Callan-Giddings-Harvey-Strominger vacuum in loop quantum and singularity resolution

Alejandro Corichi,1, 2, ∗ Javier Olmedo,3, 4, † and Saeed Rastgoo5, 1, ‡ 1Centro de Ciencias Matemticás, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3, Morelia, Michoacán 58090, Mexico 2Center for Fundamental Theory, Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, PA 16802, USA 3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA 4Instituto de Física, Facultad de Ciencias, Iguá 4225, Montevideo, Uruguay 5Departamento de Física, Universidad Autónoma Metropolitana - Iztapalapa San Rafael Atlixco 186, México D.F. 09340, México (Dated: May 9, 2018) We study here a complete quantization of a Callan-Giddings-Harvey-Strominger (CGHS) vacuum model following loop techniques. Concretely, we adopt a formu- lation of the model in terms of a set of new variables that resemble the ones commonly employed in spherically symmetric . The classical theory consists of two pairs of canonical variables plus a scalar and diffeomorphism (first class) constraints. We consider a suitable redefinition of the Hamiltonian constraint such that the new constraint algebra (with structure constants) is well adapted to the Dirac quantization approach. For it, we adopt a polymeric representation for both the geometry and the field. On the one hand, we find a suitable invariant domain of the scalar constraint operator, and we construct explicitly its solution space. There, the eigenvalues of the dilaton and the metric operators cannot vanish locally, allowing us to conclude that singular geometries are ruled out in the quantum theory. On the other hand, the physical Hilbert space is constructed out of them, after group averaging the previous states with the diffeomorphism constraint. In turn, we identify the standard observable corresponding to the mass of the at the boundary, in agreement with the classical theory. We also construct an additional observable on the bulk associated with the square of the dilaton field, with no direct classical analog.

I. INTRODUCTION the old theory broke down. Owing to the fact that working with the full Since the early days of the search for a quan- theory is, so far, intractable, it has become stan- tum theory of gravity, there has always been the dard practice to work with lower dimensional expectation that one of the results that such a models or symmetry reduced ones, since gener- complete theory will yield, would be the resolu- ally this allows more control over the analysis. arXiv:1608.06246v2 [gr-qc] 29 Oct 2016 tion of the spacetime singularities. The first and One of these systems is the well known Callan- simplest reason for this argument is that singu- Giddings-Harvey-Strominger (CGHS) model [1]. larities are in a way, places where general relativ- It is a two dimensional dilatonic model that, in ity and the continuous description of spacetime spite of being simpler and classically solvable, break down. As in other instances in the history has nontrivial and interesting properties such as of physics, this is the regime where one should a black hole solution, , etc. It look for a new theory. Obviously, any of those has been proven to be a very convenient model new theories should be able to produce the previ- for testing some of the quantum gravity ideas ously known results of the old theory and also be in the past, and it has been subject to many able to describe the physics in the regime where analyses over the past 20 years [2, 3] which has shed some light on the properties of the quantum ∗ [email protected] theory of the full 4D theory. In particular, addi- † [email protected] tional studies of the classical [4] and the semiclas- ‡ [email protected] sical [5] regimes of this model, as well as several 2 studies of its quantization [6–8], have yielded a the geometry, together with a canonical trans- deeper understanding of some of the interesting formation in order to achieve a description as physical phenomena in this toy model that can similar as possible to the one of Ref. [22] in 3+1 be expected to be valid also in more realistic sit- at the kinematical level. Then, after consider- uations, like 4D black holes. However, there are ing some second class conditions and solving the still several questions that remain unanswered, Gauss constraint classically, one ends with a to- one of them being the way in which a quantum tally first class system with a Hamiltonian and a theory of gravity resolves the classical singular- diffeomorphism constraint. Furthermore, based ity. on a proposal in Refs. [24, 25], a redefinition In this article, we study the quantization of of the scalar constraint is made, such that this the CGHS model in a new perspective, namely constraint admits the standard algebra with the within the framework of loop quantum gravity diffeomorphism constraint, while having a van- (LQG) [9–11]. This programme pursues a back- ishing brackets with itself. In this situation, we ground independent non-perturbative quantiza- can follow similar arguments to those in [20] to tion of gravity. It provides a robust kinemati- achieve a complete quantization of the CGHS cal framework [12], while the dynamics has not model, showing that the quantum theory pro- been completely implemented. The application vides a description where the singularity is re- of LQG quantization techniques to simpler mod- solved in a certain way. Additionally some new els, known as loop (LQC), observable emerge in the quantum regime, which has dealt with the question of the resolution of have no classical analogue. the singularity at different levels in models sim- ilar to the one under study —see for instance Refs. [5, 13–20] among others—. In particular, we will pay special attention to Refs. [19, 20], where a complete quantization of a 3+1 vacuum spherically symmetric spacetime has been pro- vided, and the singularity of the model is re- The structure of this paper is as follows: in solved in a very specific manner. The concrete section II, we present a very brief review of the mechanisms are based on the requirement of self- CGHS model to show that it contains a black adjointness of some observables of the model, hole solution with a singularity. Section III is and on the fact that, at the early stages of the dedicated to recall how one can derive polar- quantization, there is a natural restriction to a type variables for the Hamiltonian formulation of subspace of the kinematical Hilbert space whose the CGHS and 3+1 spherically symmetric mod- states correspond to eigenstates of the triad oper- els from a generic 2D dilatonic action, and thus ators with non-vanishing eigenvalues, from which showing the underlying similarity between the the evolution is completely determined. two models in these variables. In section IV, The purpose of the present work is to put for- we illustrate a way to turn the Dirac algebra of ward a quantization of the CGHS model, by ex- the constraint in the CGHS into a Lie algebra, tending the methods of [20] to the case at hand. and thus preparing it for the Dirac quantization. A study of the dilatonic systems in the lines of Section V is about quantization: we first intro- Poisson sigma models in LQG was already car- duce the kinematical Hilbert space of the theory ried out in Ref. [21]. The feasibility of the in V A, we then represent the Hamiltonian con- project we are considering rests on a classical re- straint on this space in V B, and in subsection sult that allows to cast the CGHS model in the V C, we argue about the resolution of the singu- so-called polar-type variables [22], similar to the larity in CGHS. Then, we put forward a discus- ones used for the 3+1 spherically symmetric case. sion about the properties of the solutions to the These variables were introduced in [23, 24] and Hamiltonian constraint in section V D. Finally were further generalized in [25]. Concretely, one we note in section V E that the same observables introduces a triadic description of the model for first derived in [19] can also be introduced here. 3

II. BRIEF REVIEW OF THE CGHS to the CGHS model is writing the latter in polar- MODEL type variables [22]. This has been mainly done in [23–25]. Here, we give a brief review of this for- The CGHS model [1] is a 2D dilatonic model. mulation and its key similarities and differences It has a black hole solution, Hawking radiation, to the 3+1 spherically-symmetric gravity. and is classically solvable. This, together with Let us start by considering the most generic the fact that it is easier to handle than the full 4D 2D diffeomorphism-invariant action yielding sec- theory or many other models, makes it a power- ond order differential equations for the metric g ful test-bench for many of the ideas in quantum and a scalar (dilaton) field Φ [2] gravity. There has been an extensive previous 1 p work on this model in the literature in the clas- S = d2x −|g| G ˆ sical and the quantum/semiclassical regime. 2   2  The CGHS action is × Y (Φ)R(g) + V (∇Φ) , Φ . (3.1)

1 2 p −2φ Sg-CGHS = d x −|g|e Within this class we choose a subclass [4, 26, 27] 2G2 ˆ that is generic enough for our purposes,  ab 2 × R + 4g ∂aφ∂bφ + 4λ , (2.1) 1 2 p Sg-dil = d x −|g| where G is the 2-dimensional Newton constant, G2 ˆ 2   φ is the dilaton field and λ the cosmological con- 1 ab × Y (Φ)R(g) + g ∂aΦ∂bΦ + V (Φ) . stant. In double null coordinates x± = x0 ± x1 2 and in conformal gauge (3.2) 1 Here, Y (Φ) is the non-minimal coupling coeffi- g = − e2ρ, g = g = 0, (2.2) +− 2 −− ++ cient, V (Φ) is the potential of the dilaton field, 1 ab the solution is and 2 g ∂aΦ∂bΦ is its kinetic term. The latter can be removed at will by a conformal trans- −2ρ −2φ G2M 2 + − 1 2 e = e = − λ x x , (2.3) formation. With the choice Y (Φ) = 8 Φ and λ 1 2 2 V (Φ) = 2 Φ λ , we obtain the CGHS model [1], where M is a constant of integration which can which is given by the action be identified as the ADM (at spatial infinity) or 1 the Bondi (at null infinity) mass. The scalar cur- S = d2xp−|g| CGHS G ˆ vature turns out to be 2 1 1 1  4G Mλ × Φ2R + gab∂ Φ∂ Φ + Φ2λ2 , R = 2 , (2.4) 8 2 a b 2 G2M 2 + − λ − λ x x (3.3) which corresponds a black hole with mass M with λ the . It coincides with a singularity at with Eq. (2.1) for Φ = 2e−φ. G M In the same way, we may notice the paral- x+x− = 2 . (2.5) λ3 lelism with 3+1 spherically symmetric gravity The Kruskal diagram of the CGHS black hole is in vacuum. By using the spherical symmetry very similar to the 4D Schwarzschild model and ansatz, 2 µ ν 2 2 2 2 is depicted in figure 1. ds =gµνdx dx + Φ (dθ + sin (θ)dφ ), (3.4) with µ, ν = 0, 1, for the metric of the 4D model, III. SIMILARITY OF THE CGHS AND its action can be written as 3+1 SPHERICALLY SYMMETRIC 1 MODELS S = d2xp−|g| spher G ˆ   As we mentioned in the section I, the key 1 2 1 ab 1 × Φ R + g ∂aΦ∂bΦ + , (3.5) point of the ability to extend the results of [20] 4 2 2 4

Singularity

Future left Future right event horizon event horizon

Past right event horizon Past left event horizon

Singularity

Figure 1. The Kruskal diagram of the CGHS black hole without matter field where G is the Newton’s constant in 4D Ein- multiplied by a Lagrange multiplier XI , to the stein’s theory. One can see that this is identical Lagrangian. Here I is a Lorentz internal index 1 2 1 to (3.2) if one chooses Y (Φ) = 4 Φ , V (Φ) = 2 , representing the internal local gauge group of the and replaces G2 with G. Note that although the theory. One then makes an integration by parts actions of both CGHS and 4D models contain such that derivatives of XI appear in the La- the variable Φ, the interpretation of this variable grangian. After ADM decomposition of the ac- is different in each of these cases. In 4D spherical tion and some further calculations, it turns out gravity, Φ is actually a part of the metric, the co- that the configuration variables are efficient multiplied by the two-sphere part of the ∗ I IJ metric as can be seen from (3.4). In the CGHS, X =  XJ , ω1 ,I,J = {0, 1} (3.6) however, it is a non-geometric degree of freedom where ω is the spatial part of the spin connec- corresponding to the scalar dilaton field. 1 tion. The corresponding momenta then will be ∂L √ A. Polar-type variables for spherically PI = = 2 qnI , (3.7) ∂∗X˙ I symmetric model ∂L 1 P = = Φ2. (3.8) ω ∂ω˙ 2 As can be seen in details in [23], one can write 1 µ (3.5) in terms of the polar-type variables. Here Here nI = nµe I is the I’th (internal) component we only explain the procedure briefly. In 3+1 of the normal to the spatial hypersurface, with µ spherically symmetric case, one first removes the e I being the tetrad, and q is the determinant dilaton kinetic term by a conformal transforma- of the spatial metric. Then by a Legendre trans- tion and then writes the theory in tetrad vari- formation one can arrive at the Hamiltonian in ables. One then adds the torsion free condition, these variables. From this Hamiltonian one can 5 get to the Hamiltonian in polar-type variables by The first equation above is just a renaming, and considering the following relation the rest of them follow naturally from (3.9). By finding a generating function for this canonical 2 2 IJ kP k = −|P | = −η PI PJ = 4q. (3.9) transformation, one can find the corresponding canonical variables {Kx,Kϕ,Qη} to the above Then we adopt the parametrization momenta {Ex,Eϕ, η} and then write the Hamil- 2 tonian in these variables. The Hamiltonian will (Eϕ) q = 1 , (3.10) be the sum of three constraints as expected (Ex) 2 based on the form of the 3+1 metric in terms of 1 1  H = dx NH + N D + ω0G (3.15) the polar-type variables. Equation (3.9) leads to G ˆ the following canonical transformation to polar- type variables where N and N 1 are lapse and shift respectively,

x ω0 is the “time component” of the spin connec- Pω =E , (3.11) tion which is another Lagrange multiplier, and ϕ 2E H, D and G are Hamiltonian, diffeomorphism kP k = 1 , (3.12) (Ex) 4 and Gauss constraints respectively. In order to 2Eϕ make things simpler, Gambini et. al. in [20] P = cosh(η), (3.13) 0 1 take η = 1 and since this is second class with the (Ex) 4 Gauss constraint, they can be solved to yield the 2Eϕ P1 = 1 sinh(η). (3.14) final Hamiltonian (Ex) 4

" √ √ ! x 0 2 ϕ √ EϕK2 x x 0 ϕ 0 x x 00 1 ((E ) ) E x ϕ E (E ) (E ) E (E ) H = dx N √ − √ − 2Kϕ E Kx − √ − + G ˆ 8 ExEϕ 2 Ex 2 Ex 2(Eϕ)2 2Eϕ # 1 ϕ 0 x 0  +N E Kϕ − (E ) Kx . (3.16)

B. Polar-type variables for the CGHS tation, and the quantization of the model can be model carried out following the ideas of loop quantum gravity. The geometric implications can be read By guidance from the procedure done in the more easily and directly. In any case, this is just spherically symmetric case, one can arrive at a choice and it is not of crucial importance. similar variables for the CGHS model. Most of It turns out that, by following the same pro- the steps are in principle similar, but there are cedure of adding the torsion free condition, writ- also some important differences. The details can ing in tetrad variables and adopting an ADM be found in [24] and, again, we will describe the decomposition, and because the kinetic term process in a brief manner. First, we should men- (and hence the time derivative) of the dilaton tion that, although almost all the studies of the is present, the configuration variables will be CGHS model have utilized a conformal transfor- ∗ I mation to remove the dilaton kinetic term in an X , ω1, Φ I,J = {0, 1} (3.17) effort to render the theory as a first class system, with the corresponding momenta we will proceeded instead with that term present. The main reason was that, in this way, the vari- ∂L √ PI = = 2 qnI , (3.18) ables will admit a natural geometrical interpre- ∂∗X˙ I 6

∂L 1 2 we also mentioned in the beginning of section III Pω = = Φ (3.19) ∂ω˙ 1 4 and due to (3.19) and (3.21), one can see that √ x ∂L q  1 0  E is classically associated to the dilaton field in PΦ = = N Φ − Φ˙ , (3.20) ∂Φ˙ N the CGHS model. It has nothing to do with the metric and is a truly distinct degree of freedom. 1 where again N and N are lapse and shift re- While, as we mentioned, it is a component of the spectively. An important consequence of these metric in the 3+1 spherically-symmetric case. are that (3.19) is now a new primary constraint Continuing with the Dirac procedure, since 1 we have a new primary constraint µ, we need to µ = P − Φ2 ≈ 0. (3.21) ω 4 check its consistency under the evolution. This leads to a new secondary constraint α, namely In the next step, by making a Legendre transfor- mation, we will get to a Hamiltonian which now 1 PΦΦ µ˙ ≈ 0 ⇒ α = Kϕ + ≈ 0. (3.29) should also contain the new primary constraint 2 Eϕ (3.21). To obtain the polar-type variables we use Preservation of α then leads to no new con- a similar relation to (3.9) which, in the case of straint. It turns out that these two new con- the CGHS model, reads straints are second class together

kP k2 = 4q = 4 (Eϕ)2 (3.22) {µ, α} 6≈ 0 (3.30) where we have used again a natural parametriza- and thus we need to follow the second class Dirac tion for q in terms of Eϕ for the CGHS model. procedure for this case. So, we solve them to get Then, we get the new variables √ µ = 0 ⇒ Φ = 2 Ex, (3.31) x ϕ Pω =E , (3.23) KϕE α = 0 ⇒ PΦ = − √ . (3.32) kP k =2Eϕ, (3.24) Ex ϕ P0 =2 cosh(η)E , (3.25) This eliminates the pair {Φ,PΦ} in the Hamilto- ϕ P1 =2 sinh(η)E . (3.26) nian. In order to simplify the process of quanti- zation, we introduce the new variable Again, they follow naturally from (3.22) with a 0 bit of educated guess. These transformations do Ax = Kx − η , (3.33) not affect the pair {Φ,P }. Once again, by find- Φ and choose η = 1 which is again second class with ing a generating function for this canonical trans- the Gauss constraint. Then, solving these second formation, we can find the corresponding conju- class constraints together yields an expression for gate variables {K ,K ,Q , Φ} to the above mo- x ϕ η Q in terms of the remaining variables. In this menta {Ex,Eϕ, η, P } and then write the Hamil- η Φ way, the pair {Q , η} are also eliminated from tonian in these variables. The Hamiltonian will η the Hamiltonian. A similar procedure has also be the sum of four constraints been done in the spherically symmetric case in 1 1  [20]. Note that we now have H = dx NH + N D + ω0G + Bµ G2 ˆ (3.27) Ax = ω1. (3.34) with B being another Lagrange multiplier. Note The Dirac brackets now become that, in this case, unlike the spherically symmet- x ϕ ric case, we have {Kx(x),E (y)}D = {Kϕ(x),E (y)}D

= {f(x),Pf (y)}D = G2δ(x − y), (3.35) Kx = ω1. (3.28) K {K (x),K (y)} = G ϕ δ(x − y), (3.36) Also note that there is an important difference x ϕ D 2 Ex here between the 3+1 spherically-symmetric case Eϕ {K (x),Eϕ(y)} = −G δ(x − y), (3.37) and the CGHS model: as a consequence of what x D 2 Ex 7 with any other brackets vanishing. These brack- by introducing the redefinition ets can be brought to the canonical from ϕ E Kϕ x ϕ Ux = Kx + x . (3.39) {Ux(x),E (y)}D = {Kϕ(x),E (y)}D E = {f(x),Pf (y)}D = G2δ(x − y), (3.38) Finally, we are left with the Hamiltonian

1 H = dx NH + N 1D G2 ˆ " ϕ0 x0 x02 x00 2 ϕ ! 1 E E 1 E E 1 KϕE ϕ x 2 = dx N −KϕUx − ϕ2 − ϕ x + ϕ + x − 2E E λ G2 ˆ E 2 E E E 2 E # 1 x0 ϕ 0  +N −UxE + E Kϕ . (3.40)

IV. PREPARING THE CGHS {H(N), D(N 1)} and {D(N 1), D(M 1)} involve HAMILTONIAN FOR QUANTIZATION structure constants and close under the bracket. But this is not the case for {H(N), H(M)}. Al- At this point, and in order to proceed though, in principle, nothing prevents us carry with the Dirac quantization of the system, we on with the study in this situation, we would will adopt an Abelianization of the scalar con- like to adopt a strategy based on strong Abelian- straint algebra. The reason is the following: ization that will allow us to complete the quan- the Dirac quantization approach involves sev- tization, since other choices are either not fully eral consistency conditions. For instance, the understood or not considerably developed. This constraint algebra at the quantum level must strategy consist in a redefinition of the shift func- agree with the classical one. It is well-known tion that anomalies in the algebra can emerge, and 1 NKϕ spoil the final quantization. Usually, this situ- N = N 1 + , (4.1) Ex0 ation is more likely to be satisfied if the con- straints fulfill a Lie algebra (with structure con- followed by a redefinition the lapse function as stants instead of structure functions of phase- EϕEx space variables). An even more favorable situa- N = N . (4.2) Ex0 tions is when (part of the algebra) is strongly Abelian. We already know that the brackets These yield

1 h 1 i H = dx NH + N D G2 ˆ " x02 2 !# 1 ∂ 1 E x 2 1 Kϕ 1 x0 ϕ 0  = dx N ϕ2 x − 2E λ − x + N −UxE + E Kϕ . (4.3) G2 ˆ ∂x 2 E E 2 E

One can check that now and thus the Dirac quantization, particularly the

0 loop quantization strategy, is expected to be sim- {H(N), H(N )}D = 0, (4.4) 8 pler and potentially successful with respect to sentation on the kinematical Hilbert space. Re- other choices considered so far. garding the diffeomorphism constraint, we will We can take advantage of this form of the adopt the group averaging technique, since, as it Hamiltonian constraint and, by making an inte- is well-known in loop quantum gravity, only finite gration by parts1, write Eq. (4.3) as spatial diffeomorphisms are well-defined unitary operators on the Hilbert space. 0 1 0 If we rename N → N, the Hamiltonian con- H = dx N straint can now be written as G2 ˆ " # x02 K2 1 1 E x 2 1 ϕ H(N) = dxN × − 2E λ − + λG2M 2 Eϕ2Ex 2 Ex G2 ˆ " x02 2 # 1 1 E 1 K + N −U Ex0 + EϕK0  , (4.5) × − 2Exλ2 − ϕ + λG M .(4.6) x ϕ 2 Eϕ2Ex 2 Ex 2 where M is the ADM mass of the CGHS black Our final step, before quantization, is to bring hole and G2 is the dimensionless Newton’s con- the above constraint in a form that will admit stant in 2D spacetimes. a natural representation on a suitable Hilbert At this point we are going to first consider the space. This is achieved by rescaling the lapse Hamiltonian constraint and prepare it for repre- function N → 2NEϕ (Ex)2 such that

" x0 2 # 1 x x 2 ϕ 2 2 ϕ ϕ x (E ) H(N) = dxNE 4 (E ) E λ + KϕE − 2λG2ME E − ϕ . (4.7) G2 ˆ E

V. QUANTIZATION the set of time reparametrizations (associated with the Hamiltonian constraint). In the loop A. The kinematical Hilbert space representation, only the spatial diffeomorphisms are well understood. Then, we must look for a To quantize the theory, we first need an auxil- suitable representation of the Hamiltonian con- iary (or kinematical) vector space of states. Then straint (4.7) as a quantum operator, and look for we should equip it with an inner product and its kernel which yields a space of states that in- then carry out a Cauchy completion of this space. variant under this constraint. Finally, one should We will then end up with a kinematical Hilbert endow this space of solutions with a Hilbert space space. Afterwards, we need to find a represen- structure and suitable observables acting on it. tation of the phase space variables as operators Here, we will adhere to a loop representation acting on this Hilbert space. In order to study for the kinematical variables, except the mass, the dynamics of the system, since we are deal- for which a standard Fock quantization will be ing with a totally constrained theory, we will fol- adopted. Our full kinematical Hilbert space is low the Dirac quantization approach. Here, one the direct product of two parts, identifies those quantum structures that are in- ! variant under the gauge symmetries generated M M g by the constraints. In this particular model, we Hkin = Hkin ⊗ Hkin-spin . (5.1) have the group of spatial diffeomorphisms (gen- g erated by the diffeomorphism constraint) and M 2 One part, Hkin = L (R, dM), is associated to the global degree of freedom of the mass of the 1 In this work, we ignore the boundary term arising from black hole M. The other part, associated to this integration by parts. the gravitational sector, is the direct sum of the 9

g spaces, Hkin-spin, each corresponding to a given Let us call the kinematical Hilbert space of a sin- g M g graph (spin network) g for which we would like to gle graph Hkin = Hkin ⊗Hkin-spin (not to be con- use the polymer quantization. This choice seems g fused with Hkin-spin). There is a basis of states to be natural in 3+1 spherically-symmetric mod- in this Hilbert space denoted by {|g,~k, ~µ,Mi}. els for the geometrical variables, and due to the Then, since now we have a measure, and thus a parallelism between that model and the CGHS Hilbert space, we can define the inner product on model, we will adopt a similar representation g H and thus on Hkin. As usual in loop quan- here. kin g tum gravity, a spin network defined on g can be To construct Hkin-spin, we first take the vec- regarded as a spin network with support on a tor space Cylg, of all the functions of holonomies larger graph g¯ ⊃ g by assigning trivial labels to along the edges of a graph g, and the point the edges and vertices which are not in g. Con- holonomies “around” its vertices, and equip this sequently, for any two graphs g and g0, we take vector space with the Haar measure to get the g¯ = g ∪ g0 and the inner product of g and g0 will gravitational part of the kinematical Hilbert be space of the given graph g. In our case these states are hg,~k, ~µ,M|g0,~k0, ~µ0,M 0i = δ(M − M 0) (5.4) ! Y Y 0 i × δ 0 δ 0 = δ(M − M )δ δ 0 . Y kj ,k µj ,µ ~k,~k0 ~µ,~µ hUx,Kϕ|g,~k, ~µi = exp kj dx Ux(x) j j 2 ˆe edges vertices ej ∈g j   Y i Obviously, the inner product can be extended × exp µjKϕ(vj) . (5.2) 2 to arbitrary states by superposition of the basis vj ∈g states. Here ej are the edges of the graph, vj are its vertices, kj ∈ Z is the edge color, and µj ∈ R is the vertex color. We indicate the order (i.e. B. Representation of operators number of the vertices) of the graph g by V . Since µj ∈ R, the above belongs to the space Now that we have a kinematical Hilbert of almost-periodic functions and the associated space, the next step is to represent the phase Hilbert space will be non-separable. space variables on it as operators. We will follow g It is evident that this Hilbert space, Hkin-spin, a similar strategy as the one of Ref. [20]. First, can be decomposed into a part associated with we choose the polymerization Kϕ → sin(ρKϕ)/ρ. the normal holonomies along the edges, which is Looking at (4.7), we note that we need to repre- the space of square summable functions `2, and sent the following phase space variables another part associated to the point holonomies, which is the space of square integrable functions x x0 ϕ 1 2 ϕ E ,E ,E , ,KϕE ,M. (5.5) over Bohr-compactified real line with the associ- Eϕ ated Haar measure, L2( , dµ ). The con- RBohr Haar Due to our polymerization scheme and the clas- struction for the mass degree of freedom is sim- sical algebra (i.e. Dirac brackets), the momenta ilar and well-known, and we will not give addi- can be represented as tional details here. Thus, the full kinematical Hilbert space can be written as ϕ ~ X ~ Ec |g, k, ~µ,Mi =~G2 δ(x − xj)µj|g, k, ~µ,Mi ! vj ∈g M M g 2 Hkin = Hkin ⊗ Hkin-spin = L (R, dM) (5.6) g x ~ ~    Ec |g, k, ~µ,Mi =~G2kj|g, k, ~µ,Mi, (5.7) M O 2 2 ⊗ ` ⊗ L ( Bohr, dµHaar) .   j j R  where ~G2 is the Planck number (recalling that g v ∈g j ~ has dimensions [LM]). The presence of the (5.3) Dirac delta function in (5.6) is due to Eϕ being a 10 density. The global degree of freedom M, corre- ~µ but with a new component ±nρ, i.e. will be sponding to the Dirac observable on the bound- {. . . , µj, ±nρ, µj+1,...}, if xvj < x < xvj+1 . 2 ϕ ary associated to the mass of the black hole, can The final term to be considered is KϕE . For be represented as it, we choose the representation proposed in [29, 30], that is we define this operator as ˆ ~ ~ M|g, k, ~µ,Mi = M|g, k, ~µ,Mi. (5.8) X Θ(ˆ x)|g,~k, ~µ,Mi = δ(x − x(vj)) To represent the last contribution in (4.7), we vj ∈g x0 1 × Ωˆ 2 (v )|g,~k, ~µ,Mi, (5.11) combine E with Eϕ , and use the Thiemann’s ϕ j trick [28], to represent it as where the non-diagonal operator Ωˆ ϕ(vj) is writ- ten as [(\Ex)0]2  X |g,~k, ~µ,Mi = δ(x − x(v )) Eϕ j 1  v ∈g ˆ ϕ 1/4 ϕ ϕ ϕ  j Ωϕ(vj) = |Ec | sgn(\E ) Nd − N[ 4iρ 2ρ −2ρ sgn(µj)~G2 2 h 1/2 × (kj − kj−1) |µj + ρ|  ρ2 ϕ [ϕ  \ϕ ϕ 1/4 + Nd2ρ − N−2ρ sgn(E ) |Ec | . (5.12) 2 vj 1/2i − |µj − ρ| |g,~k, ~µ,Mi. (5.9) This shows that we need to also represent |Eϕ|1/4 ϕ ϕ and sgn(E ). This can be achieved by means of This is due to the operator Ndnρ corresponding ϕ the spectral decomposition of Ec on Hkin as to Kϕ, which is represented by the action of the point holonomies of length ρ 1/4 ϕ ~ 1/4 ~ Ec (vj)|g, k, ~µ,Mi =|µj| |g, k, ~µ,Mi, ϕ ~ ~ 0 (5.13) N[±nρ(x)|g, k, ~µ,Mi = |g, k, ~µ±nρ,Mi, n ∈ N. (5.10) ϕ  sgn \E (vj) |g,~k, ~µ,Mi = sgn(µj)|g,~k, ~µ,Mi. 0 In this expression, the new vector ~µ±nρ either (5.14) has the same components as ~µ but shifted by ±nρ, i.e. µj → µj ± nρ, if x coincides with a Combining these yields a complete representa- vertex of the graph located at x(vj), or it will be tion of our Hamiltonian constraint on Hkin as

( )  2  [(\Ex)0]2  ˆ x ˆ 2 ϕ x ˆ ϕ x H(N) = dxN(x)Ec Θ + 4λ Ec Ec − 2λG2MEc Ec − . (5.15) ˆ Eϕ

C. Hamiltonian constraint: singularity supporting this statement for a generic 2D met- resolution and solutions ric (with generic lapse and shift). A generic ADM decomposed 2D metric can 1. Relation between volume and singularity be written as

 2 12 1  −N + N q11 −N q11 Our singularity resolution argument is based gµν = 1 (5.16) −N q11 q11 on having a zero volume at some point (or re- 1 gion) classically or having a zero volume eigen- where q11 is the spatial metric and N and N are value for the quantum volume operator in quan- lapse and shift respectively. Since we have a one tum theory. In other words, a vanishing volume dimensional spatial hypersurface, then (spectrum) at a point or region means we have a singularity there. Here we give an argument q11 = det(q). (5.17) 11

Classically we have for the volume of a region R which means that a vanishing volume in a region in a spatial hypersurface Σ, corresponds to having all the µj’s equal to zero for that region (and not for the whole spatial hypersurface). If we assume that the statement p V (R) = dx det(q). (5.18) “V (R) = 0 ⇒ singularity”, can be carried on to ˆ R the quantum level, then we can say that a region (or hypersurface) described by a state with none So if at some region we have det(q) = 0, this of its µj’s being zero is a region that does not means that we will get V (R) = 0 in that region. contain any singularity. This argument which to On the other hand, if det(q) = 0, then due to our knowledge only works generically for genuine (5.16) and (5.17), we will have for that region, a 2D spacetime metrics is the one we shall use to metric argue for singularity resolution in the next sub- section.  −N 2 0  gµν = (5.19) 0 0 2. Properties of the Hamiltonian constraint and singularity resolution independent of the lapse and shift. It turns out that the Riemann invariants of the above met- Keeping the argument of the previous sub- ric (in that region) blow up and thus we have a section in mind and having obtained a represen- singularity there. So, we conclude that in 2D, tation of the Hamiltonian constraint (5.15) on a vanishing volume in a region means existence Hkin, we shall study some interesting properties of singularity in that region. However, this does of this quantum Hamiltonian constraint. These not happen for a generic genuine 4D metric. properties will facilitate the identification of the space of solutions of this constraint, and its rela- Now, for the quantum volume operator of the tion with the singularity resolution it provides. CGHS we have Let us consider any basis state |g,~k, ~µ,Mi ∈ H . It turns out that the action of this con- X kin V|ˆ g,~k, ~µ,Mi ∝ |µj||g,~k, ~µ,Mi, (5.20) straint on it yields vj ∈g

 ˆ ~ X H(N)|g, k, ~µ,Mi = N(xj)(~G2kj) vj ∈g  h ~ ~ ~ i × f0(µj, kj,M)|g, k, ~µ,Mi − f+(µj)|g, k, ~µ+4ρj ,Mi − f−(µj)|g, k, ~µ−4ρj ,Mi , (5.21) where the functions f read G f (µ ) = ~ 2 |µ |1/4|µ ± 2ρ|1/2|µ ± 4ρ|1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] [sgn(µ ± 2ρ) + sgn(µ )] , ± j 16ρ2 j j j j j j j (5.22) ! ˆ 2 3 2 G2M 2 ~G2  1/2 1/2 2 f0(µj, kj, kj−1,M) = (~G2) λ 1 − µjkj − 2 |µj + ρ| − |µj − ρ| [kj − kj−1] 2~G2kjλ ρ G n o + ~ 2 |µ |1/2|µ + 2ρ|1/2 [sgn(µ ) + sgn(µ + 2ρ)]2 + |µ |1/2|µ − 2ρ|1/2 [sgn(µ ) + sgn(µ − 2ρ)]2 16ρ2 j j j j j j j j (5.23)

Looking at (5.21) and the form of (5.22) and (5.23), we notice some important points: 12

1. The scalar constraint admits a natural de- kj = 0. As a result this restriction, we expect composition on each vertex vj, such that it that also in the physical Hilbert space, we will can be regarded as a sequence of quantum never have any state with a singularity. operators acting almost independently on them, up to the factors ∆kj = kj − kj−1. D. Solutions to the Hamiltonian constraint In other words, there would not be cou- and the physical Hilbert space pling among different vertices if it were not for the factor ∆kj. Let us consider a generic solution, hΨg|, to 2. The number of vertices on a given graph g the Hamiltonian constraint, i.e., a generic state is preserved under the action of the Hamil- annihilated by this constraint. Assuming hΨg| tonian constraint. belongs to the algebraic dual of the dense sub- space Cyl on the kinematical Hilbert space, and 3. The constraint (5.21) leaves the sequence that it can be written as of integers {k } of each graph g invari- j ∞ X X ant. For instance, if we consider a ket hΨg| = dM hg,~k, ~µ,M|ψ(M)χ(~k) ~ ˆ0 |g, k, ~µ,Mi, the successive action of the ~k ~µ scalar constraint on it generates a sub- × φ(~k, ~µ,M), (5.24) space characterized by the original quan- tum numbers ~k. then the annihilation by the Hamiltonian con- straint dictates that 4. The restriction of the constraint to any ˆ † X ˆ † vertex vj acts as a difference operator mix- hΨg|H(N) = hΨg|NjHj = 0, (5.25) ing the real numbers µj. In this case, vj ∈g this difference operator only relates those where Hˆ j are difference operators acting on each states which have µj’s that belong to a semi-lattices of step 4ρ due to the form vertex vj and Nj = N(xj) is the lapse function of f (µ ), that vanishes in the intervals evaluated on the corresponding vertex. In this ± j ~ [0, ∓2ρ]. case the functions φ(k, ~µ,M) admits a natural decomposition of the form 5. Starting from a state for which none of V µj’s are zero (i.e. a state containing no Y φ(~k, ~µ,M) = φj(kj, kj−1, µj,M). (5.26) singularity), the result of the action of the j=1 constraint never leaves us in a state with any of µj’s being zero (also look at the One can then easily see that the solutions must details in section V D). fulfill, at each vj, a difference equation of the form The point number 5, which maybe is the most important of these, states that the subspace of − f+(µj − 4ρ)φj(kj, kj−1, µj − 4ρ, M)

Hkin containing spin networks for which no µj is − f−(µj + 4ρ)φj(kj, kj−1, µj + 4ρ, M) zero is preserved under the action of the Hamil- + f0(kj, kj−1, µj,M)φj(kj, kj−1, µj,M) = 0, tonian constraint. Simply put, if one originally (5.27) starts with a state with no singularity (in the sense of µj = 0), then one will never end up which is a set of difference equations to be solved in a state containing a singularity. Analogous together. We will provide a partial resolution of arguments could be applied to the kj quantum the problem by means of analytical considera- numbers, however as mentioned above, kj are al- tions. All the details can be found in appendix ready preserved by the constraint (unlike the µj A. Let us consider a particular vertex vj. In the valences of the vertices). following we will omit any reference to the label Thus, one can restrict the study only to the of the vertex. Due to the property 4, where µ be- subspace of Hkin for which there is no µj = 0 and longs to the semi-lattices of the form µ = ±4ρn 13 where n ∈ N and  ∈ (0, 4ρ], different orienta- • On the other hand, for k > M/2~λ, the tions of µ are decoupled. Without loss of gen- constraint is simply erality, we will restrict the study to a particular subspace labeled by , unless otherwise specified. 2 ωn(M, k, ) − ∆k = 0, (5.30) This shows that the Hamiltonian constraint only relates states belonging to separable subspaces where, again, ω is the positive eigenvalue of the original kinematical Hilbert space. n of the difference operator defined in (A12), These properties of the solutions together but this time it belongs to its discrete spec- with their asymptotic limit µ → ∞, assuming trum and is also non-degenerated. The the solutions are smooth there, will allow us corresponding eigenstates |φdsci, with n ∈ to understand several aspects of the geometrical n , emerges out of µ ' , grow exponen- operators (under some assumptions about their N tially until reach an stable regime, and spectral decomposition). More concretely, the at some µ ' µ the eigenfunction enters solutions for µ → ∞ satisfy, up to a global fac- r in a classically forbidden region and de- tor [( G )2k], the differential equation ~ 2 cays exponentially (see [31] for a related 4∆k2 − 1 treatment). Besides, the eigenfunctions − 4µ∂2φ − 4∂ φ − φ µ µ 4µ ωn form a discrete sequence of real num-   bers, all of them depending continuously G2M + 1 − ( G λ)2k2µφ = 0, (5.28) on the parameter . This dependence is 2 G λk ~ 2 ~ 2 crucial in order to have a consistent con- in a very good approximation if they are smooth straint solution, since the sequence of dis- functions of µ. The last term plays the role of the crete ∆k2 is not expected to coincide with square of a frequency of an harmonic oscillator. the sequence of ωn for a global fixed . But the sign of this term depends on the concrete Therefore, we expect that the parameter quantum numbers. Therefore, this equation ad-  must be conveniently modified according mits both oscillatory solutions and exponentially to the values of M, k and the constraint growing or decreasing ones. More concretely, this equation (5.30). differential equation is a modified Bessel equa- tion if the sign of its last coefficient is positive, These previous results have not been con- i.e. k < M/2~λ, and a Bessel equation when- firmed numerically (as well as those of [20]), ever that coefficient is negative, i.e. k > M/2~λ. though they will be a matter of future research. In Appendix A we include the details about the Let us comment, however, that they are based properties of the solutions in these two different on very robust, previous results on different sce- regimes. Let us summarize the results obtained narios already studied in the LQC literature (see there: [30–34]). Therefore, unless a very subtle point comes into play, the mentioned properties are ex- • For k < M/2~λ, the Hamiltonian con- straint takes the form pected to be fulfilled. In a final step, one should build the physi- ˆ ! G2M 2 2 cal Hilbert space. The states belonging to this ω + 1 − (~G2λ) k = 0. (5.29) 2~G2λk space are the ones that admit the symmetries of the model, i.e. the states which are invari- where ω is the positive eigenvalue of the ant under both the Hamiltonian and diffeomor- difference operator of (A28) that belongs phism constraints. As usual in LQG, one applies to its continuous spectrum and which is the group averaging technique to get these states non-degenerate. The corresponding eigen- and the induced inner product on the resultant cnt function |φω i behaves as an exact stand- subspace that is provided by this process. One ing wave in µ of frequency σ(ω) in the limit can start with the Hamiltonian constraint. Then µ → ∞. the states averaged by members of the group as- 14 sociated to the Hamiltonian constraint are of the Lie algebra g. In the case of the alge- bra member, being the Hamiltonian constraint ∞ V ∞ ˆ P ˆ H Y X X ~ H(Nj) = v NjHj, we have hΨg | = dgn dM hg, k, ~µ,M| j ˆ−∞ ˆ0 n=1 ~k ~µ × ψ(M)χ(~k)φ(~k, ~µ,M), (5.31) X ˆ g = Hˆ(Nl) = NlHl. (5.33) where vl

g = eig (5.32) Thus in this case, from (5.31) we get for the is the group member associated to the member group averaged state

∞ ∞ ∞ 1 ˆ ˆ H iN1H1 iNV HV X X ~ ~ hΨg | = V de ... de dM hg, k, ~µ,M|ψ(k, ~µ,M) (2π) ˆ−∞ ˆ−∞ ˆ0 ~k ~µ ∞ ∞ " V # ∞ 1 X ˆ X X ~ ~ = V dN1 ... dNV exp i NnHn dM hg, k, ~µ,M|ψ(k, ~µ,M) (2π) ˆ−∞ ˆ−∞ ˆ0 n=1 ~k ~µ (5.34)

The final states are endowed with a suitable inner otherwise. The final step is to construct the solu- product defined as tions to the Hamiltonian constraint which are in- variant under the spatial diffeomorphisms (gen- H 2 H kΨg k = hΨg |Ψgi, (5.35) erated by the diffeomorphism constraint). In this where the ket belongs to the kinematical Hilbert case we follow the ideas of the full theory [35]. space and the bra is the corresponding state after There, one constructs a rigging map from the being averaged with the Hamiltonian constraint. original Hilbert space to the space of diffeomor- In order to obtain explicitly the inner product, phism invariant states by averaging the initial states with respect to the group of finite diffeo- we may write |Ψgi in the basis of states of the geometrical operators involving the scalar con- morphisms. The resulting averaged states are a straint (see App. A). In this case superposition of the original states but with their vertices in all possible positions in the original 1- ∞ X dimensional manifold, but preserving the order hΨH|Ψ i = dM dω . . . dω g g ˆ ˆ 1 V of the edges and vertices. So, a physical state 0 ~ k will be V   Y ~ 2 phys X H × δ ωj − F (kj,M) |ψ(k, ~ω, M)| , (5.36) hΨ | = hΨg | (5.38) j=1 g∈[g] where F (kj,M), at each vertex vj, is given by the and the inner product is then last addend in the left hand side of Eqs. (A21) phys 2 phys kΨ k = hΨ |Ψgi, (5.39) or (A36) depending if (kj − M/2~λ) is positive or negative, respectively, i.e. where, again, the ket belongs to the kinematical Hilbert space and the bra is the physical solution. F (k ,M) =(∆k )2 if k > M/2 λ, (5.37) j j j ~ In the last product, only a finite number of finite   G2M 2 2 terms contribute, for all |Ψ i in the kinemati- F (kj,M) = 1 − (~G2λ) kj g 2~G2λkj cal Hilbert space, so the inner product is finite 15 and well defined. Let us mention that the dif- remaining ones. In [19] it was suggested that feomorphism invariance of the inner product is the reality conditions of some observables of the guaranteed since if we compute Eq. (5.39) with model provides a quantum theory free of singu- any other state related to |Ψgi by a spatial diffeo- larities. However, due to some important dif- morphism, it would yield exactly the same result. ferences of that model with the present one, we For a recent discussion see [36]. have not been able to identify such suitable ob- At the end of this process, we are left with servables in our model along those lines. a vector space of states that are invariant under both constraints, and an inner product on this E. Quantum observables space, induced by the group averaging processes, rendering this vector space a Hilbert space. The We saw in section V D that the Hamiltonian resultant Hilbert space of diffeomorphism invari- constraint does not create any new vertices in ant states are the equivalence classes of diffeo- the graph g on which it acts (and obviously nei- morphism invariant graphs [g] solutions to the ther does the diffeomorphism constraint). This scalar constraint. means that there is a Dirac observable Nˆ in the Let us conclude with some remarks. In the v bulk corresponding to the fixed number of ver- classical theory, the geometry possesses a singu- tices N = V of a graph g, larity whenever the determinant of the metric v q vanishes at some point. In this manuscript, NˆvΨphys = NvΨphys. (5.40) the vanishing of q corresponds to the vanishing of Eϕ at a given region (local singularity). In This observable is strictly quantum and has no the quantum theory we can find an analogous counterpart in the classical theory. situation, for instance if a graph g has µj = 0 On the other hand, since this model has only at one or some given vertices. Fortunately, the one spatial direction, under the action of the dif- quantum theory allows us to avoid these unde- feomorphism constraint the points can not pass sired divergences. The key idea consists in iden- each other, i.e., the order of the positions of the tifying a suitable invariant domain of the scalar vertices is preserved. This means that, associ- constraint, free of such states with nonvanishing ated to this preservation, we can identify an- µj. In this way, the solutions to the constraints other new strictly quantum observable in the will have support only on them, preventing the bulk, Oˆ(z) such that vanishing of µ (and k ) at any vertex. It is j j ˆ straightforward to prove, as a direct consequence O(z)Ψphys = kInt(zNv)Ψphys, z ∈ [0, 1] (5.41) of the previous points 3 and 4, that the subspace where Int(zN ) is the integer part of zN . To- formed by kets such that their sequences {µ } v v j gether with them, we also have the observable (and {k }) contains no vanishing components, j corresponding to the mass Mˆ , which does have remains invariant under the action of the Hamil- an analogous classical Dirac observable. tonian constraint (5.21). In particular, point 4 Besides, as it was first observed by the au- tells us that we can never reach a vanishing µ j thors of Ref. [19, 20], one can construct an evolv- by successive action of the scalar constraint, and ing constant associated to Ex from the above ob- point 3 tells us that any sequence of {k } will j servable as remain invariant. In conclusion, the restriction to this invariant domain allows us to resolve the x ˆ Ec (x)Ψphys = ~G2O(z(x))Ψphys, (5.42) classical singularity. x However, given that the sequences {kj} are with z(x) : [0, x] → [0, 1]. Since E has classical unaltered by the scalar constraint, and since they and quantum mechanically a different interpre- apparently have no significance in singularity res- tation in the CGHS model than in 3+1 spherical olution of this model, we do not see any funda- symmetry, i.e., in the former it is related to the mental argument for discriminating those with dilaton field, one should also take caution about vanishing {kj} components with respect to the its interpretation. 16

These two observables were first introduced dial direction in which the quantum theory im- for the 3+1 spherically symmetric case in [19] plements the spatial diffeomorphism symmetry and due to the similarities of the two models, we as in loop quantum gravity. can see that they exist also for the CGHS model. It is worth commenting that one can promote Particularly, the observable in (5.41) arises due the metric component Eˆϕ as a parameterized ob- to the existence of only one (radial) direction in servable. For it, we can choose the phase space both cases. So one can expect that such a quan- variable Kϕ as an internal time function (of para- tum observable will exist in many genuinely 2D metric function). Moreover, by means of the and symmetry-reduced models with only one ra- Hamiltonian constraint (on shell), it is possible to define the parameterized observable

x ∂xEc (x) Eˆϕ(x)Ψ = Ψ , (5.43) phys q 2 phys x 2 2 sin (ρKϕ) ˆ x 4[Ec (x)] λ + ρ2 − 2λG2MEc (x)

which is defined in terms of the parameter func- ence of matter (more precisely, massless scalar tions z(x) and Kϕ(x), and the observables Mˆ field), one can get a Lie algebra of constraints and Oˆ (through the definition of Ecx(x)). by strong Abelianization of the {H(N), H(M)} part of the classical constraint algebra, it is not clear whether the quantum theory is anomaly- VI. SUMMARY AND CONCLUSION free and also if one can get some useful informa- tion about the Hamiltonian constraint, as was We have shown that, with the introduction possible in the present case without matter. Fur- of polar-type variables for a CGHS dilatonic thermore, the representation of this constraint on black hole and a rewrite of its Hamiltonian the Hilbert space is expected to be much more in terms of those variables, one can follow re- involved. For a minimally-coupled scalar field (in cent LQG inspired methods, first introduced in the classical theory its dynamics reduces to the 3+1 spherically-symmetric case –also written in one of a scalar field in Minkowski) one can expect polar-type variables–, to remove the singularity a more treatable model with respect to its ana- of the CGHS model. The proposal is based on logue in 3+1 spherically-symmetric spacetimes, the assumption (proven here for the case of a 2D regarding its solubility. Nevertheless, this is an generic metric) that states with zero volume are interesting future project worth pursuing, as is those containing a spacetime singularity. Then, the study the Hawking radiation based on these singularity resolution follows if one can show that results. if one starts from a state without a zero volume In any case, the analysis here presented must present in it, one can restrict the evolution to a be viewed as a first step that requires further subspace of the Hilbert space that contains no understanding, analysis and level of precision. It zero volume states. In other words, the subspace can hopefully be further extended to give more of quantum spacetime states without a singular- insights on generic black hole singularity reso- ity is preserved under the action of the quantum lution and, more generally, on quantum gravity Hamiltonian constraint. itself. This analysis may be extended further when a matter field is present in the theory and one might then study the backreaction, but that ACKNOWLEDGMENTS analysis will certainly be more involved and is outside the scope of this paper. Although it has We would like to thank J.D. Reyes for dis- been shown recently [25] that even in the pres- cussions and comments. A.C. was in part sup- 17

ported by DGAPA-UNAM IN103610 grant, by one arbitrary vertex vj (we will omit the label CONACyT 0177840 and 0232902 grants, by j in the following). Let us recall that the local the PASPA-DGAPA program, by NSF PHY- scalar constraint of this model, once promoted 1505411 and PHY-1403943 grants, and by the to a quantum operator, acts (almost) indepen- Eberly Research Funds of Penn State. S.R. dently on each vertex. Its action on the corre- would like to acknowledge the support from the sponding states is Programa de Becas Posdoctorales, Centro de Ciencias Matematicas, Campos Morelia, UNAM and DGAPA, partial support of CONACyT grant number 237351: Implicaciones Físicas de la Estructura del Espaciotiempo, the support of the the PROMEP postdoctoral fellowship (through h Hˆ|g, k, µ, Mi =( G k) f (µ, k, M)|g, k, µ, Mi UAM-I) and the grant from Sistema Nacional de ~ 2 0 Investigadores of CONACyT. J.O. acknowledges − f+(µ)|g, k, µ + 4ρ, Mi funds by Pedeciba, grant FIS2014-54800-C2-2-P i − f (µ)|g, k, µ − 4ρ, Mi , (Spain) and grant NSF-PHY-1305000 (USA). − (A1)

Appendix A: Spectrum of geometrical operators

In this appendix we will discuss some prop- erties of the Hamiltonian constraint restricted to

with the functions G f (µ) = ~ 2 |µ|1/4|µ ± 2ρ|1/2|µ ± 4ρ|1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] [sgn(µ ± 2ρ) + sgn(µ)] , ± 16ρ2 (A2)   2 G2M 3 2 ~G2 n 1/2 1/2 2 f0(µ, k, M) = λ 1 − (~G2) µk + 2 |µ| |µ + 2ρ| [sgn(µ) + sgn(µ + 2ρ)] 2~G2kλ 16ρ o G  2 +|µ|1/2|µ − 2ρ|1/2 [sgn(µ) + sgn(µ − 2ρ)]2 − ~ 2 |µ + ρ|1/2 − |µ − ρ|1/2 ∆k2. (A3) ρ2

Here, ∆k is proportional to the eigenvalue of the on [0, ∓2ρ] respectively. Thus different orienta- operator (E\x(x))0. This operator will be diago- tions of µ are decoupled by the difference oper- nal on the spin network basis of states as well as ator (A1). We conclude that µ belongs to semi- its explicit form will depend on the definition of lattices of the form µ =  ± 4ρn where n ∈ N the operator Eˆx. and  ∈ (0, 4ρ]. Without loss of generality, we The action of the scalar constraint resem- will restrict the study to a particular subspace bles the one of a second order difference oper- labeled by , unless otherwise specified. This ator since it relates three consecutive points in shows that the Hamiltonian constraint only re- a lattice with constant step. The consequence is lates states belonging to separable subspaces of that any function φ(k, µ, M) that is solution to the original kinematical Hilbert space. the equation (φ|Hˆ † = 0 has support on lattices The solutions φ(k, µ, M) fulfill the equation of step 4ρ, as we can deduce by direct inspec- tion of Eq. (A1). Moreover, due to the functions − f+(µ − 4ρ)φ(k, µ − 4ρ, M) [sgn(µ ± 2ρ) + sgn(µ)] in (A2), f±(µ) vanishes − f−(µ + 4ρ)φ(k, µ + 4ρ, M) 18

+ f0(k, µ, M)φ(k, µ, M) = 0. (A4) formation in the functional space of solutions in order to achieve a suitable separable form of the One can straightforwardly realize that, for any constraint equation. In particular, following the choice of the initial triad section µ = , they ideas of Ref. [30], we will introduce a bijection are completely determined by their initial data on the space of solutions defined by the scaling φ(k, µ = , M). In particular, our difference op- of the solutions erator evaluated at µ =  relates the solution coefficient φ(k, µ =  + 4ρ, M) with only the ini- dcr 1/2ˆ φ (k, µ, M) = (~G2) b(µ)φ(k, µ, M), (A5) tial data φ(k, µ = , M), which can be solved easily. Therefore, the difference equation evalu- with ated at the next successive lattice points can also be solved straightforwardly, once the initial data 1 ˆb(µ) = (|µˆ + ρ|1/2 − |µˆ − ρ|1/2). (A6) φ(k, µ = , M) is provided. Without loss of gen- ρ erality, we will fix it to be real. This allows us to conclude that, since the coefficients of the corre- We might notice that the functions ˆb(µ) only sponding difference equation (A4) are also real vanish for µ = 0. But this sector has been decou- functions, the solutions φ(k, µ, M) at any triad pled, since µ belong to semi-lattices with a global section µ =  ± 4ρn will be also real functions. minimum at µ =  > 0. Therefore, the function Besides, the solutions to Eq. (A4), for con- ˆb(µ) never vanishes and the previous scaling is stant values of the quantum numbers k, M, invertible. The new functions φdcr(k, µ, M) now and the cosmological constant λ, have different fulfill the difference equation asymptotic limits at µ → ∞. Concretely, if they fulfill k < M/2 λ or k > M/2 λ, the physically dcr dcr ~ ~ −f+ (µ − 4ρ)φ (k, µ − 4ρ, M) relevant solutions either oscillate or decay expo- −f dcr(µ + 4ρ)φdcr(k, µ + 4ρ, M) nentially, respectively, in that limit. − dcr dcr We will focus now in the study of the solu- +f0 (k, µ, M)φ (k, µ, M) = 0. (A7) tions in the cases in which k > M/2~λ. In this regime, it is more convenient to carry out a trans- where the new coefficients are now

1 f dcr(µ) = |µ|1/4|µ ± 2ρ|1/2|µ ± 4ρ|1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] ± 16ρ2b(µ)b(µ ± 4ρ) × [sgn(µ ± 2ρ) + sgn(µ)] , (A8)   dcr µ G2M 2 2 1 h 1/2 2 f0 (µ, k, M) = 2 1 − (~G2λ) k + 2 2 (|µ||µ + 2ρ|) [sgn(µ) + sgn(µ + 2ρ)] b(µ) 2~G2λk 16ρ b(µ) i +(|µ||µ − 2ρ|)1/2 [sgn(µ) + sgn(µ − 2ρ)]2 − ∆k2, (A9)

This difference operator can be naively inter- by Hˆ dcr in the new one. Both are related by preted as a densitized scalar constraint, for in- ˆ dcr ˆ −1 ˆˆ −1 stance, like the one emerging after choosing the H = b(µ) Hb(µ) . (A10) ˆ −2 lapse function Nb(µ) (together a suitable fac- Now, we will study the difference operator tor ordering and a global factor ~G2). Let us denote this scalar constraint in the original scal- hˆdcr = Hˆ dcr + ∆k2. (A11) ing by Hˆ, and the corresponding scalar constraint We can deduce several properties about the spec- trum of this difference operator as well as of its 19 eigenfunctions. Let us consider, for consistency, corresponds to a difference equation similar to its positive spectrum. The eigenvalue problem equation (A7) but with functions

ˆdcr dcr dcr h |φω i = ω|φω i. (A12)

1 f˜dcr(µ) = |µ|1/4|µ ± 2ρ|1/2|µ ± 4ρ|1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] ± 16ρ2b(µ)b(µ ± 4ρ) × [sgn(µ ± 2ρ) + sgn(µ)] , (A13)   ˜dcr µ G2M 2 2 1 h 1/2 2 f0 (µ, k, M, ω) = 2 1 − (~G2λ) k + 2 2 (|µ||µ + 2ρ|) [sgn(µ) + sgn(µ + 2ρ)] b(µ) 2~G2λk 16ρ b(µ) i +(|µ||µ − 2ρ|)1/2 [sgn(µ) + sgn(µ − 2ρ)]2 − ω. (A14)

Let us assume that the solutions to this dif- 4ρ (in the previous asymptotic limit). This must ference equation has a well defined and smooth be tested carefully, but we will not deal with this limit µ → ∞. For practical purposes this limit question by now. We assume its validity, at least is similar to the limit ρ → 0, but keeping in for eigenvalues with typical scales much bigger mind that while the former is expected to be than 4ρ. well defined in our quantum theory, the latter Within this asymptotic regime and approxi- is not. This assumption involves that the solu- mation, the solutions to the previous difference dcr tions φω (µ) must be continuous functions of µ. equation (A12) satisfy in a very good approxi- But this is not true for scales ∆µ of the order of mation the differential equation

˜dcr dcr ˜dcr dcr ˜dcr dcr 0 = − f+ (µ − 4ρ)φω (k, µ − 4ρ, M) − f− (µ + 4ρ)φω (k, µ + 4ρ, M) + f0 (k, µ, M, ω)φω (k, µ, M)    2 2 dcr dcr G2M 2 2 2 2 dcr = − 4µ ∂µφω (k, µ, M) − 8µ∂µφω (k, µ, M) + 1 − (~G2λ) k µ − γ φω (k, µ, M) 2~G2λk + O(ρ2/µ2), (A15)

with γ2 = ω + 3/4. Let us recall that this dif- with ferential equation can be analogously achieved if s  instead of adopting a loop quantization, one ad- ~G2λk G2M heres to a WDW representation for this setting, x = µ 1 − . (A17) 2 2~G2λk with a suitable factor ordering. It corresponds to modified Bessel equation, where its solutions In the limit µ → ∞, the solutions I and K grow are combinations of modified Bessel functions of and decay exponentially, respectively. Therefore, the form the latter is the only contribution to the spectral decomposition of hˆdcr. In consequence, its possi- ble (positive) eigenvalues ω are non-degenerate. dcr −1/2 lim φ (k, µ, M) =Ax Kiγ (x) Besides, the functions Kiγ(x) are normalized to µ→∞ ω −1/2 + Bx Iiγ (x) , (A16) 0 hKiγ|Kiγ0 i = δ(γ − γ ), (A18) 20

2 −1 in L (R, x dx), since the normalization in this for any µ will be also real. case is ruled by the behavior of Kiγ(x) in the Eventually, the corresponding eigenfunctions, limit x → 0, which corresponds to as functions of µ, will be square summable, ful- filling the normalization condition lim Kiγ(x) → A cos (γ ln |x|) . (A19) x→0 X hφdcr|φdcr i = φdcr(k,  + 4nρ, M) ωn ωn0 ωn For additional details see, for instance, Ref. [37]. n dcr This result is fulfilled in the continuous the- 0 ×φω 0 (k,  + 4nρ, M) = δnn , (A20) ory, whenever (A15) is valid globally. But let n us recall that we are dealing with a difference recalling that these coefficients are real. It is equation possessing, in a good approximation, worth commenting that due to the scaling (A5), a continuous µ → ∞ limit, but not at all for the coefficients in the previous sum are weighted µ → 0. Therefore, the previous normalization simply with the unit. This is not the case, for in- (A18) and the asymptotic limit (A19) have no stance, in Ref. [31] where the norm of the corre- meaning in our discrete theory. In this case, and sponding eigenfunctions includes a weight func- in the absence of a meticulous numerical study of tion different from the unit, since no scalings of the solutions of this equation, we can only infer the solutions are considered. dcr some properties about φω (k, µ, M). One can The constraint in this basis takes the alge- convince oneself that our difference equation is braic form similar to the one studied in [31] for a closed FRW spacetime. In particular, the eigenfunc- ω(n) − ∆k2 = 0. (A21) tions of such a difference operator have a simi- lar asymptotic behavior for v → ∞ (or equiva- Let us now study the solutions to the con- lently µ → ∞ in our model). Nevertheless, the straint when k < M/2~λ. In this case we will spectrum of the corresponding difference opera- follow again the ideas of [30] in order to render tor turns out to be discrete (instead of continuous our equation in a suitable representation where like the corresponding differential operator) ow- it will again become separable. Let us comment ing to the behavior of its eigenfunctions at v '  that the solutions to the difference equation for (i.e. µ ' ). Therefore, we expect, following k < M/2~λ have different asymptotic behaviors the results of Ref. [31], that the eigenvalues ω of at µ → ∞ than the ones for k > M/2~λ. It in- the difference operator hˆdcr belong to a countable volves that the change of representation that will set, which we will call {ω(n)}. One cloud also ex- be considered in each case, in order to express the pect that the possible (positive) values of ω(n) constraint equation in a simple separable form, will depend on  ∈ (0, 4ρ], and for a given , they must not be the same. will also depend on k and M. Let us comment With all this in mind, let us consider this in- that the particular values of the sequence {ω(n)} vertible scaling as well as the explicit form of the eigenfunctions dcr cnt 1/2 φω (k, µ, M), to the knowledge of the authors, φj (k, µ, M) = (~G2µˆ) φ(k, µ, M). (A22) can only be determined numerically by now, un- less new analytical tools are developed. In ad- As before, µ = 0 could be problematic in order dition, a second look on the difference equation to define this redefinition properly. But let us re- (A12) tell us that the eigenfunctions are com- call that this sector has been decoupled from the pletely determined by their value at the initial quantum theory. In consequence µ has a global dcr data section φω (k, µ = , M). Therefore, the minimum equal to  > 0. Therefore, the previ- spectrum of hˆdcr will be non-degenerated. More- ous scaling (A22) can be inverted and the orig- over, let us recall that, if we choose the initial inal description recovered. The new functions dcr cnt data to be real, all the coefficients φω (k, µ, M) φ (k, µ, M) now fulfill the difference equation 21

cnt cnt cnt cnt cnt cnt cnt −f+ (µ−4ρ)φ (k, µ−4ρ, M)−f− (µ+4ρ)φ (k, µ+4ρ, M) +f0 (k, µ, M)φ (k, µ, M) = 0. (A23) but this time the coefficients are 1 f cnt(µ) = |µ|−1/4|µ ± 2ρ|1/2|µ ± 4ρ|−1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] [sgn(µ ± 2ρ) + sgn(µ)] , ± 16ρ2 (A24) 1 h f cnt(µ, k, M) = (|µ||µ + 2ρ|)1/2 [sgn(µ) + sgn(µ + 2ρ)]2 0 16ρ2µ i sgn(µ) +(|µ||µ − 2ρ|)1/2 [sgn(µ) + sgn(µ − 2ρ)]2 − ∆k2(|µ + ρ|1/2 − |µ − ρ|1/2)2, µρ2 (A25)

This version of the scalar constraint, as we men- The difference operator that will be studied now tioned previously, can be naively understood as reads a densitized version of the original classical con-   −1 ˆcnt ˆ cnt G2M 2 2 straint after the choice of Nµ as the new lapse h = H − 1 − (~G2λ) k . (A27) function (and an adequate factor ordering and a 2~G2λk global factor ~G2). Following the notation that Therefore, we have written again the original we introduced, we will denote this new scalar constraint Hˆ into a suitable separable form ac- constraint by Hˆ cnt. It is related with the original cording to the condition k < M/2~λ. one by means of We will now study the spectrum of the differ- ence operator hˆcnt, by means of eigenvalue prob- lem

ˆcnt cnt cnt h |φω i = ω|φω i, (A28)

for ω ≥ 0, which are the physically interesting values. This equation can be written in the form ˆ cnt −1/2 ˆ −1/2 H = (~G2µˆ) H(~G2µˆ) . (A26) of (A23), but with coefficients

1 f˜cnt(µ) = |µ|−1/4|µ ± 2ρ|1/2|µ ± 4ρ|−1/4 [sgn(µ ± 4ρ) + sgn(µ ± 2ρ)] ± 16ρ2 × [sgn(µ ± 2ρ) + sgn(µ)] , (A29) 1 h f˜cnt(µ, k, M, ω) = (|µ||µ + 2ρ|)1/2 [sgn(µ) + sgn(µ + 2ρ)]2 0 16ρ2µ i sgn(µ) +(|µ||µ − 2ρ|)1/2 [sgn(µ) + sgn(µ − 2ρ)]2 − ∆k2(|µ + ρ|1/2 − |µ − ρ|1/2)2 − ω, (A30) |µ|ρ2

Let us recall, again, that the coefficients be non-degenerated. Moreover, the coefficients cnt cnt cnt φω (k, µ, M) of these eigenstates are deter- φω (k, µ, M) will be real if φω (k, µ = , M) ∈ cnt ˜cnt ˜cnt mined by their initial data φω (k, µ = , M) R, since the previous functions f0 and f± are through the difference equation (A28). In con- also real. sequence, the (positive) spectrum of hˆcnt will 22

We will assume, again, that these solutions 4ρ. This continuity condition allows us to ap- have a well defined and smooth asymptotic be- proximate the difference equation for those large havior for µ → ∞. Let us recall that this involves scale eigenvalues at µ → ∞ by eigenvalues with typical scales much bigger than

˜cnt cnt ˜cnt cnt ˜cnt cnt 0 = − f+ (µ − 4ρ)φω (k, µ − 4ρ, M) − f− (µ + 4ρ)φω (k, µ + 4ρ, M) + f0 (k, µ, M, ω)φω (k, µ, M) γ2 = − 4∂2φcnt(k, µ, M) − φcnt(k, µ, M) − ωφcnt(k, µ, M) + O(ρ2/µ2), (A31) µ ω µ2 ω ω

but this time with γ2 = ∆k2 + 3/4. Let us com- µ, allow us to conclude that ment that this very same differential equation √  would have been obtained if we would have con- cnt ω lim φω (k, µ, M) = A cos µ + β , (A34) sidered a WDW representation, instead of the µ→∞ 2 loop quantization, with a suitable choice in the with A a normalization constant and β a phase ordering of the operators for the definition the that it is expected to depend on ∆k, and . This corresponding Hamiltonian constraint. asymptotic behavior is radically different in com- dcr The solutions to this differential equation are parison with the eigenstates φω (k, µ, M). In- linear combinations of Hankel functions of first stead of decaying exponentially, they simply os- H(1)(y) and second H(2)(y) kind, multiplied by a cillate as standing waves (up to negligible correc- iγ iγ √ √ factor y1/2, where y = µ ω/2. In consequence, tions) of frequency ω/2. Therefore, our experi- the asymptotic limit of the eigenstates will be ence in loop quantum cosmology [30, 32, 33] tell us that these eigenfunctions will be normalizable functions of µ (in the generalized sense) (1) lim φcnt(k, µ, M) =Ay1/2H (y) X µ→∞ ω iγ in in cnt hφω |φω0 i = φω (k,  + 4nρ, M) 1/2 (2) n + By Hiγ (y). (A32) √ √ cnt  0  × φω0 (k,  + 4nρ, M) = δ ω/2 − ω /2 . (A35) These functions have a well known asymptotic limit at y → ∞, corresponding to Eventually, the constraint in the basis of in states |φω i takes the form   r G2M 2 2 (1) 2 ω + 1 − ( G λ) k = 0. (A36) H (y) = ei(y−π/4+γπ/2), (A33) 2 G λk ~ 2 iγ πy ~ 2 It is worth commenting that these results might be modified for those “high frequency” (2) (1) ∗ with Hiγ (y) = Hiγ (y) . This asymptotic eigenvalues, where the discreteness of the lattice limit of the Hankel functions, together with in µ is important. This will be a matter of future cnt (A32) and the fact that φ (k, µ, M) ∈ R at any research.

[1] C. G. Callan, S. B. Giddings, J. A. Harvey, Phys. Rev. D45, R1005 (1992), arXiv:hep- and A. Strominger, “Evanescent black holes,” th/9111056v1. 23

[2] D. Grumiller, W. Kummer, and D. V. Vassile- [17] R. Gambini and J. Pullin, “Black holes in vich, “Dilaton gravity in two-dimensions,” Phys. loop quantum gravity: the complete space- Rept. 369, 327 (2002), arXiv:hep-th/0204253. time,” Phys. Rev. Lett. 101, 161301 (2008), [3] A. Fabbri and J. Navarro-Salas, Modeling arXiv:0805.1187 [gr-qc]. Black Hole Evaporation (Imperial College Press, [18] A. Ashtekar and P. Singh, “Loop quantum cos- 2005). mology: A status report,” Class. Quantum [4] T. Kloesch and T. Strobl, “Classical and quan- Grav. 28, 213001 (2011), arXiv:1108.0893 [gr- tum gravity in 1+1 dimensions, part i: A unify- qc]. ing approach,” Class. Quantum Grav. 13, 965– [19] R. Gambini and J. Pullin, “Loop quantization of 984 (1996), arXiv:gr-qc/9508020. the schwarzschild black hole,” Phys. Rev. Lett. [5] A. Ashtekar, F. Pretorius, and F. M. 110, 211301 (2013), arXiv:1302.5265 [gr-qc]. Ramazanoglu, “Evaporation of 2-dimensional [20] Rodolfo Gambini, Javier Olmedo, and Jorge black holes,” Phys. Rev D83, 044040 (2011), Pullin, “Quantum black holes in loop quan- arXiv:1012.0077 [gr-qc]. tum gravity,” Class. Quantum Grav. 31, 095009 [6] D. Louis-Martinez, “Dirac quantization of two- (2014), arXiv:1310.5996 [gr-qc]. dimensional dilation gravity minimally coupled [21] M. Bojowald and J. D Reyes, “Dilaton gravity, to n massless scalar fields,” Phys. Rev. D 55, poisson sigma models and loop quantum grav- 7982–7984 (1997), arXiv:hep-th/9611031. ity,” Class. Quantum Grav. 26, 035018 (2009). [7] K. V. Kuchar, J. D. Romano, and M. Varadara- [22] M. Bojowald and R. Swiderski, “Spherically jan, “Dirac constraint quantization of a dilatonic symmetric quantum geometry: Hamiltonian model of gravitational collapse,” Phys. Rev. D constraint,” Class. Quantum Grav. 23, 2129– 55, 795–808 (1997), arXiv:gr-qc/9608011. 2154 (2006), arXiv:gr-qc/0511108. [8] M. Varadarajan, “Quantum gravity effects in [23] R. Gambini, J. Pullin, and S. Rastgoo, the cghs model of collapse to a black hole,” “New variables for 1+1 dimensional gravity,” Phys. Rev. D 57, 3463–3473 (1998), arXiv:gr- Class. Quantum Grav. 27, 025002 (2010), qc/9801058. arXiv:0909.0459v2 [gr-qc]. [9] Abhay Ashtekar and Jerzy Lewandowski, “Back- [24] S. Rastgoo, “A local true hamiltonian for ground independent quantum gravity: a status the cghs model in new variables,” (2013), report,” Class. Quantum Grav. 21, R53 (2004). arXiv:1304.7836 [gr-qc]. [10] T. Thiemann, Modern Canonical Quantum [25] A. Corichi, A. Karami, S. Rastgoo, and , Cambridge Monographs on T. Vukašinac, “Constraint lie algebra and lo- Mathematical Physics (Cambridge University cal physical hamiltonian for a generic 2d dila- Press, 2007). tonic model,” Class. Quantum Grav. 33, 035011 [11] Rodolfo Gambini and Jorge Pullin, A first (2016), arXiv:1508.03036 [gr-qc]. course in loop quantum gravity (Oxford Univer- [26] T. Banks and M. O’Loughlin, “Two-dimensional sity Press, 2011). quantum gravity in minkowski space,” Nucl. [12] Jerzy Lewandowski, Andrzej Okołów, Hanno Phys. B362, 649 (1991). Sahlmann, and Thomas Thiemann, “Unique- [27] S.D. Odintsov and I.L. Shapiro, “One-loop ness of diffeomorphism invariant states on renormalization of two-dimensional induced holonomy flux algebras,” Communications in quantum gravity,” Phys. Lett. B263, 183 Mathematical Physics 267, 703–733 (2006). (1991). [13] L. Modesto, “Disappearance of the black hole [28] T. Thiemann, “Anomaly-free formulation of singularity in loop quantum gravity,” Phy. Rev. non-perturbative, four-dimensional lorentzian D 70, 124009 (2004). quantum gravity,” Phys.Lett. B 380, 257–264 [14] A. Ashtekar and M. Bojowald, “Black hole evap- (1996), arXiv:gr-qc/9606088. oration: A paradigm,” Class. Quantum Grav. [29] M. Martin-Benito, G. A. Mena Marugan, and 22, 3349–3362 (2005), arXiv:gr-qc/0504029. T. Pawlowski, “Loop quantization of vacuum [15] C. G. Boehmer and K. Vandersloot, “Loop bianchi i cosmology,” Phys. Rev. D 78, 064008 quantum dynamics of the schwarzschild in- (2008), arXiv:0804.3157 [gr-qc]. terior,” Phys. Rev. D 76, 104030 (2007), [30] M. Martin-Benito, G. A. Mena Marugan, and arXiv:0709.2129 [gr-qc]. J. Olmedo, “Further improvements in the un- [16] A. Corichi and P. Singh, “Loop quantiza- derstanding of isotropic loop quantum cos- tion of the schwarzschild interior revisited,” mology,” Phys. Rev. D 80, 104015 (2009), Class. Quantum Grav. 33, 055006 (2016), arXiv:0909.2829 [gr-qc]. arXiv:1506.08015 [gr-qc]. 24

[31] Abhay Ashtekar, Tomasz Pawlowski, Param- 77, 024046 (2008). preet Singh, and Kevin Vandersloot, “Loop [35] Abhay Ashtekar, Jerzy Lewandowski, Donald quantum cosmology of k = 1 FRW models,” Marolf, Jose Mourao, and Thomas Thiemann, Phys. Rev. D 75, 024035 (2007). “Quantization of diffeomorphism invariant the- [32] Abhay Ashtekar, Tomasz Pawlowski, and ories of connections with local degrees of free- Parampreet Singh, “Quantum nature of the big dom,” Journal of Mathematical Physics 36, bang: an analytical and numerical investiga- 6456–6493 (1995). tion,” Physical Review D 73, 124038 (2006). [36] Martin Bojowald, Suddhasattwa Brahma, and [33] Abhay Ashtekar, Tomasz Pawlowski, and Juan D. Reyes, “Covariance in models of loop Parampreet Singh, “Quantum nature of the big quantum gravity: Spherical symmetry,” Phys. bang: improved dynamics,” Physical Review D Rev. D 92, 045043 (2015). 74, 084003 (2006). [37] Claus Kiefer, “Wave packets in minisuperspace,” [34] Abhay Ashtekar, Alejandro Corichi, and Phys. Rev. D 38, 1761–1772 (1988). Parampreet Singh, “Robustness of key features of loop quantum cosmology,” Physical Review D