Interactions resolve state-dependence in a toy-model of AdS black holes

Journal of High Energy Physics, Jun 2018

Abstract We show that the holographic description of a class of AdS black holes with scalar hair involves dual field theories with a double well effective potential. Black hole microstates have significant support around both vacua in the dual, which correspond to perturbative degrees of freedom on opposite sides of the horizon. A solvable toy-model version of this dual is given by a quantum mechanical particle in a double well potential. In this we show explicitly that the interactions replace the state-dependence that is needed to describe black hole microstates in a low energy effective model involving the tensor product of two decoupled harmonic oscillators. A naive number operator signals the presence of a firewall but a careful construction of perturbative states and operators extinguishes this.

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Interactions resolve state-dependence in a toy-model of AdS black holes

Received: May Interactions resolve state-dependence in a toy-model Adam Bzowski 0 1 3 Alessandra Gnecchi 0 2 Thomas Hertog 0 3 Geneva, Switzerland 0 Celestijnenlaan 200D , Leuven , Belgium 1 Institut de Physique Theorique, CEA Saclay 2 Theoretical Physics Department , CERN 3 Institute for Theoretical Physics, KU Leuven We show that the holographic description of a class of AdS black holes with scalar hair involves dual eld theories with a double well e ective potential. Black hole microstates have signi cant support around both vacua in the dual, which correspond to perturbative degrees of freedom on opposite sides of the horizon. A solvable toy-model version of this dual is given by a quantum mechanical particle in a double well potential. In this we show explicitly that the interactions replace the state-dependence that is needed to describe black hole microstates in a low energy e ective model involving the tensor product of two decoupled harmonic oscillators. A naive number operator signals the presence of a rewall but a careful construction of perturbative states and operators extinguishes this. Black Holes; Models of Quantum Gravity - 1 Introduction 2 3 6 7 1 3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 6.1 6.2 Dynamics Tunneling and Hawking radiation Classical evolution and chaos Summary and conclusions Introduction Dual description of hairy AdS black holes Quantum mechanics in a double well and black hole microstates Canonical quantization and Hilbert spaces Black hole microstates Firewalls? Limitations of perturbation theory 4 Low energy excitations Extinguishing rewalls Perturbative states Perturbative operators 5 Low energy e ective theory: state-dependence Perturbative observables The tensor product: loss of information Operators in the e ective theory Global time evolution Time reversal and the `wrong sign' commutation relations Holography has provided a fruitful new perspective on the black hole information paradox rst formulated by Hawking [1]. Speci cally it has enabled an expression of the essence of the paradox in dual quantum eld theoretic terms. This has further sharpened some of the underlying assumptions, and it has led to novel suggestions for its resolution. Key elements that have emerged in this recent discussion include the following; Firewalls: according to [2{5] a generic black hole microstate exhibits a rewall, i.e. that its horizon is not a smooth surface. In terms of eld theory data the paradox is of the semiclassical approximation for an asymptotic observer, iii ) existence of black hole microstates visible to an asymptotic observer as states with exponentially small energy di erences, and iv ) existence of a rewall-free number operator for an infalling observer. State-dependence: in contrast with this [6{9] proposed a dual description of the black hole interior by introducing state-dependent operators. The black hole horizon remains smooth, but such operators depend on the speci c black hole microstate and hence go beyond the standard paradigm of quantum eld theory. In this way one avoids many pitfalls presented by the rewall arguments such as existence of creation-annihilation operators satisfying the `wrong sign' commutation relations. Vacuum structure: Hawking's original argument has been recast as a no-go theorem [10, 11] that states that quantum gravity e ects cannot prevent information loss if they are con ned to within a given scale and if the vacuum of the theory is assumed to be unique. This is a particularly sharp puzzle in the context of holography given the apparent uniqueness of the vacuum of dual CFTs. It can be seen as one motivation for recent investigations into a possible non-trivial vacuum structure as in [12]. The reformulation of the information paradox in terms of purely eld theoretic data elucidates the nature of the underlying issues. Devoid of a geometrical interpretation one can ask whether there exists any tractable well-de ned quantum theories modeling black holes and satisfying basic properties required by the aforementioned papers. In this paper we put forward a new holographic toy-model for AdS black holes that incorporates in a toy-model fashion a speci c proposal for the nature of non-perturbative quantum gravity corrections to black hole physics. The model consists of a quantum mechanical particle in a double well potential. Our motivation to advance this as a toy-model for black holes in AdS is twofold. First, there is a class of single-sided black hole solutions in global AdS with scalar hair outside the horizon whose dual description involves a eld theory with a double well e ective potential [13{16]. We review these black holes, which are solutions in truncations of AdS supergravity with so-called designer gravity boundary conditions, in section 2. The potential barrier in their dual description separates the perturbative degrees of freedom on both sides of the horizon, but they are coupled through multi-trace interactions. We argue that black hole microstates are states with signi cant support around both perturbative vacua. The quantum mechanical model we put forward can be viewed as a toy-model for systems of this kind since it amounts to two harmonic oscillators coupled through a `non-perturbative' interaction modeled as a double well potential. Outside the context of holography Giddings [17, 18] has studied how novel, non-local interactions (in the bulk) can resolve the information paradox. Secondly, recent work in the context of two-sided black holes in AdS has advocated that not only entanglement but also interactions between the two boundary CFTs are needed to describe the bulk [19{22]. Some implications of adding a speci c example of such interactions were explored in nearly AdS2 in [22{27]. These studies yield a di erent { 2 { motivation for our toy-model in which the two perturbative vacua are thought of as being dual to the two asymptotic regions on both sides of the horizon. The potential barrier in the dual toy-model amounts to a proposal for a speci c interaction between a single pair of left and right modes in the bulk, jnkiL and jnkiR, with xed frequency ! and xed wave vector k and related to boundary states by some form of the HKLL construction [28, 29]. For small values of nk the left and right modes are essentially separated by a potential barrier. By contrast, for su ciently large occupation numbers left and right modes interact strongly and the semiclassical approximation breaks down. In the context of holography [30] argued that such non-perturbative e ects can be su cient to resolve the information paradox. Motivated by these developments we consider a quantum mechanical particle with the of the approximate harmonic oscillators around the semiclassical vacuum states '0L and p ). The vacua in our model correspond to the left and right, or interior and exterior, semiclassical vacua in the bulk. Excited states from the standpoint of observers in one of these vacua then naturally correspond to perturbative states 'nL and 'nR. Black hole microstates nally are linear combinations of perturbative states in both vacua. We will argue that typical microstates correspond to states with signi cant support around both perturbative vacua. By contrast, bulk spacetimes without a black hole correspond to states with support around one of the vacua only. To make contact with the usual perturbative expansion in semiclassical gravity we introduce a dimensionless parameter N as Hence perturbation theory in models the usual large N expansion. The height of the potential barrier equals V = V (0) = 1=(32 ). In the limit ! 0 the barrier grows, and the two minima move apart. In the exact = 0 limit the excitations around both perturbative vacua decouple completely and the system reduces to two decoupled harmonic oscillators with frequency !. In the bulk, with designer gravity boundary conditions, this decoupling limit corresponds precisely to a limit in which the horizon of the hairy black holes becomes singular. In this paper we carefully study how states and operators in the full interacting toymodel relate to quantities in a low energy e ective theory involving the tensor product of two decoupled harmonic oscillators. At the e ective theory level our model captures many of the usual paradoxes associated with the semiclassical approximation of black hole physics in a remarkably precise manner. A major advantage of our model is that it is holes, to explore dynamical processes, and to understand in this concrete toy-model setup how non-perturbative interactions resolve the paradoxes. In particular we show explicitly that the interactions eliminate the state-dependence that is needed to describe black hole microstates in the e ective low energy dual. We also nd that a naive number operator signals the presence of a rewall, but that a careful construction of perturbative states and operators in the full model extinguishes this. Finally, when it comes to dynamical processes, we point out that tunneling near the potential maximum corresponds to Hawking radiation in the bulk, and that the scattering of classical waves nicely captures the behavior of shock waves in the bulk. We conclude this introduction with an important caveat. Evidently our model is not suitable for the analysis of properties of black holes that depend on a collection of modes. This in particular encompasses all thermodynamical properties that rely on the existence of an ensemble of modes. In this context it would be necessary to consider more complicated models such as matrix or tensor models. 2 Dual description of hairy AdS black holes In this section we review the dual description in terms of a eld theory with a double well e ective potential of a class of single-sided asymptotically AdS4 static black hole solutions with scalar hair. The perturbative degrees of freedom on both sides of the horizon correspond in the dual description to excitations around two distinct perturbative vacua. However, multi-trace interactions in the dual imply a non-perturbative coupling between both sides. As such this setup motivates the quantum mechanical particle in a double well potential as a toy-model for black holes in AdS. The black hole solutions we construct are variations of the solutions found in [13{ 16, 31{33] and recently in [34, 35]. Consider the low energy limit of M theory with AdS4 S7 boundary conditions. The massless sector of the compacti cation of D = 11 supergravity on S7 is N = 8 gauged supergravity in four dimensions. It is possible to consistently truncate this theory to include only gravity and a single scalar with action S = Z d x 4 p g 1 2 R 1 2 (r )2 + 2 + cosh(p2 ) where we have set 8 G = 1 and chosen the gauge coupling so that the AdS radius is one. The potential has a maximum at = 0 corresponding to an AdS4 solution with unit radius. It is unbounded from below, but small uctuations have m2 = 2, which is above the Breitenlohner-Freedman bound m2BF = 9=4 so with the usual boundary conditions AdS4 is stable. Consider global coordinates in which the AdS4 metric takes the form ds02 = g dx dx = (1 + r2)dt2 + dr2 1 + r2 + r2d 2 In all asymptotically AdS solutions, the scalar decays at large radius as (r) r + r2 ; r ! 1 { 4 { (2.1) (2.2) (2.3) where M0 is the coe cient of the 1=r5 term in the asymptotic expansion of grr, and where we have de ned the function which de nes the choice of boundary conditions. Consider now a speci c class of boundary conditions de ned by the following relation M = Vol(S2) [M0 + + W ] Z 0 W ( ) = ( ~)d ~ ; are functions of t and the angles. To have a well-de ned theory one must specify boundary conditions at spacelike in nity. The standard choice of boundary condition corresponds to taking = 0. However one can consider more general `designer gravity' boundary conditions [14] with 6= 0 that are speci ed by a functional relation ( ) in (2.3). The backreaction of the -branch of the scalar eld and its self-interaction modify the asymptotic behavior of the gravitational elds. Writing the metric as g = g + h the corresponding asymptotic behavior of the metric components is given by hrr = (1 + 2=2) r4 + O(1=r5); hrm = O(1=r2); hmn = O(1=r) Nevertheless, the Hamiltonian generators of the asymptotic symmetries remain well-de ned and nite when 6= 0 [13, 36, 37]. They acquire however an explicit contribution from the scalar eld. For instance, the conserved mass of spherical solutions is given by (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) HJEP06(218)7 where c1 and c2 are constants. With these, the conserved mass (2.5) is given by M = 4 M0 4 3 c1 3 + 1 6 c2 4 Both the vacuum and the dynamical properties of the theory | as well as the possible black hole endstates of gravitational collapse | depend signi cantly on W [14, 38]. In the context of the AdS/CFT correspondence, adopting designer gravity boundary conditions de ned by a function W 6= 0 corresponds to adding a potential term R W (O) to the dual CFT action, where O is the eld theory operator that is dual to the bulk scalar [39, 40]. This is generally a complicated multi-trace interaction. Certain deformations W , including those corresponding to boundary conditions of the form (2.7), give rise to eld theories with additional, possibly metastable vacua. The AdS/CFT correspondence relates the expectation values hOi in di erent eld theory vacua to the asymptotic scalar pro le of regular static solitons in the bulk. The precise correspondence between solitons and eld theory vacua is given by the following function [37], Z 0 V( ) = s( ~)d ~ + W ( ) where s( ) is obtained from the asymptotic scalar pro les of spherical soliton solutions with di erent values (0) at the origin r = 0. This curve was rst obtained in [14] for the theory (2.1) and is plotted in gure 1(a). { 5 { β Right: the scalar radial pro le of the soliton associated with the second intersection point of bc( ) with the soliton curve s( ). Given a choice of boundary condition ( ), the allowed solitons are simply given by the points where the soliton curve intersects the boundary condition curve: s( ) = ( ). For any W the location of the extrema of V in (2.9) yield the vacuum expectation values hOi = , and the value of V at each extremum yields the energy of the corresponding soliton. Hence V( ) can be interpreted as an e ective potential for hOi. This led [14] to conjecture that (a) there is a lower bound on the gravitational energy in those designer gravity theories where V( ) is bounded from below, and that (b) the solutions locally minimizing the energy are given by the spherically symmetric, static soliton con gurations. For the boundary conditions (2.7) the e ective potential is generally of the form shown in gure 2, indicating the emergence of a second, metastable vacuum.1 The AdS/CFT correspondence then suggests that the bulk theory (2.1) with such boundary conditions satis es the Positive Mass Theorem, and that empty AdS remains the true ground state.2 However the constants in (2.7) can be tuned so that the new vacuum has precisely the same energy as the AdS vacuum. For this choice of boundary conditions the e ective potential in the dual takes the form of a double well potential with two vacua at equal energy, separated by a barrier. One can also consider excitations around each of these perturbative vacua. A particular class of excitations corresponds to `adding' a black hole at the centre of the soliton. When non-linear backreaction is included, these are spherical static black hole solutions with scalar hair. Black holes of this kind were found numerically in [13, 15] for boundary conditions similar to (2.7). Regularity of the event horizon Re implies the relation 0(Re) = 1 ReV; e Re2V ( e) (2.10) 1In the bulk this new vacuum corresponds to the second intersection point of s( ) = ( ) in gure 1(a). The rst intersection point corresponds to unstable solitons associated with the local maximum of V( ). 2See [41] for a stability analysis of this theory (with more stringent conditions on W ) using purely gravitational arguments. { 6 {  stable) branch of solutions, which are associated with the second intersection point of the curves Re ( ) with bc( ), and hence have more hair. still valid solutions of the theory with boundary conditions (2.7), since the curve ( ) intersects the origin. However in addition the theory admits black holes with scalar hair at and outside the horizon. The scalar asymptotically behaves again as (2.3), so we obtain a point in the ( ; ) plane for each combination (Re; e). Repeating for all e gives a curve Re ( ). In gure 1(a) we show this curve for hairy black holes of two di erent sizes. As one increases Re, the curve decreases faster and reaches larger (negative) values of . Given a choice of boundary conditions ( ), the allowed black hole solutions are given by the points where the black hole curves intersect the boundary condition curve: Re ( ) = ( ). It follows immediately that for boundary conditions (2.7) there are two hairy black holes of a given horizon size provided Re is su ciently small. Each branch of hairy black holes tends to one of the two spherical static solitons in the limit Re ! 0. The mass (2.8) of both branches of black holes is shown in gure 2(b). The hairy black holes reviewed here are solutions where a normal Schwarzschild-AdS black hole interior solution is smoothly glued at the horizon onto a scalar soliton solution outside, slightly modi ed by the non-linear backreaction of the black hole. Hence the usual AdS vacuum outside the black hole is essentially replaced by a solitonic vacuum.3 In this way one separates to rst approximation the excitations that make up the black hole interior from the degrees of freedom outside the horizon, but without introducing a second boundary. This separation is clearly manifest in the dual description of the black holes which involve a double well e ective potential of the form shown in gure 2(a). In this description, the soliton corresponds to the ground state wave function around the new vacuum whereas the black hole degrees of freedom correspond to excitations around the original vacuum. 3The upper branch of more massive hairy black holes in gure 2(b) corresponds to black holes glued onto the unstable soliton associated with the maximum of V( ). Those black holes are, like the soliton, unstable. { 7 { However, there is evidently also an important coupling between both vacua. On the eld theory side the vacuum structure emerges from a complicated set of multi-trace interactions. In the bulk the coupling is at the semiclassical level encoded in the regularity condition at the horizon. Note furthermore that one can consider a one-parameter family of boundary conditions of the form (2.7) for which the second vacuum is always at zero energy, but is gradually taken further away. In the limit of large separation in which both vacua decouple, the hairy black holes become singular on the horizon. These features of the dual description of hairy black holes form the basic motivation to put forward a quantum mechanical particle in a double well potential as an extremely simpli ed | but solvable | toy-model for (this class of) black holes in AdS. In the remainder of this paper we study this toy-model, and its connection to black hole physics. Quantum mechanics in a double well and black hole microstates Canonical quantization and Hilbert spaces Consider the 1D quantum mechanical system of a particle in the double well potential4 (1.1). One can carry out the procedure of canonical quantization either around xL or around xR by ignoring all interaction terms. This leads to two separate Fock spaces FL and FR which come equipped with two pairs of creation and annihilation operators bL; bL+ and bR; bR+. However, while formally independent, these two Fock spaces must be related since they arise from the same system. Hamiltonian of a harmonic oscillator and HR(1) its (perturbative) correction, We rst discuss the perturbative structure around the minumum at xR. When expanded around xR, the Hamiltonian can be written as H = HR(0) + HR(1), with HR(0) the (3.1) (3.2) (3.3) HR(0) = HR(1) = p 1 p2 + 2 operators bR; bR+ satisfying where yR = x xR. Standard canonical quantization based on HR(0) around xR yields a Fock space FR with a set of basis states jniR, together with a pair of creation-annihilation bRj0iR = 0; jniR = pn! 1 (bR+)nj0iR : A similar analysis is applicable to the left minimum at xL5 and gives another Fock space FL spanned by the states jniL and with creation and annihilation operators bL; bL+. One can 4For simplicity we set ! = 1 from now on and restore ! only where it is illuminating. 5In particular, we can de ne analogous Hamiltonians HL(0) and HL(1) which di er from their xR counterparts but satisfy by de nition HL(0) +HL(1) = HR(0) +HR(1) = H. It was argued in [42] that, when modelling an eternal black hole, HR itself should be regarded as the total Hamiltonian of the theory, instead of HR HL. Our model realizes this intuition since there is only a single Hamiltonian H. This Hamiltonian can be split into its free and interacting part as in (3.1) and (3.2) to better describe an experience of the right and the left observer. But nevertheless HR = HL = H is always the same operator. { 8 { associate to these Fock spaces two observers, left and right. The right observer perceives the state j0iR as the natural semiclassical vacuum and the excited state jniR as an nparticle state. Similary, the left observer regards j0iL as the semiclassical vacuum and jniL as an n-particle state. The above canonical quantizations eliminate any relation between both Fock spaces. In particular expressions such as [bL; bR] make no sense as the operators involved act on di erent Hilbert spaces. To relate FL and FR we have to embed these into the total Hilbert space H = L2(R; C) of complex-valued, square-integrable wave functions. Consider the set f'ngn2N of normalized eigenfunctions of the Hamiltonian of a harmonic oscillator, 'n(x) = 1 1=4p2nn! Hn(x)e x2 ; 2 x 2 R ; where Hn denotes standard Hermite polynomials. We can de ne two morphisms FL and FR between the Fock spaces FL, FR and H as FR : FL : FR 3 jnRi 7! 'nR 2 H; L FL 3 jnLi 7! 'n 2 H; 'nR(x) = 'n(x xR) ; 'nL(x) = ( 'nR)(x) = ( 1)n'n(x xL) : where the CPT operator acts on elements 2 H as ( )(x) = ( x), and the asterisk denotes complex conjugation. We have introduced an additional factor ( 1)n in the de nition of 'nL which allows us to relate left and right modes as CPT conjugates of each other. The maps FR and FL are obviously isomorphisms that can be thought of as two di erent choices of basis of H associated with harmonic oscillator eigenstates around either xL or xR. Hence the total Hilbert space H is isomorphic to each Fock space FL and FR separately, H = FR = FL ; There is no tensor product. The interactions provide a non-trivial identi cation of the two Fock spaces within a single H. On the other hand, the Fock spaces FR and FL have distinct sets of creation and annihilation operators. This means that an expression such as bLjniR makes a priori no sense, since bL acts on FL, whereas jniR is a state in FR. However the isomorphisms (3.5) and (3.6) can be used to de ne new annihilation operators aR; aL constructed from bR; bL that do have a well de ned action in H. In particular, de ning the operators aR = FRbRFR 1; aL = FLbLFL 1; together with their conjugates aR+ and aL+, we get the following actions, aR'nR = p aL'nL = p aR+'nR = p aL+'nL = p n + 1'nR+1 ; n + 1'nL+1 ; In particular, the action of aL is related to the action of aR by the parity operator, { 9 { (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) HJEP06(218)7 which is the standard relation between left and right creation-annihilation operators featuring in black hole physics, e.g., [43]. Expressions such as aL'nR are now meaningful because both pairs of operators aL; aL+ and aR; aR+ act on the same Hilbert space H. Their relation to the fundamental eld operator x is simply that of a shifted harmonic oscillator, 1 2 yR = x xR = p (aR + aR+); yL = yR = xL x = p (aL + aL+): Since the two oscillators are related by a displacement (up to a sign), it follows that the creation-annihilation operators are related as well, HJEP06(218)7 1 2 N p I 2 (3.12) (3.13) (3.14) p 1 2 aL = I aR = aR: from this viewpoint every state 'nL disappears as perturbation theory [44] this limit is singular. tonian HR(1) vanishes for operators bL; bL+; bR; bR+, these operators diverge in the decoupling limit.6 A striking feature of this expression is that it does not possess a nite decoupling limit N ! 1 as an operator statement. Instead of approximating the free eld creation-annihilation As we can see, the ! 0 limit should be taken with care. The low occupancy modes 'nR and 'Lm, n; m; N nearly decouple for small and in the limit ! 0, the two sets of modes decouple completely. We end up with two separate harmonic oscillators with the tensor product Hilbert space H0 = FL FR. On the other hand the interaction Hamil= 0. From the point of view of the vacuum at xR, the second vacuum moves away and a single harmonic oscillator Hilbert space FR remains. Hence ! 0. In the language of mathematical This is also the case in the bulk for the black holes with scalar hair described in the Introduction and reviewed in section 2. The distance between the vacua in the dual is related to the boundary conditions in the bulk, which in turn determine the amount of scalar hair. Increasing the distance between both vacua, keeping the black hole mass constant, also increases the value of the scalar hair on the horizon. In the limit in which the vacua in the dual theory decouple, a curvature singularity at the horizon develops, e ectively dividing the inside and outside regions in two separate spacetimes. In black hole physics one usually assumes at the outset that the Hilbert space splits in a tensor product of Fock spaces associated with modes inside and outside the horizon. In particular this is a fundamental assumption behind much of the discussion of the information 6Another indication that the natural creation-annihilation operators aL; aL+; aR; aR+ are not to be identi ed with the perturbative creation-annihilation operators are their commutation relations. Using (3.14) we can nd that [aL; aR] = [aL+; aR+] = 0; [aL; aR+] = [aR; aL+] = 1 : (3.15) Hence aL and aR+ do not commute, in sharp contrast with the usual situation in black hole physics, where one expects the left and right creation-annihilation operators to commute as a consequence of the locality of semiclassical physics near the horizon. -10 -10 -5 5 10 x 5 10 x HJEP06(218)7 -10 -5 0+ and have slightly larger energy than even eigenstates n+ (blue). 0 (right). For energies lower than the top of the barrier V odd eigenstates n (red) paradox (see e.g., [2{5, 10, 11, 45{47]). Our toy-model shows that small, non-perturbative interactions can drastically change this structure.7 3.2 Black hole microstates Another important characteristic of our toy-model is that it is solvable. The energy eigenstates and the corresponding energies can be computed numerically to arbitrary precision by various methods, most notably the standard Ritz-Rayleigh method.8 We will say that f ( ) is non-perturbatively small if f 0 as the asymptotic expansion. Equivalently, f ( ) is non-perturbatively small if f ( ) = o( n ) for all n 0 as ! 0+. We will denote any non-perturbatively small terms by o( 1). Finally we say that two quantities are equal in perturbation theory or equal up to nonperturbative terms if they have identical asymptotic expansions. These de nitions are needed when considering, for example, the energy eigenstates of the full system, to which ! 0+, where denotes we now turn. Since the double well potential is invariant under x ! x the operator commutes with the Hamiltonian. Hence every energy eigenstate has de nite parity, and the Hilbert space can be decomposed as H = H denote energy eigenstates by + H H , where H + = H + and H = H . We n = En n ; (3.16) where n are even/odd eigenstates.9 The corresponding energies satisfy En+ < En and their di erence En = En En+ is exponentially small. In particular, the energy di erence between the ground state and the rst excited state is dominated by the 1-instanton n 2 1=2'nR. The two lowest energy eigenstates 0 can be seen in gure 3. black holes in AdS were recently analyzed in [26, 27]. lead to asymptotic expansions around 7Interactions in the form of shock waves between the left and right Hilbert spaces of two-sided eternal 8One cannot rely on perturbation theory around a single minimum, since perturbative series diverge and n to be real and normed to one. Overall signs are such that for x ! 1, 0.6 0.4 coupling . Right: the di erence between the two energies as a function of . The dashed line represents the leading 1-instanton approximation given by (3.17). λ The vacuum energy E0+ is always smaller than 1=2 | the energy of an unperturbed ground state of the harmonic oscillator. The di erence En 0 in general is a non-perturbative e ect and hence exponentially small for energies well below the maximum of the potential, En < V = 1=(32 ). For energies larger than this, non-perturbative e ects are numerically large and the di erence En ' 1=2, as one can observe in gure 3. In this sense every pair of energy eigenstates n corresponds to two microstates with exponentially small energy splitting due to nonperturbative e ects. Motivated by the dual description of the black holes with scalar hair in section 2, where the (perturbative) degrees of freedom on both sides of the horizon correspond to excitations around two di erent perturbative vacua, we interpret states with signi cant support around both minima of the potential as the dual description in our toy-model of a black hole microstate. By contrast, semiclassical states centered around one of the two vacua only are interpreted as spacetimes without a black hole.10 We consider microstates of any energy11 E V , but the lowest energy states are of a particular interest. Denote M = f + 0+ + perturbatively close to the ground state energy, h jHj i = E0+ + o( 1). Hence, in perturbation theory the ground state is degenerate. For this reason we will refer to M as 10At rst sight this description of microstates is at odds with the intuition that black holes should be high-energy states E N 2, compared to the vacuum energy, as perceived by an asymptotic observer. However for an interacting system a number operator and the Hamiltonian are in general very di erent; whereas the Hamiltonian of the system is a unique operator once the time direction is chosen, a number operator is an inherently semiclassical object that depends on a choice of a semiclassical vacuum. We return to this point below. 11This restriction on the energies is needed for a perturbative description to be meaingful and resonates with the bulk where there are no hairy black holes above a certain mass (cf. gure 2(b)). the subspace of perturbative vacua and each element 2 M will be called a perturbative vacuum. Roughly we have in mind a correspondence between the (degenerate) energy En of the states and the mass of the black holes. A relation between microstates and macrostates can be twofold. We rst discuss this from the viewpoint of a single, say right, asymptotic observer with easy access to the right portion of the wave function only. This is the natural perspective if we consider our model to be a dual toy-model description of the single-sided hairy black holes discussed earlier. In this context the right portion of the wave function speci es a macrostate, and a set of microstates di ering in the shape of the wave function around the left minimum can be considered. A second characterisation of macrostates follows from considerations of a pair of observers in two distinct asymptotic regions as e.g., in the case of eternal black holes. From the point of view of two such perturbative observers a macrostate is given by two independent pieces of the wave function: the left portion, L, and the right portion, R. We can regard a macrostate as being represented by a tensor product L R, while a set of corresponding microstates is given by all states of the form L L + R R for L; R 2 C. Each microstate is a speci c continuation through the potential barrier that eludes both observers. We will discuss the relation between such macrostates and microstates in detail in section 4. This interpretation also ts in the black hole paradigm of [49] and with a more general quantum perspective on black holes [10, 50, 51] according to which, from the point of view of a single asymptotic observer, the Hilbert space H factors as H = Hcoarse H ne. The coarse degrees of freedom Hcoarse are clearly distinguishable by the asymptotic observer within perturbation theory whereas H ne contains non-perturbative e ects to identify. In our model H ne = C2 is a two-dimensional space, which can be identi ed with the space of perturbative vacua M. The energy di erence of any two microstates is then non-perturbatively small, and hence our model satis es ne degrees of freedom that require postulate 3 of [2, 52]. also leads to the Boltzmann entropy, Finally the fact that in perturbation theory various microstates cannot be distinguished SB = log dim H ne = log 2: This is the Bolzmann entropy associated with a single pair of harmonic oscillators.12 3.3 We now explore further the implications of the above identi cation of black hole microstates in our toy-model. The model has two natural number operators that describe (perturbative) excitations from one or the other asymptotic viewpoint, NL = HL(0) = aL+aL; NR = HR(0) = aR+aR (3.19) (3.20) HJEP06(218)7 However, we are also interested in the description of observations from the viewpoint of an infalling observer with easy access to perturbative physics in both asymptotic regions, 12To get an area factor as in black holes, one should consider an ensemble of oscillators with di erent frequencies [43]. 0.20 0.15 by (3.30). Hence the plot shows excitations of a few initial modes 'nR for n of the maximum is always around n 1=(2 ). Right: matrix coe cients between the true ground state 0+ and 'nR as a function of n. Recall that in the leading order the ground state is given 0 as well as a wide peak around n 1=(2 ). In both gures inside and outside the horizon of the black hole. A rst guess for this is to consider the following number operator NA = NL + NR + O( ) = aL+aL + aR+aR + O( p ); possibly up to small corrections in .13 Indeed, if = 0 this operator counts a sum of excitations of two decoupled harmonic oscillators. One would expect that with a small coupling , the sum NL + NR recieves corrections of order O( ) in such a way that NA vanishes (or at least is small) in the new vacuum . However, to the contrary, the p expectation value of NA in a generic state turns out to be very large, p 1 2 h jNAj i & Since this diverges as N grows, in the language of [2] the microstate appears to exhibit a rewall. In particular even very low energy states including semiclassical vacua exhibit rewalls. Indeed, while as expected, relation (3.14) leads to a large expectation value h'0LjNLj'0Li = h'0RjNRj'0Ri = 0 ; h'0LjNRj'0Li = h'0RjNLj'0Ri = = N 2: h'0LjHj'0Li = h'0RjHj'0Ri = By contrast, the energy of the state remains small, 13Formally, the number operator for the infalling observer is of this form. In fact, for a xed mode, the number operator for the infalling observer is non-perturbatively close to the number operator for the asymptotic observer, since hNAi e 8 !M with M of order N . N 1 2 1 2 : 1 2 + : 3 8 (3.21) (3.22) (3.23) (3.24) (3.25) The large expectation values (3.24) are a manifestation of the fact that from the viewpoint of, say, the right minimum, the state '0L is a highly excited state. Indeed, since the full Hilbert space H is isomorphic to FR, both sets f'nRgn and f'nLgn span the entire Hilbert space H separately. We can decompose the left modes 'nL in terms of right modes 'nR as, 1 X n=0 'Lm = h'nRj'Lmi'nR: The value of the matrix element h'nRj'Lmi can be calculated by noticing that the left and right modes are related by a displacement, up to a sign, vacuum state m = 0, for which L(n) = 1, 0 where L(m ) denotes Laguerre polynomials. This expression simpli es for the semiclassical h'0Lj'nRi = ( 1)ne 41 p2n nn! : (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) Figure 5 shows numerical values of these matrix elements as a function of n. For small the distance between the minima is large and one can use Stirling's formula to nd that (3.29) attains its maximum at n = 1=(2 ). This shows that, in order to write a semiclassical vacuum state '0L as a superposition of the semiclassical states around the right minimum, 'nR, one needs to excite highly energetic states, namely those with n of order N 2. Low energy states of the left asymptotic observer are detected as highly excited states by the right asymptotic observer and vice versa. The above conclusions are directly applicable to the lowest energy eigenstates. It follows from perturbation theory that to leading order in the coupling we have 0 = p ('0R 1 2 '0L) + O( p ): Hence these states are all highly populated both with respect to NL and NR. In particular equation (3.24) implies that h 0 jNRj 0 i = h 0 jNLj 0 i = 1 4 By considering a general microstate of the form rewall expressed by equation (3.22). Thus we nd that even the ground state 2 M the de nition (3.21) leads to a 0+ is a highly populated state of very small energy as measured by the total Hamiltonian (1.2). This seems paradoxical but is in fact N 16 1=2. The dashed line represents the leading term N 2=4 in equation (3.31). just a consequence of the interacting nature of the system and related to non-perturbative e ects. Actual numerical values of the matrix elements h 0+j'nRi as a function of n are shown in gure 5, while the expectation value of the right number operator NR in the ground state 0R is presented in gure 6. To conclude, we have identi ed black hole microstates in our dual toy-model as relatively low energy states in the dual theory that are nevertheless heavily populated from the point of view of an asymptotic observer. The microstates are indistinguishable by an asymptotic observer with access to the perturbative physics only. A microstate structure emerges in our model as a consequence of non-perturbative level splitting (3.17) in the presence of interactions. This splitting can then be regarded as a source for the entropy (3.19). At rst sight the results of this section would seem to support the conclusions of [2, 3], i.e., the presence of rewalls. This, however, will turn out to be false. A caveat is that in the decoupling limit ! 0 the creation-annihilation operators aL; aL+; aR; aR+ do not approach the perturbative operators bL; bL+; bR; bR+ in any sense. In fact, as indicated by equation (3.14) such a limit is ill-de ned. In section 4, starting from the interacting model, we carefully identify the perturbative degrees of freedom from the point of view of both observers, and we construct well-de ned perturbative operators. In particular we will construct another set of creation-annihilation operators on H, which will have a wellde ned decoupling limit. 3.4 Limitations of perturbation theory Before we proceed we pause to formulate precisely the limitations of the validity of perturbation theory around one of the minima in our model. This will be important in what follows. a. Perturbation theory breaks down when the overlap between the left and right semiclassical modes, h'Lmj'nRi becomes signi cant when n m Notice that all matrix elements (3.28) are exponentially damped by a factor of e 1=(4 ). In other words one could write h'Lmj'nRi = O(e 1=(4 )) = o( 1). This is the correct behavior for an amplitude associated with the tunneling process, but it does not imply that the amplitude remains small for all states. Indeed, a degree of a Laguerre polynomial L(n )(z) is equal to n, and the leading term is ( 1)nzn=n!. Hence, for n = m the matrix element becomes h'nLj'nRi = e 4 1 (2 )nn! 1=(4 ) the Stirling's formula indicates that the denominator vanishes faster than the numerator. Therefore the non-perturbative terms become numerically b. Time-independent perturbation theory breaks down for states with occupancy numbers (3.32) (3.33) (3.34) (3.35) h'RmjHj'nRi = O( perturbative state 'nR, Since the potential (1.1) is quartic, h'RmjHj'nRi = 0 if jn mj > 4. Furthermore, p ) if m 6= n. Hence we can concentrate on the energy of the n-th 1 2 3 8 h'nRjHj'nRi = + n + (2n2 + 2n + 1): Clearly, if n is of order 1= the correction is of the same order than the unperturbed part. This is one of the many indications that the perturbative methods break down become relevant whenever applied to states with occupancy numbers n In the commutation relation (3.34) the correction is formally of order p . However, when applied to the state 'nR with n than the perturbation. 1= the unperturbed part is of the same order for states with occupancy numbers of order 1= . c. The subleading terms in the commutation relation p 2 y 2 R p2yR3 d. Time-dependent perturbation theory breaks down for times t N for any state. Breakdown of perturbation theory can also be seen in time evolution. For example, time-dependent perturbation gives a matrix element h'nRje itH j'Rmi = nm ith'nRjHR(1)j'Rmi + : : : 1=p When jn mj > 4, the matrix element in the second term is identically zero. Hence such term becomes relevant in the expansion in 1=p . Even for low occupancy states with m; n of order 1 in , when either n, m are of order , the perturbation theory breaks down after time t N . Notice that this is signi cantly shorter than the exponentially large tunneling time. In particular it agrees with the scrambling time of [23, 53] with entropy (3.19) and the mass M N . We will discuss time-dependent processes in section 6 in more detail, where we will also recover the relation M N for our toy-model. As we discussed the microstates 0 are highly excited. Hence, according to point a above, perturbation theory is expected to break down whenever `black hole microstate' e ects are probed from the point of view of one of the semiclassical vacua. A very similar conclusion was reached in [54, 55] on the basis of gravitational (bulk) arguments. However this does not invalidate perturbation theory in general, which remains valid for excitations close to the semiclassical vacuum. In this sense postulate 2 of [2, 52] holds in our model. 4 Low energy excitations In the previous section we identi ed several aspects of the holographic dictionary that relate our quantum mechanical toy-model to black hole physics in a dual bulk spacetime. In particular we established a notion of asymptotic observers, perturbative vacua, semiclassical states and their Fock spaces as well as dual black hole microstates. We have shown how non-perturbative e ects enter in the picture leading to the breakdown of perturbation theory when it comes to the ne-grained features of microstates. We have also shown that the natural (naive) creation and annihilation operators (3.8) do not possess a well-de ned decoupling limit ! 0, and therefore cannot represent creation-annihilation operators associated with the asymptotic regions. As a consequence a rewall (3.22) emerged. In this section we correctly identify perturbative degrees of freedom as perceived by the asymptotic observers. We are able to distinguish perturbative and non-perturbative physics and to de ne suitable creation and annihilation operators with a well-de ned decoupling limit. In section 3.3 we have shown that any microstate exhibits a rewall as measured by the naive number operator (3.21). We have established that the source of the rewall is the fact that the creation and annihilation operators aL; aL+; aR; aR+ act on both left and right perturbative states 'nL and 'nR. A natural resolution would seem to be to de ne a di erent set of creation-annihilation operators ^aL; a^L+ and a^R; a^R+ such that a^R and a^ R+ act only on A new number operator N^A de ned by means of the hatted operators would then act on the energy eigenstates 0 according to (3.30) as a^R'nR =? p a^R+'nR =? p ^ NA = a^L+a^L + a^R+a^R 1 2 = 0 + O( ): p N^Aj 0 i = p (aR+aR'0R aL+aL'0L) + O( p ) Thus, no rewall! (4.1) (4.2) (4.3) (4.4) The only problem with this reasoning is that the operators satisfying (4.1) or (4.2) cannot exist. Since the full Hilbert space H is isomorphic to any single Fock space associated to a minimum, the set f'nL; 'nRgn constitutes an overcomplete basis. Given an action of aR on all right modes 'nR, its action on left modes 'nL is xed by means of (3.26). One could however hope to achieve relations (4.1) and (4.2) approximately for low energy modes 'nL and 'nR with n N . In fact, according to (3.28), the overlap between 'Lm and 'nR for m; n N is exponentially small, and the subset of modes f'Lm; 'nRgm;n N constitutes an `almost' orthonormal basis. Hence, we expect that the low energy physics should be well-approximated by the tensor product FL FR, where the excitations on the left and the right become independent. We now give a speci c proposal for `orthogonalizing' the overcomplete basis f'nL; 'nRgn in such a way that the hatted operators (4.1) or (4.2) can be successfully de ned. To be more precise, we will split the total Hilbert space H into two orthogonal components, H = HL The left and right hatted annihilation operators ^aL; a^R can then be de ned as projections of the unhatted operators aL; aR onto the appropriate subspaces. The `orthogonalization' is highly non-unique, but all ambiguities are non-perturbative and hence inaccessible in perturbation theory around any minimum. In the context of black hole physics the problem of overcompleteness of the basis has been pointed out in [56]. To resolve the overcompleteness of the set f'nL; 'nRgn consider symmetric and antisymmetric combinations of all energy eigenstates, nL = p ( n+ 1 2 n ); nR = p ( n+ + n ) 1 2 and consider two Hilbert subspaces of H, spanned by nL and nR respectively, HL = spanf nLgn; HR = spanf nRgn: From (4.5) we have that h L mj nRi = 0 for all n; m and hence HL and HR are orthogonal to each other. The full Hilbert space splits into a direct sum, H = HL HR; HL ? HR; HL = HR; HR = HL: In other words is a polarization of H. We refer to HL and HR as left and right perturbative Hilbert spaces respectively.14 Furthermore by PL and PR we denote canonical orthogonal projections of H onto HL and HR respectively. De ne a^R as aR restricted to HR and similarly de ne a^L as aL restricted to HL, a^L = PLaLPL; a^R = PRaRPR: De ne a^ number operator N^A, taking into account the fact that it should count excitations both in L+ and a^ R+ as their Hermitian conjugates. We repeat this prescription to de ne a HL and HR, low occupancy states around the left vacuum. ^ NA = PLNLPL PRNRPR = PLaL+aLPL + PRaR+aRPR: 14We will discuss the precise meaning of the word perturbative in the next section. For now notice that all nR for n N are localized around the right minimum only, whereas nL are localized (4.5) (4.6) (4.7) (4.8) (4.9) |〈ΨL/R0|φRn〉| 10-8 10-18 10-28 10-38 10-48 < N > (red). On the right: the expectation value h 0+jNAj 0+i for number operators in the true vacuum state. The blue line shows the expectation value for the operator NA = aL+aL + aR+aR. This operator leads to a rewall at ^ NA = PLaL+aLPL + PRaR+aRPR. The result approaches zero at perturbative fashion and exhibits a small non-vanishing value for ! 0. The red line shows the expectation value of the operator ! 0 in a characteristic non> 0 due to the interactions. This operator is now de ned globally on the entire H. It counts excitations on top of microstates from the point of view of both perturbative vacua. In particular, we argue that its expectation value in a state = particles on the left and n particles on the right, is given by L'Lm + R'nR representing approximately m λ h jN^Aj i = j Lj2m + j Rj2n + O( p ): (4.10) in any perturbative vacuum expectation value of the number operator in the ground state ^ 2 M, h jNAj i = O( p ).16 Hence N^A is a natural candidate for a global, rewall-free number operator.15 Speci cally The numerical plot of the 0+ as a function of is given in gure 7. While mathematically we de ned hatted operators in (4.8) using projectors, it may be more physically accurate not to specify the action of a^R on HL nor a^L on HR. Similarly, we could de ne left and right number operators NL and NR on HL and HR only, as their 15There are two slightly di erent choices here. One can de ne ^aL and a^R with their images unrestricted or restricted to the corresponding subspaces HL and HR. The latter de nition is a^R = PRaRPR as we have de ned, the former means that a^R = aRPR, and similarly for a^L. The di erence, PLaRPR, is however non-perturbatively small, as we will argue in section 4.3, and hence invisible in perturbation theory. For the same reason one can consider another number operator built up with hatted creation-annihilation operators N^ A0 = a^L+a^L a^R+a^R = PLaL+PLaLPL + PRaR+PRaRPR: (4.11) While formally di erent, the fact that PLaRPR = o( 1) as well as PRaRPL = o( 1) implies that the two number operators can di er by non-perturbative terms only, N^A = N^ A0 + o( 1). Hence, in perturbation theory the two operators are indistinguishable. We will stick to the de nition (4.9), which is slightly more convenient for numerical calculations. 16In [6{9] the Authors insist on a number operator N^A which satis es N^Aj 0 i = 0 exactly. From the point of view of the QFT this seems unnecessarily strong, since interactions do create particles. Nevertheless, one can de ne the appropriate number operator N^A = A^L+A^L AL nL = pn nL 1, and A^R is de ned on HR as A^R nR = pn nR 1. ^ A^R+A^R, where A^L is de ned on HL as R 2 φ R 1 φ R 0 Ψ R 3 Ψ R 2 Ψ Ψ R 1 R 0 states nL and 'nL is symmetric and denoted by dotted lines. semiclassical states 'nR. One the other hand, the states 'nR are not entirely contained in HR, as they have a small non-perturbative overlap with states in HL. The structure of HL and coresponding nR do not align exactly with nR = 'nR + O( p ), and hence physical meaning is associated with perturbative physics percieved by the corresponding observers. We can either refuse to act with perturbative operators on non-perturbative states or accept the fact that natural perturbative observables from the point of view of a given observer become non-perturbative from the point of view of the another observer. The total number operator (3.21), however, remains globally de ned. and In order to conclude the proof of (4.10) we have to study a relation between states 'nR nR, or equivalently between HR and FR. The perturbation theory [48] implies that HJEP06(218)7 PR'nR = PR h nR + O( p i ) = nR + O( p ) = 'nR + O( p ): a^R+a^R'nR = PRaR+aRPR'nR = n'nR + O( p ) (4.12) (4.13) This implies that and (4.10) follows. scalar product (3.28) is non-vanishing. Hence 'nR 2= HR. Equation (4.12) suggests that the familiar semiclassical state 'nR is not an element of the right perturbative space HR. Indeed, if some 'nR was an element of HR, then 'nR = 'nL would belong to HL. But the two states 'nR and 'nL are not orthogonal as their While no 'nR belongs to HR, for each n a di erence between the state 'nR and its projection PR'nR on HR is non-perturbatively small. Equivalently, by following Example 6 of section XII.3 of [44] one can argue that kPL'nRk = o( 1) and kPR'nLk = o( 1) for any n. Hence, while 'nR is not an element of the perturbative Hilbert space HR, its projection PR'nR 2 HR is non-perturbatively close to 'nR. In perturbation theory, one cannot distinguish the two states. A schematic relation between various states is presented in gure 8. By de ning left and right perturbative spaces HL and HR we e ectively resolved the overcompleteness of the set f'nL; 'nRgn. While stricktly speaking no semiclassical state 'nR belongs to HR, there exist states PR'nR non-perturbatively close to 'nR lying in HR. We have found `approximate isomorphisms' up to non-perturbative terms. A notion of a perturbative state is crucial for the discussion of the information paradox. An intuitive idea is that its support is concentrated around a single minimum. As an example consider basis states 'nR. Are all these states perturbative with respect to the right minimum or only those with n N 2, so that their support is concentrated around the minimum? In the language of [6{9] one would call a state 'nR perturbative only if N 2 is small. In this paper, however, we will introduce a weaker de nition that allows for a wider range of perturbative states. n f Our de nition of a perturbative state deals with the behavior of the state as the approaches zero. Therefore, instead of a single state , we consider a family g >0 labeled by the coupling. Essentially all states we consider depend on in some implicit way. For example the perturbative states 'nR and 'nL are de ned by (3.5){(3.6) and hence they implicitly depend on . For that reason we will refer to the elements of the is perturbative with respect to the right minimum if F 1 R = ! 0+. Here FR 1 : H ! FR is the inverse of the family f g >0 as a state We say that the state converges in norm in FR when 2 H. respect to the left minimum. isomorphism (3.5) between H and FR. Analogously we de ne states perturbative with First notice that all states 'nR are mapped to jniR 2 FR, which are -independent in FR. Hence all 'nR are trivially perturbative with respect to the right minimum. Consider now a state such as a ground state , which possesses two bumps around both left and right minimum. By going to FR we may simply position ourselves at x = xR and send to zero. The right portion of the wave function then concentrates around the right minimum and approaches 'R. As the left minimum moves away to 0 function is lost in the decoupling limit = 0. Indeed, neither 1, the left portion of the wave 0+ nor 0 are perturbtive with respect to any minimum. On the other hand, all states nR and nL as de ned in (4.5) are perturbative with respect to right and left minima respectively and we have, [44], lim !0+ FR 1 and lim !0+ FL 1 nL = jniL with the convergence in norm.17 Hence every element of HL nR = jniR is perturbative with respect to the left minimum and every element of HR is perturbative 17Even if a state is non-perturbative with respect to, say, right minimum, one can still de ne its decoupling limit. We will say that a state j 0i 2 FR is a decoupling limit with respect to the right minimum of 2 H if j 0i = wlim !0+ FR 1 following decoupling limits, , where wlim denotes the weak limit. In this sense we have the w!li0m+ FR 1 nR = jniR; w!li0m+ FR 1 nL = 0; w!li0m+ FR 1 n = p12 jniR: (4.15) Numerical values of the matrix elements h 0Lj'nRi and h 0Rj'nRi as functions of n can be seen in gure 7. with respect to the right minimum. This justi es their names as left and right perturbative Hilbert spaces HL and HR. We can also sharpen our de nition of a right (left) observer by declaring HR (HL) as the Hilbert space available to the observer. Notice that our de nition of a perturbative state depends only on what happens with the state when approaches to zero. For example, for a xed value of > 0 the support of 'nR is concentrated around the right minimum only for n according to our de nition, all 'nR are perturbative. As 1 = N 2. Nevertheless, approaches zero, each 'nR concentrates around the right minimum, since for each n there exists so small that n 1 . In the language of [6{9] the space of perturbative states was nitely dimensional, as the condition n N 2 on the occupancy numbers was imposed. In particular such a space was not generated by a genuine algebra acting on a cyclic vector: the issue that led the Authors of [7] to use a concept of `algebras with a cut-o '. In our approach such issues are by a^R and a^ completely avoided. The `small algebra' AR associated with the right minimum is generated R+ and the perturbative Hilbert space is then HR = ARj 0Ri. No cut-o s of any sort are required and all states in HR are perturbative with respect to the right minimum. Perturbative operators Having de ned perturbative states, one can also de ne perturbative operators. These should be operators which: (i) preserve the decoupling of the potential wells up to nonperturbative e ects; and (ii) have a well-de ned decoupling limit. If we write an operator A in the matrix form A = ALL ALR ARL ARR ! : HL HR ! HL HR (4.16) (4.17) with AIJ mapping HI into HJ , I; J 2 fL; Rg, then: (i) ALR and ARL must be nonperturbatively small, i.e., ALR = o( 1) and ARL = o( 1); ! 0 of ALL in HL and ARR in HR must exist. We will call and (ii) the decoupling limits such an operator A as perturbative. All hatted operators such as creation-annihilation operators ^aL; a^+; a^R; a^R+ or the numL ber operator N^A (4.9) are by construction perturbative with vanishing o -diagonal terms. Their unhatted versions usually fail to satisfy condition (ii), as indicated by (3.14) or by the rewall in (3.22). On the other hand, one expects that condition (i) holds, since PRaRPR'nR = pn'nR 1 + o( 1) or equivalently PLaRPR = o( 1). Figure 9 shows the norm of the state PLaR 0R with a characteristic exponential fall-o around = 0. In this sense hatted and unhatted creation-annihilation operators agree on their corresponding perturbative Hilbert spaces. Schematically, aR = a^R + o( 1) on HR and aL = a^L + o( 1) on HL together with their conjugates. Only on HL, the complement of HR, the operators aR and a^R di er signi cantly. The unhatted creation-annihilation operators satisfy commutation relation (3.15). However, in black hole physics, locality at the level of the e ective bulk theory requires 1.0 0.8 with HL is de ned as jh ; HLij2 = kPL k2. For non-perturbative fashion. states 'nR with HL for n = 0 (blue), n = 1 (red) and n = 2 (green). The overlap of a state Hilbert space HL as function of the coupling . Right: a measure of an overlap of the perturbative HJEP06(218)7 ! 0 the overlaps approach zero in a characteristic that left and right operators commute. This is indeed the case for the hatted operators, where we nd [a^L; a^L+] = [a^R; a^R+] = 1 + o( 1); [a^L; a^R] = [a^L; a^R+] = 0: (4.18) The canonical commutation relations on HL and HR are altered by a non-perturbative factor, while left and right operators commute.18 Hence locality is maintained up to nonperturbative e ects as predicted by a number of papers, e.g., [6, 7, 49, 57, 58]. This, however, comes at the cost of the action of, say, right perturbative operators on the left perturbative states to be either unrelated to the action of unhatted creation-annihilation operators (extremely `non-local' as in [29, 59]), or simply ill-de ned [60]. In this way we have resolved the problem of ` tting' the left creation-annihilation operators into the full Hilbert space. By identifying the right perturbative Hilbert space as HR and the right perturbative operators as ^aR and a^ R+ we found the proper set of degrees of freedom and operators associated with a perturbative (asymptotic) observer in the right vacuum. In particular every state in HR is perturbative with respect to the right minimum, but non-perturbative from the point of view of the left observer. In the context of black holes a number of papers [61{64] have suggested to remove `half' of the states by considering an antipodal identi cation of the spacetime. The total Hilbert space then consists of only parity even or odd wave functions. Such an approach would remove a degeneracy within perturbation theory leading to each black hole having a single microstate. In our model we do not nd a support for such an identi cation, but we also do not nd any obstacles in its implementation. From the point of view of a single observer, the two situations are indistinguishable as long as perturbative processes of small energies are considered. Only at higher energies one would be able to notice `missing' states in the total Hilbert space. 18Slightly di erent de nitions of creation-annihilation operators as indicated in section 4.1 may result in non-perturbative corrections to the locality condition [^aL; a^R] = [a^L; a^R+] = o( 1). In perturbation theory operators are required to be de ned uniquely only up to non-perturbative terms. Finally let us point out that the original Hamiltonian H is a perturbative operator. The o -diagonal elements PRHPL = o( 1) and PLHPR = o( 1) are related to the tunneling rate and can be calculated by standard methods within the WKB approximation. With our de nition of non-perturbative e ects this statement remains true for all energies, even if actual matrix elements become numerically large. We return to time-dependent processes in section 6. 5 Low energy e ective theory: state-dependence In the previous section we have constructed states and operators in the full theory that are the natural perturbative objects from the standpoint of semiclassical (or asymptotic) observers. In this section we relate these operators (observables) in the full theory to operators (observables) in the e ective low energy theory. In doing so we will nd that the resulting operators in the e ective theory are state-dependent. Perturbative observables In semiclassical black hole physics one usually takes the Hilbert space to be a tensor product FL annihilation operators bL; bL+ in FL and bR; bR+ on HR give rise to creation-annihilation FR of the degrees of left/right freedom on both sides of the horizon. The creationoperators in the tensor product: I bR; I R b+, etc. The bulk of the Hermitian operators are of the form ALL I + I ARR. The black hole microstate is not a vacuum state, but rather an excited state. Due to the large potential barrier, low occupancy states in our toy-model exhibit an approximate tensor product structure HL HR. Indeed, states mn = m; n N resemble elements of the tensor product, as they are combinations of two, nearly Lm + nR for a number of excitations given by h mnjN^Aj mni = m + n + O(p ) as expected. decoupled states centered around, respectively, the left and right perturbative vacuum, with Beyond this, however, the two systems di er. A linear structure of the direct sum HL HR and the tensor product HL HR is vastly di erent and we cannot hope to reproduce all perturbative operators on the tensor product in the full theory. In particular the action of linear operators in the tensor product is di erent and there are signi cantly `more' linear operators on the tensor product. Consider now an operator B = ALL I + I ARR on HL HR and assume Lm and nR form a set of normalized eigenfunctions of ALL and ARR respectively, with eigenvalues Lm and R n . This means that HJEP06(218)7 B( Lm nR) = ( Lm + nR)( Lm nR): We can use the identi cation of the direct sum with the direct product HL A( Lm; nR) = ( Lm + nR)( Lm; nR): (5.1) (5.2) The operator A acts on a state Lm + nR in the same way as B acts on Lm nR. Furthermore the expectation values match, L h m nRjBj m L nRi = Lm + nR = h( Lm; nR)jAj( Lm; nR)i: In this sense A realizes the same relations on HL HR as B on HL But the operator A is not linear. It is not even very clear how to extend it to the entire Hilbert space HL HR. For instance, we may consider projections AL and AR of A on HL and HR, so that A = AL AR. Operators AL and AR can be extended separately to bilinear operators by linearity in each argument. For example, we nd that for any R 2 HR (5.3) (5.4) AR( Lm; R) = PRA( Lm; R) = ( LmI + ARR) R: This is clearly a linear operator with respect to the right portion of the state, R 2 HR. The appearance of L , however, amounts to a form of `state-dependence': the value m of the right operator AR depends on the `hidden' left portion of the state. In this state-dependent sense we can think about AR as an operator on HR only, and we can write ARL( R) = AR( L; R). That is, we can regard ARL as an observable of the right observer. Its value on R, however, depends on the `black hole microstate' L, i.e., on the shape of the wave function on the other side of the potential barrier. In section 5.3 we relax our assumption that the action of the operator A takes the form (5.2). Starting from the weaker requirement that expectation values of operators in the full theory agree with those of the corresponding operators in the e ective theory, up to non-perturbative terms, we will show there that the corresponding global operators again cannot be de ned in a `state-independent' manner. However, the `state-dependence' then no longer pertains to the entire portion of the `hidden' wave function L, but instead is restricted to a choice of perturbative vacuum state 2 M. More generally, regardless of the details, we nd one cannot realize perturbative observables in the full theory as linear operators. Mathematically this is a consequence of the incompatibility of the linear structure of the direct sum and that of the tensor product. Physically, this means that perturbative interpretations of various operators depend on microstates of the system, as argued in [6, 7]. 5.2 The tensor product: loss of information R 2 HR an e ective state e as In the previous section we have argued that it is impossible to represent all perturbative observables on the tensor product HL HR by linear operators in the full theory HL On the other hand, the low energy physics in our model does e ectively take place on the tensor product. One therefore expects that an e ective theory based on the tensor product should capture the low energy physics, up to non-perturbative e ects. Let s : HL HR denote the canonical bilinear map s( L; R) = L HR = HL HR we can assign to any state = L + R, with L 2 HL and = L + R 7 ! e = N s( ) = N L R; (5.5) where N is a normalization. If s( ) 6= 0, we choose N 2 = k L + k L Rk2 = k kLkL2k2+k kRRk2k2 ; Rk2 so that the norms of and e are equal. First notice that if either L = 0 or R = 0, then 2 H is perturbative with respect to any minimum, then e = 0. In other words if a state = 0. We will say that a state is typical if it is represented by a non-vanishing e ective state in the e ective theory. Clearly, every typical state = L + R is generic in the sense that upon `random choice' HJEP06(218)7 of L and R it is unlikely to end up with an atypical state. Only typical states can be represented in the e ective theory. The image of the map (5.5) consists of a set Hpert of all simple tensors L R 2 HL normed state HR. A pre-image of a given simple tensor L 2 Hpert can be written as L R, however, is not unique. Every R with k Lk = k Rk = 1 and an irrelevant phase . The most general form of 2 H mapped onto e is = L L + R R; with j Lj2 + j Rj2 = 1; where arg L + arg R = whereas all corresponding + 2 n. We will refer to any e ective state e as a macrostate are microstates. Many di erent states are mapped onto the same macrostate in the e ective theory. In our toy-model, all possible microstates corresponding to a given macrostate e are parametrized by a unit vector ( L; R) 2 C2. Equivalently, this ambiguity amounts to a choice of a normed perturbative vacuum 2 M. Physically this means that knowledge of the left and right portions L and R of the wave function is not su cient to reconstruct the full wave function L L + R R. Instead a prescription for the continuation of the wave functions through the potential barrier is needed. This is clearly a non-perturbative e ect, and hence invisible in the e ective theory. R = TrL j e ih e j. L state R 7! the density matrix Given a simple tensor L R 2 Hpert, one can de ne its projection on, say, HR as R. From the point of view of the right vacuum these are pure states since, when traced over HL, they lead to the pure state density matrix j Rih Rj. Every other HR that does not belong to Hpert can be regarded as a mixed state with Note also that from the point of view of the right asymptotic observer, only states that lie in HR FR are perturbative. States beyond HR lack any perturbative interpretation. Hence states in FL FR that do not belong to HL HR lack a perturbative description from the standpoint of both asymptotic regions, even as mixed states. 5.3 Operators in the e ective theory Given an operator A in the full theory, we would also like to construct an operator Ae in the e ective theory such that 1 Z h e i + o( 1); (5.6) (5.7) (5.8) assuming and e are normalized to one. The proportionality factor Z should correspond to the number of microstates represented by an identical macrostate e . The relation (5.8) is known as the equilibrium condition [42]. It is in this sense that the full theory is realized by the e ective theory up to non-perturbative terms. Note that in order for (5.8) to hold, one must have e 6= 0 if is non-zero, i.e., the state must be typical. Secondly, o -diagonal elements of A must be non-perturbatively small, i.e., the operator A must be perturbative. Given a perturbative operator A a natural guess for its e ective counterpart Ae would be B = ALL I + I ARR: (5.9) (5.10) (5.11) (5.12) then reads while the right hand side is Many operators considered in the e ective theory are expected to be of this form. Unfortunately, B does not satisfy relation (5.8). Indeed, consider a normed state = L L + R R with k Lk = k Rk = k k = 1. The e ective state is L R with an irrelevant overall phase , which drops out from the expectation value. The left hand side of (5.8) h jAj i = j Lj2h LjALLj Li + j Rj2h RjARRj Ri + o( 1) h e jBj i = h LjALLj Li + h RjARRj Ri: The mismatch is not surprising: the correlator h j j i clearly distinguishes speci c miA crostates of the system, whereas h e jBj i depends on the overall macrostate only. In order for the condition (5.8) to hold one possibility is for the system to be in a special `equilibrium' state, and with an operator A that does not distinguish between microstates. Mathematically, if and ALL = ARR , then indeed Ae = B and we nd 1 2 h e jAe j i + o( 1): The Z factor accounts for the degeneracy of the macrostate, Z = eSB = 2, where SB is the Boltzmann entropy (3.19). What are equilibrium states in our model? Notice that the condition = means R = L = 2 1=2. Hence every energy eigenstate n is an equilibrium state. Furthermore every state of xed parity, even or odd, is also an equilibrium state. Operators satisfying ALL = ARR are those that act on both sides of the potential well `in the same way' regardless of a speci c microstate. This is closely related to the de nition of operators satisfying the Eigenstate Thermalization Hypothesis, e.g., [65{67]. In particular the total number operator and the Hamiltonian are of this form. By contrast, if the microstate is not an equilibrium state, then (5.10) depends on the speci c values of L and R. Given a global operator A, we can construct a class of e ective operators Ae , which depend on these parameters. Indeed, by comparing (5.10) with (5.11) we see that we need A e = ALL j Rj2I + j Lj2I ARR: (5.13) In this case Z = 1, as the right hand side of (5.8) produces the expectation value of A within a single, given microstate . The family of the operators Ae is parametrized by a unit vector ( L; R) 2 C 2 = M, or equivalently, by a perturbative vacuum state In particular every state of the form 1 X L (a^L+)n n pn! R (a^R+)n + n pn! 1 X n=0 1 X n=0 L 0L + R 0 R . j nLj2 = j nRj2 = 1 (5.14) is characterized by the same vector ( L; R) with = It is a de ning property of an e ective theory that it should give an approximate description of the full theory within its region of validity. In the context of quantum theory this means that the correlation functions calculated within the e ective theory should approximate those in the full theory. Given an operator A in the full theory, the operators Ae de ned in (5.13) satisfy this condition. This is a family of operators together with a `fake' type of state-dependence as described in [42]. The parameter can be thought of as parametrizing degenerated perturbative vacua. Indeed, each Ae is a perfectly wellde ned linear operator on HL HR, since the numbers from the point of view of the e ective theory. L and R are xed parameters HJEP06(218)7 One can however revert this last construction. Starting from an operator B as given in (5.9), one can try to construct a global operator A such that (5.8) is satis ed. This is a weaker condition than what we have considered in section 5.1, since here we only demand the agreement between the expectation values of the operators (up to non-perturbative e ects). It is obvious that a de nition of A must depend on the speci c microstate of the full system. We can construct a family of operators A in the full theory such that their action on a microstate L L + R R with L 2 HL, R 2 HR and k k = k Lk = k Rk = 1 reads A jALLLj2 + ARR j Rj2 : (5.15) With this de nition (5.8) holds with Z = 1. The functions A are now non-linear, i.e., `state-dependent', as they implicitly depend on the parameters L and R of the state L L + R R they act on. In other words they depend on the choice of the microstate within a given macrostate L This is an explicit construction of state-dependent operators in the theory in the spirit of [6{9]. Operators of the form (5.9) corresponding to naive perturbative observables become non-linear functions or, equivalently, state-dependent operators. In the context of [68] their matrix elements represent certain conditional probabilities. Furthermore, since the time evolution generically mixes various microstates, comparison of their matrix elements at di erent times seems problematic. 5.4 Global time evolution It is common in the context of quantum eld theory to calculate time-dependent eld operator correlation functions such as h0j (t1) (t2) : : : (tn)j0i. To consider such correlators in our toy-model, we must decide what are the corresponding state j0i, the eld operator , and the Hamiltonian driving the time evolution. From the point of view of the full theory, j0i is the vacuum state the eld operator, and time evolution is governed by the full Hamiltonian H. In the context of black hole physics, however, one typically considers correlation functions in the perturbation theory corresponding to the viewpoint of a single, say right, asymptotic and observer. In perturbation theory j0i corresponds to the right perturbative vacuum corresponds to the right perturbative eld operator y^R = PRyRPR = (a^R + a^R+)=p2. , What remains to be analyzed is the time evolution governed by the unitary operator U (t) = eitH . Its e ective version U e (t) should be such that the evolution of the states and the operators in the full theory matches with that in the e ective theory. Assume that R 0 , U (t) satis es the condition (4.17), i.e., ULR(t) = o( 1) and URL(t) = o( 1), and that HJEP06(218)7 the diagonal elements ULL(t) and URR(t) remain unitary, at least up to non-perturbative e ects. Since the Hamiltonian H satis es (4.17), this is the case for times t su ciently small for the tunneling e ects to be insigni cant. Now assign U (t) = ULL(t) + URR(t) 7 ! U e (t) = ULL(t) Since ULL and URR are unitary, they preserve the norm of the states and hence for any (U +)e (t) = (U e )+(t): This means that the states evolve in the same way both in the full and in the e ective theory. In particular Ae (t) = U +(t)A U (t) e = (U e )+(t)Ae U e (t): with Ae de ned as in the previous section. Therefore the condition (5.8) holds for timedependent operators as well, 1 Z h h jA(t)j i = e jAe (t)j e i + o( 1) With (5.16) the e ective theory correctly describes the expectation values of timedependent operators as long as the perturbation theory remains valid. 5.5 Time reversal and the `wrong sign' commutation relations In the previous section we have de ned a notion of time evolution in the e ective theory that is speci ed by the full theory, based in particular on the global time t inherited from the full Hamiltonian H in the Hilbert space H. From the perspective of a single observer (say the right one), however, the common practice is to consider the doubled Hilbert space built out of the right Hilbert space HR. In this case, the e ective theory as constructed by the right observer lives on HR HR = ( HL) HR, with e ective states de ned as = L + R 7 ! e = N ( L) R 2 HR HR where N is the same normalization as in (5.5). This does not change the analysis of previous sections in any signi cant way. Simply, every operator acting on HL must now be accompanied by a conjugation by . For example, equation (5.13) would read A ~e = ALL ARR (5.16) (5.17) (5.18) (5.19) (5.20) (5.21) and so on. For the Hamiltonian H of the full theory we have HLL = HRR and hence we nd its e ective version H~ e = HRR Consider now the time evolution operator U (t) = eitH . Due to the additional factor = URR( t). Hence, in order to maintain (5.17), we have to include an additional conjugation by in (5.16), i.e., to de ne U~ e (t) = ULL(t) URR(t) = URR( t) h~ of U~ e (t) satisfying U~ e (t) = eith~ is From the point of view of the right observer, in the e ective theory on HR HR time directions are opposite in both component spaces HR. In particular the Hermitian generator HJEP06(218)7 h~ = HRR I + I HRR: This is a Hermitian generator of time translations in the e ective theory on HR but it is not bounded from below. Furthermore, it is not proportional to any H~ e , the HR, family of e ective operators corresponding to the full Hamiltonian. Finally, h~ exhibits the famous `wrong' commutation relations [2, 5{7, 49] with left creation-annihilation operators satisfying, [h~; a^L+ I] = L I + O( ): p (5.25) This is in no contradiction with unitarity or any other property of a well-de ned quantum theory. The e ective theory is merely designed to mimic the full theory within the regime of its validity. It recognizes the full Hamiltonian as H~ e de ned in (5.22). The time evolution of the e ective theory is however driven by h~ in such a way that (5.19) holds. This is di erent from the e ective theory on HL HR (or HL HL as perceived by the left observer). Di erent e ective viewpoints realize observables in di erent ways, despite describing the same theory. However expectation values of corresponding operators in corresponding states are equal in all e ective theories we have constructed here. (5.22) (5.23) (5.24) 6 With all elements of our toy-model in place we now turn to a number of dynamical processes that are both tractable and have a clear dual interpretation in terms of black hole physics. Before we proceed, let us point out the obvious: our toy-model with Hamiltonian (1.2) is unitary. Since it also has a unique vacuum state, 0+, it would seem that our model violates 0 Hawking's theorem, as stated in [11]. However, the vacuum state in our e ective theory is doubly degenerate, , and, since the theorem describes the semiclassical situation, its assumptions are not satis ed. First we investigate tunneling through the potential barrier. Depending on the energy range, this can be viewed either as the decay of one of the perturbative vacua, the Hawking radiation process, or scattering of waves o of the black hole. Then we consider the evolution of a classical particle. We identify signatures of the chaotic behavior and scrambling. V* E I where where x1 are the turning points as illustrated in gure 10, and E < V = V (0) = 1=(32 ). The tunneling time and the tunneling rate are then 1=2 e . In our toy-model the tunneling probability (6.3) within the WKB approximation can be expressed in terms 1 (6.1) (6.2) (6.3) II -x2 -2 1λ x1 2 1λ x2 -5 Right: the shape of the real part (blue) and imaginaty part (red) of the (generalized) energy eigenfunctions representing waves incident from the left and scattered by the potential of the inverted harmonic oscillator. The functions correspond to 2 = 1=2 and energies = 1; 0; 1. 6.1 Tunneling and Hawking radiation the left and right states 0L and 0R, as The evolution operator eitH restricted to the space of perturbative vacua M slowly mixes eitH = eitE0 L ! 0 R 0 cos 12 i sin 12 t E0 t E0 i sin 12 cos 12 t E0 t E0 L ! 0 R 0 1 2 E0 = (E0+ + E0 ); E0 = E+ 0 E0 : Assume that at t = 0 the system is in the state R. As the system evolves, the wave function 0 slowly leaks into the left minimum, evolving into `more typical' black hole microstates. This is reminiscent of known instabilities of perturbative vacua [69, 70] evolving into black holes and envisioned e.g., in the formation of fuzzballs [51, 71]. On the other hand, as the wave functions builds up on the left, the right observer perceives the in ow of highly excited particles. Indeed, according to the results of section 3.3, when decomposed in terms of semiclassical modes 'nR, 0L is a highly-excited state. We can interpret these particles as the Hawking radiation. The same remains true for any low energy black hole state . We can identify some features of the Hawking radiation in our model by tracing back these high occupancy modes through the potential barrier. In the WKB approximation the tunneling probability for a particle of energy E in a double well potential V is given by = Z x1 p2(V (x) x1 E)dx ; of complete elliptic integrals, 1 q 1 + p E s 1 1 + p p ! K s p !# 1 1 + p where = E=V = 32 E with 0 < < 1, and K denotes the complete elliptic integral of the rst kind. For energies E close to V we have 1 and then (6.4) reduces to ( ; ) = p 2 + 3 2 2 + O( 3); = V E: The rst term in (6.5) corresponds to tunneling in an inverted harmonic oscillator with the potential Viho = !2 x42 , for a particle of energy E = ! . The tunneling rate becomes iho(E = ! ) = exp 2 2 6 2 2 + O( 3) : p in a black hole state of mass M 1=p = N . This resonates with [72] where it was shown that Hawking radiation can be viewed as a tunneling e ect between regions near both sides of the horizon. Furthermore, speci c deviations from an exact thermal spectrum were found in such a process, in a range of di erent black hole backgrounds. A comparison with [72] shows that the Hawking radiation in our toy-model corresponds roughly to a single pair of modes of a eld of frequency Building on the analysis of [72] it has been shown in [73] (see also [74]) that the scattering matrix of an inverted harmonic oscillator also appears in the investigation of scattering of shock waves from in the black hole background. Indeed, this is precisely the conclusion from our toy-model. Around x = 0 energy eigenfunctions are approximated by wave functions of the inverted harmonic oscillator with energy E. The approximate Hamiltonian reads and in our speci c toy-model 2 = !2=2 = 1=2. To analyze quantum scattering, we consider oscillating wave functions satisfying the Schrodinger equation with the Hamiltonian (6.7), where = E V and 2 x. A pair of solutions exists for each energy value , which can be expressed in terms of hypergeometric functions, see [75]. With an appropriate normalization the two solutions represent waves incident from the left and from the right, as shown in gure 10. By expanding the wave functions in the asymptotic regions transmission and re ection coe cients can be found, T = e 2 1 2 i R = e i p 2 =2 1 2 i which satisfy jT j2 + jRj2 = 1. Loosely speaking these coe cients describe the behavior in our toy-model of excitations near a black hole horizon. It is remarkable that the same form Hiho = 2 2 2 1 2 2 2 1 2 + = p ( ) = 0 (6.4) (6.5) (6.6) (6.7) (6.8) (6.9) HJEP06(218)7 of the transmission and re ection coe cients (6.9) was recovered there from the analysis of the dynamics of shock waves traveling in the black hole background in [73]. To leading order the result follows simply from the repulsive nature of the inverted harmonic oscillator and depends on a single parameter in (6.7). Classical evolution and chaos Consider now a classical particle of energy E 0. Such a particle will stay on a closed orbit. If E < V = V (0), then the orbit remains `outside the black hole', i.e., it does not cross the maximum of the potential at x = 0. The period of the classical motion is then HJEP06(218)7 (6.10) (6.11) Ttrapped = 8 + p1 + p 0s 1 + p 1 1 1 A ; When expanded around = 0 one nds as expected, Ttrapped = 2 + O( ). On the other hand the period diverges logarithmically when E ! V , Ttrapped = p 2 log 2 + O( ); = V E: This is a sign of a critical behavior. When analyzed from the point of view of the timedependent position x(t) of the particle, it is a manifestation of chaos. To see it explicitly, assume that is small enough for the particle to be located close to the tip of the potential for a long time. Hence we can neglect interaction terms and consider the potential of the inverted harmonic oscillator only. The solution to the equations of motion is then simply x(t) = x0 cosh( t) + v0= sinh( t), where is the `frequency' of the inverted harmonic oscillator as de ned in (6.7). Parameters x0 and v0 are initial position and velocity at t = 0 and under their variation de nition chaotic behavior with the Lapunov exponent e t( x0 + v0= ) for large times. This is by = !=p2 = 1=p2. As in the previous section the chaotic behavior is driven by the inverted harmonic oscillator and parametrized by a single parameter . The same conclusion arises from the analysis of shock waves in a black hole background in [23, 24, 26, 27]. Perhaps unsurprisingly, our analysis shows a connection between the classical chaotic behavior and quantum Hawking radiation from the point of view of the tunneling process as analyzed in the previous section. Both processes are described by the same parameter characterizing the `frequency' of the inverted harmonic oscillator (6.7). One can also analyze the quantum evolution of the system numerically and show how the classical evolution becomes inaccurate when E approaches V . To do so, we consider the evolution of suitable initially coherent states. The resulting evolution is illustrated in gure 11. As we can see, at low energies the original coherent state (blue curve on the left) bounces back and retains its Gaussian shape after a full period (green). A small change in the shape is caused by interactions of order into the left minimum, creating barely visible ripples in the density j j2. On the other hand a coherent state of energy only slightly lower than V rapidly turns into a highly quantum oscillating wave. The original classical particle (blue curve on the right) starts p . The wave function leaks slightly V/100.7& |ψ|^2 0.6 V/50.&7 |ψ|^2 0.6 x -10 -5 10 x state. Plots on the left show the evolution of a low energy wave packet on a classical path starting at x = 5:5 (classical energy E = 0:14). Plots on the right present the evolution of a wave packet of energy just below the energy V of the tip of the potential (E = 2:88, V = 3:125), with the initial position at x = 7. In both cases = 1=100. The outlines of the potential in the 3D plots are placed at times: t = 0 (blue), quarter of the classical period (purple, only on the right), half of the period (red), and the full classical period (green). The 2D plots show the shapes of the wavefunctions at the corresponding times. The dashed lines indicate the classical energy of the wave packet. moving towards the maximum of the potential. At the quarter of the period the peak widens (purple curve). By the time it bounces back at half the period it already breaks into a highly oscillating quantum wave (red curve). After the full classical period passes, it becomes completely scrambled and spread out over the entire domain (green curve). It cannot be viewed as a classical, localized state any more. 7 Summary and conclusions Motivated by the holographic description of certain classes of black holes in AdS we have put forward a new and solvable dual toy-model of black holes in terms of a quantum mechanical particle in a double well potential. The e ective low energy description involves the tensor product of two decoupled harmonic oscillators representing the degrees of freedom on both sides of the horizon, or in both asymptotic regions. At this level our model captures many of the usual paradoxes of semiclassical black hole physics expressed here in quantum eld theoretical language without explicit reference to geometry. The e ective low energy description of the system as a pair of decoupled oscillators is altered drastically by non-perturbative interactions. We have carefully explored how states and operators in the e ective theory emerge from and relate to corresponding quantities in the full model. This elucidates how holographic black hole models involving nonperturbative interactions between two decoupled low energy theories resolve some of the paradoxes of semiclassical black hole physics. Our key ndings are the following: Firewalls: black hole states in our model are represented by wave functions with signi cant support in both minima.19 At rst sight the low energy theory predicts a rewall. This is because the expectation value in black hole states of the naive number operator (3.21) is large. This number operator has the same form in the decoupling limit and hence it is usually assumed that it should represent the number operator for a small coupling as well. However, by carefully identifying and disentangling the perturbative degrees of freedom we have shown that this is incorrect. We have constructed a di erent number operator (4.9) that is well-de ned in the full model and perturbative in a precise sense. In the decoupling limit this operator correctly reproduces (3.21). We found this does not predict a rewall. Hence our model satis es all four postulates of [2]. State-dependence: our toy-model describes both sides of the horizon in terms of a single well-de ned, local, unitary quantum theory. Hence it does not require any `state-dependent' operators to describe behind the horizon physics. However we have shown that a clear notion of `state-dependence' emerges when one relates perturbative operators in the full theory to observables in the e ective low energy model on the tensor product. Mathematically this is because the linear structure of the direct sum is very di erent from that of the tensor product. Physically, the state-dependence accounts for the dependence of perturbative operators in the full theory on the black hole microstate, i.e., on the shape of the portion of the wave function behind the barrier that is inaccessible to a given asymptotic observer. Vacuum structure: our model circumvents Mathur's no-go theorem [10], based on Hawking's original calculation, that states that the information paradox cannot be resolved by exponentially small corrections to correlation functions. This is because this theorem assumes there is a unique vacuum. In our toy-model the perturbative vacuum is degenerate, and the degeneracy is only broken by non-perturbative e ects leading to a unique ground state 0+ in the full theory. In our model the vacuum of an infalling observer is not represented by a semiclassical vacuum, but rather by a superposition of two semiclassical vacua in line with e.g., [76, 77]. Phrased di erently, one could say our model takes seriously the doubled copy of the system and in fact realizes an ensemble of states in the full interacting theory that are to some extent similar to the thermo eld double state. 19This resonates with the model in [76] which was also argued to remove the rewall. description of the system from the standpoint of an observer in one of the asymptotic regions. This is not in contradiction with unitarity of the full model since the Hamiltonian of the full theory is represented by a di erent operator (5.22) than the generator (5.24) of time translations in the e ective theory around one of the perturbative vacua. A major advantage of our toy-model is that it is solvable. This has enabled us to analyze the role of non-perturbative interactions in a number of interesting dynamical processes. The appearance of an inverted harmonic oscillator potential separating both wells as a toy-model for non-perturbative interactions leads to features, such as chaotic behavior, which have a natural analog or `dual interpretation' in black hole backgrounds where similar behavior was obtained e.g., in the analysis of shock waves. Moreover, the breakdown of classical evolution of initially coherent states in our model shows that the evolution of an infalling object becomes highly quantum from the viewpoint of an external observer, in line with the principle of black hole complementarity [52]. It would be interesting to investigate the role of non-perturbative e ects and the emergence of the e ective theory in more complex models. These could include matrix and tensor models [78{83], where many features discussed in this paper emerge. Another direction includes the CFT analysis in the context of holography [30, 84, 85], where non-perturbative e ects also become essential in the understanding of unitarity and locality. Acknowledgments It is a pleasure to thank Ramy Brustein, Jan De Boer, and Kyriakos Papadodimas for helpful discussions. This work is supported in part by the European Research Council grant ERC-2013-CoG 616732 HoloQosmos, the C16/16/005 grant of the KU Leuven, and by the National Science Foundation of Belgium (FWO) grant G092617N. AB is supported by the CEA Enhanced Eurotalents Fellowship. The work of AG is supported by a Marie Sklodowska-Curie Individual Fellowship of the European Commission Horizon 2020 Program under contract number 702548 GaugedBH. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 2460 [INSPIRE]. 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Adam Bzowski, Alessandra Gnecchi, Thomas Hertog. Interactions resolve state-dependence in a toy-model of AdS black holes, Journal of High Energy Physics, 2018, 167, DOI: 10.1007/JHEP06(2018)167