Interactions resolve statedependence in a toymodel of AdS black holes
Received: May
Interactions resolve statedependence in a toymodel
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 toymodel 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 statedependence 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: statedependence
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 rewallfree number operator for an infalling
observer.
Statedependence: in contrast with this [6{9] proposed a dual description of the
black hole interior by introducing statedependent 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
creationannihilation operators satisfying the `wrong sign' commutation relations.
Vacuum structure: Hawking's original argument has been recast as a nogo
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 nontrivial 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 wellde ned quantum theories modeling black
holes and satisfying basic properties required by the aforementioned papers.
In this paper we put forward a new holographic toymodel for AdS black holes that
incorporates in a toymodel fashion a speci c proposal for the nature of nonperturbative
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 toymodel for black holes in AdS is twofold. First,
there is a class of singlesided 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 socalled 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 multitrace 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 toymodel
for systems of this kind since it amounts to two harmonic oscillators coupled through a
`nonperturbative' interaction modeled as a double well potential. Outside the context of
holography Giddings [17, 18] has studied how novel, nonlocal interactions (in the bulk)
can resolve the information paradox.
Secondly, recent work in the context of twosided 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 toymodel 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 toymodel 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 nonperturbative 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 toymodel setup
how nonperturbative interactions resolve the paradoxes. In particular we show explicitly
that the interactions eliminate the statedependence 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 singlesided 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, multitrace interactions in the dual imply a nonperturbative coupling between
both sides. As such this setup motivates the quantum mechanical particle in a double well
potential as a toymodel 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 BreitenlohnerFreedman 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 wellde 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 selfinteraction 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 wellde 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 multitrace 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
nonlinear 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 SchwarzschildAdS
black hole interior solution is smoothly glued at the horizon onto a scalar soliton solution
outside, slightly modi ed by the nonlinear 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 multitrace
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 oneparameter 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  toymodel for (this class of) black holes in AdS. In the remainder
of this paper we study this toymodel, 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 creationannihilation
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 nparticle 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 complexvalued, squareintegrable 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 nontrivial 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 creationannihilation 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
creationannihilation 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 creationannihilation
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 creationannihilation operators aL; aL+; aR; aR+ are not to be
identi ed with the perturbative creationannihilation 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 creationannihilation 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 toymodel shows that small, nonperturbative
interactions can drastically change this structure.7
3.2
Black hole microstates
Another important characteristic of our toymodel 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 RitzRayleigh method.8
We will say that f ( ) is nonperturbatively small if f
0 as
the asymptotic expansion. Equivalently, f ( ) is nonperturbatively small if f ( ) = o( n
)
for all n
0 as
! 0+. We will denote any nonperturbatively 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 1instanton
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 twosided 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 1instanton 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 nonperturbative e ect and hence exponentially
small for energies well below the maximum of the potential, En < V
= 1=(32 ). For
energies larger than this, nonperturbative 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 toymodel 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
highenergy 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 toymodel description of the singlesided 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
nonperturbative e ects to identify. In our model H ne = C2 is a twodimensional space,
which can be identi ed with the space of perturbative vacua M. The energy di erence
of any two microstates is then nonperturbatively 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 toymodel. 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 nonperturbatively 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 nonperturbative
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 toymodel 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 nonperturbative 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 creationannihilation 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 illde 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 wellde ned perturbative operators. In particular we will
construct another set of creationannihilation 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 nonperturbative terms become numerically
b. Timeindependent 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 nth
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. Timedependent perturbation theory breaks down for times t
N for any state.
Breakdown of perturbation theory can also be seen in time evolution. For example,
timedependent 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 timedependent processes in section 6 in more detail, where we will also recover the relation M N for our toymodel.
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 toymodel 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 nonperturbative e ects enter in the picture leading to the breakdown of perturbation
theory when it comes to the negrained features of microstates. We have also shown that
the natural (naive) creation and annihilation operators (3.8) do not possess a wellde ned
decoupling limit
! 0, and therefore cannot represent creationannihilation 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 nonperturbative physics
and to de ne suitable creation and annihilation operators with a wellde 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 creationannihilation 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
wellapproximated 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 nonunique, but all ambiguities are nonperturbative 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/R0φRn〉
108
1018
1028
1038
1048
< 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 nonvanishing 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, rewallfree 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
nonperturbatively 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 creationannihilation 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 nonperturbative 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 nonperturbative 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 nonperturbative
states or accept the fact that natural perturbative observables from the point of view of
a given observer become nonperturbative 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 nonvanishing. 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 nonperturbatively 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 nonperturbatively 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 nonperturbatively close to 'nR lying in HR. We
have found `approximate isomorphisms'
up to nonperturbative 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 nonperturbative 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 cuto '. 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 cuto 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 wellde 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 creationannihilation 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 fallo
around
= 0. In this
sense hatted and unhatted creationannihilation 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 creationannihilation 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
nonperturbative 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 nonperturbative
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 creationannihilation
operators (extremely `nonlocal' as in [29, 59]), or simply illde ned [60].
In this way we have resolved the problem of ` tting' the left creationannihilation
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 nonperturbative 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 creationannihilation operators as indicated in section 4.1 may result in
nonperturbative 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 nonperturbative 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 nonperturbative e ects this statement remains true for all energies, even if
actual matrix elements become numerically large. We return to timedependent processes
in section 6.
5
Low energy e ective theory: statedependence
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 statedependent.
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 creationannihilation
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 toymodel 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 `statedependence': the value
m
of the right operator AR depends on the `hidden' left portion of the state. In this
statedependent 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
nonperturbative terms, we will show there that the corresponding global operators again
cannot be de ned in a `stateindependent' manner. However, the `statedependence' 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 nonperturbative 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 nonvanishing 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 preimage 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 toymodel, 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 nonperturbative
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 nonperturbative terms.
Note that in order for (5.8) to hold, one must have
e 6= 0 if
is nonzero, i.e., the
state
must be typical. Secondly, o diagonal elements of A must be nonperturbatively
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 statedependence 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 nonperturbative
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 nonlinear, i.e.,
`statedependent', 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 statedependent operators in the theory in the spirit
of [6{9]. Operators of the form (5.9) corresponding to naive perturbative observables
become nonlinear functions or, equivalently, statedependent 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 timedependent eld
operator correlation functions such as h0j (t1) (t2) : : : (tn)j0i. To consider such correlators
in our toymodel, 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 nonperturbative
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 creationannihilation 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 wellde 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 toymodel 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 toymodel 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 toymodel
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 highlyexcited 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 toymodel 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 toymodel. 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 toymodel 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 toymodel 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 toymodel 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 nonperturbative 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 wellde 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].
Statedependence: our toymodel describes both sides of the horizon in terms of a
single wellde ned, local, unitary quantum theory. Hence it does not require any
`statedependent' operators to describe behind the horizon physics. However we have
shown that a clear notion of `statedependence' 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 statedependence
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 nogo 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 toymodel the perturbative
vacuum is degenerate, and the degeneracy is only broken by nonperturbative 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 toymodel is that it is solvable. This has enabled us to
analyze the role of nonperturbative interactions in a number of interesting dynamical
processes. The appearance of an inverted harmonic oscillator potential separating both wells
as a toymodel for nonperturbative 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 nonperturbative 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 nonperturbative 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 ERC2013CoG 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
SklodowskaCurie 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 (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
2460 [INSPIRE].
[1] S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976)
[2] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or
rewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
[3] A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for rewalls,
JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
[arXiv:1207.5192] [INSPIRE].
[4] R. Bousso, Complementarity is not enough, Phys. Rev. D 87 (2013) 124023
[5] D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev.
Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
[6] K. Papadodimas and S. Raju, Statedependent bulkboundary maps and black hole
[7] K. Papadodimas and S. Raju, Black hole interior in the holographic correspondence and the
[8] K. Papadodimas and S. Raju, Remarks on the necessity and implications of statedependence
in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
[9] K. Papadodimas and S. Raju, Local operators in the eternal black hole, Phys. Rev. Lett. 115
[10] S.D. Mathur, What exactly is the information paradox?, Lect. Notes Phys. 769 (2009) 3
[11] S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26
[12] S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116
(2015) 211601 [arXiv:1502.06692] [INSPIRE].
[arXiv:0803.2030] [INSPIRE].
(2009) 224001 [arXiv:0909.1038] [INSPIRE].
(2016) 231301 [arXiv:1601.00921] [INSPIRE].
005 [hepth/0503071] [INSPIRE].
[grqc/0608075] [INSPIRE].
[INSPIRE].
[13] T. Hertog and K. Maeda, Black holes with scalar hair and asymptotics in N = 8
supergravity, JHEP 07 (2004) 051 [hepth/0404261] [INSPIRE].
[14] T. Hertog and G.T. Horowitz, Designer gravity and eld theory e ective potentials, Phys.
Rev. Lett. 94 (2005) 221301 [hepth/0412169] [INSPIRE].
[15] T. Hertog and G.T. Horowitz, Holographic description of AdS cosmologies, JHEP 04 (2005)
[16] T. Hertog, Towards a novel nohair theorem for black holes, Phys. Rev. D 74 (2006) 084008
[17] S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070]
[18] S.B. Giddings, Nonviolent unitarization: basic postulates to soft quantum structure of black
holes, JHEP 12 (2017) 047 [arXiv:1701.08765] [INSPIRE].
D 22 (2013) 1342011 [arXiv:1305.6343] [INSPIRE].
[19] B.D. Chowdhury, Black holes versus rewalls and thermo eld dynamics, Int. J. Mod. Phys.
[20] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61
(2013) 781 [arXiv:1306.0533] [INSPIRE].
[INSPIRE].
[21] L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690]
[22] V. Balasubramanian, M. Berkooz, S.F. Ross and J. Simon, Black holes, entanglement and
random matrices, Class. Quant. Grav. 31 (2014) 185009 [arXiv:1404.6198] [INSPIRE].
[23] S.H. Shenker and D. Stanford, Black holes and the butter y e ect, JHEP 03 (2014) 067
[arXiv:1306.0622] [INSPIRE].
[INSPIRE].
[24] S.H. Shenker and D. Stanford, Multiple shocks, JHEP 12 (2014) 046 [arXiv:1312.3296]
[25] P. Gao, D.L. Ja eris and A. Wall, Traversable wormholes via a double trace deformation,
JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
[26] J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys.
65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
wormholes, arXiv:1708.09370 [INSPIRE].
[27] R. van Breukelen and K. Papadodimas, Quantum teleportation through timeshifted AdS
[28] A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a
boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hepth/0506118]
[29] A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local
bulk operators, Phys. Rev. D 74 (2006) 066009 [hepth/0606141] [INSPIRE].
[30] A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS3/CFT2, JHEP
05 (2016) 109 [arXiv:1603.08925] [INSPIRE].
[31] C. Martinez, R. Troncoso and J. Zanelli, Exact black hole solution with a minimally coupled
scalar eld, Phys. Rev. D 70 (2004) 084035 [hepth/0406111] [INSPIRE].
[32] A. Acen~a, A. Anabalon, D. Astefanesei and R. Mann, Hairy planar black holes in higher
dimensions, JHEP 01 (2014) 153 [arXiv:1311.6065] [INSPIRE].
[33] A. Anabalon, Exact black holes and universality in the backreaction of nonlinear models
with a potential in (A)dS4, JHEP 06 (2012) 127 [arXiv:1204.2720] [INSPIRE].
[34] F. Faedo, D. Klemm and M. Nozawa, Hairy black holes in N = 2 gauged supergravity, in
Proceedings, 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and
Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14),
[35] A. Anabalon, D. Astefanesei, A. Gallerati and M. Trigiante, Hairy black holes and duality in
an extended supergravity model, JHEP 04 (2018) 058 [arXiv:1712.06971] [INSPIRE].
[36] M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Asymptotically antide Sitter
spacetimes and scalar
elds with a logarithmic branch, Phys. Rev. D 70 (2004) 044034
[hepth/0404236] [INSPIRE].
[37] T. Hertog and K. Maeda, Stability and thermodynamics of AdS black holes with scalar hair,
Phys. Rev. D 71 (2005) 024001 [hepth/0409314] [INSPIRE].
[38] I. Papadimitriou, Multitrace deformations in AdS/CFT: exploring the vacuum structure of
the deformed CFT, JHEP 05 (2007) 075 [hepth/0703152] [INSPIRE].
[39] E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence,
hepth/0112258 [INSPIRE].
[40] M. Berkooz, A. Sever and A. Shomer, `Double trace' deformations, boundary conditions and
spacetime singularities, JHEP 05 (2002) 034 [hepth/0112264] [INSPIRE].
[41] T. Hertog and S. Hollands, Stability in designer gravity, Class. Quant. Grav. 22 (2005) 5323
[hepth/0508181] [INSPIRE].
(2014) 055 [arXiv:1405.1995] [INSPIRE].
(2016) 015002 [arXiv:1409.1231] [INSPIRE].
Academic Press, U.S.A., (1978).
[42] D. Harlow, Aspects of the PapadodimasRaju proposal for the black hole interior, JHEP 11
[43] D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88
[44] M. Reed and B. Simon, Analysis of operators, Methods of modern mathematical physics 4,
[51] T. Hertog and J. Hartle, Observational implications of fuzzball formation,
[52] L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole
complementarity, Phys. Rev. D 48 (1993) 3743 [hepth/9306069] [INSPIRE].
[53] Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096]
[45] J.M. Maldacena, Eternal black holes in antide Sitter, JHEP 04 (2003) 021
[hepth/0106112] [INSPIRE].
[46] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler quantum gravity,
[54] S. Ghosh and S. Raju, Breakdown of string perturbation theory for many external particles,
Phys. Rev. Lett. 118 (2017) 131602 [arXiv:1611.08003] [INSPIRE].
[55] S. Ghosh and S. Raju, Loss of locality in gravitational correlators with a large number of
insertions, Phys. Rev. D 96 (2017) 066033 [arXiv:1706.07424] [INSPIRE].
[56] D.L. Ja eris, Bulk reconstruction and the HartleHawking wavefunction, arXiv:1703.01519
[57] D. Kabat and G. Lifschytz, Finite N and the failure of bulk locality: black holes in
AdS/CFT, JHEP 09 (2014) 077 [arXiv:1405.6394] [INSPIRE].
[58] S. Raju, Smooth causal patches for AdS black holes, Phys. Rev. D 95 (2017) 126002
[arXiv:1604.03095] [INSPIRE].
[59] A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a
holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum
ibid. D 75 (2007) 129902] [hepth/0612053] [INSPIRE].
[60] D. Kabat and G. Lifschytz, Does boundary quantum mechanics imply quantum mechanics in
the bulk?, JHEP 03 (2018) 151 [arXiv:1801.08101] [INSPIRE].
[61] N.G. Sanchez and B.F. Whiting, Quantum
eld theory and the antipodal identi cation of
black holes, Nucl. Phys. B 283 (1987) 605 [INSPIRE].
[62] A. Chamblin and J. Michelson, Alphavacua, black holes and AdS/CFT, Class. Quant. Grav.
[48] J. ZinnJustin and U.D. Jentschura, Multiinstantons and exact results I: conjectures, WKB
expansions and instanton interactions, Annals Phys. 313 (2004) 197 [quantph/0501136]
[49] K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212
[arXiv:1601.03447] [INSPIRE].
Phys. 47 (2017) 1503 [arXiv:1612.08640] [INSPIRE].
[64] G. 't Hooft, The rewall transformation for black holes and some of its implications, Found.
[chaodyn/9511001] [INSPIRE].
eld theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].
JHEP 01 (2016) 008 [arXiv:1506.01337] [INSPIRE].
Phys. 21 (2017) 1787 [arXiv:1610.01533] [INSPIRE].
HJEP06(218)7
5042 [hepth/9907001] [INSPIRE].
mechanics, JHEP 11 (2016) 131 [arXiv:1607.07885] [INSPIRE].
oscillator, Annals Phys. 333 (2013) 290 [arXiv:1206.4519] [INSPIRE].
[arXiv:1403.5947] [INSPIRE].
nitedimensional, Int. J. Mod. Phys. D 26 (2017) 1743013 [arXiv:1704.00066] [INSPIRE].
JHEP 02 (2010) 073 [arXiv:0808.0530] [INSPIRE].
complementarity , Phys. Rev. D 89 ( 2014 ) 086010 [arXiv: 1310 .6335] [INSPIRE].
information paradox , Phys. Rev. Lett . 112 ( 2014 ) 051301 [arXiv: 1310 .6334] [INSPIRE].
Class . Quant. Grav. 29 ( 2012 ) 235025 [arXiv: 1206 .1323] [INSPIRE].
[47] M. Van Raamsdonk , Evaporating rewalls , JHEP 11 ( 2014 ) 038 [arXiv: 1307 .1796] [63] G. ' t Hooft, Black hole unitarity and antipodal entanglement , Found. Phys . 46 ( 2016 ) 1185 [65] M. Srednicki , Thermal uctuations in quantized chaotic systems , J. Phys. A 29 ( 1996 ) L75 [69] H. Ooguri and C. Vafa , Nonsupersymmetric AdS and the swampland , Adv. Theor. Math. [70] B. Freivogel and M. Kleban , Vacua morghulis, arXiv: 1610 .04564 [INSPIRE]. [71] S.D. Mathur , Tunneling into fuzzball states , Gen. Rel. Grav . 42 ( 2010 ) 113 [72] M.K. Parikh and F. Wilczek , Hawking radiation as tunneling , Phys. Rev. Lett . 85 ( 2000 ) [74] G. Horowitz , A. Lawrence and E. Silverstein , Insightful Dbranes, JHEP 07 ( 2009 ) 057