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Callan-Giddings-Harvey-Strominger Vacuum in Loop Quantum Gravity and Singularity Resolution

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Callan-Giddings-Harvey-Strominger Vacuum in Loop Quantum Gravity and Singularity Resolution Callan-Giddings-Harvey-Strominger vacuum in loop quantum gravity and singularity resolution Alejandro Corichi,1, 2, ∗ Javier Olmedo,3, 4, y and Saeed Rastgoo5, 1, z 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 quantum gravity 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 loop quantum gravity. 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 dilaton 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 black hole 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, Hawking radiation, 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 ∗ corichi@matmor.unam.mx theory of the full 4D theory. In particular, addi- y jolmedo@lsu.edu tional studies of the classical [4] and the semiclas- z saeed@xanum.uam.mx 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 quantum cosmology (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 −|gj G ˆ sical and the quantum/semiclassical regime. 2 2 The CGHS action is × Y (Φ)R(g) + V (rΦ) ; Φ : (3.1) 1 2 p −2φ Sg-CGHS = d x −|gje 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 −|gj 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.
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