JHEP12(2019)031 Springer October 7, 2019 : December 4, 2019 November 21, 2019 : : Received Published Accepted , phase space and a Schwarzian action. Published for SISSA by https://doi.org/10.1007/JHEP12(2019)031 2 b ) acting as Mobius transformations in proper time. R ) Lie algebra. On the other hand, this approach al- , R 1 , (2 sl . 3 and Etera R. Livine a 1909.13390 The Authors. c Global Symmetries, Conformal Field Theory, Space-Time Symmetries

We show that the simplest FLRW cosmological system consisting in the homo- , [email protected] . We show that the CFT two-points correlator is realized as the overlap of the evo- 1 E-mail: [email protected] Center for Gravitational Physics,Kyoto Yukawa Institute University, for 606-8502, Theoretical Kyoto, Physics, Japan ENS de Lyon, Universit´eLyon 1,F-69342 CNRS, Lyon, Laboratoire France de Physique, b a Open Access Article funded by SCOAP ArXiv ePrint: with the cosmological vacuumnew annihilated perspectives by in the classical scalar and quantumthe constraint. cosmology, conformal among These bootstrap which results the program possibility suggest to to quantize apply cosmological backgrounds. Keywords: Preserving this conformal structure atthe the Wheeler-de quantum Witt level quantization fixesa and the CFT allows ordering to ambiguities formulate in lution FLRW quantum in cosmology proper as time offunction is cosmological built coherent from wave-packets. a In vacuum particular, state the which, two-points although not conformally invariant, coincides under the 1D conformalThis group invariance SL(2 is made explicitmechanics. through the On mapping of the FLRWas one cosmology combinations onto hand, of conformal we the identifyvolume, Hamiltonian the which scalar corresponding constraint, form conformal the alows extrinsic Noether closed to curvature charges, write and FLRW the cosmology 3D in terms of a AdS Abstract: geneous and isotropic massless Einstein-Scalar system enjoys a hidden conformal symmetry Jibril Ben Achour Cosmology as a CFT JHEP12(2019)031 8 29 21 26 35 5 28 15 30 37 ) structure R 16 , ]. This striking feature of cosmological 22 (2 10 sl 9 5 11 – 36 6 22 1 20 ) flow 18 R , ) flow – 1 – R 36 , 35 13 ) quantization 12 R 18 , geodesics to physical cosmological trajectories 2 ) irreducible representations and lowest weight vector geometry as the phase space metric R , 2 2 1 32 as the cosmological phase space 2 ) unitary representations ) symmetry of FLRW cosmology R R , , 3.2.1 The3.2.2 two-point function from the SL(2 3.2.3 Vaccuum state and quasi-primary operator 2.4.2 AdS 2.4.3 From AdS 2.2.1 The2.2.2 CVH algebra Charges algebra and their infinitesimal action 2.4.1 AdS 2.1.1 Mobius2.1.2 covariance of the action Conformal2.1.3 transformation and Schwarzian action The for2.1.4 associated Cosmology Noether’s charges Mapping to conformal mechanics A.1 The deformedA.2 CVH algebra Physical trajectories from SL(2 3.1 Wheeler-de Witt3.2 SL(2 CFT correlators as wave-packet overlap in quantum cosmology 2.3 Cosmological trajectories2.4 from SL(2 AdS 2.2 Hamiltonian formulation 2.1 Lagrangian formulation: symmetries and Noether charges entropy and a temperature, just like black holes [ It is well knownmathematical that features. cosmological Classically, information andor escaping black falling beyond inside holes the the spacetimes black cosmological holehole share horizon cannot horizons many be radiate recovered. physical at Moreover, and cosmological both a cosmological spacetimes temperature and can black given be by considered their as surface thermodynamical gravity. objects As equipped a with consequence, an 1 Introduction B SL(2 C Time overlap of thermal dilatation eigenstates 4 Discussion A Conformal structure of the homogeneous (A)dS universe 3 Quantum cosmology and CFT correlators Contents 1 Introduction 2 SL(2 JHEP12(2019)031 2 ], which aim at 31 ) representations. R , ] for the near-horizon 20 – 16 ]. These conformal structures ] for reviews). 13 33 , – 32 11 ] for different investigations on this ) structure. Quasi-normal modes and ] (see also [ R 40 , , 15 , 39 14 ]. The presence of such conformal symmetry – 2 – 35 ] and [ ]. This body of results obtained so far suggests that 38 – 30 – 36 21 [ 1 ], test probes enjoy indeed a SL(2 ]. Yet, most of the results obtained so far concern the three-dimensional version of de 34 45 – 41 ]. Do hidden conformal symmetries exist in cosmological backgrounds? If yes, can thefrom thermodynamical an features effective of conformal cosmological description spacetimes based on be CFT reproduced techniques? 47 , It is well known that similar conformal symmetries exist in the de Sitter geometry. Over the last decades, important efforts have been devoted to understanding the state This set of results led to the Kerr/CFTOne conjecture could and its also black mention hole/CFT the cousins [ on-going efforts to understand the thermodynamical properties of the • • 46 2 1 building a consistent holographic description of black holesde (see [ Sitter space fromconjecture an [ holographic pointSitter. of More view, recently, leadingin asymptotic [ eventually symmetries to associated the to well cosmological known horizons de have been Sitter/CFT investigated ary backgrounds, motivating thecosmological development perturbations. of the Seepoint. cosmological [ bootstrap Up program to topropagating on now, classical these cosmological backgrounds. investigations A have natural focused question on is whether the a quantization more of test fields As shown in [ propagators of these testMoreover, fields conformal can symmetries then of begrounds test obtained have probes from also propagating the been onprovides SL(2 a more investigated powerful general in tool FRW [ to back- bootstrap the correlation functions of test probes on inflation- and finally their state counting the local physics near blackof hole the horizons near is horizon encoded gravitational in field.holes the and From underlying this cosmological perspective, conformal spacetimes structure the rise common a features of natural black question concerning the latter: of extremal black holes).test Moreover, fields additional propagating conformal on symmetrieswere the were used also near-horizon to found region re-deriveas for in many their features various spectroscopic black of properties, holesground, these the the spacetimes black dynamics low holes and of energy super-radiant from quasi-normal modes a modes on spectrum, CFT the the perspective, Kerr emission such of back- gravitational waves counting of black holes. Oneformal major symmetries outcome emerge was in theallow the discovery to near-horizon that import region some conformal [ unexpected fieldmal con- theory (CFT) description results of and the builddetails thermodynamical a of universal properties specific effective of confor- quantum black gravity proposals holes, [ without relying on the relativity is only adegrees statistical of freedom description of of the underlying micro-statesmicro-states, quantum and associated geometry. therefore Understanding to the the entropy more nature ofone of fundamental cosmological major these and challenge black of holes modern geometries, theoreticalof physics remains and gravity. an ultimate goal of a quantum theory and black holes spacetimes suggests that their classical mechanics as described by general JHEP12(2019)031 ]. 65 ] for ]. The 57 – 88 – 52 ], as well as 77 51 – 48 conjecture [ 1 CFT 1) hidden structure in loop quantum / ] for examples, and has recently , 2 76 (1 ] for FLRW cosmology and clarify – su 60 72 – ]. This structure is very similar to the ∼ ) 58 60 R – , (2 ]. As such, conformal mechanics also plays 58 [ sl 65 3 – 3 – ]. 64 – 61 ] for more details on this system. This simple mechanical model have found 71 ) than the well known dAFF model [ – R 66 , More precisely, we focus our attention to the homogeneous and isotropic Einstein- As a matter of fact, it turns out that recent investigations in canonical quantum cos- From the point of view of quantum gravity, the quantization of cosmological back- Related works in quantum cosmology based on a 3 existence of this newFLRW cosmology, symmetry associated implies to the themations translation, existence of dilatation the of and time special three coordinate. conformal new transfor- From the conserved gravitational charges point in of view, thesecosmology new can conserved be found in [ Scalar cosmological system. Surprisingly,invariance we under shall time find reparametrization, that,invariant under additionally the Mobius to action transformation the of offorms standard the this as proper system a time turns primary providedform the field. out scale of to As factor a also trans- a Schwarzian be consequence, action the for cosmological cosmology. action can Moreover, be following recast Noether’s in theorem, the the SL(2 a crucial role for classicalvestigate deeper and the quantum conformal cosmology. structure Inthe found the concrete in following, mapping [ we with shall the therefore dAFF in- model. See [ applications in a wide rangeregained of attention physical due systems, to see its [ goal potential of role in the the presentsystem the work AdS enjoys is indeed to the show very that same the conformal homogeneous symmetry and under isotropic the Einstein-Scalar 1d conformal group mology have revealed an unexpectedgeneous conformal and isotropic structure Einstein-Scalar in system theone appearing phase in space the of simplest example themechanics of homo- introduced a one-dimensional by CFT, de the well-known Alfaro, conformal Fubini and Furland (dAFF) in the mid seventies [ devoted to provide suitable initialto discuss conditions the for fate the of the quantumreviews). classical universe Big In [ Bang this singularity at work,properties the we of quantum would level cosmological (see like spacetimes [ to cancosmology point indeed certainly be that, provides reformulated the if in simplest the CFT framework thermodynamics terms, to and quantum study quantum these questions. quantization schemes. Performing a quantizationintegral using one, the canonical one approach obtains orhomogeneous a the region path simple of quantum space. theory Despiteit encoding the is the apparent worth simplicity quantum pointing of properties that suchof of quantum the one cosmology, gravitational a still field, faces such thethe as crucial realization the timeless challenges of nature inherent the of to quantum the the covariance. Wheeler-De quantization Witt Most equation efforts and in quantum cosmology have been allowing the bootstrap its quantization. grounds represent the simplestdegrees of models freedom one but can theas a scale build. scalar factor, field, one and obtains By considering a simple a freezing symmetry-reduced non-trivial all model matter which the allows content to such gravitational test different fundamental conformal structure exists at the level of the background spacetime itself, JHEP12(2019)031 ] ] . 1 60 81 . In – 58 ] within 2.1.2 ], the ho- 89 59 , ] about preserv- 58 , we establish the 60 – 58 2.1.4 and the scalar Hamiltonian v )-invariant conformal mechanic ) charge algebra in a canonical R R , , , we present the Mobius invariance (2 sl ) Lie algebra. This clarifies the physical 2.1.1 R , , we investigate the Hamiltonian realization (2 ]. Indeed, a simple reformulation of the grav- sl 2.2 – 4 – suggests that one could solve the model based 65 , the 3D volume C 1 geodesics. 2 metric, which allows to identify the physical trajectories on 2 ) symmetry associated to bulk conserved currents. This provides a R , ) structure for alternative quantization schemes). In the quantum theory, R , and show that they form an , we derive the associated Noether charges. In section , H (2 sl 2.1.3 ], it is still possible to identify non-normalizable states together with a quasi-primary This paper is organized as follows. In section At the quantum level, this newly found conformal structure turns out to have far- At the classical level, these findings show that, as anticipated in [ This unexpected structure of the homogeneous and isotropic Einstein-Scalar system 81 of the homogeneous andand isotropic write symmetry the reduced cosmological action actionsection of in the term Einstein-Scalar of system themapping Schwarzian with derivative the in dAFF section model. In section in [ operator such that bothtechnics conspire to will lead be to usedquantum the desired cosmology here. form in for From termsolely the a correlator. on of more symmetry The a general same arguments CFT quantum perspective, and cosmology. import the the reformulation conformal of bootstrap this program [ packets evolving in time actuallyrelator. reproduces In the universal order form tofor of derive the conformal this two-point quantum result, CFT mechanics. we cor- recovered will in A follow the crucial closely standard point the waythe is approach because present proposed that there one-dimensional in the are setup. [ no CFT conformally correlator However, invariant as cannot vacuum shown state be for in conformal quantum mechanics quantization scheme, `ala Wheeler-dequantization Witt, ambiguities and provides determine a theing powerful operator the criterion ordering to (see narrowthe [ the natural question isquantum then cosmology how framework. do We the will usual show CFT that correlation the functions overlap appear between in coherent the wave indeed an exact SL(2 very simple example of anGeneral exact Relativity. conformal symmetry in a symmetry-reduced sector ofreaching 4d consequences, as expected. Preserving the the cosmological phase space as AdS mogeneous and isotropic symmetryThis reduced is Einstein-Scalar rather systemever, is surprising a it as covariant appears GR CFT that is for well a known to specific not choice of be foliation, conformally invariant. this gravitational How- system enjoys allows to build a straightforwardof mapping de with the Alfaro, SL(2 Fubini anditational Furlan system (dAFF) shows [ thatwritten the in dAFF proper mechanics time. correspondsrally Additionally, endowed to we with the show the cosmological that AdS action the cosmological phase space is natu- bulk currents. At theconditions Hamiltonian for) level, we the identify extrinsicconstraint these curvature Noether charges asmeaning (the of initial the so-calledand CVH provides algebra a for more FLRW complete cosmology picture introduced of earlier this in structure. [ charges have two surprising properties: they are time-dependent and they correspond to JHEP12(2019)031 ) 3 – / is R 1 ◦ 58 , V (2.1) (2.2) 2.3 = ◦ ` , we present . Here we do t d 3.1 reads: φ/ ◦ ]. This unexpected V = d 65 , 0 φ , the symmetry reduced  φ 2 0 N ) Lie algebra. Section j and aa x R t d , d i 3 πG ) structure of the charge algebra. x (2 8 a/ d R sl , ij − δ (2 ) = d 2 0 t N sl 0 ( φ 2 2 a 3 a a  + t 2 derives the cosmological trajectories from the t d – 5 – R )d t Z ( model and the homogeneous and isotropic sector ◦ 2 2.4.2 1 V N − tackles the quantum theory. In section ] = ˙ φ = 3 , we underline that this conformal structure extends beyond 2 ˙ a, s A d N, a, [ S ) structure on the phase space, and the cosmological dynamics can be R , presents the derivation of the CFT 2-point function in the quantum (2 sl 3.2 ) symmetry of FLRW cosmology R , ) group structure while section geometry. Finally section R , This action depends on two lengths scales. On the one hand, the scale 2 ]. We then show that the Noether charges form a encodes the size of the fiducial cell to which we restrict the spatial integration to avoid where the primes standnot for discuss the the time (Gibbons-Hawking) derivative, fiducial boundary cell. terms living They on shouldshould the not become spatial play relevant boundary any in of important a the role more in general the setting. homogeneous dynamics, but For an homogeneous minimallyaction of coupled the massless Einstein-Scalar system scalar integrated field over a cell of volume this simple system, including the (A)dS cosmology2.1 and the Bianchi I Lagrangian universe. formulation:Let symmetries us and consider Noether the FRLW charges geometry in the isotropic and homogeneous flat slicing Fubini-Furland conformal mechanics developedmapping in between the theof simplest seventies General CFT [ Relativity allows toformal recast field classical theory cosmology and as pave arepresentations. the simple way In for one appendix a dimensional straightfroward con- quantization in term of SL(2 dinates. The covariance underto Mobius leads transformation to of three the conserved Noether’s cosmologicalcharges form charges. action an At is the shown hamiltonianfully level, we reproduced show by that integrating these theresult, flow this generated simple by these cosmological three system Noether’s appears charges. as As a a covariant version of the de Alfaro- In this section,mology we filled review with a theThen, massless we lagrangian show scalar and that field,reparametrization, this hamiltonian setting enjoys simple an formulation our cosmological additional notation system, of invariance underfactor on for FRLW the transforming top the 1d as cos- of a conformal rest quasi-primary the group, of field the invariance under scale this under Mobius time work. transformation of the time coor- Finally, section cosmology seting. We concluderesults by on a the discussion conformal invariance on of the FLRW research cosmology. directions2 opened by our SL(2 60 dedicated to showing thatSL(2 the cosmological dynamicsAdS can be fullythe Wheeler-De integrated Witt using quantization the preserving the of the conformal symmetry and show its relation to the CVH algebra introduced in [ JHEP12(2019)031 ) τ 2.4 (2.4) (2.5) (2.6) (2.7) (2.3) signals , πG  . 2 8 t  f √ d 2 d / τ = 3 d a P d ) = ` t (  0 associated to the fidu- f 1 ◦ πG ≡ ` 6 ) t − ( , h 2 ˙ φ , 2 3 = 0 a ¨ φ  = 0 . 3 τ a 2 ˙ d φ + 2 = 0 R ˙ Z φ 2 ˙ ˙ a ◦ φ 2 3 V πGa a a = but not on the scale + 4 – 6 – # 3 πG 2 2 ) = 3 4 a with the Jacobian ˙ ˙ a πG φ a πG 8 3 + ˙ − 8 a 3 √ 2 ¨ ( a ˙ τ a a − = a ) t 2 ˙ P ( = 2 = d = φ ` 2 3 a N φ a N ) t E E t E " ( ) ) )d ). More precisely, time reparametrization corresponds to the ) . Then the variations with respect to the scale factor, the scalar 1 t t t t τ t τ ( ( ( − ( ( d , which manifest through the invariance under arbitrary rescaling d φ f h h a t N R Z φ/ . The action then reads: = = t ◦ ˜ ˜ t t ) = ) = ) = d V ˜ ˜ ˜ t t t d = d ( ( ( N ˜ ˜ a ˙ φ ˜ φ N ] = = ˙ φ τ ˙ a, and 7−→ 7−→ 7−→ 7−→ 7−→ τ t d N a φ t d

N, a, a/ [ As it is well known, this theory enjoys a symmetry under arbitrary reparametrization S = d ˙ symmetry. This invariance is clearer written in proper time. Starting from the action ( 2.1.1 Mobius covariance ofOn the action top of theturns out invariance under that the 1D cosmological diffeomorphism action given admits by an time unexpected reparametrization, time-dependent it conformal The first one correspondstion to while the the acceleration third equation, one thedepends is second on the to standard the the first Planck Friedman continuitycial length equation. equa- cell. Notice that the dynamics where the dota notation now standsfield for and the the derivative lapse with function lead respect to to the standard the Friedman proper equations: time, Due to this invariance, it isand convenient physically write to the absorb action thedefined lapse and in by the the d resulting time coordinate equations of motion in terms of the proper time of the lapsetransformations function role when working with suchinhomogeneities finite homogeneous can region develop. of space,the On setting high the the energy scale other UV at hand, regime which the Planck length of the time coordinate divergences. It corresponds to an infrared cut-off of the system. This scale plays a crucial JHEP12(2019)031 can (2.8) (2.9) τ (2.11) (2.12) (2.10) and ˜ j τ x  3 2 d / i 4  ) x 2 δ d  / . τ 3 ij τ + F d˜ a δ d ) d d˜   τ γτ δ  ( ( + , 2 3 i + a ) 1 a πG τ = 1 6 + , and absorb all the Mobius ( γτ ˙ t φ 2 −  ) , ). All these implementations in βγ ) δ = 2 τ ˜ 2 τ d N . − ˜ t d  ( τ + 3 , the relation between ˙ ˜ t, ˜ a γ τ d φ t a , d˜ d αδ d ) ) γτ − ( τ δ  N 2 ( 3 ) to ( − a 2 , +  ˜ a γ = h 2 3  = / / L τ with γτ 3 τ 2 t, N τ 2 (  ) a d s d˜ ) , δ f d d τ τ ◦ . – 7 – ( φ τ Z d + πG V ) a , d ◦ 6 d δ β  τ 2 Z V 2 2 ( γτ ) ) ) + + ( φ δ δ δ = = 7−→ ) = τ + i γτ τ + + ατ d ) ( ) = ) = τ j τ F τ = γτ (˜ γτ γτ x (˜ (˜ ˙ ( ˜ φ ˜ ˜ d ˜ a φ τ , i ) = ( = = ( x τ d 2 τ (˜ ˜ φ τ ˙ 7−→ 7−→ 7−→ ij ˜ a d˜  d˜ d δ , 2 ) ) a τ φ / τ τ 3 τ ( (˜ a d˜ 2 ˜ a d ˜ a h  L + τ 2 d˜ τ d Z − = 2 s d It is interesting to translate this Mobius transformation in proper time in terms of These transformations can be understood as conformal transformations on the metric: coordinate time and lapse.be Indeed, achieved since through different the mappings d proper from ( time are related tois each assuming other by that time the reparametrizations. coordinate The time simplest realization does not change, It is worth pointingrather that unexpected such property new since continuous general symmetryof relativity this for work is homogeneous is not to cosmology conformally discussed is in invariant. detail a This this goal new conformal structure of the cosmological action. tive and the equationThis shows of that the motion homogeneous are andMobius isotropic therefore covariance Einstein-Scalar on unaffected action top enjoys by of an the the additional time above reparametrization transformations. freedom of the action. with the total derivative term is given by: Therefore, while the Lagrangian is not invariant, the action is modified by a total deriva- The latter is the key identity to prove the invariance of the action: In particular, this implies the following Jacobian and transformation of the time derivatives: given above, we perform a Mobius transformation of the proper time: JHEP12(2019)031 N 0 and t . t R 2 γ / (2.14) (2.15) (2.16) (2.17) (2.13) 3 = τ ˙ ha 2 / being related 3 δ − h , 1 2 and γ + !# , 3 2 β / a τ 3 ˙ , ¨ f f d a α d

. 2 . / 2 d 1 dτ ) conformal transformations, − ! = 1 . R h ˙ − h h , ) 3 t =

( a (2 βγ ] , ) 3 2 explicitly involve the proper time sl N f 2 − 3 / / 2 4 − 3 2 ) ) = 0. . a ) αδ δ 0 ¨ h h τ t ] = 0, which is true only for Mobius Sch[ ) 2 δ t t / d ( f " 1 + ( 1 τ , = + a τ h ) 2 d t γτ ( d( γτ ( ! with ( φ ˙ Z ¨ 1 h f f – 8 – ) as an integral in proper time, we perform an

- which provides the missing parameter. This corresponds t , = ) = ◦ 0 ) = t ) = πG t 3 2 t V t ( 2 2.4 = δ ( β ( / ˜ 12 τ ˜ 3 ˜ ˜ t a φ N − + + d˜ a ˙ d˜ f f ... ] = γτ ˙ ατ φ 7−→ 7−→ 7−→ 7−→ t ˙ (so that the 3D volume is of weight 1): ] = a, a φ , N ) in coordinate time seem only to depend on the two parameters τ , f ) = ) 1 3 a, τ [ τ )d ( ( 2.13 τ f a Sch[ ( ) − S h = 1. Nevertheless, the definition of the proper time as an integral, τ ] ( ˙ ˜ 3 φ ≡ / βγ , 1 ˙ τ ˜ a, ) − h d τ ˙ a, ( f [˜ αδ f S as a function of the coordinate time ) = = τ = τ (˜ ] denotes the Schwarzian derivative, defined as: ˜ ˜ · τ d˜ a t N d 7→ 7→ 7→ R τ Starting with the FLRW action ( τ d a The transformation laws (

) = 4 t ( , although the actual Mobius transformation is defined by three parameters, on top of the 1D diffeomorphisms defined by timeδ reparametrizations. by the constraint involve a further parameter —to the choosing initial the time event at which the proper time vanishes, This means that theseonly transformations if are a the symmetrytransformations: Schwarzian of derivative the vanishes, cosmological Sch[ action if and This confirms that FLRW cosmology is invariant under where Sch[ A straightforward calculation allows to show thatby the a variation Schwarzian of action the action up is to given exactly a total derivative: as a primary field of weight Before computing the Noether charges associatedto with the consider Mobius general invariance, we conformal wouldto transformations like the and Schwarzian show action. how the FLRW action isarbitrary related transformation of the proper time and assume that the scale factor transforms An important remarkτ is that these transformations 2.1.2 Conformal transformation and Schwarzian action for Cosmology transformation in the transformation of the lapse function: JHEP12(2019)031 (2.24) (2.23) (2.18) (2.19) (2.20) (2.21) (2.22) . )  − = 0. = N E , γ is simply displaced τ = . = 1 φ E  δ 2 ˙ a gives us the corresponding = = , a  , a , E , α )  ˆ=0 3  πG 2 ˙ φ . ˙ ˙ 8 a , a     = 0 , a a ) ) − 1 + ) E − − τ τ β +  ( ( 2 ˙ 3 a √ ) can be decomposed into combina- πG ˙ a φ τ φ = 8 ) = 3 ) = = 3 2.8 a τ = πG τ 1 − ( ) are simply reparametrized. This repro- ( , β 8 1 2 ) = ) = ˜ − τ τ φ 2 a ( τ τ ˙ δ  φ − (˜ (˜ φ − 3 − = 0 ˜ ˜ a τ φ φ = d a ) ) d E τ 1 2 τ ˙ , γ – 9 – φ ( ( τ  , α ˜  a φ d 7−→ 7−→ 7−→ ◦ ) and ◦ = 1 V τ = ˜ = a φ τ Z = 0 V ( δ

◦ a a = φ γ t  t  = = δ δ V − = − α − Q ˙ Q β = S  ) and thus vanishes on-shell: δ 2.6 translation ( dilatation ( special conformal transformation ( under an infinitesimal translation, the proper time ) and ( 2.5 2  τ τ + + + τ τ τ motion ( where the equality ˆ=assumes the equations of motion which turns out to becheck exactly that the its Hamiltonian time as derivative we can will see be in entirely the expressed next in section. terms We of the equations of Computing the variation of the actionNoether at charge: leading order in The associated Noether charge is thus given by The infinitesimal field variations are: Translation: with the offset, while theduces fields a straightforward infinitesimal time reparametrization: = = = • ˜ ˜ ˜ τ τ τ transformations. We shall therefore derive the Noether’s charges associated to each of these elementary Thanks to the Noether’stion is theorem, associated we toThe known the Mobius that existence transformations of any conserved of continuoustions charges. of the symmetry translations, Let proper of dilatations us and time the specialactions now ( conformal ac- read: derive transformations, these whose infinitesimal charges. 2.1.3 The associated Noether’s charges JHEP12(2019)031 (2.34) (2.31) (2.32) (2.33) (2.25) (2.26) (2.28) (2.29) (2.30) (2.27) 1 which    ˙ a 3 2 a a , ) with 1 ◦ πG τ πG V 6  ( 4 . a +
+ ˙ , a ˆ=0 τ 2 − 2 3  a   = 1 + a a , ) ˙ a  τQ − . ) 2 3 τ τ 1 λ πG ( 2  1  = 2 a − − ˙ ( 3 ˙ φ ˆ=0 aτ τ  ˙ / a 3 a O ˙ ∼ τ 1 0 a τ τ − ˙ − ) ) 2 ˙  φ , )   Q + a −  2  3 τ a  τ ) 2 ( 2 2 τ τ , followed by a translation and finally a  a τ ˙ a 1 ( πG τ 3 ) = a ) = − φ 4 . ∼ /τ / ∼ τ 3 τ 2 − 1 πG 3 ( ) − ( ) = (1 + 8 + − 3 a ˙ φ τ τ πG a τ ) Q ( ( ) = (1 + ) = 8 ˜ 2 τ  ◦ + φ − a − ' τ τ τ . 2 τ 7→ − πG V (˜ (˜ consider now the special conformal transforma- ) – 10 – ˙ φ − ) a ) 6 ˜ ˜ − a − φ τ τ τ τ a = τ 2 ˙ τ ( ) ( φ ) ˙ ( φ + ˜ a φ τ τ 1+ τ φ + 3 ( 3 ( 0 τ  πG a ˜ 7−→ 7−→ 7−→ Q a φ = ˜ = d 8 ◦ = 1 2 d V τQ a φ τ a φ = ˜ = ) = (1 + ) = ˜  τ − d  2 d

 τ τ δ 2 = a δ 2 φ (˜ (˜ s ˙  s τ −  ˜ ˜ φ a φ 0 δ δ 3 τ − − Q a d  d Q 2 1 ◦ 2 7−→ 7−→ 7−→  V τ a φ τ τ

= =  ◦ + V consider now the dilatation by a factor Q = 0 Q The conservation ofof this motion: charge can be checked directly assuming the equations from which one can derive the associated Noether charge: One obtains the following infinitesimal field variations: second inversion. Under such infinitesimal special conformal transformations, one has Special conformal transformations: tions which consist in an inversion which leads to the following Noether charge: The conservation of this charge can be checked by a direct calculation: Under such infinitesimal transformations, the field variations are; transforms the proper time and the fields as Dilatation: • • JHEP12(2019)031 ) 2.2 . ), we a (2.40) (2.35) (2.36) (2.38) (2.39) ]. From 2.6 65 ]: 65 , ] suggests that one − 65 2 Q 2 ], which is now involved ! τ ˙ φ 65 2 3 − a 0

τQ , 2 πG 1  − − 2 12 β q 2 2 τQ q − associated to the translations should − − ] for several applications to black holes. = 2 = ˙ − = 0 (2.37) ˙ q q q 76 Q   3 β – 1 2 2 q seems to be the evolving constants of motion 2 τ β q ] It has been studied intensively over the last 72 d + = − 65 ] = 1. From the continuity equation ( + R Q , β q – 11 –  β Z ) 2 2 2 ˙ q τ β = ¨ q (  and 2 q ] = ˙ / + E q 2 0 and [ 3 τ 2 a Q q, 2 2 ˙ q [ / + β 1 ] as well as [ correspondence. ◦ ,  ˙ q S πG L V 1 2 τ 71 6 β ) algebra as expected. This will turn out to have far reaching q τq – − R r . In the next section dedicated to the Hamiltonian analysis of , is a constant amounts to solving the equations of motion for the scalar field ] = ˙ + − 66 q 3 q q β 2 2 (2 a /CFT 2 2 ˙ q 1 2 q 2 ) = sl τ ( = = = and q 0 − + ˙ a Q Q Q 2 a to: 5 is a constant of motion. In term of these variables, the FLRW cosmological β Nevertheless before proceeding to the canonical analysis of the cosmological action,we Effectively assuming that 5 and inserting the on-shell solution back in the action to obtain an action for the sole scale factor φ Its equation of motion reads simply The Noether charges associated to the conformal invariance are given by [ This recovers the exact actionthis of point conformal of mechanics view, introducedknown by that the dAFF this analogy in system with [ an enjoys example the an of invariance dAFF a under model 1D thedecades, conformal is 1d see field for conformal made theory examples group, [ explicit. [ and provides Now, it is well with physical dimension [ know that action reduces could map the two systems.a Indeed, change it of turns variables. out Let that us this analogy therefore can introduce be the made new explicit set by of variables in the study of the AdS 2.1.4 Mapping toThe conformal close mechanics analogy ofconformal the mechanics introduced above by conformal de symmetry Alfaro, of Fubini the and Furland action [ with the well known charges indeed form an consequences both classically and quantum mechanically. would like to show how thisof conformal structure variables. becomes apparent This through aconformal will mechanics simple developed change provides by de a Alfaro, direct Fubini and Furland mapping [ between FLRW cosmology and to the leads to threeat time-dependent their charges structure, whichcorrespond it are the appears Hamiltonian, conserved while that inrespectively the time. for charge Lookingthe carefully system, we will confirm this interpretation and further show that these three Noether As expected, the new conformal symmetry under Mobius transformations of the action ( JHEP12(2019)031 (2.42) (2.41) ] that 92 ) at the – 2.2 90 ): . q 2.2 . Its Hamiltonian E t
)-invariant, coincides , τq 0 R , , − φ  ˙ ◦ q 2 0 V 2 N 3 τ N ) structure of the phase that a aa R , = = 3 πG ) generators. However, as shown 0 (2 + 8 L R τ sl , ∂ Q ∂φ d − d ]. In the second part of this paper, = 2 0 N ) algebra. This is a special property to φ φ 98 2 R 3 , , a q  E t ) d , π ˙ q ) can be mapped to the dAFF model. We point 0 – 12 – R τ Z 2 N aa 2.2 ◦ − V ◦ q V πG ( 3 4 ] = 1 2 0 − = , φ 0 = 0 0 τ ): L Q d ∂ ∂a d a, φ N, a, a = [ a S π , q E q denote the derivative with respect to the coordinate time 0 = ˙ φ − τ Q d and d 0 algebra, a subsector of which is the Virasoro algebra. However, because the system a ], it is still possible to find a suitable operator such that its two points function admits ∞ Having presented the conformal invariance of the cosmological system ( At the quantum level, another subtlety of this system is that contrary to 2d CFT, Before concluding this section, let us discuss a subtlety of the conformal mechanics. 81 w where form is obtained byconfiguration a variables ( Legendre transform. We compute the conjugate momenta to the 2.2 Hamiltonian formulation Let us now proceed to the canonical analysis of the cosmological action ( Lagrangian level, we turncompute the now charge to algebra and thewe exhibit Hamiltonian shall the use underlying formulation, as which a will guiding line allows through us the to quantization. the standard form of thepoints CFT two functions point correlator. have The alsodevoted construction to been of the the investigated quantum three in theory,constructed and in we four [ quantum will cosmology, and show the that vacuum,with while the not the desired SL(2 two cosmological pointsnatural vacuum function interpretation annihilated can in be by this cosmological the context. scalar constraint, finding therefore a would be interesting toour understand cosmological how model. this We infinite leave set this of for charges future can work. there be is no constructed vacuum in statein annihilated [ by the three SL(2 an contains only a finite numberthe of charges degrees forming of this freedom, infinitethe contrary set three to are conserved the not charges two independent generating dimensionalkeep and the case, in boil SL(2 mind down when to dealing combination whith of such one dimensional conformally invariant system. It the Einstein-Scalar system reduces to such one-dimensional conformal fieldIn theory. two dimensions, aof CFT conserved is associated charges toconformal generating mechanics a the is Virasoro also 2d symmetry equipped and diffeomorphism. with thus an an infinite It set infinite of was set conserved shown charges which in form [ vanish on-shell: Consequently, the cosmological action ( out that it is nevertheless quite surprising that the homogeneous and isotropic sector of such that their time derivatives are all proportional to the equation of motion and thus JHEP12(2019)031 which (2.50) (2.47) (2.48) (2.44) (2.46) (2.49) (2.45) of the . While b 1 O − L ) Lie algebra R , ] = H (2 sl . . H} , the scalar constraint = 1 , t . In terms of this pair of
} . 3 {O H − # 2 a L b, v − = a { π 0 0 φ ] = O φ 3 b πG π 1 2 N = 1 (2.43) ◦ # . The symplectic structure is given + } . 2 V − = 0 L φ 3 a 3 vb a 2 φ a τ 2 O a = 0 π d π 2 κ φ, π such that d ] = = { φ " − γ t = ◦ π H} 0, with physical dimension [ 1 d . So, for an arbitrary observable , V 2 √ φ ˙ v τ O φ ) to the Hamiltonian framework. π x , ], the volume forms a closed Z 3 π , = – 13 – " 0 d { 2.6 H' 59 a 2 , H N Na ] = = Z = 1 , for this simple system, it is then easy to see that its φ 58 φ = } } = 1 ˙ φ π πG a ] = 1 and [ 4 v H , a ,H , φ, π a a, π = H is not a dynamical variable and plays the role of a Lagrange { {O a 2 N ] = [ a π N = a ◦ = π 1 V t is a constant of motion: N, a, π O [ 3 d H d φ S − π = = 0 b O . Its associated canonical variable is given by the Hubble factor proportional to the Planck length. 3 L πG ] = 12 v √ = κ Let us now look at the evolution induced by this cosmological Hamiltonian. In particu- It is convenient to introduce a new set of canonical variables. We define the rescaled generates the evolution in proper time This translates the continuity equation ( 2.2.1 The CVHLet algebra us look atdoes the not depend evolution on ofconjugate the the momentum scalar cosmological field variables. First, since the Hamiltonian lar, we will see that, astogether introduced with in [ the scalarwill constraint identify and them the as integrated the Noether extrinsic charges curvature. for Moreover, the we Mobius transformations. canonical variables, the Hamiltonian becomes with such that [ reads Its physical dimension is as expected an inverse volume, [ 3d volume as: As usual, the lapse function multiplier enforcing the scalar constraint the Hamiltonian generates theH evolution in thecosmological phase coordinate space, time we have: with the Hamiltonian given by: by the canonical Poisson brackets between pairs of conjugated variables: The cosmological action can then be written as: with physical dimensions [ JHEP12(2019)031 . It b (2.56) (2.57) (2.52) (2.53) (2.54) (2.55) (2.51) ), its scaling 2.8 ) Lie algebra by  R H on the geometrical , 2 6 (2 , σκ z sl v . bv . j 2 − . − = . κ
2 z } 2 2 φ = j = v κ π σκ } , v a 2 − y − . a  x {C π , k k = 1 κ 2 x . 3 2 = 0 κ k C = { ˙ 2 − 1 H σκ z , κ 2 ]. = , ) algebra: = x H − C = 60 −H , j R a k v – , , 3˙ 2  Na − H 2 = H} κ 3 (2 58 H a = 2 v, sl − ◦ −H ) algebra: { H} form a closed Lie algebra: } V – 14 – , R σκ = y , = , = H = 2 y
, k {C + z k (2 v τ z j 2 j H} d d sl − , { γK and v + 2 σκ x = √ {C x , 2 k C ˙ , v x k  observables to the usual basis of the , C 3 y = − 2 v 1 κ d k 3 ˙ 2 κ z H C ◦ j V = = can be identified as the dilatation generator σκ = Z = x } C and = x C := H} v v , k , z C , k v, j C 2 ) Lie algebra, which we shall refer to as the CVH cosmological algebra. { C { κ R , = = 0. is the speed of the 3D volume in proper time: φ y (2 C k sl ]. It is the proportionality factor in front of the 3d volume and the scalar is an arbitrary parameter, also present in the quantum realization of conformal 65 σ is actually the dilatation generator on the whole phase space. Notice that it nevertheless does not We can map the Now, in order to derive the evolution of the geometrical sector, it is useful to introduce C 6 contain matter contribution. The reasonunder the is Mobius that transformation, the but scalardimension remains field being invariant does under ∆ not a conformal transform transformation as ( a quasi-primary field constraint realized as null generators of the We can compute the Casimir of this with the standard commutation relations: The factor mechanics [ alternative models of quantum cosmology in [ This forms an This algebra encodes howtransform the under basic scale physical transformations. quantities, the This Hamiltonian structure and was previously the noticed volume, and used in Then the time derivative of the dilatation generator is the Hamiltonian itself: The system of differentialbrackets equations of closes the and three can variables be integrated. Moreover, the Poisson This phase space function sector parametrized by the 3Dturns volume out and that its conjugated variable, the Hubble factor the trace of thespatial extrinsic hypersurface Σ: curvature Tr(K), integrated over the fiducial cell living in the JHEP12(2019)031 , and which (2.59) (2.60) (2.58) 2.1.3 C 2 φ , π ) unitary v for more R , A (2 ]. The vacuum sl 59 , 58 . 0 . Let us focus on the scale Q  − = 2 2.1.3 Q = 0, we can read the trajectories } is also a constant of motion. H 2 2 − τ µ C ˙ τ Q − ,Q 0 + C − Q τ τQ { 2 − − + 2 H v τ τQ + 2 , – 15 – ) symmetry is the Hamiltonian counterpart of − Q + −   κ R − 0 2 H C , Q 2 κ Q Q  (2 = = = 2 ) algebra: sl 6= 0, in which case the Hamiltonian constraint becomes = − 0 + R ) = } , ) = ) = Q Q Q τ  τ τ ( (2 ( ( sl H C v ,Q 0 ’s are constant of motions, Q { Q ). It is straightforward to compute their Poisson brackets with the 2.8 = 0, would then correspond to a null representation with vanishing Casimir = 0, and we are in the special case where φ H π = 0. For physical trajectories of the Einstein-scalar system, the scalar constraint τ Finally, we would like to show that the Noether charges indeed generate the Mobius We would like to point out that the CVH algebra also applies in the presence of a non- We will see below that this at vanishes, transformations ( basic variables andunder check conformal that transformations they derived earlier reproduce in section the infinitesimal variations of the fields In particular, the NoetherH charges are the initial conditions, i.e. the values of Furthermore, since the for the 3D volume and the integrated extrinsic curvature: These expressions allow to computethat the they Poisson also brackets form of a the Noether closed charges and show admit a simple expression in terms of the canonical variables: show that CVH generatorstransformations. can be mapped onto the Noether2.2.2 charges for Charges the algebra conformal The and Noether their charges infinitesimal for the action Mobius transformations, derived in the previous section vanishing cosmological constant Λ a time-like or space-likedetails), generator and depending in on the presence the of sign anisotropies of as Λ inthe (see the appendix Bianchi invariance I of model. the action under Mobius transformations. More precisely, we will corresponds to the kinetic energymeans of that the cosmological scalar system field. gravity plus Uponrepresentations matter quantization representations would of from be this the described system, continuouscase, this by principal i.e series [ (up to possible quantum correction due to operator ordering). The Casimir is negative and turns out to be proportional to the constant of motion JHEP12(2019)031 − 1) , z j (1 z (2.65) (2.61) (2.62) (2.66) (2.64) (2.63) η su ) group = R ), dilata- , ~ J · 2.20 ~η  ) group elements z R ik . ij , . 2 matrix formed by ∓ x . × ! iλ = = 2 1) i , 0 , and satisfy the } − I } − i  ] = 2 − 0 z − ], this allows to exponentiate , k ) Lie algebra, it is possible to SU(1 , k = a + z R , λ

58 s  ∈ j k , y 2 y , δ { { λ λ λ ~ =
· (2

[ † ~η = y a sl i 2 λ = ~ ˙ d  a ) as described in the previous section. e , δ 2 2 x , λ 1) Lie algebra vectors, M z λ , = , ) flow τ = − a iλ ! y 2.31 (1 t R  ˙ a 2 G + δ , ik but su − τ 1 0 = λM ~ τa  I 0 1 −  − as ˙ a x 3 with ] = 2 i  = 2 k – 16 – y

3 H a form a closed − − 2 z † = = , λ = λ x = = = H }  x λ } } } k and [ ], we introduce the Hermitian 2 , λ ,M , a , a , a C and GMG 58 0 ~ , + − , J ! C y v Q = { Q Q , 1 { { { with ). In particular, as shown in [ iλ  v   − M R , 1 0 0 •} , : ] = 2

~ x J ! · M ~η = , λ z { − z j z e k λ λ [ z + j k

, the Poisson brackets can be expressed in a compact manner as a matrix = y k ) and special conformal transformation ( generators: 1) matrices. Following [ y M , ; we obtain: η R a , 2 2.26 − To this purpose, it is convenient to use the representation of SL(2 This shows how the CVH Lie algebra is directly related to the invariance of the cos- sl x k x This formulation leads tovectors on a the simple matrix formula for the exponentiated action of Lie algebra These matrices square tocommutation the relations: identity, η multiplication: in terms of Lorentzian Pauli matrices, Writing linear combinations of the scalar constraint and integrateelements, the leading evolution to in a proper group time theory in terms description of of SL(2 theas cosmological SU(1 trajectories. the 2.3 Cosmological trajectoriesSince from the SL(2 Poisson brackets of exponentiate this CVH algebra andflows derive in the the flow Lie generated group by these SL(2 three observables as and recovers thetion ( infinitesimal variations respectively under translationmological ( action under conformalformations transformations in of proper the time. metric realized as Mobius trans- factor JHEP12(2019)031 C ) ) , v (0) z (0) z j j (2.70) (2.68) (2.69) (2.67) − − . Since ) algebras. ·} (0) x (0) , x ) R k k , ( ( (0) z {H 2 2 (2 j 2 2 τ η η sl − − − e − (0) x (0) y (0) y k ( ηk η ) as we derive using ηk 1) transformations: , and are thus clearly , + − + ) are the realization of both form . . (0) 2.60 τ 0 (0) (0) x y (0) z v ( τ , j k k (0) Q gives the time evolution of v ) , v τ τ 1 z ,  C −H v λ − τ + ) = and ) = ) = Q . Indeed the Noether charges = 0 and so on. This allows to (0) G η η η = = = τ − ( ( ( τ y x C z H ) and } (0) } x (0) j k k 2 } 2 τ τ τ C λ

τ κ ( κ ( H M H 2.2.2 , 2 2 C , , v , τ , = τ τ τ τ † (0) G τ κσ v ),  { {C {C 4 ) τ − H

i z ( . λ , ) = , v φ − − H (0) = 1): τ κ x π ( C I ), we obtain null SU(1 (0) λ 2 (0) ( N R  2 η H C = M , i H τ τκ ) − 2 – 17 – = z = (2 e κ , 2 : − λ 2 +  C sl τ C − ) Lie algebra, just as the original phase space variables κσ (0) ! x (0) 2 (0) R λ , ( H C v , C − (0) − (0) z τ/ (2 2 j k τ κσ H sl 4 = τ τκ i = 0 ) = η (0) + (0) ) = ) = z − j + k τ in the present cosmological setting. τ τ e ( observables and the Noether charges ( ( v H C − (0) 8

v C H = ) H H = = = z ) generators under the Hamiltonian flow generated by the scalar constraint

= 0, the Hamiltonian always vanishes and the expression of the , τ λ R C τ = − G τ from which we get the trajectories of the cosmological observables τ , , x v H C (0) v 7 λ (2

H ( ], still form a H sl 2 η i ·} , − e {H ) is a null generator in = τ z ) is a nilpotent matrix with vanishing square. Acting by conjugation with j z − ! λ ) − are identified as the initial conditions ) η denotes the initial condition for the volume at η x − ( 0 ( k z − 1) generators, x Q ( j k , (0) λ v ) 1 ) κσ (1 η . We refer to it as the dynamical CVH algebra for FLRW cosmology: 2 η ( ( Restricting our attention to physical trajectories, thus imposing the scalar constraint and su H z + The evolving cosmological observables, generated by the exponentiated Poisson action of the scalar The evolution of the j 8 7 = k is quadratic in the flow parameter 

and This explains why both the constraint exp [ H trajectories further simplifies. The dilatationphysical generator cosmological becomes trajectories a constant and of gives motion along the speed of the linear growth of theThe volume: value of the dilatationby solving generator the is Hamiltonian fixed constraint: by the scalar field momentum, up to a sign, Q constant of motions. Thenevolving the constants of expressions motion for on initial conditions, where recover the exact same formulathe for conformal the Noether cosmological charges trajectories in ( the previous section the in proper time (or equivalently in the gauge H where ( this group element on the generator matrix, We apply this formula to the Hamiltonian flow of the scalar constraint JHEP12(2019)031 R ∈ t (2.73) (2.72) (2.71) ]. This 65 metric component used on an internal ) structure and thus rr φ g R , . While we only briefly , , the integrated extrinsic (2 2 v 2 sl r , the , d ∞ 1 . (0) − + 0 v φ  C 2 2 → + κ + ` r . r τ  )  C . 0 2 (0) 1 + v κ φ 3  − + geometry. Using global coordinates 0 φ geometry. As we are going to see, a rather = ( v + τ 2 κ 2 φ 2 C v  t π 2 e d κ – 18 – = , endows the cosmological phase space and tra-   (0) 2 2 v H ` r ln H} ) structure of the cosmological system. We have seen = 1 κ R φ, v , 1 + {   (2 = = sl − φ φ τ = ∂ 2 s d ) group structure generated by the Noether charges resulting from metric reads R 2 , geometry that the cosmological action can be recast into the dAFF model [ correspondence as a potential candidate for the holographic dual of the two 2 as the cosmological phase space and the scalar constraint ) structure is a powerful tool at both classical and quantum level. Indeed, it , the AdS 1 2 C R + 2.1.4 , is the AdS curvature radius. Contrary to higher dimensional AdS spaces, this 2d R (2 ` has known a renewed interest over the last years with respect to the low dimensional CFT ∈ sl / 1 r To summarize, the CVH algebra formed by the 3D volume 2 where version has two time-like boundaries. Indeed, taking relates the cosmological singularitymention this to link, the it would null be infinity interesting of to2.4.1 investigate AdS it further AdS in theLet future. us review theand basic properties of the AdS CFT AdS dimension gravity side. Frombetween this our relation, cosmological model one andintriguing could the relationship wonder AdS if can there indeed is be any established relationship at the level of the phase space, which 2.4 AdS Before closing this section devotedinteresting to side the product classical of theory, the wein would section like to stress one more the invariance of theThis original action underentirely characterizes Mobius the transformations model. inthe In the conformal particular, preserving invariance proper this at time. quantization the ambiguities, as quantum we level will will see provide in a section stringent criteria to fix all This describes the cosmological evolutionclock, in with terms two of regimes the corresponding scalar to field contracting or expandingcurvature trajectories. jectories with a SL(2 We can write it inthe a invariance of deparametrized the fashion theory by under getting time rid reparametrization: of the time variable, reflecting does not provide themotion by evolution hand, of the scalar field. We have to integrate its equation of The sign decides if we are in a contracting or expanding phase. Unfortunately, this method JHEP12(2019)031 : τ = 0 r . The (2.79) (2.76) (2.77) (2.74) (2.75) (2.78) 2 2 X . , d ∞ − + 2 1 X → d r − 2 0 X , = d , t ` 2 t ` s geometry is defined as the one- . cos , , sin 2 , 2 2  2 2

 ` r 2 ` r 2 2 2 2 ` u η η u d = for massive particles for massless particles 1 + − + 1 + 0 corresponds to the and Poincar´epatch 2 2 + 2 2 r r ` ` = 0 for massive particles 2 X ` = 0, the geometry reduces to a Minkowski- ` η ), the AdS 2 u > 2 2 r r d + 2 ` = =
2 u u 2 1 τ t ` u t ` = 0 for massless particles + ,X 1 − r , – 19 – X 1 − − 2 2 r r 1 + 1 ` 2 2  τ τ = sin cos   + ) with ,X d d d d 2 2 ρ ρ 0 ρ 1 1 u u 0 ` `uη ,

X X η, u = = = = 1 sin − ) is obtained by using the coordinates 2 1 2 0 sinh cosh cosh s ` ` ` − X X X d , this provides again a 1d time-like line. 2 r 2.73 = = = E and 0 1 2 τ √ X X X ` E 1 ) = ) = 0 is an arbitrary constant. The solutions, with initial condition − τ τ ( ( >  r r 2 2

E ` r , while massive particles following time-like geodesics always follow bounded geometry is to start with its embedding in the three-dimensional Minkowski 1 + r  2 E = t τ = 0, are given by Furthermore, an efficient method to explore the various foliations and parametrizations The geodesic equations gives the propagation of test particles on this 2d geometry and d d τ leading to the line element is defined by Another useful coordinates system ( with the induced metric inheritedstandard from form the of flat the 3D metric metric, ( d trajectories and cannot approach the time-like boundary at asymptotic of the AdS space. Calling the 3Dsheet coordinates hyperboloid: ( So, only massless particleslarge following radius light-like geodesics can propagate towards arbitrary where the energy at like line element and at fixed are easily written in terms of the proper time (or affine parameter in the massless case) vanishes, leaving a 1d time-like line. Then, at JHEP12(2019)031 2 2 2 sl . . For (2.83) (2.81) (2.82) (2.80) H b and ). Putting and , we have C φ v , π v , b, v φ ) group transfor- , -axis at vanishing R b y φ, π , k = , valid on the half-AdS C y ). The k , 1 b − ∂ v , 2.79 ) using the triplet of variables 
= z 2 . j . b, v v with the hyperboloid radius given , 2 AdS . Lorentzian metric and discuss the b ` 2
2 ). y − parametrizations is simply given by 2 2 b 2 φ 2 b k + z d κ π x 2 d 2 AdS 2 2 k ` and b geometry, we now show that the cos- − v −  + d − x 2 2 x − = k 2 − 1 k σκ = 2 2 η v d 2 presented in ( v + d 2 y d k z 9 − − bv∂ = j 2 2 – 20 – 2 z − j v 2 AdS H H − C 2 = 2 2 x ` , ` v z k b v 2 metric 2 = d = κ b∂ = − 2 2 2 2 2 z , − s s − j s
v d d d z j , from which we can deduce the relation with instead, it takes the simpler form of a conformally rescaled y + k , v∂ x b /u = k ∂ 2 3 = 1 X z σκ aside and working at fixed scalar field momentum = and φ v x . The correspondence with the AdS 0, gives the 2D metric: k as the phase space metric /κ 2 = φ v > 1 π = 0 represents the cosmological singularity, but also defines the configuration X = v , ) satisfying the quadratic condition: z ) algebra. j H R We can write this in terms of metrics on the phase space. We start with the flat The cosmological phase space for FRW cosmology coupled to a homogeneous scalar AdS , = We can compute its Killing vectors and check that this is a maximally symmetric 2D manifold with ` 9 , v, 0 (2 C space of a quantization in terms of wave-functions Ψ( three Killing vectors: which we recognize as the Hamiltonian vector fields for the three observables respectively patch with which is nothing elsevolume than the AdS and impose the Casimir equation by X instance, re-introducing the variables the quadratic conditionsl above is simply understood as3D metric: the Casimir equation for the Moreover, we have shownmations that on this the triplet cosmological ofgenerators phase variable in generates the space. standard SL(2 basis, Mapping this triplet of variables with the the scalar field parametrized the two-dimensional gravitational( sector ( mological phase space caninterpretation be of endowed the with physical the cosmologicalpoint trajectories of AdS view. and the singularity from the AdS field is a four-dimensional manifold parametrized by the four variables ( two-dimensional Minkowski space-time: 2.4.2 AdS Having review the basic properties of the AdS Using the coordinate JHEP12(2019)031 ] 82 (2.84) ): H ) algebra , R C , . can be iden- v, 1 + (2 φ , u π − sl . First focusing − 2 b . + u − C H u 2 . φ − κ AdS π u 1 + α correspondence. ` γ ! 1 + = = γ = − − y − − ` coordinates are the ones that − k 0 is mapped to the bound- . β α βu  δ δu /CFT metric is conformally equiv- − u − 2 − ,B → 2

u 1 2 − v geodesics corresponding to the − − u 7−→ = v 2 AdS 2 1 − = = = ` − } M , = } − t } , 2 − z − ) , u − + + u metric leads to a mapping of the cosmo- , u − H u u d u C, u v, u 2 manifold with constant curvature, defined 2 − + − { { { u + u u

2 − C v metric is given by the light-cone variables (see e.g. [ space-time). These + with φ 4d κ 2 u , 2 π 1 + − (1 + – 21 – = = 2 + , = u ) + x , + 1), with the AdS ` 2 R u − k ,  s , is interpreted as the lapse. The second term cou- 2 d . The first term generates time evolution according = = 1 = B C H } d C , u SL(2 η SO(1 } + } η δ β / making explicit that the AdS ∈ + − + ) , u 1 + + + 2 ! R H − z and + , v, u 2 C, u + v v d ) group elements. More precisely, the geodesics are drawn by { { { v 

γu γ δ α β η αu R H , ,

2 + + , u . Among the different coordinates choice, the Poincar´epatch is − into z u 2 AdS geodesics to physical cosmological trajectories u = 7−→ z ` y 10 − v j H k + 2 u − M  H u = − + η x u 2 ` k 1 + s d = = at null infinity. z  ` ~ )-action is generated by the Poisson brackets with our triplet of observables ( ) symmetry of FLRW cosmology in the context of the AdS j ) action generated by exponentiating the Hamiltonian flow of J ≡ u 2 R R · R , , , is proportional to the cosmological constant, while the third term generates scale ~η v η So the gravitational phase space at fixed matter conjugate momentum Different coordinates choices allow to provides alternative intuitive picture of the cos- Another interesting compactification of the AdS 10 This pair of light-cone variablesof would the be SL(2 especially relevant when investigating the possible reformulation These SL(2 In these variables, the metric takes an extremely simple form: which is invariant under the Mo¨ebiustransformation: for a review oftransform various under parametrization Mo¨ebiustransformations: of the AdS transformations. 2.4.3 From AdS Endowing the cosmological phase withlogical the trajectories AdS generated by the scalar constraint to geodesics of AdS flows generated by SL(2 the SL(2 vectors to the scalarpling constraint In these Poincar´ecoordinates, the cosmological singularity ary of AdS tified to the Lorentzianas 1+1-dimensional AdS the group quotient SL(2 mological singularity defined by switching alent to the 2d Minkowski metric: JHEP12(2019)031 ) 2 t 1 − ` using the tan( . This fits ` 2 . B 2 = = r z = 0. These geodesics B , depending on whether becomes = z

A B 2 s = z , or equivalently 2 AdS ` ). Another way to derive this is to see = 2 b 2.75 2 v . This null ray ). Then these physical cosmological trajectories – 22 – AdS ` 2.73 6= 0. As shown in appendix ) quantization as given by ( R 2 , = 0 implies that ) algebra. Following the previous discussion, the phase space phase. It is interesting to notice that, because of the infinite R H 2 , , such geodesic can never reach the boundary of AdS . From this perspective, it would be interesting to see whether 2 2 (2 sl ]. parametrization ( up to the cosmological singularity at null infinity in AdS constant equal to ) algebra, so physical trajectories for such universe should correspond 81 2 2 R H , X (2 = 0, the scalar constraint can be re-written as a space-like, (respectively time- sl y > k = C 0 or Λ As any canonical quantization scheme, the definition of the quantum theory faces Let us now consider the case Λ It is tempting to view a nearly singular universe as a massless test field approaching < for systems invariant underthe time-reparametrizations: quantum theory, the onetimeless then time observables has coordinate and to operators. disappears reconstruct in a time flow, orambiguities. internal However, clock, the from unexpected the conformal symmetry of the FLRW action provides The construction of thestraint quantum theory is is quantized straightforward. intoKlein-Gordon-like The equation the Hamiltonian relating Wheeler-de scalar the variation con- Wittgravitational of degrees (WdW) the of wave differential freedom function andtion equation, with the (with respect which no matter to explicit is degrees the time of a derivative) freedom. reflecting the It well-known is problem a of timeless time equa- occurring cedure to implement the newlywe found conformal then symmetry. show With how thisoverlap simple the for quantization, the standard evolution CFTclose of correlators analogy coherent with emerge wave-packets. [ in This quantum last cosmology proof3.1 as will the be Wheeler-de developed Witt in SL(2 3 Quantum cosmology andWe CFT turn correlators now toquantization the of quantum this theory. simplecanonical cosmological Different Wheeler-de system. Witt approaches (WdW) can In quantization be scheme the and used following, revisit to we the perform shall quantization the pro- focus on the like geodesics on thepotential AdS barrier in AdS well with the factconstant that curvature spacetimes. AdS or dS flat cosmologies are actually free of singularity, being universe approaches the singularity and the fate of theΛ cosmological singularity. like) generator in the trajectories of such AdS or dS flat cosmologies would correspond to space-like or time- in the original AdS follow the flow of the null-infinity of AdS numerous investigations and results(among which concerning chaos properties) the could physics be relevant near to discuss the the regime boundary where a of contracting AdS erator in the to geodesics of photonsthat on the AdS scalar constraint Poincar´ecoordinates, in which case the line elementare vanishes at d on the case of a vanishing cosmological constant Λ = 0, the scalar constraint is a null gen- JHEP12(2019)031 (3.2) (3.3) . We + , which 0 b and scalar → v  with  ) Lie algebra, ensuring ' R ] and that this suggests , , into finite translations on and find a suitable factor v 93 in the scalar constraint. In b (2 ˆ Ψ . Let us quantize this in the φ H 1 v , sl i∂ Ψ , which generates rescalings of πG φ − i . and vb 12 v √ = 1 (3.1) ] = , , φ } = C 2 ˆ φ π Ψ = κ ) depending on the volume vb Ψ = φ ˆ φ, ˆ 2 b φ π [ 2 φ, π κ v, φ { − becomes a legitimate quantum operator acting 1 2 φ v is not a trick to avoid the singularity. Indeed, – 23 – − 2 π v + ∗ = = 1 , R } H , Ψ ∈ v Ψ b, v v v i∂ i , { ] = ˆ v , to the dilatation generator v Ψ = Ψ = b, ˆ ˆ v b [ ˆ [. The inverse volume ∞ + is natural as we switch our focus from the canonical observable , ]0 + ∗ ∈ R . The absence of singularity would amount to showing that no semi-classical v + ) structure and study its reformulation in CFT terms. Let us nevertheless remind . The canonical variables are raised to differential operators: 0 φ R )-polarization i.e. using wave-functions Ψ( A remark is in order before going any further. The restriction of the volume to the Let us start with the FLRW phase space, provided with its canonical brackets, , → (2 . It is interesting to note that switching between momentum variable and dilatation v, φ underline that the goalto here identify is a not Wheeler-de tosl Witt discuss quantization the of fate the ofthat FLRW the the cosmology cosmological Wheeler-de compatible singularity, Witt with but resolution, quantization the and is more typically refined not quantization schemes enough have to to capture be the used to singularity discuss the quantum generator is the simplesta way shortcut to to derive derivingalso the the mention Unruh temperature that radiation associated restrictingit [ to to does the not FLRW remove space-time.v the We singularity should sincewave divergence packet and can accumulation evolved can such that still it occur ends as up peaked around the real line. half-line generates translations in v that the algebraic structurelevel. of The the subtle conformal technical pointorder Noether is to charges avoid the that is inverse this volume carried becomeshalf-line factor to an issue, the we quantum workby with multiplication, wave-functions on but the we positive give real up exponentiating the operator The next step isordering to ensuring quantize that the the CVH three operators observables still form a closed with the Planck length (up( to a numerical factor) field and the scalar constraint for flat FLRW cosmology coupled a massless scalar field: WdW quantization in the lighttransformations. on this This new invariance providesfunctions of a FLRW of cosmology preferred the under factor Mobius inner universe ordering product. solution and to leads the to WdW physical equation wave- and provided with a physical a useful criteria to narrow these quantization ambiguities. Below, we revisit the standard JHEP12(2019)031 ) R for , (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) = 0 is 11 µ . on their . φ = 0  p Ψ , ] = 0  b b H v H and in , ∂ v φ . + π . b C 2 2 v , , ) iκ v v∂ = 0 2 v ] = [ˆ  b ln C Lie algebra as wanted: 2 v∂ ψ , 2  2 2 κ 2 φ 2 p κ ] = + κφ π sl ( + leads to the normal ordering for the b . + + H 0 ip 2 and in particular properly define φ 2 φ e . ψ ∂ ] discusses a polymer quantization ∂ v, ] = [ˆ v = 0 [ˆ ¯ ΨΨ H v v b v Ψ 1 1 2 58 2 ψ φ , 2 φ ). The fact that the “physical” case v∂ ) = ) d with plane waves, we are left with a φ − ∂ − φ v π µv + and ( µv [ˆ v φ = d 2 , p 2 = 2 1 1 b v π v ψ n √ χ b κ − 2 v , H b b ) R H H (2 i 2 ∂ C v p 2 ( ip = − , 2 v  p n  1 4 Ψ = ) for the geometrical sector: + ( i I ψ ] for details on the representation theory of the , 0 v – 24 – v Ψ = 0 is a singular case of the more general eigenvalue and form a closed ] = , ψ v ( ∂ Ψ − ∝ b v | 0 96 b H  H ψ ) – 2 2 φ Ψ ) Lie algebra. ) = , 1 2 iv∂ v v∂ h ˆ κ ( π b R : C ¯ 94 ΨΨ [ = , + Ψ + ψ on + b , , − 2 H v v, φ φ n (2 ψ p v b ( C 2 v 12 d sl = v∂ ∂  p or [ v v∂ 2 iκpφ 2 Ψ  v d b e κp χ C i B ˆ v , R Ψ = 0 now becomes a 2nd order Wheeler-de Witt equation: i 1)). − = v , = b , . In the general case, the differential equation reads: ) = H = i p b = C C , φ ˆ v i χ 0 ( n algebra is given by the energy-momentum of the scalar field: v b ∈ p b φ v H ] = + Ψ 2 ln ˆ v b | SU(1 χ π µ , +

sl ip Ψ ˆ C ∼ h b [  ), and diagonalizes the two differential operators in v , H ) e φ ˆ v Ψ for ( R h = µ = , χ 2 2 b v ) 1 κ κ ip v (  − v ψ Ψ = = b H 2 term fits well with the Casimir formula for the continuous series of unitary SL(2 ) = ) = sl v 1 4 C ( The scalar constraint Let us now quantize the three observables The choice of scalar product of course changes the operator ordering ensuring that the operators are The case of a vanishing scalar constraint  p v, φ due to quantum corrections. 11 12 ψ b problem This is solved by modifieda Bessel singular functions limit begs theH question of the relevance of possible energy off-set to the scalar constraint operator Hermitian. For instance, theCVH scalar operators: product whose commutators also form a closed This Euler differential equation is easily solved by complex power laws: The standard way to solveΨ( such a Klein-Gordon-like equationown. is to Diagonalizing separate the thesecond scalar variables, order field differential equation momentum representations (see appendix Lie group SL(2 The Casimir of this The This factor ordering isthe the scalar unique product choice ensuring that the operators are Hermitian of FLRW cosmology, preserving thea conformal resolution symmetry of at the quantum singularity. level and leading to a Hamiltonian constraint operator UV completion of gravitational system. For instance, [ JHEP12(2019)031 ] ) ) = R , 3.5 i 59 0 , (2 (3.12) (3.13) (3.10) (3.11) Ψ 58 | Ψ = 0, sl Ψ b . On the H h  p = 0. H and the negative . , and to shift the + p ) b ) commutators ( R v, φ , ( and 0 while the scalar product (2 v , , sl + p )Ψ ) ) 0 0 ). One would thus quantize p p v, φ 3.6 − − are now either ill-defined or not , p p . This path described in [ Ψ( b v ( ( ) b ) δ δ H p 0 v = = − 0 0 0 − . Having to distinguish between positive  p  p and p 0 ( ) is still well-defined and Hermitian. This v Ψ Ψ b δ C 1 2 κφ  p  p , : − ) representation determined by the value of = : Ψ Ψ + appears to be redundant. In this situation, ) action on the Hilbert space then gives the b v φ ) ) v − p R ve R 0 0 φ , ( , v – 25 – φ v∂  phys ( i φ δ i − − 0 d  p is not a Dirac observable. In particular, it does not and v φ v = ) Casimir equation ( ( or over Ψ ( v d | v ˆ R δ C  p v , 1 v Z Ψ φ κδ (2 h φ provide a basis for the physical cosmological states solving κ d sl d . The SL(2 v v v φ d d = 2 is adapted to the positive modes Ψ π Z Z  phys and ln + phys are by definition orthogonal. A less ad hoc prescription based on i 0 φ p − , and the Ψ | h·|·i b H Ψ h is classically a Dirac observable on physical trajectories at C and to a physical inner product suited to the solutions of the WdW equation. One b C 0 , b v Ψ generally does not vanish, as we can check for all basis solutions Ψ is adapted to the negative modes Ψ ¯ ΨΨ b v φ b H d This realizes the canonical quantization of FLRW cosmology, respecting the Let us conclude with a remark on the action of the CVH operators on the space of The original scalar product is sill-defined on those solutions of the WdW equation. − phys v d focus entirely on the CVHbetween variables. One would startthe with system the directly asthe an scalar irreducible field SL(2 momentum integrated flow of the three basic operators then other hand, the dilatation operator works because symmetry induced by the invariance under Mobiusof transformations. the A different quantization perspective would be to forget the original variables physical wave-functions. Operators suchHermitian as anymore the with volume respect toonce the physical we scalar remember product. that This thelegitimately is volume act not a on deep the problem space of solutions to the Wheeler-de-Witt equation: if which does not depend on theand arbitrary constant negatives modesmodes is when similar solving the to Klein-Gordon distinguishing equation. between positive and negative energy The scalar product h·|·i where we assume that theby positive wave-functions sector Ψ spanned bythe wave-functions Ψ classical analysis ofto symmetries, the equations integration over of the motion classical and trajectories: Dirac observables amounts typically fixes the integration over the Hamiltonian scalar constraint.contracting The and two expanding possible phases signs of reflect the the classical existence cosmological of evolution. Actually the the two two integrationsit over isR understood that one should switch from the kinematical scalar product These plane waves in JHEP12(2019)031 (3.14) (3.16) , which i Ω | and focus on φ , ) satisfying the b v b π τ . )-invariant states. ( R ∆ , and O (2 ) at two different times b ] = 0 C τ sl ∆ , ( ) operators, their action b O H R O , ]. In a conformally invariant b v, [ , (2 81 ], such a vacuum state can not sl 2∆ 81 | ) and there does not exist such a 1 τ , i 1 = 0 (3.15) 3.6 ). − ∆ i v 2 ( O Ω τ | | ψ b v = = i ] = ∆ i Ω ∆ | Ω ) – 26 – O 1 b Ψ = 0, will be especially relevant when the scalar C| , τ b b ( C H [ = O i ) 2 Ω τ ) and consider all the operators as acting as differential ( b v H| ) Casimir equation ( , i ( |O R ψ ∆ , φ Ω ˙ h φ O (2 iπ on the wave-functions sl e v ] = , and in particular how CFT correlation functions emerge from the ∆ 1 ) = O = 0. This symmetry applies to the flow of all observables on the full , b H v, φ [ H i evaluated on this vacuum state are given by 1 τ In the following, in this whole section, we will fix the eigenvalue of Let us start by describing how quantum cosmology could be recast in CFT terms, Here we would like to show how the Hilbert space for FLRW cosmology carries a and 2 time evolution of the dual of coherent cosmological wave-packets. the dynamics of the geometricalwave-functions Ψ( sector of the wave-function.operators This in amounts the to volume considering and so on the higherexist order due correlators. to As the underlinedprimary non-null in operator. [ A method wasfrom nevertheless quasi-primary outlined operators how acting to on recoverWe non-normalizable the follow and CFT non- the correlators same approach here and further show that it corresponds to looking at the then the two-points correlation function ofτ the primary operator following commutations relation with the conformal generators Now imagine starting with a normalizedtherefore and satisfies conformally-invariant vacuum state representation of CFT quantum cosmology setting. following what was done for conformalfield quantum theory, mechanics [ a primary operator would be defined as an operator operator. Been ablesolution to to the study present WdW theconstraint equation will whole acquire space extra terms ofinteraction terms either states, such by gravity and adding modifications extra not or matter focusing a field potential only or for by on the adding scalar the extra field. It is important to keepis in mind valid the off-shell, conformal invariance withoutscalar of constraint the necessarily action restricting of FLRW tophase cosmology space “physical” of trajectories the satisfyingcharges theory. act the At on the the quantum whole level, kinematical this Hilbert means space that before the imposing conformal Noether the scalar constraint provides with the wholeand toolbox eigenvectors. of It is thein actually standard this the basis purely next of algebraic section pointand in of realize order view the to that CFT connect we 2-point with will function the take as below conformal3.2 a field cosmological theory wave-function overlap. (CFT) CFT framework correlators as wave-packet overlap in quantum cosmology is ultimately equivalent to the WdW quantization described above, but it nevertheless JHEP12(2019)031 . C and ∈ (3.17) (3.18) (3.19) (3.20) (3.21) b v 2-point vaccuum 1 = 0. This 1 i Ω b H | , = 0 (∆) ), we can diagonalize , (∆) the interpretation of i ζ Im 3.4  − (∆) , 3 ζ (∆) | , = b ) algebra, which will lead to C ζ v i vaccuum state. b 3 σκ . H R ∆ v ) 2 , 1 3 σκ 1 + : b τ Ci v − (2 σκ 3 h − C 2 e 2 sl τ b v 1 − ∆ + , ∈ ( σκ i e − i 2 e 1 2 − | , − (∆) = 1 2 (∆) ∆ ζ † (∆) | Re  v b 2 ζ ). (∆) A b h H v 1 + ∝ τ iτ Re ] is the speed of the volume in proper time. ) = ∞ e v − 3 ) Lie algebra structure of the CVH operators b + (∆) i H ( – 27 – 0 0, i.e. if the expectation value for the volume is , 2 ζ R := τ v Z , σκ 2 b v, b C (∆) [ τ (∆) i , giving the states > i i ζ (2 b v∂ ζ | C − − = 2 sl i ∝ for a specific choice of ∆ defining the CFT (∆) τ 1 ζ 3 i (∆) τ (∆) 1 | = ⇒ i ζ (∆) − b v h | ζ σκ b as differential operators given in ( (∆) Re C | i ⇔ 1 2 (∆) ζ τ b | C ζ | b v (∆) (∆) | − ) := := ζ ζ 2 | 2 h (∆) b τ and A (∆) , τ | ζ h 1 ζ b 0. v h τ = ∆ ( (∆) stands respectively for the real and imaginary parts of ∆ > i = G ∆ Im ∆ i (∆) i b v ζ b v | h for an arbitrary complex eigenvalue ∆ h b A A . Then we choose the highest weight for the (∆) and H Re The proof relies mostly on the We will show below that this quantum state overlap is exactly the CFT We are interested in the time evolution of those coherent wave-packets: Let us consider the operators physical state turns out to play the crucial roleand of on the CFT their resultingoperator time evolution underthe the expected result. flow generated by the scalar constraint which depends only the time difference ( function and decrease as state. Moreover we will formulatelike this operator two-point acting function on as the the state time correlation solving of the vertex- Hamiltonian scalar constraint and more precisely in the overlap between the same state evolved to different times: which converges if andstrictly only positive if 2 where Note that that states are not necessarily normalizable, The eigenvalue ∆the actually integrated givescoherent extrinsic the wave-packets: curvature expectation values of both the volume the operator where the dilatation generator Using the expression of JHEP12(2019)031 , b H ) Lie σκ (3.22) (3.26) (3.23) (3.24) (3.25) R , − ) Cartan 3 (2 R sl b v , σκ 2 (2 . , i sl . = z . ∆  j ˆ z . | i ˆ j 2 k ˆ y b H ∆ ˆ k −  | i , , σκ b H operator, i.e. evolution b C b . H 2 , . +2 z ] = σκ j ˆ b ] = e z z H − iκ j j ˆ ˆ σκ ] = +  2 +2 ˆ k gives its time evolution. In = x ˆ k e κ , ) Lie group flows. This gives + ˆ k , = = b i 2 b H + 2 ] = A , z R 3 τ ) ˆ k j ˆ z , b 1 b [ [ H H j ˆ b τ (∆) b v H [

: σκ ζ − σκ † | 2 b v, 2 b , 2 σκ C − [ b τ A 2 ( − b i ) structure = H +2 e e τ , τκ † R x b , H b b , ˆ x k , , and A e A , + b with ˆ k σκ H b b b i b (2 b b H H b b H C − v H A , H +2 i sl y σκ σκ σκ e = = = σκ 2 ˆ k − | i + − +2 b ] = + + H ∆ b e e b = † , our specific choice of operator ordering ensures H H z ∆ – 28 – h operators to eigenvectors of the | iτ b b ] = j ˆ b A A iτ C iτ , − b b = − H − A x e 3.1 ) = , e = = i ˆ k , ) = b [ b C b = ∆ b H C v e ) [ are thermal evolutions of the − + 2 z i b b b (∆) σκi τ † ˆ ˆ k k H H H j ˆ σκi ζ 2 b

∆ | A iτ iτ iτ | − − e e e , z x 1 j ˆ z (∆) τ ˆ k = j ˆ b v , ζ ( and = +2 i i h , τ ) = b ) = ) = ( τ − A y τ 1 σκ ( † τ τ ( ˆ k 2 ( b b ] = ( i i ⇔ A A b b ) structure of the CVH observables carries on to the quantum level. ) := b v b C v H ] =

2 ,

=  y R b (∆) C , ˆ k , τ x [ ζ b , H | 1 ˆ k (2 x τ ( ˆ k sl = [ , : G ) commutators: ) = ∆  b z A R i ˆ k j ˆ ) Lie algebra standard basis is a simple linear map: , R + : (2 (∆) , z x ζ j ˆ sl | ˆ k (2 ( b A 3 sl We are now ready to compute the overlap resulting from the time evolution of the σκ = b eigenstates of This allows to mapoperator eigenstates of the Plugging these expressions inparticular, the this definition shows of that thein operator imaginary proper time: This Lie algebra structure isthe immediate to same integrate proper to time SL(2 evolution for the quantum operators as the classical trajectories: It is convenient to writealgebra, in in order terms to of connect the with lowering the and operators raising operators of the to the v with the As we have seen in thethat previous the section classical The mapping of the CVH algebra, 3.2.1 The two-point function from the JHEP12(2019)031 ) R , (3.32) (3.27) (3.28) (3.29) (3.30) (3.31) . . i φ ). κ ∆ . π | i b H 3.28 ∆ := | σκ s + +2 ˆ k i e b H b , we compute its time = 0 leads to the elegant , C ) i i (∆) i i ζ b σκ H | with , . ∆ ∆ (∆) τ | b i i iτ H +2 ζ j, 1 | e | + , iτ , iτ b ˆ ( k . H e . b − + H e  (∆) b
ˆ k H 1 4 | σκ ) i ζ 1 4 b ∆ + 1 ∆ ) Casimir, as expected from the H σκ h σκ (∆) τ ∆ − ) leads to the result ( +2 ( R σκ j, j, + − (∆) e +2 b | | Re , operator gives a simple expression: 2 C e | z 2 2 ζ G, +2 s 1 2 1 2 j ˆ b s (2 − e 3.27 τ A b ∆ i − h C| − ˆ k ) := −  1) sl b + b C − H τ ∆ † − (∆) | = b ( − b H + 1) − iτ i ∝ A † | b 2 e b i − h G H j j A σκ b ) Re | = = C b (∆) H ∆ τ (∆) +2 i | b − σκ ζ A ζ − | b σκ (∆) H h ∆ b – 29 – i i ζ A e +2 +2 (∆ + (∆ h ) generators on eigenstates of the Cartan element ˆ = +2 v σκ j, b (∆) − e i 0 iτ | 1 2 1 2 H − , = ( b R ζ H ) ) i e +2 G h , i = = j j σκ e τ ∆ − i − + b (2 b b C H H + + 1) ˆ − i k (∆) τ∂ j, b sl ) = z | iτ j b H | σκ ζ is a highest weight vector, H j (∆) ˆ leads to: | ( τ e ˆ v ∆ ζ i ( j b +2 σκ h | b H = C e 13 b i 2 ∆ G b H iτ b C +2 | 1 = (∆ + = = ∆ = (∆ C − h i κ b e i e H iτ i i i i ) leads to a negative b e − H G b σκ , this determines an irreducible unitary representation of SL(2 ∆ ∆ ∆ ∆ C τ ) Casimir of the CVH algebra is determined by the scalar field | | 3.6 = j, j, j, j, B +2 ¯ ∆) | | | | R e 2 y z , C − + (∆) (∆) | ˆ k j ˆ ˆ ˆ ζ ζ k k (2 ∆ h h − i h i sl 2 x ˆ k = = = ) irreducible representations and lowest weight vector R , it also determines the lowest weight of the corresponding representation as − G = +(∆ + , φ 2 τ and the action of the z π j ˆ G τ∂ τ is = + τ∂ 2 sl 2 1 C − Expressing the dilatation generator back in terms of the are: 13 = z which leads to: Plugging these into the derivative of the two-point function ( j As reviewed in appendix from the principal continuousj series. Such representation are labeled by a complex spin momentum we review below. 3.2.2 SL(2 The Casimir equationclassical ( theory: which integrates the expected simple behavior for the two-point function: Finally, since the A few algebraic manipulations Thus assuming that the state equation: derivative: Writing this in terms of a single time label ˆ JHEP12(2019)031 ) R , i ∝ = 0, i (3.36) (3.35) (3.33) (3.34) (∆) 0 ζ | ∆ j, . | (∆) ζ + φ h κ ˆ k π i ) representations ,  i R 0 , 1 2 ∆ − | ) operators in SL(2 b R = H , is . 0 σκ (2  i +2 sl ) admits eigenfunctions for all e 1 2 0 b H ) to the overlap of the evolution z ) − (∆ . 2 j ˆ 1 ζ τ 3 | = with ∆ , τ − v 1 b . This vector satisfies 2 σκ 0 H τ τ 2 , we get: 1 ) ( ( 1 φ i − τ e π G e − , b 1 . H φ 2 κ i τ − π 1 0 ( τ i σκ i iκ e ∆ , reflects that the state is non-normalizable. | 1+ | ∝ +2 + ) + e 0 − ) 0 0 | v 1 2 τ 0 – 30 – ( (∆ ], the operator − + 1) i.e. ∆ → ζ ∆ G = ∆ h h 97 j ) = τ i = ( v is self-adjoint means that the eigenfunctions for real 0 = = ( 0 ) − z i , we have already computed the expression of the state ∆ 0 | j ˆ 2 does not affect the scaling of the time overlap function. v τ z (∆ ζ or j 0 and is actually a non-normalizable state ˆ | two-point function ζ ] obtained for irreducible unitary SL(2 /κ 1 j 1 τ φ < 81 ζ π h . 3 . These are generally non-normalizable states, they live in the ) and leads to the expected decreasing behavior for the overlap: , = + = C i ∞ σκ 0 ) s ) = . In order to truly understand the mapping of quantum FLRW 0 ∈ 2 i − 3.30 ∆ 2 ). Choosing ∆ (∆ ] = , τ τ 3 ζ 1 ζ = | | τ span the Hilbert space of quantum states and form a decomposition of 1 ( 0 3.25 τ ∆ ζ i R G h b v v/σκ h ∈ = ∆ − i ) 0 exp[ (∆ 2 ζ earlier in ( for the mapping of FLRW cosmology to a CFT | − i i b v ) A 0 As for now, we have written the two-point function as: Merging with the result of the previous section, this determines the eigenvalue ∆ In this context, our proposal is to use the (non-normalizable) highest weight vector 0 wave-functions. The fact that ∞ ∆ (∆ 2 + 0 ζ j, As a wave-function of the| volume evaluated on the states: We have to identified the CFT in proper time cosmology to the CFTtwo-point framework, correlation we of an still observable need insertion to evaluated write on a this vaccuum correlation state. in terms of the from the principal discretesion series, depends with explicitly positive on the Casimir.ducible considered unitary representation. In representations Here that from in the case, our principalthe the case, continuous representation series, considering scaling with label irre- dimen- negative Casimir, 3.2.3 Vaccuum state and quasi-primary operator The divergence at theThis initial fits time, as with the results of [ Let us pointume out operator thatR this leads to an a priori negativeentering expectation the equation value ( for the vol- the identity. | which determines the eigenvalue: unitary representations presentedcomplex in eigenvalues [ ∆ space of slowly increasingL wave-functions, which is much largereigenvalues ∆ than the Hilbert space of As underlined in the thorough analysis of the eigenstates of all JHEP12(2019)031 ) – R 90 ) as , R (2 (3.38) (3.37) (3.39) , sl ), which vanishes, R , φ π ) but in a null R , (2 is not a true primary sl . σ . 3 b i ) Noether charges due to the O on the Wheeler-de Witt b v R σκ Ω . | 2 , | ) − (2 3.1 2 2 e . sl τ τ 1 ( φ κ 1 − σ − π b v i b 1 O + τ ) | = 1 v of FLRW quantum cosmology as a τ σ ( b σ 14 O i ∝ ) = b O v Ω | ( | Ω i Ω ) h 2 with τ = ( – 31 – i σ 2 involves non-trivial operators. However, it realizes b τ O , ψ ζ ) b i C | 1 1 τ Ω τ | ( = 0 ζ σ h σ i and b b O O Ω b h : H ) symmetry. An interesting remark is that it is possible to identify a ) = b b = H 2 R H | can be obtained from the physical state by the action of the | T , iτ i , τ Ω ) ], the caveat of this approach is that b − 1 A h 0 e τ 81 ( σ ) algebraic structure to write down differential equations entirely (∆ b ζ G O R | , b H (2 iτ e sl ) = )-invariant vacuum state but over the cosmological vacuum annihilating the τ R ( ). , σ b O C . As explained in [ 1 Finally, we would like to comment that we have not constructed the quantum states More work is required to reformulate the 3-point function and higher correlations in We have focused on the SL(2 , we do work with a unitary representation of the 1D conformal group SL(2 ]. These charges are nevertheless entirely determined by the three 14 φ whole ladder of charges92 forming a Virasorofinite algebra number of for degrees conformalcosmology, of quantum this freedom mechanics, of would as the paveHilbert theory. shown the space, If in way providing this a to [ Virasoro first implement realization structure unitarily of can quantum the also covariance time-diffeomorhphism for be such at realized very in the simple FLRW gravitational level system. of the a symmetry but not asπ a gauge symmetry. For eachallows value of to the use matter the energy-momentum determining the correlationthe functions pure as gravity in case a does conformal not bootstrap. live in When the trivial representation of time overlap of statesappendix (see e.g. the overlap of thermal dilatationover eigenstates a presented SL(2 in Hamiltonian operator. This can be considered as natural from the perspective of SL(2 the intuitive idea thatHamiltonian scalar the constraint. vacuum of the theorycosmological is terms, the especially physical toquasi-primary analyze state operators. and annihilating get In the the a meanwhile,algebraic deeper the structure, understanding toolbox allows of introduced quite the here, generally notion using to the of compute the time evolution of operators and This establishes theCFT first step ofoperator. a Its reformulation commutator with The time ordered versiontwo-point function: of this operator correlation fits exactly with the desired CFT equivalent of a vertex operator in our simple setting: This means that theoperator two-point function is obtained by the insertion of the time-evolved of quantum cosmology. Asquantization, the analyzed physical in state the reads: previous section Clearly the eigenstates of It seems natural to use the physical state solving the scalar constraint as the vaccuum JHEP12(2019)031 . )- H R , ) descends ] which form 3.34 60 – 58 ) for the two points function 3.28 . ) symmetry. This conformal structure, 1 R and the Hamiltonian scalar constraint , v (2 ], showing that the FLRW cosmology action sl ) invariant scalar product. 65 – 32 – R = 0. It means that one can identify an operator, , ) for the two points function, provided one works i (2 0 sl ∆ 3.28 | ], it is nevertheless possible to construct a SL(2 + 65 ˆ k ) algebraic structure of the system, and as such, is com- R , , the 3D volume ) Lie algebra and generate the infinitesimal translation, dilata- (2 C R sl , ) with vanishing quadratic Casimir. Following earlier work on (2 R sl , ), and therefore a state, which reproduces the standard expression of (2 sl 3.17 only from this keywith equation the ( lowest eigenstate given by ( the CFT two-points functions. Then,a we physical have interpretation used of the this WdW quantization operator to and obtain the associated eigenstate, and make a On the other hand,correlator can we be have reproduced shownthat in how the FLRW derivation the of quantum the standard cosmology. mainrelies form Let differential solely equation us of ( on emphasize the the that pletely two-point CFT independent from the WdW quantization. The final result ( On one hand, we havequantization shown in how order to adapt tothrough the preserve the standard the Wheeler-de CVH Witt algebra,guities, canonical is such a as powerful the criterionstructure factor to requires ordering narrow to the ones. use quantization of Moreover, an ambi- the presence of this conformal At the quantum level, the conformal symmetry allows to bootstrap the quantum theory. Finally, we have provided a precise mapping with the conformal mechanics introduced • • written in proper timelevel, this coincides shows that with the theinvariant homogeneous field and conformal theory isotropic mechanics in Einstein-Scalar action. system one is dimension, a i.e At conformal a the CFT Let classical us summarize the main results and underline the two sides of our computation. the hamiltonian realization of thecisely conformal this symmetry. CVH It structure issymmetry which worth initially, reported pointing suggested here. that the it existence is of pre- the hidden conformal by de Alfaro, Fubini and Furlan (dAFF) [ are not surface chargesthese but charges form are an associatedtion to and the special bulk conformal region. transformations.for the They At turn extrinsic the out curvature Hamiltonian toThese level, provide three to phase initial space conditions functions form the CVH algebra unveiled in [ reduced action of the Einstein-ScalarMobius system, transformation enjoys of a the hidden proper time,field, conformal with symmetry. the the Under dynamics scalar factor of transformingsuch General as symmetry a Relativity reduced primary minimally situation, remains coupledorem, invariant. to this As a conformal expected massless symmetry from scalar of the field, Noether’s the the- in action is related to three conserved charges which conformal quantum mechanicsinvariant [ density matrix, interpreted as a vacuum mixed state with4 thermal properties. Discussion We have shown that FLRW cosmology, defined as the homogeneous and isotropic symmetry representation of JHEP12(2019)031 , i Ω | ) invariant annihilated ], might be R i , 98 Ω (2 | corresponding to sl i 0 ∆ ζ | ) can be viewed as a coherent ), which can be rewritten as a 3.36 2.12 ). Therefore, even if once cannot realize the ] for some proposals concerning holographic – 33 – 101 3.37 – can find a holographic interpretation remains to ) evaluated on the cosmological vacuum state 1 τ 99 ( ˆ O , which furthermore, can be expressed as the two-points 1 τ and 2 τ is obtained from the cosmological vacuum state ]. In this work, we have obtained a rather new type of conformal is 35 ,  ). 1 2 34 − 3.2.3 = 0 shown that this correlator is naturallywave realized packet as at the time overlap of cosmologicalfunction coherent of a vertex operatorgiven by ( quantization, we have shown thatwave packet. the eigenstate Moreover, we ( have∆ shown = how the ∆ specific state by the scalar constraint,two-points given function by on ( a conformally invariant vacuum state as in 2d CFT, we have link with the standard mini-superspace approach. Using the WdW From a more general point of view, the results presented above might also be relevant Moreover, let us point that the conformal symmetry reported here is realized in the Let us now underline some crucial difference with previous works. As already men- This first computation of the two-points function suggests that the results and technics when trying to construct aquantum phase cosmological transition spacetimes interpretation of canderlying quantum only cosmology. quantum emerge Indeed, geometry, through sinceout. a most coarse-graining The of idea of thethe that the degrees underlying quantum un- of quantum cosmological freedom geometry, geometries have and correspond been thus to integrated to phase some transitions fixed of point of the renormalization gravitational surface charges discussedlation in of the FLRW cosmology literature.be as shown. a Whether CFT the Ifa present this FLRW/CFT reformu- is correspondence. possible,quantum See this cosmology [ models. might provides an interesting new path to construct investigate deeper the relationship betweentest conformal fields. structure of the background and the bulk and is notdefined tied as to integration some on boundary the conditions. bulk, The and conserved Noether’s are charges therefore are quite different from the standard test fields, through thetransformation, existence the of cosmolgical conformal metric Killingglobal transforms vectors. time-dependent conformal as Indeed, rescaling ( by under antransformation our appropriate might choice Mobius of be lapse related function.a to Such crucial conformal role Killing in vectors, the which conformal properties also of appear test to fields. play Hence, it would be interesting to tioned in the introduction, it isbackground, now such well as known de that Sitter test orformal fields FRW backgrounds propagating symmetry enjoy on [ some cosmological exact orsymmetry, approximative as con- it applies toderstand the how dynamical the background conformal itself. symmetry It of would the be background interesting is to translated un- at the level of the developed for the conformalthe mechanics at three-points the and quantum fourapplicable level, points to such as functions FLRW the using quantumcomputation computation of the cosmology. of the conformal higher It structure order correlation [ would functions be following this interesting bootstrap to strategy. push further the JHEP12(2019)031 – .A 107 A metric 2 ] for details. 116 , 115 ) symmetry. However, at this stage, R , (2 sl geodesics and physical trajectories of ], while the deformation of the CVH 2 58 – 34 – ] is different in that it includes both a self-interacting potential 106 ]. Interestingly, it might shed some light on the chaotic ] have a common ground, but this requires more works to be fully 15 106 114 – 112 [ 2 ]. A new input suggested by our work would be to use the so called ] for details. 105 – 106 102 ], the conformal symmetry reported here acts at the level of the action. Moreover, it seems 106 As a final comment, we point out that the conformal structure found here is not tied Moreover, the possibility to rewrite the cosmological action as a Schwarzian calls for ad- In this work, another type of conformal structure was found by mapping the cosmological dynamics ) for the scalar field, as well as homogeneously curved backgrounds. In our case, adding these two ] concerning the Schwarzian theory. Additionally, the appearance of the AdS φ 15 ( it is hard tomotion make contact in with [ this finding.that the Indeed, conformal while structure theV found conformal symmetry in affects [ theingredients equation would of break thetwo structures CVH found algebra here andunderstood. and thus in the [ conformal symmetry. Yet it might be that the The work of J.BAin-Aid was for supported Scientific by Research Japan No. Society 17H02890. for the Promotiononto of a Science quintic Grants- non-linear Schrodinger equation which enjoys an presence of perturbative inhomogeneitiesIn would particular, provide it another is interestingfound crucial development. conformal to symmetry. understand how cosmological perturbations break the newly Acknowledgments been shown to holdalgebra for due the to Bianchicomplete the I investigation of inclusion model the of fate insystems of a [ this is conformal cosmological structure mandatory. constant inScalar a is On-going system wider discussed set efforts will of be in concerning gravitational presented appendix the in spherically a future symmetric work. Einstein- Moreover, the fate of this symmetry in in the context ofbehavior AdS of the universe nearin the particular singularity discussed in in the various Bianchi cosmological IX models, (or and mixmaster) universe.to See the highly [ symmetric framework of homogeneity and isotropy. The CVH structure have on the cosmoloigcal phaseter space understanding begs of also the forthe relationship further universe between investigations. in AdS In phase particular,structure space extends a could to bet- open more complex interesting backgrounds,interesting directions. such to as For discuss the example, Bianchi the provided IX cosmological model, this interpretation it would of be the chaotic behavior investigated quintic non-linear Schrodinger equationsstead (NLS) of as the Gross-Pitaevskii’s mean one, field insymmetry. order approximation See to schemes [ take into in- account the newly found conformal ditional investigations. It would be111 interesting to make contact with the recent works [ this fixed point. Hence,ingredient the to conformal construct structure presented suchsense here scenario. of might provide this It the phase would missing transitiontum be picture gravity. interesting An of to interesting cosmological discuss well spacetimecondensates developed if in [ framework one known to models can test of make this quan- idea would be the GFT flow, is not new. However, one needs to identify the conformal symmetry associated to JHEP12(2019)031 = 0. (A.5) (A.3) (A.4) (A.1) (A.2) Λ H . Indeed the C 1) generators: , v . 2 ) Lie algebra. It , (1 κ R 2Λ 3  su , . 3  a + (2 z j sl Λ Λ πG 1) Lie algebra is slightly  , 8 2 −H κ (1 − 2 = j 3 su 2 σ 2 0 x } , 1) structure extends to a non- d Λ N i ,
2Λ aa , z 0, or a time-like vector generating x v . H 0 j (1 d 2 , − 3 > κ πG ij Λ su ∼ + 1 δ 3 8 {C and dilatation generator ) x  t ) k + v ( − Λ − 2 2 3 2 a x H 2 0 but , k vb N σκ φ + 4 2 2  2 2 κ κ 3 = 2Λ 2 t 3 a – 35 – κ  − )d 2 − t 3 t v , σ 2 ◦ φ ( v 2 d 2 , v 2 V π b = 2Λ y N N r } = K R − v Z Λ , v 1 + ( = ◦ = H V {C C 2 s   d , is no more a constant of motion along the Hamiltonian trajec- ] = 1 σκ ˙ φ C 2 2 vb ˙ a, κ = = = Λ 0. C } H N, a, < Λ [ S H v, { The dilatation Consider again the FLRW metric tories. The replacing constant of motion is: the equality holds on-shell, assuming that the Hamiltonian constraint is satisfied, So the Hamiltonian constraintbecomes is a not space-like anymore vector a generatingrotations boosts null-vector when of when Λ Λ the The CVH algebra ismodified. still closed We but keep the its same identification expressions to for the the dilatation generator and thebut volume we change the relation between the Hamiltonian constraint and the This changes the Poisson-algebra with thecosmological volume constant comes in frontdoes a not volume scale term as in the the other Hamiltonian terms: constraint, which Performing the canonical analysis,space with the a flat non-vanishing cosmological homogeneoustonian constant and scalar is constraint: isotropic given by cosmology a phase modification of the Hamil- Let us now showconstant. how the CVH algebra is deformedA.1 by the presence The deformed of CVH the algebra cosmological and the symmetry reducedstant action integrated of over the a Einstein-Scalar cell of system volume with a cosmological con- For the sake ofvanishing completeness, cosmological we constant. describediscussion, how in Although particular the this the cosmological willcosmology constant not does onto not conformal be enter quantum the relevantthe mechanics, mapping role to of it of quantum the seems the rest cosmological instructive constant to of whenever take the possible. into account A Conformal structure of the homogeneous (A)dS universe JHEP12(2019)031 2 C q ∈ (B.6) (B.1) (B.2) (B.3) (B.4) (B.5) (A.6) j 3], and sim- / 2Λ , − i p ) group elements, as sinh τ , j, m R | , ). It is nevertheless in- 2Λ , where the spin 3 i i C i 1 − sinh N A.2 + 1 − 1) Lie algebra: − r  1 , . j, m | − z (0) (1 j (0) Λ j ˆ M C from the Hamiltonian flow gen- j, m j, m 2 | | H su 2 3] and sinh[ ∈− − 2 / C κ ), as: m ∼ κ − − 2Λ 3 ) Lie algebra are obtained by making ) 2Λ ˆ k = + 1) + 1) 0, this leads to: ] = R R − and + j j − − , − , j ( ( ˆ k < ˆ k v cosh j j p D (2 , (2 (0) τ + + i sl v sl − − (0) ˆ k + , [ ) flow v  ˆ k 1) i R . j, m − 0. These expressions are of course linear in the | , 2 , + 1) − ˆ k – 36 – 2Λ 3 i − ( >  j, m q 1 2 − | m m ˆ k ( ( + 1) C − j, m cosh) +  r | m m j N 2 2 z . Inverting them to get the initial conditions in terms ( 1 and integrated in terms of SL(2 − j ˆ κ m p p j ] = (0) Λ − (1 = +1+  = = = = j H M H j ˆ i i i i C ∈ , (0) Λ z m , j ˆ cosh H [ and 2 j, m j, m j, m j, m = | | | | (0) Λ κ (0) : we distinguish the positive and negative series, which consist 2Λ z + 6= 0 corresponds to a modified potential with an extra-term in j 3 b C + − H C (0) j ˆ for the case Λ = 0. For Λ ˆ ˆ k k C D , ) symmetry according to Noether. We postpone to future study the ] = ] = ] = 2.3 : R τ τ τ (0) 2 [ [ [ N . This leads to three classes of unitary representations: , v v Λ C − ∈ ˆ k ) with Λ H at arbitrary time will give time-dependent conserved charges, which should j ) unitary representations = R H A.3 † + , ˆ k and The discrete series respectively in lowest and highestintegers weight representations. They are labelled by half- C , • v The unitary (irreducible) representations of the sure that These commutators can begives represented the value as of acting the on Casimir states B SL(2 At the quantum level, the commutators of the non-vanishing cosmological constant at theteresting level to of notice that, the according actionconstraint to ( ( the map tobesides conformal mechanics, the the conformal modified potential scalar term in ilarly with trigonometric functions forinitial Λ conditions of generate the SL(2 derivation of the corresponding precise action of Mobius transformations in the case with a where cosh and sinh respectively stand for cosh[ We can compute the evolutionerated in by the proper scalar time constraint explained of in section A.2 Physical trajectories from SL(2 JHEP12(2019)031 + R (B.7) (B.8) (C.1) (C.2) ∈ ) flow 0 and for an . s R b v i , ∆ ϕ | ). b 0. It can be H < j < R , iτ 1 2 e < | i 1) representations − ∆ ϕ
, | 1 4 ϕ ) h i b i H + ϕ . 2 iβ b β C| s j, m ) = b − | H − b τ C 0. These representations do iτ ( − β C b e g C 1 2 | = < ( = ϕ + † β b . Using the explicit formula for h , derived from the SL(2 C H Z b = 0 and the principal continuous C functions on SL(2 i . Notice that the computation of b . ∈ M C R iτ H j i < 2 e m τ | (∆) ∈ L 4 i − ϕ / = h ϕ Im β , j, m 1 i | 2 | b C − . These are the SU(1 − b ) s 0. These can be constructed from the b H − H C is iτ = iβ i − Z iβ e > 2 + ∈ ϕ : b b C H | ,P | 1 2 M + 1 2 m − b iτ i H ϕ b C − τ − h − iτ = – 37 – g , i ( e + 1) e τ c b = j = C j, m ) j = ∝ | b ) = P b ( + 1), with b H j H H i j ) j iβ∂ : they are labelled by a real positive number C β j ( iτ τ ϕ ( | − Z iβ ( + j e = b ∈ e g Re H + 2 b + C M m : they are labelled by a real number = iτ g b C β H e while odd parity representations are spanned by half-integer β = C − ) = b H b + C τ Z (∆) + + 1) with τ ( e ( : s | | i b j C ∈ P ϕ ϕ ( = ∆ Im h h . Even parity representations are spanned by states with integer j i i ϕ m β |  b : C = = = = 2 1 2 = C g + for an arbitrary real parameter  = ∆ τ i Z b ] to define cosmological coherent states. H τ∂ ∆ β ∈ 63 ϕ − | e m b β C b C b H β + e not appear in the Plancherel decomposition for series on the imaginary line The Casimir is given by The quadratic Casimirunderstood is as now strictlythat negative, we use in the present workThe to quantize complementary the series cosmological phaseinterpolate space. between the discrete series starting at states canonical quantization of the pairused of harmonic in oscillators [ in the spinorial formulation The principal continuous series and a parity magnetic moment The Casimir is given by • • := β b which implies the scaling inthis overlap time, does not require thefunction choice of computed a in highest or the lowest body weight vector, of as the the two-point paper, which involves the volume operator This allows toeigenstate compute the behavior of the time overlap C the time-evolved operator generated by the CVH operators, we easily get: C Time overlap ofIn thermal this dilatation section, we eigenstates would like to consider eigenvectors of the “boosted” dilatation generator, JHEP12(2019)031 , ]. ]. 16 ]. 88 , SPIRE SPIRE IN IN (2003) 009 SPIRE ]. ][ ][ IN ]. (2011) 1355 05 Phys. Rev. ][ Phys. Rev. Lett. , ]. AIP Conf. Proc. , 13 SPIRE , ]. IN SPIRE [ JCAP ]. IN SPIRE Phys. Rev. Lett. , , IN ][ Class. Quant. Grav. SPIRE , ]. ][ Entropy ]. IN , SPIRE hep-th/0501055 ][ IN arXiv:1310.3727 [ [ hep-th/0609128 [ SPIRE (1973) 2333 [ SPIRE IN ]. IN ][ ][ D 7 (1974) 30 Black hole versus cosmological horizon (2005) 050 SPIRE arXiv:0809.1554 (2014) 014 IN [ ]. NHEG mechanics: laws of near horizon ]. [ 02 astro-ph/0305121 248 03 The four laws of black hole mechanics hep-th/9905099 ]. [ (2007) 084003 [ ]. Near-horizon symmetries of extremal black holes Phys. Rev. SPIRE ]. , SPIRE JHEP IN – 38 – ]. IN SPIRE JHEP , Nature D 75 ][ [ arXiv:0705.4214 SPIRE , , IN Hawking radiation of apparent horizon in a FRW arXiv:1306.2517 [ Cosmological event horizons, thermodynamics and IN (1977) 2738 SPIRE ][ [ The extreme Kerr throat geometry: a vacuum analog of (2009) 155018 ][ IN (2003) 2753 SPIRE ][ Classification of near-horizon geometries of extremal black IN 26 (1999) 104030 20 Inflation and de Sitter thermodynamics ]. D 15 ][ ), which permits any use, distribution and reproduction in ]. (1973) 161 Phys. Rev. (2013) 8 Thermodynamic behavior of Friedmann equations at apparent First law of thermodynamics and Friedmann equations of , (2007) 4169 D 60 31 16 SPIRE gr-qc/9504004 SPIRE 24 IN [ IN ][ Phys. Rev. ][ Black holes and entropy arXiv:1207.1488 CC-BY 4.0 hep-th/9812013 , Black hole explosions [ gr-qc/0203001 [ [ Thermodynamics of space-time: the Einstein equation of state This article is distributed under the terms of the Creative Commons gr-qc/9906126 Phys. Rev. [ , Entropy from conformal field theory at Killing horizons Near horizon conformal symmetry and black hole entropy Effective conformal descriptions of black hole entropy Effective conformal descriptions of black hole entropy: a review Black hole entropy from conformal field theory in any dimension Class. Quant. Grav. 2 Class. Quant. Grav. (1995) 1260 , S , Living Rev. Rel. (2012) 54 × 75 , 2 (1999) 2828 arXiv:1107.2678 hep-th/0212327 Class. Quant. Grav. holes extremal geometry (thermo)dynamics 1483 AdS (1999) 3327 (2002) 241301 [ horizon of FRW universe universe 82 entropy Friedmann-Robertson-Walker universe Lett. particle creation [ Commun. Math. Phys. H.K. Kunduri and J. Lucietti, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, S. Carlip, J.M. Bardeen and G.T. Horowitz, H.K. Kunduri, J. Lucietti and H.S. Reall, S. Carlip, S. Carlip, S. Carlip, R.-G. Cai, L.-M. Cao and Y.-P. Hu, S. Carlip, T.M. Davis, P.C.W. Davies and C.H. Lineweaver, R.-G. Cai and S.P. Kim, M. Akbar and R.-G. Cai, G.W. Gibbons and S.W. Hawking, A.V. Frolov and L. Kofman, J.D. Bekenstein, J.M. Bardeen, B. Carter and S.W. Hawking, S.W. Hawking, T. Jacobson, [7] [8] [9] [5] [6] [1] [2] [3] [4] [18] [19] [15] [16] [17] [12] [13] [14] [10] [11] Attribution License ( any medium, provided the original author(s) and source are credited. References Open Access. JHEP12(2019)031 , , D ]. , ]. Phys. Nucl. (2012) , , ] ] ]. ]. 15 SPIRE ]. (2015) 443 SPIRE IN IN ]. ][ Phys. Rev. SPIRE ][ SPIRE , (2012) 475401 IN SPIRE IN B 749 IN ][ ][ SPIRE ][ IN A 45 ][ ]. ]. arXiv:1109.4942 arXiv:1205.1523 Living Rev. Rel. [ [ , ]. Phys. Lett. ]. Extremal rotating black holes hep-th/0112055 , SPIRE SPIRE J. Phys. Logarithmic corrections to black [ , IN IN arXiv:1103.2355 ]. [ ][ ][ SPIRE Black hole superradiance from ]. SPIRE IN ]. arXiv:1501.03515 (2012) 492 The Kerr/CFT correspondence Cargese lectures on the Kerr/CFT IN arXiv:1612.04833 arXiv:1012.2219 ][ [ SPIRE [ [ ][ (2012) 075002 IN SPIRE arXiv:1701.05462 ][ SPIRE [ 29 Conformal field theory interpretation of black ]. IN Static patch solipsism: conformal symmetry (2011) 194 B 864 IN (2002) 151301 ][ ][ Hidden conformal symmetry of the Kerr black Conformal structure of the Schwarzschild black 88 (2015) 531 216 (2017) 090 (2011) 204 SPIRE – 39 – IN arXiv:1004.0996 arXiv:1106.0999 63 04 [ [ Universal low-energy dynamics for rotating black holes Gravity waves from the Kerr/CFT correspondence ][ arXiv:0907.3477 Nucl. Phys. B 699 arXiv:1401.3746 [ , (2017) 1700023 ]. [ hep-th/9702015 JHEP ]. [ , ]. ]. 65 Generalized hidden Kerr/CFT High energy physics signatures from inflation and conformal Conformal symmetries of FRW accelerating cosmologies Operator product expansion of inflationary correlators and Inflation and conformal invariance: the perspective from radial arXiv:0809.4266 Class. Quant. Grav. Quasi-normal modes of extremal black holes from hidden arXiv:1309.3671 SPIRE [ Phys. Rev. Lett. , [ , IN SPIRE Fortsch. Phys. (2010) 024008 (2012) 064018 SPIRE (2010) 019 SPIRE Hidden conformal symmetry and quasi-normal modes IN , Phys. Lett. ][ IN IN , The cosmological bootstrap: weight-shifting operators and scalar seeds [ ][ 04 arXiv:1009.1010 ][ (2014) 044038 (1997) 4975 [ D 82 D 85 Nucl. Phys. Proc. Suppl. , (2014) 547 Fortsch. Phys. The Kerr/CFT correspondence and its extensions , JHEP (2009) 124008 D 90 D 56 , ]. ]. B 884 Phys. Rev. Phys. Rev. D 80 , , arXiv:1203.3561 (2010) 126013 SPIRE SPIRE [ IN IN arXiv:1112.1431 arXiv:1503.07861 symmetry of de Sitter quantization arXiv:1910.14051 Phys. conformal symmetry of de[ Sitter 11 of the de Sitter[ worldline hole entropy from Kerr/CFT Rev. correspondence [ Phys. Rev. 82 hole hole hole quasinormal modes conformal symmetry in the near-horizon limit:[ phase space and symmetry algebra Phys. Rev. Kerr/CFT A. Kehagias and A. Riotto, D. Baumann et al., A. Kehagias and A. Riotto, A. Kehagias and A. Riotto, A. Kehagias and A. Riotto, G. Comp`ere, D. Anninos, S.A. Hartnoll and D.M. Hofman, M. Guica, T. Hartman, W. Song and A. Strominger, I. Bredberg, C. Keeler, V. Lysov and A. Strominger, D.A. Lowe and A. Skanata, A.P. Porfyriadis and A. Strominger, A. Pathak, A.P. Porfyriadis, A. Strominger and O. Varela, A. Castro, A. Maloney and A. Strominger, S. Bertini, S.L. Cacciatori and D. Klemm, D. Birmingham, I. Sachs and S.N. Solodukhin, B. Chen and J.-j. Zhang, B. Chen and J. Long, J.M. Maldacena and A. Strominger, I. Bredberg, T. Hartman, W. Song and A. Strominger, G. Comp`ere,K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, [38] [39] [35] [36] [37] [33] [34] [31] [32] [28] [29] [30] [26] [27] [23] [24] [25] [21] [22] [20] JHEP12(2019)031 08 ]. , D JCAP ] , ]. D 98 SPIRE (1983) JHEP Class. , in the IN , , century: a [ SPIRE . , November st (2012) 213001 (2018) 01007 IN Phys. Rev. , D 28 21 , ][ 29 ]. Phys. Rev. 168 (1995) 846 , (2019) 126013 (1982) 25 hep-th/0106113 74 (2015) 023901 [ SPIRE , January 7–10, IN 78 Phys. Rev. , in the proceedings of the D 99 ]. , ][ ]. B 117 The cosmological bootstrap: arXiv:1811.00024 Liouville theory beyond the ]. , (2001) 034 century arXiv:1108.5735 SPIRE SPIRE EPJ Web Conf. [ IN st , IN 10 [ Class. Quant. Grav. ][ SPIRE symmetry in quantum cosmology 21 Phys. Rev. Lett. Phys. Rev. , ]. ) IN , [ , Phys. Lett. R ]. , , Quantum cosmology for the JHEP (2 Higher spin realization of the dS/CFT Rept. Prog. Phys. ]. ]. SPIRE , , arXiv:1411.7873 Lorentzian quantum cosmology [ IN SL [ SPIRE (2017) 015009 Is there really a de Sitter/CFT duality? IN – 40 – ]. SPIRE SPIRE Marcel Grossmann Meeting on General Relativity ][ 34 IN IN th ][ ][ arXiv:1005.2471 Nishinomiya-Yukawa Memorial Symposium gr-qc/9701022 arXiv:1903.09271 SPIRE Polymer quantum cosmology: lifting quantization Protected (2015) 158 Wave function of the universe [ th IN ]. ]. ]. Tunneling wave function of the universe conformal symmetry ]. ]. 03 ][ 11 Cosmological horizons, Noether charges and entropy ) ]. R gr-qc/0208018 , SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN JHEP IN IN , SPIRE Spacetime structure near generic horizons and soft hair SL(2 [ ][ ][ ][ century: arXiv:1703.02076 ][ IN (2019) 165005 [ st Class. Quant. Grav. ][ birthday. Proceedings, Workshop and Symposium The dS/CFT correspondence arXiv:1808.02032 arXiv:1904.06149 The future of theoretical physics and cosmology: Celebrating Stephen , 21 The Interpretation of quantum cosmology and the problem of time Quantum cosmology: effective theory Quantum cosmology: a review ]. 36 [ [ th Predictions from quantum cosmology Creation of universes from nothing Quantum cosmology: problems for the Towards causal patch physics in dS/CFT hep-th/0202163 60 [ ]. SPIRE , in the proceedings of the 12 IN [ (2019) 012 (2017) 103508 SPIRE arXiv:1806.09290 gr-qc/9406010 arXiv:1501.04899 arXiv:1209.3403 arXiv:1710.05682 IN 7–8, Nishinomiya, Japan (1996), 09 ambiguities using a [ Cambridge, U.K. (2002), [ Physics in the debate July 12–18, Paris, France (2009), proceedings of Hawking’s (2018) 066003 [ [ 2960 95 Quant. Grav. arXiv:1908.09833 correspondence cosmological horizon [ inflationary correlators from symmetries and singularities [ (2002) 045 J. Ben Achour and E.R. Livine, J. Ben Achour and E.R. Livine, A. Vilenkin, J.B. Hartle, M. Bojowald, C. Kiefer and P. Vargas Moniz, J.J. Halliwell, M. Bojowald, M. Bojowald, A. Vilenkin, J. Feldbrugge, J.-L. Lehners and N. Turok, A. Vilenkin and M. Yamada, L. Donnay and G. Giribet, D. Grumiller et al., J.B. Hartle and S.W. Hawking, G. Comp`ere,L. Donnay, P.-H. Lambert and W. Schulgin, Y. Neiman, A. Strominger, L. Dyson, J. Lindesay and L. Susskind, D. Anninos, T. Hartman and A. Strominger, N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, [58] [59] [56] [57] [54] [55] [52] [53] [49] [50] [51] [46] [47] [48] [44] [45] [41] [42] [43] [40] JHEP12(2019)031 ] B , (2009) Nuovo ]. D 96 , ] 04 ] ]. ]. (1999) 021 SPIRE Phys. Lett. 11 IN , JHEP (2001) 1997 SPIRE ][ SPIRE ]. , , ]. Phys. Rev. IN IN arXiv:0710.2956 ]. , [ ][ ][ ]. JHEP ]. A 16 , ]. SPIRE SPIRE SPIRE hep-th/0303166 IN IN [ hep-th/0312194 IN SPIRE ][ coherent states on the LQC ][ SPIRE [ ][ IN IN SPIRE 1) ][ , IN ][ hep-th/0004107 quantum gravity black holes ][ (2008) 064005 [ 2 2 SU(1 hep-th/0411008 (2004) 066 [ arXiv:1706.10225 AdS AdS Mod. Phys. Lett. D 77 (2003) 125013 [ , ]. 03 Rational deformations of conformal Near extremal black hole entropy as ]. ]. arXiv:1707.07357 D 68 arXiv:1705.03772 arXiv:1204.0539 hep-th/0105237 [ SPIRE [ [ ]. (2000) 086002 JHEP [ gr-qc/0703144 IN , [ arXiv:1506.05596 2 Phys. Rev. Superconformal black hole quantum mechanics – 41 – Conformal invariance in quantum mechanics ][ [ (2005) 104029 SPIRE A realization of the infinite-dimensional symmetries SPIRE , 1 IN IN SPIRE D 62 Group theoretical quantization of isotropic loop Anomaly in conformal quantum mechanics: From AdS Black hole thermodynamics from near-horizon conformal (2017) 1750157 ][ IN ][ [ Coarse graining as a representation change ]. Phys. Rev. On the geometry of conformal mechanics Central charge for Renormalisation with Vacuum states for Thiemann complexifier in classical and quantum FLRW D 71 ]. Note on correlation functions in conformal quantum , (2001) 741 /CF T 2 Conformal mechanics on rotating Bertotti-Robinson (2018) 026017 A 32 (2012) 124052 (2017) 066025 (2016) 227 ]. ]. ]. SPIRE (2007) 123512 AdS SPIRE Phys. Rev. IN B 619 IN 367 , D 98 ][ D 85 D 96 [ hep-th/0412322 SPIRE SPIRE SPIRE Phys. Rev. [ D 75 , IN IN IN [ Quantum conformal mechanics emerging from unitary representations of ][ ][ A matrix model for arXiv:1811.02792 arXiv:1704.00286 Dynamical coherent states and physical solutions of quantum cosmological [ arXiv:1904.13269 [ Black holes and conformal mechanics , Implications of conformal symmetry in quantum mechanics (1976) 569 Nucl. Phys. Mod. Phys. Lett. Phys. Rev. Phys. Rev. Phys. Rev. (2005) 017 , , , , , Annals Phys. Phys. Rev. ]. ]. ]. , ), 11 R A 34 arXiv:0803.3621 , (2019) 69 [ SPIRE SPIRE SPIRE hep-th/9904143 IN IN IN hep-th/0104175 [ entanglement entropy via [ 026 quantum mechanics [ [ space-time JHEP mechanics [ molecular physics to black holes (2017) 066030 of conformal mechanics mechanics arXiv:1108.1299 SL(2 cosmology bounces Cim. cosmology Hilbert space 792 T. Azeyanagi, T. Nishioka and T. Takayanagi, T. Hartman and A. Strominger, H.E. Camblong and C.R. Ordonez, A. Strominger, M. Spradlin and A. Strominger, G. Clement and D. Gal’tsov, D. Gaiotto, A. Strominger and X. Yin, J.F. Cari˜nena,L. Inzunza and M.S. Plyushchay, S. Mignemi, H.E. Camblong and C.R. Ordonez, M. Cadoni, P. Carta and S. Mignemi, S. Khodaee and D. Vassilevich, K. Andrzejewski and J. Gonera, K. Andrzejewski, T. Okazaki, M. Bojowald, V. de Alfaro, S. Fubini and G. Furlan, N. Bodendorfer and D. Wuhrer, N. Bodendorfer and F. Haneder, E.R. Livine and M. Martin-Benito, J. Ben Achour and E.R. Livine, [79] [80] [76] [77] [78] [74] [75] [71] [72] [73] [69] [70] [66] [67] [68] [64] [65] [61] [62] [63] [60] JHEP12(2019)031 1 ] C 1) , (2005) 1 ]. 04 ] SU(1 /CFT 2 SPIRE Phys. Lett. B JHEP ]. IN , , ]. ][ arXiv:1711.08482 ]. (2017) 203 [ ]. (1999) 006 (2004) 073 ]. SPIRE ]. 10 SPIRE IN [ 04 07 (2012) 089905] IN SPIRE arXiv:1703.08749 SPIRE IN , ][ arXiv:1704.03666 SPIRE ]. IN [ ][ JHEP IN ][ , JHEP D 86 JHEP ][ , , SPIRE duality: the case of massless scalar IN 1 arXiv:1805.04405 ][ [ , Mathematical physics monograph (2017) 808 arXiv:1712.4180 PoS(Modave2017)001 On the unitary representations of /CFT algebras, conformal mechanics and black Conformal quantum mechanics as the CFT , 2 arXiv:1711.08169 ) arXiv:1106.0726 , ]. Conformal quantum mechanics and holography holographic duality Erratum ibid. ∞ [ B 772 hep-th/9910065 Modular discretization of the AdS [ ( d (1965) 13. [ arXiv:1803.06827 1 [ W / 3 boundary conditions arXiv:1707.04816 – 42 – (1970) 27. d SPIRE (2019) 015002 Horizon temperature on the real line [ 2 2 The conformal bootstrap: theory, numerical techniques dilaton gravity from reductions of some nonrelativistic IN 13 On the quantum resolution of cosmological singularities A ]. (2012) 043513 [ 2 91 ][ arXiv:1306.5670 (2011) 503 [ Phys. Lett. ]. Towards a big crunch dual AdS Holographic description of AdS cosmologies , (2000) 1731 ]. representations (2012) 045017 SPIRE ]. D 86 ) ]. ]. IN 17 (2018) 046012 Continuous bases for unitary irreducible representations of R [ B 701 , SPIRE Non-commutative AdS (2017) 066019 SPIRE IN Conformal blocks for the 4-point function in conformal quantum D 86 SPIRE (2014) 109 IN ][ SPIRE SPIRE f SL(2 IN D 98 Menagerie of AdS ][ IN IN 02 ][ D 96 Rev. Mod. Phys. ][ ][ Phys. Rev. holography and the SYK model , , arXiv:1804.10550 2 Phys. Lett. [ Ann. I.H.P. Phys. Theor. , A note on the infinite dimensional symmetries of classical Hamiltonian JHEP 2 Phys. Rev. Notes on Conformal mechanics and the Virasoro algebra AdS , 1), , The Lorentz group and harmonic analysis Ann. I.H.P. Phys. Theor. , Phys. Rev. hep-th/0004150 ]. ]. ]. , , Phys. Rev. Class. Quant. Grav. 1), , , , hep-th/0503071 (2019) 82 SU(2 [ SPIRE SPIRE SPIRE hep-th/0406134 arXiv:1205.0443 hep-th/9901139 arXiv:1708.08471 IN IN IN [ 005 using AdS/CFT SU(1 mechanics [ and series, W.A. Benjamin, U.S.A. (1970). holes systems 788 and applications [ [ theories [ [ in noncommutative space–time [ dual to AdS holography fields T. Hertog and G.T. Horowitz, B. Craps, T. Hertog and N. Turok, G. Lindblad and B. Nage, R. Jackiw and S.Y. Pi, T. Hertog and G.T. Horowitz, L. C. Biedenharn, J. Nuyts and N. Straumann, W. R¨uhl, A. Kitaev, S. Mignemi, M. Arzano and J. Kowalski-Glikman, D. Poland, S. Rychkov and A. Vichi, J. Kumar, S. Cacciatori, D. Klemm and D. Zanon, K.S. Kolekar and K. Narayan, G. S´arosi, M. Mezei, S.S. Pufu and Y. Wang, K.S. Gupta, E. Harikumar and N.S. Zuhair, D. Grumiller et al., M. Axenides, E.G. Floratos and S. Nicolis, A. Pinzul and A. Stern, C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, [97] [98] [99] [94] [95] [96] [92] [93] [89] [90] [91] [87] [88] [84] [85] [86] [82] [83] [81] [100] [101] JHEP12(2019)031 20 , (2018) (1998) ]. 18 ]. 81 120 SPIRE IN SPIRE ][ IN ][ (2016) 110 (2016) 224001 Class. Quant. Grav. 12 Phys. Rev. Lett. Phys. Rev. Lett. , ]. ]. 33 , , ]. (2018) 036 (2016) 111601 JHEP 05 SPIRE SPIRE Comptes Rendus Physique , Shockwave S-matrix from gravity , SPIRE IN IN 117 arXiv:1804.09834 2 [ IN ][ ][ arXiv:1602.08271 [ ]. ][ JHEP AdS The Schwarzian theory — A Wilson line , Solving the Schwarzian via the conformal ]. SPIRE IN (2018) 182 Bouncing cosmologies from quantum gravity Emergent Friedmann dynamics with a quantum Class. Quant. Grav. Cosmological billiards ][ , ]. ]. 11 – 43 – Correlation functions in the Schwarzian theory SPIRE Phys. Rev. Lett. ]. IN (2017) 04LT01 , [ arXiv:1311.1238 Homogeneous cosmologies as group field theory SPIRE arXiv:1806.07765 SPIRE Chaos in quantum cosmology [ 34 IN JHEP [ arXiv:1705.08408 IN ]. , SPIRE ][ [ ][ ]. ]. ]. ]. IN On CFT and quantum chaos Fine grained chaos in ][ holography SPIRE 2 SPIRE SPIRE SPIRE SPIRE IN (2014) 013 (2018) 022 arXiv:1804.00424 IN IN IN IN ][ [ (2017) 136 06 ][ ][ ][ ][ 12 08 arXiv:1802.09186 , The Schwarzian theory — Origins hep-th/0212256 JHEP Class. Quant. Grav. Inflationary cosmology, diffeomorphism group of the line and Virasoro [ JHEP arXiv:1612.09521 , , Chaos in AdS , [ (2018) 036 JHEP The universe as a quantum gravity condensate , arXiv:1712.04963 11 [ gr-qc/9708046 [ arXiv:1605.06098 arXiv:1603.03020 arXiv:1801.09605 arXiv:1602.05881 3571 (2003) R145 121601 [ [ Schwarzian quantum mechanics JHEP [ perspective bootstrap (2017) 235 coadjoint orbits condensates condensates bounce from quantum gravity condensates [ N.J. Cornish and E.P.S. Shellard, T. Damour, M. Henneaux and H. Nicolai, K. Jensen, G. Turiaci and H. Verlinde, H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, V.V. Belokurov and E.T. Shavgulidze, F.M. Haehl and M. Rozali, A. Blommaert, T.G. Mertens and H. Verschelde, T.G. Mertens, G.J. Turiaci and H.L. Verlinde, D. Oriti, J.E. Lidsey, T.G. Mertens, D. Oriti, L. Sindoni and E. Wilson-Ewing, D. Oriti, L. Sindoni and E. Wilson-Ewing, S. Gielen, D. Oriti and L. Sindoni, [115] [116] [113] [114] [110] [111] [112] [108] [109] [105] [106] [107] [103] [104] [102]