to the simplified form of the Hamilto- corresponding quantum Hamiltonian (71) acts in the Hilbert nian (31) which now reads 2 space L (R+ × R+, dν(x1, x2)) corresponding to . ( , , , ) It is not difficult to show that the mapping (67) is covariant H0 t q2 p1 p2 in the sense that one has − − ( , , ), ( , , )> := q2 ln F0 t p1 p2 for F0 t p1 p2 0 , ( , , )< 1 −  0 for F0 t p1 p2 0 U(ξ ) fUˆ †(ξ ) = dμ (ξ)|ξ f (ξ 1 · ξ)ξ|=LL f , 0 0 L 0 ξ0 A G1 (74) (69) with limF →0− H0 = 0 and limF →0+ H0 =+∞. In fact, − 0 0 where LL f (ξ) = f (ξ 1 · ξ) is the left shift operation (37) the condition ξ0 0 −1 −1 q−q0 p and ξ · ξ = (q0, p0) · (q, p) = ( , ). 0 p0 p0 F (t, p , p )>0 (75) The mapping (67) extended to the Hilbert space H = 0 1 2 2(R × R , ν( , )) L + + d x1 x2 of the entire system and applied defines the available part of the physical phase space for the to an observable fˆ reads classical dynamics, defined by (29), which corresponds to the approximation (73). Equations (73)–(74) define the approx- ˆ( )= 1 μ (ξ ,ξ )|ξ ,ξ  (ξ ,ξ )ξ ,ξ |, f t d L 1 2 1 2 f 1 2 1 2 imation to our original Hamiltonian system, defined by Eqs. A1 A2 G (70) (14)–(19), to describe the dynamics in the close vicinity of the singularity. where dμL (ξ1,ξ2) := dμL (q1, p1)dμL (q2, p2). After long, otherwise straightforward, calculations we get the Schrödinger equation (72) in the form ∂ 4 Quantum dynamics i (t, x , x ) = Hˆ (t, x , x ), (76) ∂t 1 2 0 1 2 The mapping (70) applied to the classical Hamiltonian (14) where reads ∂ B Hˆ := i − i − K (t, x , x ), 0 ∂ 1 2 (77) ˆ 1 x2 x2 H(t) = dμL (ξ1,ξ2)|ξ1,ξ2H(t,ξ1,ξ2)ξ1,ξ2| , A A G 1 2 and where (t, x1, x2) := x1, x2|(t). The functions B (71) and K are defined to be where t is an evolution parameter of the classical level. ∞ ∂ ( ) := dp 2 p ∗( ), B A2 2 p (78) The quantum evolution of our gravitational system is 0 p ∂p defined by the Schrödinger equation: and ∂ i |(s)=Hˆ (t)|(s) , (72) ∞ ∂ 1 dp1 s K (t, x1, x2) := A A p2 where |∈H, and where s is an evolution parameter at the 1 2 0 1 ∞   quantum level. dp2  ( , p1 , p2 ) | ( )|2| ( )|2, 2 ln F0 t 1 p1 2 p2 (79) In general, the parameters t and s are different. To get 0 p2 x1 x2 the consistency between the classical and quantum levels we 123 Eur. Phys. J. C (2019) 79:45 Page 11 of 16 45 where where tH > 0 is a parameter of our model. This condition is   t < t  p1 p2 consistent with (82) and for H gives ln F0(t, , ) ∞  ∞ x1 x2 tH |η( , )|2   ( )|( )= dx1 + x1 x2 p1 p2 p1 p2 t t dx2 ln F0(t, , ) , for F0(t, , )>0 x x := x1 x2 x1 x2 (80) 0 1 t tH 2 , ( , p1 , p2 )< ∞ ∞ |η( , )|2 0 for F0 t x x 0 dx1 x1 x2 1 2 = dx2 , (87) 0 x1 tH x2 −  = +  =−∞ with limF0→0 lnF0 0 and limF0→0 lnF0 . Thus, B and K become known after the specification of the which is time independent so that the quantum evolution is unitary. fiducial vectors |1 and |2. It results from the definition ∗ ˆ The condition (86), which restricts x ∼ 1/p ,see(88) of B that one has B = 1 − B. The requirement of H0 2 2 and the text below it, implies that the probability of finding being Hermitian, leads to the result that 2(x) ∈ R and < B = 1/2. However, these results can be obtained in the case the system in the region with x2 tH vanishes. the following conditions are satisfied: 4.1 Elementary quantum observables ˜ ˜ ˜ 2(x) =: x(x), lim (x) = 0, lim (x) = 0, x→0+ x→+∞ The affine coherent states quantization procedure introduces (81) 2 2 the carrier space L (R+, dν(x1, x2)). The elementary quan- and tum observables, qˆk and pˆk, can be expressed in terms of √ x ∈ R+ (k = 1, 2). Namely, it is not difficult to find that ˜ ˜ k (t, x1, x2) =: x2 (t, x1, x2), lim (t, x1, x2) = 0, x →0+ 2 pˆk ψ(t, x1, x2) ˜   lim (t, x1, x2) = 0. (82) ∞ x2→+∞ 1 dpk 2 1 = pk |(pk)| ψ(t, x1, x2) A p2 x Due to the above, the Eq. (76) reduces to the equation k 0 k k  = 1 1 ψ( , , ), = , , ∂ ∂ i t x1 x2 k 1 2 (88) A x i (t, x1, x2) = i − − K (t, x1, x2) k k ∂t ∂x2 2x2 where the fiducial vector (pk) is normalized to unity. (t, x1, x2). (83) The multiplication operators representing the generalized The general solution to Eq.(83) is found to be momenta (4) determine the physical meaning of the variables  xk used in the description of quantum states, as proportional x (t, x , x ) = η(x , x + t − t ) 2 to the inverse of the generalized momenta. Similarly, 1 2 1 2 0 x + t − t   2 0 ∂ t ˆ ψ( , , )= − + i ψ( , , ), = , , × ( , , + − ) , qk t x1 x2 i t x1 x2 k 1 2 exp i K t x1 x2 t t dt ∂xk 2xk t0 (89) (84) 2 2 where ψ ∈ L (R+, dν(x1, x2)). One can show that both, where t ≥ t0, and where η(x1, x2) := (t0, x1, x2) is the qˆk and pˆk, are symmetric operators on the subspace of this initial state. Hilbert space, which consists of the functions satisfying In what follows, we extend the range of the time variable to include t = 0 (we quantize the sector t > 0 of classical 1 1 0 lim √ ψ(x , x ) = 0 = lim √ ψ(x , x ). (90) + 1 2 + 1 2 dynamics). x1→0 x1 x2→0 x2 To get insight into the meaning of the solution (84), let us consider the inner product 4.2 Singularity of dynamics

|η( , + )|2 x2 x1 x2 t According to Sect. 2, the singularity of the classical dynamics (t)|(t)= dν(x1, x2) 2 + R+ x2 t is defined by the conditions:

∞ ∞ 2 dx1 |η(x1, x2)| + + = dx . (85) q1 →−∞, q2 − q1 →−∞, F0 → 0 as t → 0 . x 2 x 0 1 t 2 (91) Thus, the norm of the solution decreases in time. To get an = unitary evolution, we choose the initial state in the form It means that the singularity may only occur at t 0, and one cannot see the reason for the classical dynamics of not η(x1, x2) = 0forx2 < tH , (86) being regular for t > 0. 123 45 Page 12 of 16 Eur. Phys. J. C (2019) 79:45

If for the  satisfying the Schrödinger equation (83)we where get ⎧ ⎨0, for x2 ≤ tH lim (t)|ˆq1|(t)=−∞, lim (t)|ˆq2 −ˆq1|(t)     → + → + v(x1, x2) = x2 ( − )2 t 0 t 0 ⎩ √1 1 x2 tH , > . (β + )3 (β − + )3 for x2 tH =−∞, (92) Nv 1 x1 2 tH x2 (98) and in addition (θ) ( )| ˆ |( )= , and where 1,2,β1,β2 ∈ R (Nv denotes normalization lim t F0 t 0 (93) t→0+ constant). our quantization fails in resolving the singularity problem of Direct calculation gives the classical dynamics.7 ˆ (θ) (ts, x1, x2)|ˆqk|(ts, x1, x2)=k, k = 1, 2, (99) The operator F0 , which occurs in (93), is of basic impor- tance and is found to be and (θ) (θ) ˆ ( )( , , ) = ˘ ( , , )( , , ), (θ) F0 t t x1 x2 F0 t x1 x2 t x1 x2 (94) ( , , )| ˆ ( )|( , , ) > , ts x1 x2 F0 ts ts x1 x2 0 (100) where ∞ as the integrand of (100) is positive definite due to (95). ˘ (θ) 1 dp1 F (t, x1, x2) := One can also verify that we have 0 A A p2  1 2 0 1 ∞ (ts, x1, x2)|ˆpk|(ts, x1, x2) dp2 (θ) p1 p2 2 2 F t, , |1(p1)| |2(p2)| , (95) ∞ ∞ |v( , )|2 2 0 x x dx1 x1 x2 0 p2 1 2 = C p dx2 < ∞, k = 1, 2, 0 x1 t xk(x2 − t) with H (101)     (θ) p1 p2 p1 p2 p1 p2 F t, , := θ F t, , F t, , , where C p is a constant. 0 x x 0 x x 0 x x 1 2 1 2 1 2 Therefore, the state (97) is regular at ts. (96) where θ is the Heaviside step function defined as follows: 4.3.2 Initial state obtained in backward evolution θ( ) = > θ( ) = ≤ ˆ (θ) x 1forx 0 and x 0forx 0. Therefore, F0 is a multiplication operator. The state (97) is an example of the state that can be presented, due to (84) and (86), as follows 4.3 Resolution of the singularity problem (ts, x1, x2) = 0, for x2 ≤ tH − ts, where t < t , (102) In what follows, we first define an example of a regular state s H  t = t > x2 at s 0. Next, we make generalization. Afterwards, we (ts, x1, x2) = η(x1, x2 + ts) = x + t map the general regular state to the initial state at t 0, by  2 s ts inverting the general form of the solution defined by Eqs.(84) × ( , , + − ) , = exp i K t x1 x2 ts t dt (with t0 0) and (86). Finally, we argue that the initial state 0 = is regular at t 0 due to the unitarity of the quantum evolu- for x2 ≥ tH − ts. tion. This way we get the resolution of the initial singularity (103) problem of the underlying classical dynamics. The above state can be inverted to get the initial state at 4.3.1 Regular state at fixed time t = 0 via the backward evolution:

η(x1, x2) = 0, for x2 ≤ tH , (104) Let us define a state defined at t = ts > 0, that is “far away”  from the singularity, as follows x2 η(x1, x2) = (ts, x1, x2 − ts) x2 − ts , ≤  0 for x2 tH ts (ts, x1, x2) = i(1x1+2x2) × exp −i K (t , x , x − t )dt , for x > t . v(x1, x2)e , for x2 > tH , 1 2 2 H 0 (97) (105)

7 The precise meaning of Eq. (93) will become clear in the next sub- Since the “forward” evolution is unitary, the “backward” evo- section. lution is unitary as well. 123 Eur. Phys. J. C (2019) 79:45 Page 13 of 16 45

4.3.3 Regularity of the initial state so its complex conjugate changes the sine of the time variable in a state vector. It is easy to check that ∂ i (˜ t, x , x ) ∂ 1 2 η( , )|ˆ |η( , ) < ∞, t  x1 x2 qk x1 x2 (106) ∂ i = −i + − K (−t, t, x , x ) (˜ t, x , x ). ∂ 1 2 1 2 and x2 2x2 (110) η(x , x )|ˆp |η(x , x ) 1 2 k 1 2 The general solution to Eq. (110), for t < 0, is found ∞ ∞ |( , , − )|2 = dx1 ts x1 x2 ts < ∞, to be C p dx2  0 x1 t xk(x2 − ts) H ˜ x2 (107) (t, x1, x2) = η(x1, x2 +|t|−|t0|) x +|t|−|t |  2 0 t where k = 1, 2 and where C p is a constant. It is so because the × exp i K (−t , x1, x2 − t + t ) dt , integrands of (106) and (107) are positive definite functions, t0 and due to (90). (111) Now, let us address the issue presented by Eq. (93). To where |t|≥|t |, and where η(x , x ) := (˜ t , x , x ) is the show that this equation cannot be satisfied, we should prove 0 1 2 0 1 2 initial state. that for all tH > ts = 0wehave The unitarity of the evolution (with t0 = 0) leads to the η( , )| ˆ (θ)( )|η( , ) condition x1 x2 F0 ts x1 x2

∞ ∞ ˘ (θ) 2 η( , ) = < | |, dx F (ts)|(ts, x1, x2 − ts)| x1 x2 0forx2 tH (112) = 1 dx 0 > . 2 − 0 0 x1 tH x2 ts which corresponds to the condition (86). (108) For |t| < |tH | we get

∞ ∞ |η( , )|2 We exclude the case ts = 0, because for p1 = p2, due ˜ ˜ dx1 x1 x2 (θ) (θ) (t)|(t)= dx2 , (113) = ˆ ( ) = ˆ x | | x to (96), we have F0 0, which means F0 0 0. The 0 1 tH 2 “zero operator” cannot represent any physical observable as which shows that the norm is time independent. its action does not lead to any physical state. Namely, it maps Comparing Eqs. (83) and (110) we can see that the dynam- |ψ ˆ|ψ any state into the zero vector 0 . Such a vector cannot ical equation fails to be time reversal invariant because the be normalized. Thus, it cannot be given any probabilistic ˆ Hamiltonian H0 does not have this symmetry. However, the interpretation. solutions to these equations have only different phases. Thus, Since the integrand defining Eq.(108) is positive definite, the probability density is continuous at t = 0 (that marks the equation is satisfied, which completes the proof. the classical singularity) due to Eqs. (84) and (111) (with Thus, the initial state is regular, i.e., does not satisfy Eqs. |t0|=0), which we call the quantum bounce. (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 quan- 5 Conclusions tum evolution is well defined for any t ≥ 0. This is a direct consequence of the unitarity of considered quantum evolu- Near the classical singularity the dynamics of the general tion. Bianchi IX model simplifies. Due to the symmetry of the physical phase space of this model, we can apply the affine 4.4 Quantum bounce coherent states quantization method. The quantum dynamics, described by the Schrödinger equation, is devoid of singular- Let us examine the issue of possible time reversal invariance ities in the sense that the expectation values of basic operators of our quantum model. In what follows, we examine the time are finite during the quantum evolution of the system. The reversal invariance of our Schrödinger equation and its solu- evolution is unitary and the probability density of our system tion. is continuous at t = 0, which marks the classical singularity. The operator of the time reversal, Tˆ : H → H, is defined We name the state defined by Eqs.(102)–(103)therescue to be state. It is chosen to be regular because one expects that the quantum state far away from the singularity is a proper quan- ˆ ψ( , , ) = ψ(˜ , , ) T t x1 x2 t x1 x2 tum state. The Schrödinger evolution does not lead outside ∗ := ψ(−t, x1, x2) , where ψ ∈ H, (109) the space of such states. 123 45 Page 14 of 16 Eur. Phys. J. C (2019) 79:45

The choice of the fiducial state as the rescue state leads, Data Availability Statement This manuscript has no associated data or under the action of the affine group, to another rescue state the data will not be deposited. [Authors’ comment: The article concerns entirely theoretical research.] with changed parameter tH . Namely, Open Access This article is distributed under the terms of the Creative U(q, p)Kt = Kt /p, p ∈ (0, +∞), (114) H H Commons Attribution 4.0 International License (http://creativecomm K ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, where tH denotes the rescue space with tH parameter (for simplicity we consider only one half plane). In general, the and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative fiducial vector is known to be a free “parameter” of the coher- Commons license, and indicate if changes were made. ent states quantization formalism. However, our final conclu- Funded by SCOAP3. sion does not depend on the fiducial vector. ˆ The quantum Hamiltonian, H0(t, x1, x2) =−ˆq2 − ˆ K (t, x1, x2), is not invariant under the affine group action Appendix A: Alternative affine coherent states for half- as we have plane

U (q˜ , p˜ )U (q˜ , p˜ )Hˆ (t, x , x )U (q˜ , p˜ )−1U (q˜ , p˜ )−1 1 1 1 2 2 2 0 1 2 1 1 1 2 2 2 The phase space 1 may be identified with the affine group ˆ 1 ˆ Aff(R) by defining the multiplication law as follows = H0(t, p˜1x1, p˜2x2) =− qˆ2 − K (t, p˜1x1, p˜2x2) p˜2  q = Hˆ (t, x , x ). (115) (q , p ) · (q, p) = + q , p p , (A1) 0 1 2 p Therefore, the affine coherent states quantization does not with the unity (0, 1) and the inverse introduce the affine symmetry as the symmetry of our quan-  tum system. However, the quantum evolution restricted to ( , )−1 = − , 1 . q p q p (A2) the space of all possible rescue spaces is unitary and devoid p of singularities. Thus, the rescue space, which spans the sub- The affine group has two, nontrivial, inequivalent irreducible space of our Hilbert space, is of basic importance in our unitary representations [18] and [19,20]. Both are realized in quantization scheme. the Hilbert space L2(R+, dν(x)), where dν(x) = dx/x is The non-diagonal BIX underlies the BKL conjecture the invariant measure on the multiplicative group (R+, ·).In which concerns the generic singularity of general relativity. what follows we choose the one defined by Therefore, our results suggest that quantum general relativity ( , )ψ( ) = iqxψ( / ), is free from singularities. Classical singularity is replaced by U q p x e x p (A3) quantum bounce, which presents a unitary evolution of the where |ψ∈L2(R+, dν(x)). quantum gravity system. For simplicity of notation, let us define integrals over the Since general relativity successfully describes almost all affine group Aff(R) as follows: available gravitational data, it makes sense its quantization to +∞ ∞ get the extension to the quantum regime. The latter could be 1 dp dμL (q, p) = dq , (A4) (R) 2π −∞ p2 used to describe quantum gravity effects expected to occur Aff 0 near the beginning of the Universe and in the interior of black +∞ ∞ μ ( , ) = 1 dp, holes. d R q p dq (A5) Aff(R) 2π −∞ 0 p Our fully quantum results show that the preliminary results +∞ ∞ 1 obtained for the diagonal BIX within the semiclassical affine dμU (q, p) = dq dpρ(q, p). (A6) (R) 2π −∞ coherent states approximation [29,30] are correct. Appendix Aff 0 B presents the affine coherent states applied in these papers, The last one is intended to be used as invariant measure in which define another parametrization of our coherent states. respect with the action U(q, p). As far as we are aware, we are pioneers in addressing the Fixing the normalized vector |∈L2(R+, dν(x)), issue of resolving the generic singularity problem of gen- called the fiducial vector, one can define a continuous family eral relativity via quantization. Therefore, trying to confirm of affine coherent states |q, p∈L2(R+, dν(x)) as follows our preliminary results within different quantization scheme |q, p=U(q, p)|. (A7) would be interesting. In particular, the choice of different time and corresponding Hamiltonian in the reduced phase As we have three measures, one can define three opera- space approach (see, Eq.(13)) would be valuable. tors which potentially can leads to the unity in the space L2(R+, dν(x)): Acknowledgements We would like to thank Katarzyna Górska for the derivation of Eqs. (25)–(27), Vladimir Belinski and John Klauder for BL = dμL (q, p)|q, pq, p|, (A8) helpful correspondence, and Claus Kiefer for discussion. Aff(R) 123 Eur. Phys. J. C (2019) 79:45 Page 15 of 16 45

∞ dx ∞ B = dμ (q, p)|q, pq, p|, (A9) = | f (x)|2 dp|(x/p)|2 R R x2 Aff(R) 0 0 ∞ dx ∞ dp = μ ( , )ρ( , )| ,  , |. = | f (x)|2 |(p)|2 BU d U q p q p q p q p (A10) x p2 Aff(R) 0 0 ∞ dp Let us check which one is invariant under the action U(q, p) = |(p)|2 (A17) p2 of the affine group: 0

+∞ if  f | f =1. † U(q , p )BU U(q , p ) = dq In the derivation of (A17) we have used the equations: −∞ ∞ ∞ dx dpρ(q, p)|q/p + q , p pq/p + q , p p| (A11) x|x =xδ(x − x ), |xx|=I, 0 0 x ∞ dx One needs to replace the variables under the integral: δ(x − x ) f (x) = f (x ). (A18) x q˜ = q/p + q and p˜ = p p, (A12) 0 ˜ = ( ˜ − ) = p . q p q q and p (A13) p Appendix B: Orthonormal basis of the carrier space ∂(q,p) = Calculating the Jacobian ∂(q˜,p˜) 1 one gets: The basis of the Hilbert space L2(R+, dν(x)) is known to be p˜ [25] dμ (q, p) = ρ(q, p)dqd˜ p˜ = ρ(p (q˜ − q ), )dqd˜ p˜ U p  (α) n! − / ( +α)/ (α) (A14) e (x) = e x 2x 1 2 L (x), (B1) n (n + α + 1) n This implies, the transformed weight should be equal to p˜ the initial one, ρ(p (q˜ − q ), ) = ρ(q, p) for every (α) α>− p where Ln is the Laguerre function and 1. One can (q , p ). The simplest solution is ρ(q, p) = const, so we ∞ (α) (α) (α) verify that en (x)em (x)dν(x) = δnm so that en (x) is μ ( , ) = 0 get d U q p dq dp. an orthonormal basis and can be used in calculations. It also implies that the operators BL and BR do not com- mute with the affine group. The action (A3) is not compatible neither with the left invariant, nor with right invariant mea- References sures on the affine group. The irreducibility of the representation, used to define 1. V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz, Oscillatory the coherent states (A7), enables making use of Schur’s approach to a singular point in the relativistic cosmology. Adv. lemma [23], which leads to the resolution of the unity in Phys. 19, 525 (1970) 2. V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz, A general solution L2(R+, dν(x)): of the Einstein equations with a time singularity. Adv. Phys. 31, 639 μ ( , )| ,  , |= I, (1982) d U q p q p q p A (A15) 3. V. Belinski, M. Henneaux, The Cosmological Singularity (Cam- (R) Aff bridge University Press, Cambridge, 2017) where the constant A can be calculated using any arbitrary, 4. S.L. Parnovsky, Gravitation fields near the naked singularities of the general type. Physica A 104, 210 (1980) normalized vector | f ∈L2(R+, dν(x)): 5. S.L. Parnovsky, A general solution of gravitational equations near their singularities. Class. Quant. Gravit. 7, 571 (1990) A = dμU (q, p)  f |q, pq, p| f . (A16) 6. S.L. Parnovsky, W.Piechocki, Classical dynamics of the Bianchi IX (R) Aff model with timelike singularity. Gen. Relat. Gravit. 49, 87 (2017) This formula can be calculated directly: 7. N.J. Cornish, J.J. Levin, The Mixmaster universe is chaotic. Phys. Rev. Lett. 78, 998 (1997)

∞ 8. N.J. Cornish, J.J. Levin, The Mixmaster universe: a Chaotic Farey

A = dμU (q, p) × dν(x ) tale. Phys. Rev. D 55, 7489 (1997) Aff(R) 0 9. V.A. Belinskii, I.M. Khalatnikov, M.P. Ryan, The oscillatory ∞ regime near the singularity in Bianchi-type IX universes. Preprint × ν( )( ( ) iqx ( / ))( −iqx( / ) ( )) d x f x e x p e x p f x 469, (1971), Landau Institute for Theoretical Physics, Moscow 0 ∞ ∞ (unpublished); published as Secs. 1 and 2 in M. P. Ryan, Ann. = dx dx Phys. 70, 301 (1971)

0 x 0 x 10. V.A. Belinski, On the cosmological singularity. Int. J. Mod. Phys. ∞  +∞  1 D 23, 1430016 (2014) dp dqeiq(x −x) π 11. R.T. Jantzen, Spatially homogeneous dynamics: a unified picture. 0 2 −∞ arXiv:gr-qc/0102035. Originally published in the Proceedings of

f (x ) f (x)(x /p)(x/p) the International School Enrico Fermi, Course LXXXVI (1982) on 123 45 Page 16 of 16 Eur. Phys. J. C (2019) 79:45

Gamov Cosmology, edited by R. Ruffini and F. Melchiorri (North 21. J.R. Klauder, Private communication Holland, Amsterdam, 1987), pp. 61–147 22. J.Q. Chen, J. Ping, F. Wang, Group Representation Theory for 12. C. Kiefer, N. Kwidzinski, W. Piechocki, On the dynamics of the Physicists (World Scientific, Singapore, 2002) general Bianchi IX spacetime near the singularity. Eur. Phys. J. C 23. A.O. Barut, R. R¸aczka, Theory of group representations and apli- 78, 691 (2018) cations (PWN, Warszawa, 1977) 13. E. Czuchry, W. Piechocki, Asymptotic Bianchi IX model: diagonal 24. A. Perelomov, Generalized Coherent States and Their Applications and general cases. arXiv:1409.2206 [gr-qc] (Springer, Berlin, 1986) 14. V. Belinski, Private communication 25. J.P. Gazeau, R. Murenzi, Covariant affine integral quantization(s). 15. E. Czuchry, W. Piechocki, Bianchi IX model: Reducing phase J. Math. Phys. 57, 052102 (2016) space. Phys. Rev. D 87, 084021 (2013) 26. H. Bergeron, J.P. Gazeau, Integral quantizations with two basic 16. P.Małkiewicz, A. Miroszewski, Internal clock formulation of quan- examples. Ann. Phys. 344, 43 (2014) tum mechanics. Phys. Rev. D 96, 046003 (2017) 27. M. Reed, B. Simon, Methods of Modern Mathematical Physics 17. P. Małkiewicz, What is Dynamics in Quantum Gravity? Class. (San Diego, Academic Press, 1980), Vols I and II Quant. Gravit. 34, 205001 (2017) 28. J.R. Klauder, Enhanced Quantization: Particles (Fields and Grav- 18. I.M. Gel fand, M.A. Naïmark, Unitary representations of the group ity, World Scientific, Singapore, 2015) of linear transformations of the straight line. Dokl. Akad. Nauk. 29. H. Bergeron, E. Czuchry, J.P.Gazeau, P.Małkiewicz, W.Piechocki, SSSR 55, 567 (1947) Smooth quantum dynamics of the mixmaster universe. Phys. Rev. 19. E.W. Aslaksen, J.R. Klauder, Unitary representations of the affine D 92, 061302 (2015) group. J. Math. Phys. 9, 206 (1968) 30. H. Bergeron, E. Czuchry, J.P.Gazeau, P.Małkiewicz, W.Piechocki, 20. E.W. Aslaksen, J.R. Klauder, Continuous representation theory Singularity avoidance in a quantum model of the Mixmaster uni- using unitary affine group. J. Math. Phys. 10, 2267 (1969) verse. Phys. Rev. D 92, 124018 (2015)

123