# State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

###### Abstract

We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in arxiv:1304.6483 and arxiv:1307.4706, in favour of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the “frozen vacuum” objections of arxiv:1308.3697. Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.

[]; a,b]Kyriakos Papadodimas c]and Suvrat Raju \affiliation[a]Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG, The Netherlands. \affiliation[b]Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland. \affiliation[c]International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, IISc Campus, Bengaluru 560012, India. \emailAdd \emailAdd \keywordsAdS-CFT, Information Paradox, Black Holes

## 1 Introduction

In a previous paper [1], we proposed a holographic description of the interior of black holes in anti-de Sitter space (AdS). In this paper we expand on several aspects of our proposal and address the information paradox for black holes in AdS in the light of the extensive recent discussion on the firewall proposal [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59].

The central point that we wish to make in this paper is that the assumption that gravity can be described in a unitary quantum mechanical framework is consistent with the existence of operators labeled by a point that can be interpreted as a spacetime point, and low point correlation functions , in the black hole state that can be understood as coming from effective field theory. These low point correlators are the natural observables for a low-energy observer. However, if we take the number of points to scale with the central charge of the boundary CFT, , or take two points to be very close (comparable to ), then this effective spacetime description may break down. Nevertheless, this breakdown is not consequential for a low-energy observer, and does not imply the existence of firewalls or fuzzballs, or require any other construction that radically violates semi-classical intuition.

A key feature of our description of local operators in this paper is that mapping between CFT operators to the bulk-local operator depends on the state of the CFT. This is not a violation of quantum mechanics: the operator is an ordinary operator that maps states to states in the Hilbert space. However, it has a useful physical interpretation as a local operator only in a given state. Said another way, the analysis in this paper relies on the assumption that to obtain a convenient description of the physics, in terms of a local spacetime, we need to use different operators in different states. This issue is related to the issue of whether it is possible to have “background independent” local operators in quantum gravity. If one gives up the idea of “background independence”, one is naturally led to the “state-dependent” constructions that we discuss here.

Nevertheless, granting this assumption, we show that our construction resolves all the arguments that have been advanced to suggest that the black hole horizon has structure, or that AdS/CFT does not describe the interior of the black hole.

In our previous paper [1], we had proposed a construction of interior operators by positing a decomposition of the CFT Hilbert space into “coarse” and “fine” parts. In this paper, we present a refinement of our proposal that does not rely on any such explicit decomposition, although it reduces to our previous proposal in simple cases. The feature of state-dependence of the interior operators carries over from [1]. But our refined construction removes some of the ambiguity inherently present in our previous proposal, and allows us to write down an explicit formula for interior operators in the CFT, without necessarily understanding the detailed structure of its Hilbert space at strong coupling.

The thrust of our paper is rather simple to summarize. First, we point out that the issue of whether there is structure at the horizon of the black hole, and the related issue of whether the black hole interior is visible in the CFT, can be translated to a simple question about CFT operators. It is well known that local-operators outside the black hole horizon can approximately be mapped to modes of single-trace operators on the boundary, which we call , where labels the conformal primary and are its modes in frequency-space and the angular momentum on the spatial sphere. To describe a smooth interior, we need to effectively “double” these modes and find another set of operators , which not only commute with the original operators, but are entangled with them in the state of the CFT. So, within low-point correlators, where the number of insertions of single-trace operators does not scale with the central charge ( in theory), we require

(1) |

Several authors have pointed out that the CFT does not seem to have enough “space” for the existence of the operators. However, our punch-line is as follows. In a
given state , the equations above must hold provided
we do not have too many operator insertions and . The set of
all possible such insertions is finite, and loosely speaking,
scales like . So, demanding that
has the correct behaviour within low point correlators computed
in a given state simply leads to a set of linear equations for the -operators,
which can be solved in the large Hilbert space of the CFT, which has
a size that scales like for energies below .
Moreover, as we discuss in detail, these equations are consistent precisely when is close to
being a thermal state.^{1}^{1}1In this paper, by “thermal state” we mean a typical pure state in the high temperature phase of
the gauge theory.

This analysis leads to our conclusion that it is possible to find state-dependent local operators in the bulk that commute with the local observables outside the horizon. We then proceed to show that this construction resolves all the recent paradoxes associated with black hole information.

First, we describe how our construction of interior operators resolves the strong subadditivity paradox. The resolution is simply that the operators inside and outside the black hole are secretly acting on the same degrees of freedom. One of the objections to this idea of black hole complementarity, has been that naively, measurements outside the black hole would not commute with those inside. As we describe in great detail, our construction is tailored to ensure that the commutator of local operators outside and inside the black hole—and all of its powers— vanish exactly when inserted within low-point correlators.

We turn our attention to some of the more recent arguments of [57, 59], which suggest that the black hole interior cannot be described within the CFT. The authors of [57] pointed out that the operators behind the horizon appear to satisfy the usual algebra of creation and annihilation operators, except that “creation” operator maps states in the CFT to those of a lower energy. If this were really the case, it would lead to a contradiction since the creation operator of a simple harmonic algebra always has a left-inverse, and the number of states of the CFT decrease at lower energy.

Our construction resolves this issue, because the operators behind the horizon behave like ordinary creation and annihilation operators, only when inserted within low-point correlators. Since they satisfy the algebra only in this effective sense, and not as an exact operator algebra, there is no contradiction with the “creation” operator having null vectors.

We also address the argument of [59], which we call the argument. The authors of this paper pointed out, that assuming that the interior operators were some fixed operators in the CFT, the eigenstates of the number operator for a given mode outside the horizon would not necessarily be correlated with the eigenstates of the number operator for the corresponding mode inside the horizon and so the infalling observer would encounter energetic particles at the horizon. However, this conclusion fails for state-dependent operators. Our interior operators are precisely designed so that, for a generic state in the CFT and its descendants that are relevant for low-point correlators they ensure that the infalling observer sees the vacuum as he passes through the horizon. We describe this in more detail in section 4.4.

After having addressed these issues, we then turn to the “theorem” of [60] that small corrections cannot unitarize Hawking radiation. We point out that our construction evades the theorem because of two features: the interior of the black hole is composed of the same degrees of freedom as the exterior, and the operators inside that are correlated with those outside depend on the state of the theory.

This brings us to a final objection that has been articulated against this state-dependent construction: the “frozen vacuum” [61, 58]. Although our construction suggests that the infalling observer encounters the vacuum for a generic state, it is true that there are excited states in the CFT, in which we can arrange for the infalling observer to encounter energetic particles. Our equilibrium construction already allows us to analyze such time-dependent processes. For example, we can consider a time-dependent correlation function in an equilibrium state, and our prescription provides an unambiguous answer. However, in section 5, we discuss how to adapt our construction to build the mirror operators directly on non-equilibrium states. This extension takes advantage of the fact that it is always possible to detect deviations from thermal equilibrium by measuring low-point correlators of single-trace operators. To perform our construction on a state that is away from thermal equilibrium, we “strip off” the excitations on top of the thermal state, and then perform our construction in this base state. Low point correlators in the excited state are now simply equated with slightly higher point correlators in the base state. We describe this construction in section 5.

In section 6 we discuss a beautiful and intriguing connection of our construction with the Tomita-Takesaki theory of modular isomorphisms of von Neumann algebras. We start this section by reviewing our construction, but from a slightly different physical emphasis. We then show how our construction can be compactly phrased in the language of Tomita-Takesaki theory. In this section, we also clearly show how our construction of the interior in this paper reduces to our previous construction [1] in simplified settings. We hope to revisit this interesting topic again in future work.

This paper is organized as follows. In section 2, we show that the issue of whether AdS/CFT describes the interior in an autonomous manner reduces to the issue of finding operators, which we call the “mirror” operators, with certain properties in the CFT. After outlining these constraints, we then explicitly construct operators in 3 that satisfy them, when inserted within low-point correlators. This central section also contains multiple examples of our construction. We show how our construction works in a general theory, in the CFT, in a toy-model of decoupled harmonic oscillators, and also in the spin chain. In section 4, we then apply this construction to the recent discussions of the information paradox, and find that it successfully addresses each of the recent arguments that have been raised in favour of structure at the horizon. In section 5, we show how to extend our construction to non-equilibrium scenarios, and thereby also resolve the issue of the “frozen vacuum.”. In section 6, we explore the link between our construction and Tomita-Takesaki theory. Section 7 contains a summary, and some open questions. The Appendices contain several other details, including a discussion of one of the first “measurement” arguments for firewalls articulated in [2].

Appendix E may be particularly interesting to the reader, who wishes to quickly get a hands-on feel for the properties of the mirror operators that we describe. This documents a computer program (included with the arXiv source of this paper) that numerically constructs these mirror operators in the spin-chain toy model. The essential ideas of this paper are summarized in [62], and the reader may wish to consult that paper first, and then turn here for details.

## 2 Bulk Locality: Need for the Mirror Operators

In [1], we discussed how to construct local operators outside and inside the black hole, by using an integral transform of CFT correlators. We review this construction briefly, and explain the need for the mirror operators.

Consider a generalized free-field operator in the conformal field theory at a point in time and on the sphere . By definition this is a conformal primary operator of dimension , whose correlators factorize at leading order in the expansion.

(2) |

where runs over the set of permutations.

In this paper, we will be interested in fields with a dimension that is much smaller than . We remind the reader that, as in our last paper [1], by , we are referring to the central charge of the CFT, and if the reader wishes to think about supersymmetric theory, then she may take .

Now, we take the CFT to be in a state that is in equilibrium and and has an energy . We write here, but to be precise, we need to take the energy to be much larger than the central charge so that the theory is unambiguously in the phase corresponding to a big black hole in AdS.

The same generalized free-field now factorizes about this energetic state as well. Moreover, at leading order in , we expect that correlators in this state will be the same as thermal correlators

(3) |

where is the partition function of the CFT at the temperature .

As we showed in [1] we can use the modes of this operator to construct another CFT operator that behaves like the local field outside the black hole. The formulas of [1] were written for the case of the black-brane in AdS, but here we can write down the analogous formulas for the CFT on the sphere to avoid some infrared issues in discussing the information paradox.

(4) |

Here are the modes of the boundary operators in frequency space and on the sphere respectively, while the sum over goes
over the spherical harmonics.^{2}^{2}2These need
to be suitably regulated in frequency space, and we discuss this
carefully in the section 3.2, although this issue is unimportant here.
What this means is that if we consider the CFT correlators

(5) |

then these CFT correlators behave like those of a perturbative field propagating in the AdS-Schwarzschild geometry.

The analogue of (4) in empty AdS had previously been
discussed extensively in the literature [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. However, in writing (4), we pointed out that, in momentum space, it was possible to extend this construction in pure-states close to the thermal state. This relies on the fact
that thermal CFT correlators have specific properties at large spacelike momenta,
and this observation allowed us to sidestep some of the complications that were encountered in [68].^{3}^{3}3It is somewhat delicate to write down the position space version of (4). This is because the position space “transfer function” must account for the fact
that it can only be integrated against valid CFT correlators. So, the transfer function must be understood as a distribution that acts as a linear functional on the restricted domain of multi-point CFT correlators. This leads to subtleties in writing it as a simple integral transform. This observation has led to recent claims that the transfer function does “not exist” in the black hole background or, indeed, in any background with a trapped null geodesic [75, 76].
This statement—which simply refers to the fact explained above — does not have any significant physical implication;
the mapping between degrees of freedom between the bulk and the boundary
continues to exist.

Turning now to the region behind the horizon, effective field theory tells us that in the analogue of (4), the CFT operator describing the interior must have the form

(6) |

Here are the analytic continuations of the left-moving modes from outside, to inside the black hole, while are right-moving modes inside the black hole.

These right-moving modes can be understood in several ways. In Hawking’s original calculation [77], these modes were the very energetic modes in the initial data that can be propagated through the infalling matter using geometric optics. In terms of solving wave-equations, the modes can also be obtained by analytically continuing the modes from the “other side” (region III) of the eternal black hole, as we discussed in [1].

However, we should caution the reader that while these physical interpretations are useful as mnemonics, they are both fraught with ultra-Planckian problems. This is clear in Hawking’s original interpretation, but we also note that while the analytic continuation from region III is easily performed in the free-field theory, mapping the modes at late-times in the black hole, back to region III requires us to go through the ultraviolet regime.

We emphasize that neither of these ultra-Planckian problems are relevant to our discussion. Here, our statement is simply about effective field theory in the patch that is shown in figure 1. In this patch, we can locally expand the field in modes, and we find that to get a local perturbative field, we need both left and right moving modes.

What is important here, though, is the appearance of the modes . First, we need these operators to effectively commute not only with the ordinary operators of the same species , but with other “species” of operators that enter the fields outside the horizon as well

(7) |

The in (7) indicates that this equation must hold when this commutator (or a power of this commutator) is inserted within a low-point CFT correlator like (5), as we discuss in more detail below. As we have mentioned, and will discuss again below, if we consider a correlator with insertions, then we should not expect a semi-classical spacetime, or an equation like (7) that expresses locality in such a spacetime to hold.

For the horizon of the black hole to be smooth we require that within a low-point correlator evaluated in a pure state that is close to a thermal state

(8) |

where is the partition function of the CFT at temperature . The reader should note that the analytically continued operators, which appear with the index and primed coordinates, have been moved to the right of all the ordinary operators, and moreover their relative ordering has been reversed.

In momentum space, the equation (8) can be translated to

(9) |

In Fourier transforming from (8) to (9), we should keep in mind that while the modes of are defined by , where is the spherical harmonic on the sphere, the modes of are defined by . This convention simply tells us that the modes have the opposite energy and angular momentum to the modes .

To emphasize again, we require operators that when inserted within a state automatically achieve the ordering within the thermal trace that we have shown here: both in terms of moving to the right of ordinary operators, and in terms of reversing their relative positions.

The reader may wish to consult section 5 of our previous paper [1], where we showed how the condition (8) leads to smooth correlators across the horizon. This is clear, because in this case, the calculation of correlators across the horizon reduces to the calculation in the eternal black hole geometry, which is clearly smooth. In fact, the converse also holds: correlators are smooth across the horizon if (8) holds, at least at leading order in .

#####
corrections

We should point out that the status of the condition (9) (or equivalently (8)) is quite different from that of (7) with respect to corrections. When these are included, we would like (7) to continue to hold at all orders in the expansion and its violations, if any, should be suppressed exponentially in . On the other hand (8) can receive corrections at the first subleading order in . We can see that such corrections will come about, purely because of differences between correlators in the state and the thermal state. Another source of corrections, comes from interactions in the CFT which, in the bulk, corresponds to the back-reaction of the Hawking radiation on the background geometry.

##### Charged States

In writing (8) we have tacitly assumed that the state does not have any charge. In fact, the CFT contains several conserved charges, which we will generically call . Just as we can associate a temperature with the state using correlation functions (or the growth in entropy with energy), we can also associate a chemical potential with a charged state.

In this paper to lighten the notation, we will not write the charge explicitly. But the reader should note that our entire analysis below goes through with the replacement of .

### 2.1 Comparison with Flat Space Black Holes

We briefly mention why these mirror operators are also important in the context of flat-space black holes. The modes in the background of a flat-space black hole have a slightly different structure. Roughly speaking, we can divide the modes into those that are “ingoing” and “outgoing” near the horizon of the black hole, and those that are “ingoing” and “outgoing” at infinity.

For the familiar case of a scalar field propagating in the 4-dimensional Schwarzschild black hole of mass , we can make this precise by introducing tortoise coordinates outside the horizon, and by introducing a second Schwarzschild patch just behind the horizon. Effective field theory tells us that, in the free-field limit, near the horizon, and at infinity, we can write

(11) |

where “just inside” and “just outside” refers to just inside/outside the horizon. We have taken the field to be massless, which allows both ingoing and outgoing modes to exist at infinity for all frequencies. Note the the presence of the potential barrier between and implies that the oscillators and commute whereas the pairs and have non-trivial commutators. Starting with the Schwarzschild vacuum, which is defined by

the Unruh vacuum is defined by allowing the “ingoing” modes at infinity to remain in their ground state and by entangling the “outgoing” modes at the horizon with their corresponding tilde-partners in a thermofield doubled state

which leads to .

### 2.2 Summary

In this section, we have tried to argue that the issue of whether the horizon of the black hole is smooth or not has to do with the issue of whether we can find operators in the CFT that satisfy (7) and (9). All the recent discussions of the information paradox can, essentially, be phrased as questions about whether such operators exist. We will make this more clear when we discuss these arguments below. In the next section, we describe how to find operators that satisfy these properties.

We should mention that, in the argument above, we have pointed out the necessity of the mirror operators for generalized free-fields in the CFT that enter the modes of perturbative bulk fields. However, we will actually succeed in finding mirror operators, for observables in a large class of statistical-mechanics systems. In the case of the CFT, we will succeed in “doubling” not only the generalized free-fields but a much larger class of operators.

We should point out that there are powerful (although, in our opinion, not conclusive) arguments that suggest that one cannot find fixed (i.e. state-independent) operators that have the correct behaviour specified by (7) (8) (or (9)) for an arbitrary given state . However, if we allow the mapping between CFT operators and local bulk operators to depend on the state itself, then one can indeed find such operators as we show explicitly below.

Moreover, these operators then resolve all the recent paradoxes that have been formulated to suggest the presence of a structure at the horizon.

## 3 Constructing the Operators behind the Horizon

In this section, we will explicitly construct operators behind the horizon. We will perform this construction in three steps so as to make this section maximally pedagogical. We start with a description of our idea in a general setting. It is well known that given a limited set of observables, almost any pure state drawn from a large Hilbert space looks “thermal” or equivalently looks as if it is entangled with some environment. In the first part of this section, we show how, in this single Hilbert space, it is possible to construct operators that behave as if they were acting on the environmental degrees of freedom.

In fact, the operators behind the horizon that we have described above are precisely of this form. So, in the second and central part of this section, we go on to describe our construction of these operators in the CFT. This case comes with a few quirks, including the fact that the CFT has conserved charges, and so some properties such as the charge and energy of the mirror operators is still visible outside the horizon.

Finally, we descend from this complicated situation and discuss two toy models in detail. The first is a toy-model of decoupled harmonic oscillators. This captures our ideas in a concrete setting, and has many of the essential features of the CFT, without some of the technical complications. The second is a simple spin-chain, which is a popular model — and probably the simplest available one — for considering the information paradox. We describe how the mirror operators can be constructed in this setting as well.

The reader may choose to read this section in any order, or even jump directly to the toy models.

### 3.1 Defining Mirror Operators for a General Theory

Let us say that we have some system, which is prepared in a pure state drawn from a large, but finite-dimensional, Hilbert space . We are able to probe the system with a restricted set of operators. Let us call the

(12) |

As we have written explicitly above, is a linear space and we can always take arbitrary linear combinations of operators in . However, it is important that may not quite be an algebra. It may be possible to multiply two elements of to obtain another operator that also belongs to . In fact, we will often discuss such products of operators below. However, we may not be allowed to take arbitrary products of operators in this set. In particular, if we try and take a product of operators, it may take us out of the set .

We wish to consider states that satisfy the following
very important property
^{4}^{4}4Later, in the discussion
on the CFT, we will consider situations where may be
an eigenstate of a conserved charge, in which case (13) does not hold for certain operators but, for the current discussion, this is an unimportant technicality.

(13) |

Note that this statement holds for all elements of , or equivalently for all possible linear combinations of the basis of observables written in (12). An immediate corollary of this statement is that the dimension of be smaller than the dimension of the Hilbert space of the theory:

(14) |

Equation (13) also means that if is a state of finite energy, then the energy of our probe operators in the algebra is also limited.

We wish to emphasize that these conditions on the observables we can measure and the state under consideration are physically very well motivated. For example, if the reader likes to think of a spin-chain system, then could consist of all local operators— the Pauli spins on each site—bi-local operators—which comprise products of local operators at two sites—all the way up to -local operators, as long as — the length of the spin chain. Generic states in the Hilbert space of the spin-chain now satisfy (13). We work this spin chain example out explicitly in section 3.4

However, more generally, as the reader can easily persuade herself, if we place a large system in a state that appears to be thermal, and consider some finite set of “macroscopic observables” (for example, those that obey the so-called “eigenstate thermalization hypothesis”), then the condition (13) is easily satisfied. In fact, we can consider a larger class of states, which are excitations of thermal states that are out-of equilibrium.

Now, it is very well known that, given such a set of observables , and a pure state , we can construct several density matrices , corresponding to mixed states, which are indistinguishable from , in the sense that we can arrange for

Such a density matrix is not unique but the correct way to pick it, assuming that the expectation values of are all the information we have, is to pick the density matrix that maximizes the entropy: [78, 79]. In fact, this maximum entropy is what should correspond to the thermodynamic entropy of the system. For a generic state , we expect to find

(15) |

up to corrections, where is the partition function.^{5}^{5}5In an equilibrium state, in any case, we expect off-diagonal terms in the
energy eigenbasis in the density matrix to be strongly suppressed, although
the eigenvalues may be corrected from the canonical ones. For the significance of such corrections, see appendix A, and for non
equilibrium states, see section 5.

It is also well known that the statements above imply that even though the system is in a pure state, it appears as if the system is entangled with some other heat-bath.
This pure-state in the fictitious larger system is called the “purification” of .
This purification is not unique,
even given but given a generic state in which the density
matrix is thermal as in (15), we will pick it to
be the thermofield doubled state [80].^{6}^{6}6In appendix A, we
discuss other choices of the purification which are, in fact,
required at and this issue of the lack of uniqueness.

(16) |

where the sum runs over all energy eigenvalues of the system. Note that the subscript tfd emphasizes that this state is distinct from the pure state , and lives in a (fictitious) larger Hilbert space.

The new point that we want to make here is as follows. In the pure state , we can also effectively construct the operators that act on the “other” side of the purification. So, for all practical purposes the thermofield doubled state and the doubled operators may be realized in the same Hilbert space!

More precisely, we want the following. For every operator acting on the Hilbert space of the system

(17) |

we have an analogous operator that acts on the fictitious environment

(18) |

The complex conjugation is necessary to ensure that this map remains invariant if we, for example, decide to re-phase the energy eigenstates of the system by and those of the environment by under which the state (16) is obviously invariant.

The operator has two other important properties. First, it clearly commutes with the operators , since these act on different spaces

(19) |

Second, with some simple algebra (see appendix A) we can see that

(20) |

We now desire the existence of operator that act in the single Hilbert space and mimic the action of (20) and (19) while acting on the state . Naively, this may seem impossible. For example, if we consider a spin-chain and the set comprises the set of Pauli-matrices acting on each site, then there is no operator in the Hilbert space that commutes with all the .

However, as we describe here, given a state , there is an elegant and almost unbelievably simple definition of these operators! First, we need to expand the set of observables a little so that for each , we adjoin to the element may not be closed under the multiplication of arbitrary pairs, if the product , we may also want to include the products and . We will call this expanded set of observables . If , then the elements of this expanded set also satisfy (13). . Next, as we mentioned above, while

We want to emphasize that the reader should not get lost in the technicalities of this “expanded” set. In fact, in the interesting case of the CFT below, we will see that coincides with . This is because in the situation where the have some definite energy , these factors simply simply invert the energy, and insert a factor of .

Now, we simply define the mirror operators by the following set of linear equations

(21) |

where . In a given state , these two lines together just correspond to equations. Note, that we can write these two lines as the single compact equation

(22) |

but we have written them separately because, as will become clear below, the two lines of (21) have different physical interpretations.

Note that are linear operators in a Hilbert space of dimension that we are interested in. The equation (22) makes it clear that we are specifying the action of these operators on a linear subspace, , produced by acting with all elements of the set on the set . Equivalently, we are specifying the action of on basis vectors. It is always possible to specify the action of an operator on a set of linearly independent vectors that is smaller in size than .

So, the only constraint we have to check is that is that the vector produced by acting on are linearly independent i.e. that we cannot find some coefficients . However, (13) tells us that there is no such linear combination.

So, we conclude that, provided (13) is met, we can always find an operator that satisfies (21). In fact, it is easy to write down an explicit formula for this operator. Consider the set of vectors

(23) |

where and the operators run over any basis of the set . Now, define the “metric”

(24) |

and its inverse satisfying . This inverse necessarily exists, because the are linearly independent by the conditions above. Now, an operator that satisfies the condition (21) above is given by

(25) |

where the repeated indices are summed, as usual. Of course, the operator , where is any operator that satisfies also satisfies (21). In (25), we have simply taken , but this ambiguity is physically irrelevant.

Furthermore, note that the rules (21) also allow us to build up the action of products of the mirror operators recursively. For example, notice that these rules lead to

(26) |

Here in the first equality, we use the first rule of (21). In the next equality we use the second rule to commute to the right, and then we use the first rule again to obtain our final expression! Notice in particular that

(27) |

Next, note that the rules (3.1) lead to the result that acting on the state , the mirror operators commute with the ordinary operators. For example, consider the commutator of an ordinary and mirror operator within some product of ordinary and mirror operators acting on

(28) |

Here, the key point is that the second line of (21) allows us to move through and any other occurrences of operators till the first occurrence of another operator. In writing these equations, we have tacitly assumed that we can take the product of the operators , while remaining within the set . This is justified as long as .

Now, we make a few remarks about correlation functions. First, note that by construction we have . Within mixed-correlators involving both and , we see that we have the following properties

(29) |

To show this involves only a small amount of additional work. First, we see that

(30) |

where we have used the second line of (21) to move the to the right, and then used the first line to substitute its action on . Now, given the right hand side of (30), we can use the same procedure to move to the extreme right and then substitute for its action. Continuing this, we see that finally

(31) |

Now, we we discussed above, correlators of ordinary operators in the set in the state are the same as those in the thermofield doubled state. So, we find that

(32) |

where the reader can easily use the property (20) to verify the second equality.

##### The Notation:

This feature, where the properties of the operators hold only within correlation functions evaluated on a particular state is important enough that we will introduce some special notation for it, which we have already used above, and will use extensively later. We will write

(33) |

to indicate that (28) holds, but the operators and may not commute as operators. It is just that this commutator annihilates and its descendants produced by acting with elements of the algebra .

##### The space :

Before we conclude this subsection, let us make a comment about solving the linear equations (21). We have carefully argued above that it is possible to find a set of solutions to these equations. In constructing such solutions, we do not even actually need to consider the full vector space . In fact, it is convenient to consider a slightly smaller vector space

(34) |

which is just the space formed by the action of the set on the state . In all cases of interest that we will study below, coincides with , and in these cases we can also write . We we see (21) is a statement about the action of the operators on the domain and the action of these operators outside this space is unspecified. In fact, we could even choose to annihilate states in the space of vectors orthogonal to without affecting low-point correlators. Note that the definition (21), and the fact that may not be closed under arbitrary pairwise multiplication implies that the range of may differ slightly from , even in this case. These “edge effects” are usually unimportant, and the physically relevant subspace is .

Our construction, as we have presented it here, applies to any statistical mechanics system. We now specialize to the CFT which, as we will see, has a few new ingredients.

### 3.2 Mirror Operators in the CFT

We now discuss the construction of the tilde-operators in an interacting CFT. Our construction follows the general method that we outlined above, but this section is written so as to be self-contained. We will find two new features in the CFT. One is technical and, in our view, not so important: we have to regularize the modes of the CFT to obtain a finite set of observables . The second is also somewhat technical, but a little more interesting. The operators that we are constructing are not gauge-invariant, and so, while they commute exactly with almost all operators, within correlation functions, they do not commute with the global charges or the Hamiltonian.

To be concrete, we will consider a CFT on . The black hole is dual to a state in the CFT, with an energy that is much larger than, but of the same order as . In this section, we will show how to construct the tildes on this state .

#### 3.2.1 Regularizing the Space of Operators

First, let us discuss the operators that we can use to probe the black hole geometry — this is the set above. We have some number of light operators in the CFT that correspond to the supergravity fields. In addition, we could probe the black hole geometry with excitations corresponding to stringy-states, and perhaps even with brane-probes. In the CFT, all of these can be represented by conformal primary operators with a dimension that is much smaller than . We remind the reader that is the central charge. So, in maximally supersymmetric theory, and even a giant graviton operator has dimension .

It will be convenient for us to discuss the modes of these operators, which are defined by

(35) |

where is the spherical harmonic indexed by the integers in the array .

Now, the relevant spacing of the energy levels around energies of order is actually . So, the spectrum of modes of low-dimensional conformal primaries is almost continuous even when the CFT is on a sphere.

Now, consider two energy levels and . We can consider the precise mode that causes transitions between these levels. However, if the differences between energies are non-degenerate, as we expect on general grounds for a “chaotic” system, then this mode will have a zero matrix element between any other states.

So, we need to “coarse-grain” these modes a little to come up with a useful set of operators. We will do this, by introducing a lowest infrared frequency , and bin together the modes of in bins of this width. More precisely, we define

(36) |

These regularized modes have a smooth behaviour in the Hilbert space, and we might reasonably expect them to obey the ETH, as we show in more detail in section 5. We will often use

(37) |

and correspondingly also write .

We can take to go to zero faster than any power of , but it must be much larger than . So, for example, we could take . So, the reader may wish to think of the theory, with an infrared cutoff that scales like . This is certainly adequate for all purposes of constructing perturbative fields in the interior.

We have now regulated both the maximum dimension of allowed probe operators, and their modes in the manner above. Let us call these various operators where refers to the conformal primary, and specify the mode. We now consider the set formed by taking the span of arbitrary products of up to numbers of these operators

(38) |

The set is limited by the constraint that each product occurring in satisfies which limits the total energy that can appear in this set.

Note, that as we emphasized in [62], taking the linear span of the products of operators above is exactly the same as thinking of as the set of all polynomials in the modes of the operators

(39) |

with the constraint that

(40) |

We also require that the set cannot be too large:

(41) |

The second constraint is automatically satisfied if we also limit the number of insertions in the polynomials

(42) |

and do not take to be too large. In fact, there is an interplay between the value of , ,and so that (41) can be preserved. For example, if we take , then we must take in order to preserve (41). If we take to scale just as an inverse power of , we can take to be larger.

Note that these polynomials, are polynomials in non-commutative variables, since the operators do not commute with one another. However, there may be operator relations within the CFT, and as a result it may happen that some particular set of polynomials vanish because of these relations. In taking the set of polynomials above, we must mod out by these relations. For example if for three operators that appear above: , then, the polynomial must clearly be identified with the polynomial .

This set consists of all possible probes that we are allowed to make in the black hole geometry. We emphasize that the set of operators in is essentially the largest set of operators, for which one might hope to make sense of a semi-classical geometry. For example, if we start including products of up to of the conformal primary modes, then there is no reason at all that expectation values of such operators should be reproducible by calculations in a semi-classical geometry.

In this concrete setting, the reader can also see another feature that we discussed in the section above. The set is not quite an algebra, because of the cutoff (40) that have imposed on the energy of the operators that can appear. On the other hand, it is often possible to multiply elements of together to obtain another member of .

Before we proceed to the definition of the mirror operators, we must impose a final technical constraint on the set . We do not take the Hamiltonian itself, or any conserved charge (by which we mean any operator, which commutes with the Hamiltonian) to be part of this set. This is equivalent to excluding the zero-modes of conserved currents. These zero-modes to not correspond to propagating degrees of freedom in the bulk and, in any case, we will deal with them separately below.

#### 3.2.2 Defining the Mirror Operators

We now describe how to define the mirror operators. The CFT in a generic thermal state has the following property.

(43) |

This is simply the statement that the insertion of a small number of light operators cannot annihilate the generic thermal state. We will work with states that satisfy (43). States that do not satisfy this condition are a measure-0 subset of the set of all states, and as we discuss below, they may not have a smooth horizon.

We will now define the tilde operators, by specializing the rules that we gave above. The mirror operators are defined by two very simple rules.

(44) | ||||

(45) |

As advertised, we do not need to expand the set of allowed observables to in the CFT to define the mirror operators.

Note that (44) and (45) together give us linear equations for the . However, can operate in a space that is dimensional! These equations are all internally consistent because of the condition (43). So, there are many possible solutions to these constraints. One explicit solution is shown in (25).

All these solutions are equivalent for our purposes, since they do not show any difference at all, except when inserted in very high-point correlators. As we pointed out above, there is also an, in principle, difference between (44) and (45). While (44) needs to be corrected order by order in , (45) is already correct at all orders in the for the correlators that we are interested in.

#### 3.2.3 Choice of Gauge: Hamiltonian and Abelian Charges

We now turn to the issue of a choice of gauge. We are willing to consider cases, where is an energy eigenstate, and certainly it may be possible to put in an eigenstate of some other conserved charge. We first discuss the inclusion of the Hamiltonian, which corresponds to zero-modes of the stress-tensor, and other Abelian charges, then turn to other kinds of conserved charges including non-Abelian charges in the next subsection.

If is an energy eigenstate, or the eigenstate of some other charge, we still expect it to appear thermal. However, in such cases, we see that we might have

(46) |

where is the charge operator and is the corresponding eigenvalue. This is the reason that we cannot include in the set . If, with this inclusion, we were to also demand (45), we would get an inconsistency.

However, this is quite simple to fix. We set to have a non-zero commutator with the zero-mode of the corresponding conserved current. In fact, this zero-mode is not of any interest, except for the fact that it includes the charge itself. So, we append the charge to the set and add an additional rule to the set of rules above.

First, since the position space operator is Hermitian, we need to re-organize its modes into operators that transform simply under the charger under consideration. If this charge is just the Hamiltonian or the angular momentum on , then the modes already transform in a simple manner. But, in any case, we can construct linear combinations , which have a well defined charge so that . The action of the mirror operators on the original linear combinations can be constructed by using the anti-linearity of the mirror-map. We now add the following rule to the set of rules above

(47) |

In the appendix B, we discuss this issue further. We show how a choice of gauge results in these commutation relations, and how they may be interpreted in terms of Wilson lines. We also explore the fact that these relations already seem to lead to some interesting physical implications. We note, that by virtue of this rule we see that does not really correspond to a local field on the boundary, since such a field would have non-zero commutators for other modes of the current as well. Here, this is not a difficulty, since the bulk fields constructed from cannot ever be taken close to the boundary to obtain any kind of contradiction. But this also provides a criterion for when the fields can enter bulk operators, and explains why they cannot be used in bulk fields below the Hawking Page transition.

Second, notice that since the charge and energy of the can be measured by the CFT Hamiltonian, this tells us that there is not really any “other side” of the collapsing geometry. We return to this at greater length in Appendix B.

#### 3.2.4 Non-Abelian Charges

We now describe how the mirror-operators act on descendants of the state produced by acting with various non-Abelian charges.^{7}^{7}7We thank Rajesh Gopakumar for a discussion on this issue. The main difference with the analysis for the Hamiltonian and Abelian charges above, is that in this case, we can have other kinds of null-vectors. The analysis of the subsection above is subsumed in the more general analysis of this subsection.

For example, we might want to consider a Schwarzschild black hole, and consider a corresponding ensemble in the CFT, where the states transform in a small representation of some non-Abelian charge, but are yet not charge eigenstates. Now, we may have , for some “raising operator” . We wish to ensure that our definition of the -operators is correct in this case. Below, we will denote any polynomial in the charges by . The space of physical states is produced by acting with all such polynomials on the base state , and then modding out by the null vectors. The action of must be correct on this quotient space, in that it must annihilate all null vectors.

##### The Set of Null Vectors

First, the condition that the action by an observable does not annihilate the state must be refined in the presence of such charges. We will impose the following condition. Consider a set of charge-polynomials . Now, we demand

(48) |

Translated into words, this means that we get various “descendants” by acting on the base state with the charges. If these descendants are linearly-independent, then by acting on them with with our observables, we cannot “make” them linearly dependent. This is a very natural generation of (43) above, and more formally speaking the states that do not satisfy (48) form a measure-0 space in the Hilbert space. Of course, we can also phrase (48) as

(49) |

Now, we want to consider the structure of the quotient space that we can get by acting both with the