Holographic cameras: an eye for the bulk

We consider four-point correlators in an excited quantum state of a field theory. We show that, when the theory and state are holographic, a judiciously applied Fourier transform produces high-quality images of point-like bulk particles, revealing the geometry in which they move. For translation-invariant states, the bulk Einstein's equations amount to local differential equations on correlator data. In theories or states that are not holographic, images are too blurry to extract a bulk geometry. We verify this for gauge theories at various couplings and the 3D Ising model by adapting formulas from conformal Regge theory.


Introduction
The AdS/CFT or gauge-gravity correspondence shows that the dynamics of certain strongly interacting quantum systems may be better understood in terms of a dual "bulk" system, which enjoys one additional dimension of space and dynamical gravity [1]. Gravity is not part of the original system but only emerges in the bulk. The new coordinate is loosely correlated with the size of an excitation, but, remarkably, small relative changes -1 -

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in it can represent sharply distinct bulk points: the bulk dynamics is local down to scales much shorter than its curvature radius. There has been much progress in understanding which quantum systems admit local duals (building on the celebrated large-N large-gap criterion [2]) and on how to reconstruct various aspects of the bulk. Yet, the emergence of a bulk and its associated entanglement structure from the boundary dynamics remain largely mysterious. Most basically, if a system holographically encodes a bulk universe, why should it hide it?
The purpose of this note is to explore a decoding tool which comes with any holographic system: its own Hamiltonian. A straightforward Fourier transform applied to four-point correlators will produce easily-interpretable images of point particles moving in a non-trivial bulk geometry. We dub the resulting devices holographic cameras.
Since the cameras are relatively straightforward to describe (theoretically), let us introduce them immediately and give more context below. Half of the device is an "holographic canon" which shoots a particle into the bulk. Starting from an arbitrary state |Ψ , we perturb it by acting with a local operator O integrated against a wavepacket cannon: |Ψ → The wavepacket will be assumed to create high-energy excitations with relatively well-defined energy and momenta (p 0 L 1); the position uncertainty L should be small compared with intrinsic time and length scales of the state. If the field theory admits long-lived quasi-particles with momentum near p µ , eq. (1.1) would be a familiar way to excite them by driving them at resonance. Rather, we will consider momenta p µ that do not correspond to boundary quasi-particles. If the system admits a dual description in terms of bulk quasi-particles, eq. (1.1) will instead excite them.
Concretely, we will mostly study relativistic conformal field theories, where p µ is a timelike vector. We will imagine that O(x) is a simple operator such as the electromagnetic current J µ (x) or the stress tensor T µν (x), and treat it like a light scalar for simplicity.
To confirm that a projectile has indeed been fired, one could measure one-point functions in the state (1.1). Removing an inessential integration, this boils down to three-point functions The only integration here is over the separation δx µ between the operators that create and absorb the excitation; this suffices to control its energy and momentum. Such correlators have been much studied and will be further discussed below. If O is the stress tensor, for example, then (1.2) measures the energy and momentum density at y. In holographic theories in the vacuum, this reveals an essentially spherical expanding shell of energy [3]. One could also replace the state |Ψ by a density matrix, ie. a thermal state (to keep our notations light we won't indicate this). In a thermal state, correlators like (1.2) with boosted p µ can probe jet-like excitations of a strongly interacting plasma (see [4] and references therein). In addition to the energy and charge spreading out with time, among other features, one observes the jet center-of-energy slowing down and eventually stopping as it  Figure 1. Two models of holographic cameras. In both, an energetic projectile is fired from a boundary point x µ with initial momentum p µ x . The plane represents the R 1,d−1 boundary spacetime in which we can do measurements, and the thick blue paths are null geodesics in the (arbitrary) bulk metric. (a) In the radar camera (1.3), one sends a pulse from a point z and records at y its reflection off the projectile. (b) In the active camera (1.4), one records how the projectile interferes with the reverse time evolution of a second one reaching y with momentum p y . In a theory with local bulk dynamics, this signal will be sharply peaked around vanishing bulk impact parameter b bulk . dissipates into the medium. These observations are all compatible with a dual picture of a point-like (or stringy) projectile reversibly moving in a bulk gravitational field. However, they are admittedly rather blunt probes of the bulk, and no smoking guns.
To better image the bulk itself, we propose two variants of holographic cameras. The first is inspired by a venerable technology: the radar. The second ("active") camera is more exotic and requires the ability to evolve small perturbations backward in time.
In the "holographic radar", the projectile's position is imaged by sending in a pulse from a point z, whose reflection is then recorded at a later point y. This can be modelled by the correlator: radar: We are imagining that x and z are spacelike from each other and y is in their common future, see figure 1a. The operator ordering is that of a two-point function in the state O|Ψ . The device is meant (ideally) to operate in the regime of bulk geometrical optics. For suitable choice of z, the forward lightcone of z intersects the projectile's trajectory at a point P , and its reflection from there is visible at any point y along the future lightcone of P . The signature of a bulk point-like projectile is a sharp singularity or peak as y approaches that lightcone.

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cuts [14,15], where one attempts to fire energetic excitations from the boundary onto a common bulk point. At a technical level, we will add little new: the correlators (1.3)-(1.4) differ from earlier ones by tweaks on operator ordering and Fourier wavepackets. Qualitatively, these tweaks are significant. They enable to focus on a given bulk point P using fewer boundary points and a much smaller boundary footprint, just using two (or three) small open sets that intersect the past and future lightcones of P . The images also seem easier to interpret, since only two (or three) bulk geodesics need to be visualized, in any spacetime dimension. Finally, they show that simple correlators can simultaneously reveal the local dynamics of bulk excitations and the geometry they evolve in. As mentioned, our key formulas will be adaptations of existing ones (from eikonal approximations and conformal Regge theory). However, since their derivations may not be widely known and have many subtle points, we felt that reviewing them before adapting them might not do proper justice to the original papers. Instead, we opt to give a selfcontained presentation from a hopefully uniform perspective, referring to original sources along the way.
This paper is organized as follows. Section 2 is devoted to the eikonal approximation (2.1) for the out-of-time-ordered correlator in a general background, including useful limits in subsection 2.6. Most of the discussion is based on technical examples, while the role of symmetries and large-N expansions is discussed more qualitatively in subsection 2.2. section 3 explains how the camera correlators, for conformal theories in the vacuum states, is captured by conformal Regge theory. In particular, relations between various perspectives are explained in 3.1. The extensive literature on Regge theory enables us to plot the correlators in theories that go beyond gravity, including supersymmetric Yang-Mills theory at weak and strong coupling and the critical 3D Ising model. The importance of the large-N large-(higher-spin)-gap criterion to obtain sharp images is emphasized. Section 4 shows the images obtained in a thermal state of a (strongly coupled) CFT 2 , and how these images straightforwardly reveal a bulk metric. In particular, for any translationinvariant state in CFT d , the bulk's Einstein equations translates into a simple differential equation (4.12) on correlator data. Section 5 summarizes our results and discusses some open questions. Three appendices record useful formulas on: Fourier transforms and harmonic analysis, shockwaves in AdS space, and conformal Regge theory.

Review and extension of eikonal approximation
When the theory and state |Ψ admit a holographic description, we expect to find a regime where the correlator (1.4) is dominated by tree-level graviton exchange between the null geodesics in 1b. The purpose of this section is to elaborate on eq. (2.1) below, which expresses this. The reader satisfied by this formula could in principle skip to section 3, since this section and the next one are largely complementary to each other.
Let us state directly the key formula which expresses the correlator (1.4) in the bulk tree-level approximation, at the same time setting up notation: (2.1) The integrand is a bulk-to-bulk propagator for metric perturbations; X(u) and Y (v) are coordinates along null geodesics with initial conditions (x, p x ) and (y, p y ), ie. the blue curves in figure 1b, and the times u and v are suitably normalized affine times (see (2.30) and (2.39) below). Intuitively, (2.1) arises from the classical coupling between a point particle and a metric perturbation, where we write perturbations as g µν = g Ψ µν + κh µν with g Ψ µν is the background. The bulk gravitational constant is We work in mostly-plus metric and sometimes use the following abbreviation for normalized timelike vectors, and reflection tensor: Finally, we will sometimes omit the argument L x , L y in formulas like (2.1) to indicate that the finite width of wavepackets was ignored. Conceptually, we propose to view (2.1) as a kind of Operator Produced Expansion expressing the limit δx → 0 of products O(x+ 1 2 δx)(· · · )O(x− 1 2 δx) in terms of simpler objects like duh uu . This expansion is conceptually similar to how shockwaves appear in the Regge limit of correlators. Our presentation will thus overlap greatly with [16][17][18][19][20], adapted to a less familiar operator ordering. We introduce the OPE viewpoint starting with three-point functions in perturbative φ 4 theory, and gradually move on to gravity in AdS space and to four-point functions.

Folded OPE in φ 4 theory
Consider massless λφ 4 theory in d ≈ 4 dimensions, with action: We will work perturbatively in small λ; a mass term could be added and would not modify the short-distance effects to be discussed. Let us begin by studying a vacuum three-point function analogous to (1.2): We would like to understand the coincidence limit δx → 0 when y is timelike from the origin. Crucially, the operators are not time-ordered and the two φ's are not next to each other. If that were the case, we could simply apply the standard Operator Product Expansion to approximate φ(δx)φ(0) as a series in local operators. Instead, we will find a more interesting picture where energetic φ quanta propagate a finite distance before being measured by O(y).
For lack of established terminology, we refer to this as the folded OPE limit.

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To evaluate (2.6), it is helpful to temporarily revert to a Schrödinger picture: where U t 2 ←t 1 = e −iH(t 2 −t 1 ) is the time-evolution operator. Standard arguments allow to rewrite U as a path integral over field configurations φ(x, t). The same can be done for U † , of course. The quirk is that the path integral now requires a doubled set of fields: where Z C is the result of the path integral without insertion. The time contour C, usually called Schwinger-Keldysh contour, can be visualized as: Subscripts in (2.8) indicate which branch a given field lives on. 3 Projection onto the vacuum state |Ω can be implemented by the usual i0 prescriptions: the contours are slightly tilted so that Im(t) → ±∞ at the start and end, respectively. More generally, an arbitrary initial state or density matrix could be inserted as boundary conditions at the left ends of the contour.
We briefly review the essential rules for perturbation theory, which can be deduced from the path integral (2.8), or by analytically continuing the usual rules for Euclidean quantum field theory (see [22] for a more detailed discussion and further references). Each interaction vertex can now be inserted on either branch, and there is an overall sign change on the second branch. Propagators between fields on the first branch are the usual time-ordered ones, and those on the second branch are anti-time-ordered. Those connecting different branches are Wightman functions. In position space: (2.10) The i0 prescription in the last case is easy to guess from the contour (2.9): the second branch is shifted by a small negative imaginary time. In general we denote by x ± a vector with a small imaginary time shift: The formalism manifests the fact that points on different branches of (2.9) are not "close": a particle cannot connect them without going around the fold.  Folded OPE limit of three-point functions, in perturbative field theory and in perturbative gravity. Singularities as δx → 0 come from paths which leave x on the first timefold and return nearby on the second timefold, approximately following null geodesics (thick blue lines). The two panels need not be continuously connected: boundary and bulk perturbation theory don't normally overlap. They look technically similar because the Landau analysis applies to any perturbative description, whether it exists in the original spacetime or in a dual one.
Nonetheless, the correlator (2.6) is singular as δx → 0. Let us see this explicitly for the connected diagram at order λ. Accounting for the contributions from each branch, we have (2.11) We kept the numerator unevaluated, anticipating a passive spectator role for O 1 (y).
A standard Landau-(Coleman-Norton) analysis indicates that singularities in position space come from integration regions representing classical paths of particles with arbitrary high energy (see for example [23] or section 2 of [14]). The defining feature of the folded OPE limit is that infinitely many such paths exist: any lightray leaving the origin on the first fold will return to it on the second fold. 4 The δx → 0 singularity comes from the full lightcone! This can be confirmed by expanding (2.11) near the cone, as depicted in figure 2a. We parametrize it in terms of a null vector n µ where v → 0 is a retarded time near the cone, u is an affine time along lightrays, and n ∈ S d−2 is a unit vector in d − 1 dimensions. (Normalized so that n·n = −1.) The 4 On the second branch, positive-frequency excitations propagate backward in time.
where we dropped subdominant terms as δx → 0. (Explicitly, ". . ."= +δx·(δx − 2vn).) The salient feature is that as δx → 0, the v contour gets pinched between branch points: v 0+i0 We drew the picture for u > 0. For u < 0, the two cuts are on the same side and there is no pinch. Thus only the future lightcone contributes a singularity. Assuming that the numerator of (2.13) is regular as v → 0, we can neglect its v-dependence and analytically integrate over v, see I ∆ φ ,∆ φ in (A.6). The contribution from the second timefold gives the same up to sign, so the result is 5 (2.14) Let us interpret this result. Crucially, O 1 (y) plays a passive role, as anticipated. The result is thus really a statement about products of φ's, in the sense that inserting either side in a path integral gives the same result as δx → 0. We defined the "light transform" along the Schwinger-Keldysh contour C: where the integral starts from the origin u = 0 on the first branch and ends at the origin on the second branch. The inclusion ⊃ in (2.15) indicates that the full OPE contains many other terms, notably the identity (proportional to φ 2 (δx) φ 1 (0) ) and bilinears in φ (so-called doubletwists), which exist already in the non-interacting theory. The matching calculations necessary to disentangle all those are in fact particularly subtle in φ 4 theory (φ 2 being itself a double-twist). This will be discussed below and needs not concern us yet: here we are simply describing a way to rewrite the δx → 0 limit of one diagram in some simpler way.
Eq. (2.15) captures the singular part of correlators like (2.7) as δx → 0. It represents physically a φ excitation moving along the lightlike direction n µ and interacting with the external world through a coupling du −iλφ 2 2 along its worldline. The angular distribution of the excitation depends on how the limit δx → 0 is approached. This is described by 5 For reference, the overall constant isP For example, if δx is purely in the time direction, then the distribution is spherically symmetric.
The integrand in (2.14) is a retarded product supported within the past lightcone of y. This makes the u range effectively finite: whatever happens after the measurement can't affect the measurement.
It is easy to see that (2.15) holds even if the quantum state of the theory is not the vacuum. The essential requirement was that neither the state |Ψ nor O(y) create sharp features within a distance v ∼ n·δx of the lightcone, so as to not invalidate the expansion leading to (2.14). 6 It is instructive to evaluate (2.15) for a concrete three-point function in the vacuum, say with O = φ 2 . The diagram shown in figure 2a, for one value of n, gives the integral (2.17) The final integration over n is generally nontrivial, but notice that the above vanishes in d = 4, where the theory is conformal (at this order in perturbation theory). This is related to the statement that light transforms annihilate the vacuum [25]: in a conformal theory we only expect the expectation value of L n,C [· · · ] to be nonvanishing in the presence of other operators both inside and outside the cone. Alternatively, the contribution (2.15) had to vanish in a conformal theory because the shape of vacuum three-point functions is fixed by symmetries, and one can check that it does not contain any term with exponent ∼ 1/δx 2∆ φ −1 . Regardless, eq. (2.15) remains valid, and nontrivial, in excited states.
We conclude this φ 4 discussion with a technical comment. Calling (2.15) an OPE is a slight abuse of language: L n,C [O] is not an "operator" acting on the standard Hilbert space of the theory. It is only an operator in the sense that its insertion makes sense inside the (Schwinger-Keldysh) path integral. This subtlety makes a nonperturbative classification of "operators which can be inserted on a null cone" more challenging than that of local operators, which are simply states of the theory on a small Euclidean sphere in radial quantization.

More loops, more fields, more theories
Pinching contour arguments of the sort just reviewed give a clear picture of perturbative singularities in terms of classical particles. We can use it to anticipate how the OPE gets modified for other operators, at higher orders in λ or δx, or in other theories.
A simple generalization is to replace each φ in (2.6) by φ m . A contribution with m copies of (2.15) and (m−m ) direct Wick contractions between the φ's then yield an integral over m distinct lightrays: (2.18) 6 In a conformal field theory, the OPE of any two operators acting on the vacuum is convergent for any δx µ [24]. We do not know whether the OPE that starts with (2.15) is convergent or only asymptotic.

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This demonstrates a basic fact: configurations with multiple particles on the null cone do contribute to folded OPEs. Single-particle excitations cannot always give the full answer. Of course, we expect the same phenomenon when (2.15) itself is expanded to sufficiently high order in perturbation theory, since a generic diagrams contains many multi-particle intermediate state. These may however be suppressed by powers of δx, as happens in the example of (2.18): the m th term is suppressed by a relative |δx| m . It will be useful to understand and classify these factors by symmetry arguments. As explained in the next section, in conformal theories in the vacuum state, the limit under consideration is equivalent to the Regge limit. The present discussion greatly benefits from hindsight from the extensive literature on this topic.
Consider another system: scalar QED. The Lagrangian density is The OPE of Φ 2 Φ * 1 can now contain line integrals of the gauge field A u du. Its calculation is essentially identical to (2.11) above, with the crucial difference that the ΦΦ * A vertex contains a derivative so we end up with a more singular denominator. To be precise, we should only record results for gauge-invariant local operators. For O ≡ (Φ * Φ) we get terms with two integrals A u du: (2.20) Before discussing the dependence on δx, let us discuss a simple improvement. The above is only the first term in an expansion in small A, and one may worry that the series breaks down if L n,C [iA u ] is not small in the state under consideration. Indeed, diagrams with multiple insertions of A u along each lightray also contribute at the same order in δx. However, it is relatively easy to show that these simply sum up to Wilson lines. Thus, it is better to rewrite (2.20) in terms of Wilson lines: (2.21) Since U n is a unitary, its expectation values are bounded. The advantage of (2.21) over (2.20) is that the latter remains valid even the U 's are not parametrically close to identity. The omitted terms have coefficients suppressed by e 2 or higher powers of δx; they can contain more Wilson lines. These statements naturally generalize to non-abelian gauge theories.
No terms involving direct Wick contraction should to be added to (2.21). This is because O's are made of charged excitations, which cannot connect the two local operators without dragging a Wilson line along the contour C. Rather, the vacuum expectation value of O 2 O 1 is automatically accounted for by (2.21) by taking the weak-field limit U n → 1 and integrating over angles. 7 7 If the gauge group is G, operators on the right-hand-side of (2.21) must be invariant under independent G × G gauge transformations at the two endpoints. "Weak field" really means that Un is n-independent, thus spontaneously breaking this group to its diagonal: G×G → G. Small perturbations can be parametrized as Un = e igW a n Ta where Wn is called the Reggeized gluon, see [26] and references therein.

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Extrapolating these examples, one might guess that exchange of a field of spin J comes with a factor 1/(−2n·δx) 2∆ φ +J−1 . This is nicely formalized by the light transform of [25]. It shows that in conformal theories, if H is an operator of dimension ∆ and spin J (represented as a J-index symmetrical traceless tensor H µ 1 ···µ J ), then the integral L n,C [H u···u ] transforms like an operator with transformed quantum numbers: Applied to the operator φ 2 which has (∆, J) = (2∆ φ , 0), this gives (∆ L , J L ) = (1, 1 − 2∆ φ ). This explains precisely the δx dependence in (2.15). (The leading coefficient of L in the OPE of two scalars of dimensions ∆ 1 and ∆ 2 by definition takes the form .) This also explains why multi-particle contributions in scalar field theory must be power-suppressed, as we saw in (2.18).
Conformal symmetry does not play a role here: J L is the Lorentz symmetry of the tangent space near x µ (of which δx µ is an element) while ∆ L acts by rescaling δx.
As mentioned below (2.15), the full OPE in φ 4 theory starts with double-twist operators at tree-level. One might be tempted to do a derivative expansion (φ 2 (0) + δx µ ∂ µ φ 2 (0) + . . .)φ 1 (0), however these products are not really well-defined and the more robust way to write the free-theory OPE is to integrate over nonlocal lightray operators which analytically continue φ∂ J φ, as discussed recently in [27]. A peculiarity of this theory is that the leading trajectory with ∆(J) ≈ d − 2 + J mixes with its shadow with ∆ → d − ∆ already in the free theory. This makes the renormalization of L[φ 2 ], which lives precisely at their intersection, particularly subtle. The upshot, according to [27], is that at small but finite coupling, the contribution (2.15) can be subsumed into an O(λ) correction to double-twist OPE coefficients. This will be seen quite explicitly in eq. (3.31) below. Since operators like (2.21) do not lie near tree-level intersections, we do not expect them to be affected by such subtleties (nor the operators we study in gravity below).
For gauge theories, the relation (2.22) gives ∆ L [U ] = 0 which explains why operators with arbitrary many Wilson lines in (2.21) are not suppressed by δx: Wilson lines are marginal operators in the folded OPE. Quantum corrections however change ∆ L . This is captured by the famous BFKL evolution equation, which has been recast in terms of Wilson lines in many references, see for example [28,29]. Operators with different numbers of Wilson lines mix at the cost of a loop factor g 2 . (In the t' Hooft planar limit, the parameter is really g 2 ∼ 1/N c not g 2 N c : Wilson line number is effectively conserved at large-N c , see [26].) Crucially, the quantum corrections increase J and make the δx → 0 limit more singular. This is the famous fact that the BFKL Pomeron has intercept This means that in non-abelian gauge theories, contributions with more Wilson lines, although formally suppressed by powers of g 2 , grow faster with energy. This ultimately causes the structure in (2.21) to break down at exponentially small δx. (In some situations, this breakdown is necessary to satisfy nonperturbative bounds on correlators, like the Froissart-Martin bound.) In other words, the form (2.21) may not give the true asymptotic limit δx → 0, but only a transient "pre-saturation" regime at intermediate δx. This is still an interesting regime -13 -

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and we will try to stay within it. Large-N c helps by parametrically delaying saturation (sometimes called scrambling), to |δx| J * −1 ∼ 1/N 2 c . A similar comment will apply to tree-level graviton exchange below.

Scalar coupled to gravity in AdS
We now move on to our main interest: theories with a perturbative description in a dual, asymptotically AdS d+1 , spacetime. Consider a scalar field coupled to gravity AdS . We would like to understand the OPE of boundary operators O in the presence of other operator insertions. From the preceding discussion, we anticipate graviton exchange to dominate whenever it is nonvanishing, so let us study three-point functions like at tree level, where there is a single diagram shown in figure 2b. We start with a bulk geometry that is pure AdS and generalize shortly. The AdS metric in Poincaré coordinates is where the field theory is at the r → 0 boundary and η µν is the flat Minkowski metric. We are interested in the singular part of the diagram as δx → 0. From a Landau analysis similar to (2.15) we anticipate that the region near the future lightcone of the origin will dominate as δx → 0, so we directly zoom onto this region. The base of the cone is now a (d − 1)-dimensional hyperbolic space H d−1 . It can be naturally parametrized by a timelike unit vector θ µ (θ 2 = −1), which gives the bulk null vector θ µ ∂ µ + ∂ r . We parametrize a neighborhood of the cone by variables (θ, u, v) (see figure 2b) with u is an affine coordinate and v is a retarded time away from the null cone (v = 0). Note that null geodesics take an infinite amount of affine time but a finite coordinate time to reach the boundary. This is why the boundary is reached at u = −∞. Using affine time makes the parametrization slightly awkward but it makes the calculation more similar to (2.15) and will also simplify things below. Finally, we need the scalar bulk-to-boundary propagator (with canonical normalization in the field theory OO = 1/( .

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Accounting for the two derivatives at the vertex, the diagram to leading order in v ∼ δx → 0 is given by the bulk analog of (2.13) 8 (2.28) Neglecting as before the v-dependence of the correlator, we can integrate over v using (A.6) to get the OPE The exponent in the denominator agrees with the formula 2∆ θ is the canonical measure on the unit hyperboloid and L is further discussed below.
Eq. (2.29) will be our main tool to study the OTOC and its Fourier transform. It will be useful to write the bulk light-transform covariantly. Consider a metric which asymptotically approaches the AdS form (2.25) as r → 0. Sufficiently close to the boundary, the (boundary) momentum g µa dX a du becomes independent of affine time. We normalize the affine time u by equating this momentum to a timelike vector p µ . The light transform L x,p is then defined as follows: first, we find the null geodesic in the background geometry with affine parameter u and appropriate initial momentum; then we integrating the pullback of metric perturbations h uu du along this geodesic: (We will use the same notation L x,θ when θ µ is a unit vector instead of a "momentum".) The turning point u 0 is any value sufficiently large to cover all regions with nonvanishing integrand, for example to cover the past lightcone of y in the current example.
With these definition, p µ transforms like a cotangent vector at the boundary point x. It parametrizes the direction in which a null geodesic is fired into the bulk, together with its energy. From the boundary perspective, we do not control what the geodesic finds along its route, but we can control how it starts.
We claim that with this definition, the OPE (2.29) holds in any background geometry. This can be established by studying the propagator in a neighborhood of the null cone from x. In general this can be conveniently parameterized as (eq. (5.5) of the lecture notes [34]): Here we are simply extending the boundary Schwinger-Keldysh timefolds into the bulk, following [30,31].
Special prescriptions near black hole horizons have been discussed [32,33]. They are not needed here since our observables only probe the bulk's perturbative density matrix at the past edge of the causal diamond between x and y.

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where Q, g ij and ξ i can depend on all coordinates (u, v, θ) but Q is independent of u when v = 0. Null geodesics on the cone are paths with v = 0 and constant θ. The scalar field's equations of motion near the cone take the form The idea is to expand near the null cone and look at the most singular term φ ∼ φ 0 (u, θ)v −∆ ; we see that the equation of motion setsg 1/4 φ 0 to be u-independent. This means that the most singular part of the propagator simply follows geodesics in such a way that the graviton source term The effects of spatial derivatives (or of mixing between the scalar and graviton, neglected in (2.32)) are suppressed by powers of v. Note that the v → 0 expansion is subject to similar limitations as any WKB-like expansions; it may break down for example if the effective energy ∼ 1/v is not sufficiently large compared with variations of the background geometry. Note that at early times when the excitation is close to the boundary, the locally measured energy is not large and a WKB approximation is not justified; the small-v expansion of (2.32) nonetheless makes sense.
In the vacuum state, the three-point function φT φ obtained from (2.29) vanishes for timelike y by a mechanism similar to (2.17). Yet the three-point function is nontrivial: there is a δ-function at lightlike y, given by the shockwave discussed in appendix B. This is the expanding shell of energy described in conformal collider physics [3]. The shell has a small thickness δx in retarded time which becomes a δ-function in the OPE under consideration.

Focusing using Fourier transforms and wavepackets
We can now see how the Fourier transform described in introduction localizes around a single bulk lightray. Let us first ignore the Gaussian envelope, and consider a momentum p µ = (ω, 0) purely along the time direction. The time integral against (2.29) gives Integrating this over δx would give a δ-function that sets θ = 0, thus localizing the OPE (2.29) to a single lightray fired at a square angle into the bulk. More generally, one could include a Gaussian envelope of width L for x and get: We see that the shooting angle θ is Gaussian distributed with width 1/(ωL). This is the standard diffraction limit for the angular resolution of an optical lens of size L. We conclude that the optical aspects of looking into the bulk are very similar to conventional optics. 9 If the bulk theory contains a dilaton-like scalar ϕ, such that scalar kinetic term is multiplied by e −ϕ , the propagator will be such that the source term d d−1 θ √g e −ϕ (∂vφ 0 ) 2 is unaffected by the dilaton. The results of this section thus apply in the Einstein frame metric.

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Let us define a standard set of wavepackets. For p an arbitrary positive-frequency timelike vector, we can covariantly boosting the above. Additionally, we can include independent widths L and L 0 for the space and time components in the rest frame. A natural normalization is provided by the vacuum two-point function: . We will be interested in the regime L|p| 1 where the shooting angle is well-defined, so we ignore here 1/(L|p|) corrections to the normalization. In the limit that L → ∞, we find that the omitted proportionality constants in (2.33) nicely cancel the factors in (2.29), so its transform gives simply as anticipated in eq. (2.2). The omitted terms, which are the same as in (2.29), give insight into the limits of this approximation.
To account for the Gaussian envelope, we can now use that multiplication by a Gaussian in x is equivalent to convolution by a Gaussian in p. Thus (2.36) needs to be convolved with a Gaussian of width 1/(L|p|) in the orientation of p, and with a Gaussian of width 1/L 0 in its magnitude. We will typically be interested in situations where the energy dependence of correlators is mild, so the latter is a negligible effect. We thus omit L 0 from our notations. The angular uncertainty 1/(L|p|) is important however.
The reader should be aware that if p µ is highly boosted, then the footprint of our covariant wavepackets can become much larger than L along the longitudinal direction. A highly boosted source with a finite footprint would create waves that rapidly disperse, and do not follow a unique bulk geodesic [4]. In this paper we will only consider momenta p µ that are not parametrically boosted.

Four-point correlators
The out-of-time-order correlator featured in (1.4) can now be obtained using two copies of the OPE. Technically, it involves a four-sheeted timefold For clarity we have exaggerated the infinitesimal separation between the folds, but it will be important that they are actually all atop each other (this is unlike the late-time limit of [10]). The OPE between the O is still given by (2.29), while that between the O is a time-reversed version of it: where the light transform is defined analogously to (2.30), for a geodesic that now reaches the boundary at v = +∞: (2.39) Note that the excitation has positive energy (θ 0 > 0 in (2.38)) but it propagates backward in time, since it starts on the second branch of the contour (2.37), whence the sign in −h vv,2 .
The four-point OTOC is now reduced to a graviton two-point function: , as depicted in figure 3. The relevant combination turns out to be a retarded propagator: (2.40) The retarded propagator depends only on the bulk geometry dual to the state Ψ and is insensitive to the presence of bulk gravitons (which could come, for example, from Hawking radiation if there is a black hole in the bulk). It is useful to define a shorthand for its integral against null geodesics: (2.41) The normalization choice will become clear below. The OPE (2.29) thus gives us a formula for the OTOC at tree-level in a general geometry

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Equivalently, its Fourier transform (1.4) using the wavepacket normalized precisely in (2.35) gives, in the geometrical optics regime,  [17] and (2.43) to the impact parameter representation of [35] (see also [36,37]). The propagator Π Ψ entering (2.42) was normalized so it reduces to the transverse propagator, also called Π in that reference, up to a factor of energy, as discussed shortly. They are meant to capture the contribution to the correlator with the fastest energy growth while the right-hand-side of (2.43) is still smaller than unity.
The eikonal approximation (2.29) also controls the radar-like correlator (1.3) in the regime where the projectile has a much higher energy than the probe. This is further discussed in section 3.3 below.

Useful limits: near coincidence and vacuum AdS
Let us finish this section with an important special case of (2.41): when two geodesics nearly intersect at a point. The singular behavior in this limit is controlled by the graviton retarded propagator in a small neighborhood of this point. It takes the flat space form is a projector, and the + subscript indicates the small imaginary shift p 0 → p 0 + i0 appropriate to the retarded boundary condition. Suppose the metric is given locally in the form (2.31). The integrals over null times u and v in (2.41) set two momenta to zero, and the remaining (d − 1) integrals give the Green's function of the transverse Laplacian. Reinstating factors of the local metric g * ab near the collision, the singular behavior is thus: (see eq. (A.1)). The various factors have simple interpretation in terms of the center-of-mass energy and impact parameter in the bulk. For example, in vacuum AdS, a short calculation yields  1b: if the cannon shoots towards the right, the eye will see the signal arriving from left.) Thus b bulk is the chordal distance between points proportional to p x and −I (x−y) ·p y . The collision occurs at the halfway point in time, at a radius determined from the radial velocity. (In the regime relevant to (2.45), the impact parameter is small, b bulk R AdS .) In vacuum AdS, we can say more. The intersection between the future and past lightcone of x and y is geometrically an hyperboloid H d−1 , and the chordal distance b bulk in (2.46) defines a natural impact parameter even when it is not small. Exploiting symmetries, one can in fact calculate Π Ψ exactly in this case. It turns out that the u integral in (2.41) sources a shockwave supported on the null cone from x, and the propagator (2.44) becomes exactly that on the transverse hyperboloid: ). The calculation, presented in appendix B, is related to one in [17] by a conformal transformation.

The AdS vacuum and conformal Regge theory
The goal of this section is to better understand the camera by taking first images in the vacuum state of various conformal field theories. The folded OPE limit is then equivalent to the well-studied Regge limit, which will enable us to numerically study the correlator (1.4) beyond Einstein's gravity. For spacelike separated points, conformal correlators take the form where the nontrivial dependence factors through two cross-ratios defined by For configurations in Euclidean space, z is the complex conjugate of z and G is a single-valued function with no branch cut ambiguity. Let us specialize to the kinematics in (2.42). With no loss of generality, in this section we fix x = 0 and y = e where e is a unit timelike vector (e 2 = −1), and parametrize coordinates as In the limit of small δx, δy, the cross-ratios (3.2) are small and approximate to Recall that I µν e = η µν − 2e µ e ν /e 2 is geometrically a reflection in the e direction.

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Even though the numerical value of cross-ratios are small, the out-of-time correlator is not given simply by (3.1): the nontrivial ordering is only reached after an analytic continuation. This is reviewed in appendix C.1, where it shown that the desired correlator (3.3) in a "reference" region δx 0, δy≺0 (where z, z are small, real and positive) is reached after taking z counter-clockwise around 1, holding z > 0 fixed: (3.5) Other regions of δx, δy (spacelike, past timelike, etc.) can then be reached without leaving the region of small z, z, paying appropriate care to the i0 prescriptions in (3.4). Examples are given in table 1. In particular, when δx and δy are both spacelike with δx ∝ −I e ·δy, the "bulk-point limit" of [14] is approached.
We briefly discuss the radar-like system (1.3), which fits in a Schwinger-Keldysh two-fold: radar camera: There many kinematical regions one could explore with this system. In this paper we will assume that x and z are spacelike separated (note that this is not apparent in the drawing), and restrict attention to the case δx → 0. Physically this is a "ballistic" limit where the particle fired from x has a high energy, compared with the pulse sent from z and recorded at y. Cross-ratios vanish in this limit: In a reference region with δx spacelike and U y,z ·δx > 0 (U y,z is always spacelike), z, z are small with a positive real part and the correlator is proportional to G . 10 Thus, in the vacuum state of a conformal theory, the ballistic limit of the radar camera is equivalent to the active camera up to a continuation sign and the replacement −I e ·δy − → U y,z . (When the correlator becomes singular, such as near the bulk-point limit, small imaginary parts from z → z + and y → y − should be added, which adds a negative shift (U y,z ) − .)

Relating Regge correlators, detectors, and OTOCs
The correlator G(z, z) is exactly the same one which controls the Regge limit (see for example [17,38,39]). In this limit, a large relative boost is applied between two pairs of points. An intuitive way to understand this relation is shown in figure 4 in terms of the conformal theory on the Lorentzian cylinder R × S d−1 . Our usual spacetime R 1,d−1 is embedded as a single Poincaré patch, which consists of the set of points spacelike-separated from a reference ∞. The key point is that correlators that span multiple patches are 10 The second term in the commutator (1.3) yields (3.6) with z inserted on the second branch. This second term will be negligible when a positive energy is absorbed by the wavepacket at y, and we will ignore it here.    equivalent to correlators involving multiple timefolds on a single patch, up to a predictable overall phase (see [25]). The figure illustrates how the same configuration of four points on the cylinder can be interpreted in various ways, depending on how one chooses to place them relative to an (arbitrary) Poincaré patch. The dynamics of these limits are clearly similar: in all cases, the Landau singularities populate a complete null cone. In the folded OPE of section 2, the two tips of the cone are at the same "point" but on different Schwinger-Keldysh timefolds. In the Regge limit, rather, the two tips are at the "same" point at past and future null infinity, respectively. The limit creates a shockwave on the x − = 0 null sheet, with excitations moving at different transverse positions x ⊥ ∈ R d−2 ; this null sheet is nothing but (a stereographic projection) of the null cone from infinity.
The correlators dubbed "detectors" in [27] combine these limits: operators created at the origin cross a shockwave at I + , and are then absorbed back near the origin on the second Schwinger-Keldysh branch. The shock represents physically different ways to weight final states at infinity (for example, energy flux measurement), while the two branches respectively compute the amplitude and its complex conjugate entering a cross-section. In all cases, where lines intersect is where particles can collide.
The relation between the Regge limit and detectors are relatively well explored, since perhaps the work of [3] on conformal collider physics. In perturbative QCD, it is sometimes called timelike-spacelike correspondence, see for example [40][41][42][43]. The relation between OTOCs and the Regge limit was noted in the context of the bound on chaos [10,44]. It will be further explored in this section.  Figure 5. In empty AdS, all lightrays from x 1 arrive at x 2 whether they pass through the boundary or bulk (the interior of the cylinder). When a macroscopic amount of energy (wavy black line) is added to the bulk, bulk geodesics like the thick blue one experience a time delay and may not even return to the boundary, thus washing out the Regge singularity at (x 2 − x 1 ) 2 → 0. In contrast, the folded OPE singularity between x 1 and x 1 + δx exists for any bulk state.
We stress that these equivalences hold only around the vacuum. The limits become qualitatively distinct in excited states. For example, if the bulk contains matter (or a black hole), geodesics that go through the bulk will no longer simultaneously arrive at the antipodal point due to Shapiro time delays. They may even never make it through. Thus the Regge signal gets washed out when the bulk background geometry is far from AdS, as depicted in figure 5. This phenomenon is known as saturation in high-energy scattering, or scrambling.
In contrast, the geodesics in the folded OPE limit perfectly follow each other on the different Schwinger-Keldysh branches, no matter how far the background geometry is from the vacuum. We only need the probes to not significantly backreact on each other. This can be ensured by keeping the center-of-mass energy below the Planck energy, at the impact point. This state-independence of the folded OPE is crucial for our story. It is why the cameras can image non-vacuum states.

Conformal Regge Theory
Conformal Regge Theory describes the Regge limit of correlators in terms of exchanges of Regge trajectories J i (ν). Assuming for notational simplicity that a single trajectory dominates, its contribution takes the form [38] The measure ρ(ν) and harmonic functions P are generic kinematical objects defined in appendix A.1. A specific formula for graviton exchange in AdS is obtained by inserting the Regge trajectory J = 2 and coefficient functions from eq. (100) of [38]: (3.9) -23 -

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Here and below, Ω d = 2π d/2 Γ(d/2) is the volume of the unit sphere S d−1 and (3.10) The expression (3.9) can be understood as follows: the quantum number ν is related to the scaling dimension of exchanged operators as ∆ = d 2 + iν; the Regge trajectory represents a J-plane singularity of OPE coefficients c(∆, J) in the Lorentzian inversion formula [39,45]. The explicit poles in (3.9) thus correspond to exchange of a CFT operator with ∆ = d (and its shadow at ∆ = 0), ie. the stress tensor, while the γ factors have poles at double-twist exchanges ∆ − J = 2∆ O + 2m for m ≥ 0 and J = 2. Loosely speaking, eq. (3.9) is the simplest function with all these poles and no other singularities.
The overall normalization is fixed by the residue on the stress tensor pole, proportional to the inverse central charge 1 as reviewed in appendix C.2. We used this constraint to write eq. (3.9) in general d (it was only explicitly quoted for d = 4 in [38]).
It is significant that α GR (ν) ∼ e −π|ν| at large ν. This makes the correlator (3.5) a rather smooth and featureless function of the angle between δx and δy (if ∆ O , ∆ O are not parametrically large), in the reference kinematical region. However, it ensures that the representation (3.8) converges even when analytically continued to other regions. For example, for spacelike δx, δy, where (z, z) ∝ (e iϕ , e −iϕ ), we get P 2−d 2 +iν (cos ϕ) ∼ e ±ϕν . In the bulk point limit ϕ → π, the integral becomes only distributionally convergent.
Thus the fast decay of α enables to continue (3.8) (a priori defined only for z, z > 0) to spacelike regions, which are much better at probing bulk physics (see also [20,46]). To our knowledge, it has not been established in an abstract CFT, only in examples.
To access those more explicitly, we rewrite the result (3.5) in a form that eschews cross-ratios and separates the dependence on δx and δy. This can be achieved by integrating over angles (parametrized by a null direction n µ ) using the identity (A.11): (3.11) Assuming the mentioned decay of α(ν), eq. (3.11) can be safely continued from the reference region δx 0, δy≺0, where the parentheses in the second line are positive.
It is now straightforward to integrate (3.11) against wavepackets since we can transform with respect to δx and δy separately. The dependence on n and p 2 is then determined by symmetries. Inserting (A.16)-(A.17) (twice) into (3.11) and reverting the n integral (A.11), one thus obtains a concise formula for the contribution of a Regge trajectory to (1.4): (3.14) This is essentially the "impact parameter amplitude" [35][36][37], with an extra Gaussian factor accounting for the optical resolution discussed below (2.34). Our normalization ofα was chosen to simplify the relation to flat space scattering amplitudes, see (3.21) below. The main feature of (3.13) is that most factors in the gravity expression (3.9) neatly cancel, in particular all the double-trace poles, leaving only the graviton exchange: For gravity, the ν-integral in (3.12) can then be done analytically as it is merely a fancy way of writing a propagator on the transverse hyperboloid H d−1 . Specifically, as L → ∞ one finds simply the function Π d−1 (p·I e ·p ) in (A. 15), in precise agreement with section 2.6. Conceptually, this works because the ν-representation can be used already at the level of the OPE (2.29). The lightray L x,θx is a function of an angle θ ∈ H d−1 , and a complete basis for such functions can be labelled by a null vector n and a momentum ν, such that the following transform is invertible (see (A.8)): Because the OPE (2.29) is Lorentz covariant, its transform must be proportional to L x,θx and we find indeed: .
(3.17) Intuitively, the operator L x;n,ν creates a bulk shockwave with momentum ν in the transverse H d−1 , and the last factor is the transform of the bulk-to-boundary propagator. We see from the γ factor, which decays rapidly at large ν, that the scalar's bulk-to-boundary propagator (when δx is timelike) has a smoothing effect which obstructs access to large momenta. The Fourier transform, which commutes with the harmonic decomposition, cancels this γ factor, as above. This is why it nicely localizes to a bulk geodesic.
All this was, in fact, discussed in the classic papers [17,35]. The only novelty here is that conformal symmetry plays no role: we are only using the Lorentz symmetry of the tangent space near x. This makes the folded OPE (3.17) also apply in nontrivial bulk geometries far from the AdS vacuum.

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Instead of Fourier-transforming (3.17), for our purposes one could equivalently convolve it with any kernel which approximately cancels γ. The wavepackets with finite support |ν| < ν max ≈ ∆ higher−spin gap used in [47] for example, enable to do everything without leaving positive-timelike δx.

Relation to bulk S-matrix
To plot the singular behavior of correlators as realistically as possible, below we will account for stringy corrections to pure gravity. This can be readily done by relating the OTOC to a local scattering process in the bulk.
Intuitively, such a relation exists in a holographic theory in the approximation where one neglects the interactions between the bulk particles created by O (x 4 )O(x 1 )|Ω , up to near the point marked as a "star" in figure 4. Then we have two "asymptotic" excitations hitting each other in a locally flat region. This approximation is certainly valid at large-N . More generally, it captures the local bulk interactions which we expect to be responsible for small-angle singularities. The bulk two-particle state is characterized by center-of-mass energy and impact parameter given in (2.46), namely: After the collision, its products freely evolve back to the boundary of AdS. Notice from figure 4c that the scattering occurs backward in time since it is on the second timefold. Its amplitude is thus given by the complex conjugate of the 2 → 2 flat space S-matrix: Let us first discuss the intended regime of operation of the camera: 1/|p| L |x − y|, where the wavepackets have well-defined position and shooting angles. The fact that the shooting positions are well-separated (L |x − y|) implies that the bulk scattering angle is small. (Otherwise, the particle fired from x ± O(L) would not return near x after scattering and reverse time evolution.) This is why shooting angles control the bulk impact parameter in (3.18), as opposed to a bulk scattering angle. The eikonal graviton exchange formula in (2.45) can thus be replaced, for a generic scattering process, by the transform of 1 − iM † to impact parameter: A similar reasoning led to eq. (3.11) of ref. [14] in the bulk point limit. However, our setups differ in important ways. First, we only see small-angle scattering with a fixed impact parameter, rather than fixed angle. Second, here our 2 → 2 scattering is localized around a unique bulk point, in any spacetime dimension, as opposed to a (d − 2)-dimensional manifold. Third, the center-of-mass energy at the collision is fixed, rather than integrated over. These differences are caused by our use of out-of-time-order correlators as well as to our use of wavepackets.

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To compare with (3.12), we simply need the fact that at large ν and small impact parameter (and νb fixed), the harmonic transform reduces to a transverse Fourier transform: (3.20) Equating (3.19) with (3.12) then gives the desired approximate relation, which is most simply stated in the case that M is controlled by a single Regge trajectory: ) is obtained by the same map. This is exemplified below for the Veneziano-Shapiro amplitude.
The finite-size of wavepackets modifies this discussion through the uncertainty (2.34) in the shooting angle, which amounts to a Gaussian convolution in the impact parameter b bulk . This is precisely accounted for by the Gaussian factor in (3.12). Note that a Gaussian uncertainly ∼ L in the shooting origin should not be added. This effect is absent because in (1.4) we only integrate over the relative displacement δx between creation and annihilation operators; the effect would be present if we had instead inserted independent wavepackets for each.
Let us briefly comment on the radar-like camera. Integrating δx against a wavepacket in the correlator (3.6), the γ ∆ O +1 factor gets canceled and the analog of (3.12) is of the form The argument of the harmonic function is a peculiar dot product of a timelike and spacelike vector. The angle cosh −1 (p x ·(Û y,z ) − ) is imaginary and approches iπ/2 when the dot product vanishes. This effectively cancels the second γ factor at large ν and creates a singularity (or peak) when Mathematically this is similar to the bulk point singularity of the correlator in the small angle regime. The condition (3.23) means that the future lightcone of z and the past lightcone of y intersect the geodesic (0, p x ) at the same point. This is precisely the expected signal from the radar camera (1.3), which was intended to record at y the reflection of a pulse sent from z. There is a signal when z and y satisfy (3.23) even before we Fourier transform with respect to y (or z). The transform would reveal more information about the angle of arrival. (The holographic radar is further illustrated in figure 11 in the next section.) It may seem surprising that the pulse arrives and leaves at independent angles. A good analogy is Compton scattering where a soft photon hits an ultrarelativistic electron sideways. In the lab frame, the scattered photon can leave at any angle, and will simply experience a correlated Doppler-like energy shift (related here to the ratio of ∂ ∂y (p x ·U y,z ) and ∂ ∂z (p x ·U y,z )). From the center-of-mass frame perspective, the deflection is however small and the Mandelstam invariants generically satisfy −t/s 1. The microscopic interaction is thus dominated by t-channel graviton exchange. This is why the eikonal approximation of the last section (for the (0, p x ) geodesic) makes sense for p x sufficiently large.

First images: N = 4 SYM at weak and strong coupling and 3D Ising model
We are now ready to experiment with the AdS camera in various theories, not only pure AdS gravity. We numerically study N = 4 super Yang-Mills (sYM) at strong and weak coupling, as well as the critical 3D Ising model. In the sYM model, we consider the OTOC of two half-BPS operators in the stress-tensor multiplet: O ∼ ReTr [XX] and O ∼ ReTr[Y Y ], where X, Y are two complex scalar fields. At large λ, sYM is dual to string theory where the string length is related to the 't Hooft coupling λ as As λ → ∞, states on the leading trajectory are long rotating strings [48]. As long as ν √ λ, the string remains short compared with the AdS radius, and therefore looks like one in flat space. This means that the trajectory is approximately linear over a broad range extending to ν ∼ √ λ, J ∼ √ λ: The corresponding amplitude can be obtained from the flat space Virasoro-Shapiro amplitude, which we only need in the Regge limit: (3.26) Using (3.21), this provides a high-momentum completion of the gravity result (3.15): This expression is now valid for any ν √ λ, including small ν. 11 Note that this expression should only be used when ν 2 s bulk R 2 AdS . In practice, we satisfy this constraint by terminating the ν-integral in (3.12) where the factor e −ν 2 σ 2 o /2 already makes the integrand negligible.
At weak coupling, we borrow the following formulas from [38] (see also [16,49]): (3.28) In figure 6 we show the numerical integration of the right-hand-side of (3.12) in these models, where we fix the projectile's momentum p µ x to be purely in the time direction 11 We believe it should be equivalent to eqs. (106) and (112) of [38] but we have not compared in detail. ) inside the unit disk, normalized by peak value. A narrow peak requires good optics and strong coupling. Above the diagonal, the camera's angular resolution is limited by poor optics; below the diagonal, by dynamics; along the diagonal, these limitations are roughly commensurate. and we display the amplitude of the signal for different angles of p µ y . A sharp peak is interpreted as a signature of local bulk dynamics. The two most important parameters are the optical quality |p x |L x = |p y |L y = |p|L ∼ 1/ √ eff and coupling g 2 ≡ λ 16π 2 . We used the weak-coupling formulas for g = 0.1, and strong-coupling for other values. (These approximations may not be great at intermediate g. We expect the trends and qualitative features to be robust, though.) Evidently, one sees that if σ o ∼ 1/|p|L is not small, the camera has poor optics and any putative bulk object will seem blurred. This is the familiar uncertainty relation: by observing the wave in a small region we can't know its momentum (ie. angle of arrival) accurately. This effect can be mitigated by increasing the energies p x , p y or by making the wavepackets wider, within the constraint that L x , L y |x − y|. The next cause of blur is dynamical. At strong coupling, this can be understood from the transverse broadening described by the Veneziano-Shapiro amplitude: ∆b 2 ∼ α log s. A crude approximation to the observed signal is a Gaussian which adds optical and dynamical effects in quadrature: . (3.29) We see that even if one uses energies above the string scale, and ideal optics, the camera The DGLAP dimensions and BFKL exponent are still determined by the same vertex operator, so the relation discussed in the previous section should hold for all λ, consistent with the weak coupling result. That is, j 0 is determined by ∆(j 0 ) = 2. In the large-λ limit we have studied the ∆-j relation only near j = 2, where it takes the form (3.40).
In the weakly coupled limit the (∆, j) locus has a complicated structure [45]. In the normal operator analysis one has ∆ = 2 + j + O(g 2 ), i.e. twist two in the free limit. BFKL identify another branch to the solution, where j = 1 + O(g 2 ). The inversion symmetry ∆ → 4 − ∆ implies a third branch ∆ = 2 − j + O(g 2 ). It follows that at zero coupling the Pomeron physical state condition must be in order to capture all three branches of the solution. At one loop we would expect a correction Let us analyze the solution near the point (∆, j) = (3, 1) where the BFKL and DGLAP lines meet. We assume that a(∆, j) is nonsingular there and approximate it by a constant a ≡ a(3, 1); we can also approximate ∆ − 2 + j = 2. The intersection of the BFKL and DGLAP lines is then resolved into a smooth hyperbola, one branch of which is shown in Fig. 2.
If we approach this point along the BFKL branch, the physical state condi--30 - can never resolve bulk impact parameters shorter than the string scale. (The last term is only meaningful when the bulk center of mass energy exceeds the string scale, so that the logarithm is positive. For lower energies, this term is in any case negligible compared with the others since L x , L y |x − y|.) The large difference between weak and strong coupling is perhaps best explained from the sketch of Regge trajectories shown in figure 7. The last factor in (3.29) is essentially the curvature of the leading (single-trace) Regge trajectory. Good resolution is possible at strong coupling because the trajectory is nearly flat. In general the curvature is controlled by the higher-spin gap which gives the lower bound σ eff 1/∆ higher−spin gap . At weak coupling, the high curvature of the leading trajectory obstructs access to large transverse momenta; no amount of optical wizardry would bring "bulk objects" into focus. 12 Let us now turn to the 3D Ising model. Its leading trajectory is well understood for integer spins, from various analytic and numerical methods [51][52][53]. Less is known about its analytic continuation to the principal series, where ∆ = 3 2 + iν with real ν. Ref. [54] gave numerical estimates of the trajectory near the origin, with intercept j * ≈ 0.8, as reproduced in figure 8a (see [55] for a related discussion of the O(N) model). Here we will also need functional forms for the trajectory and coefficient functions. We do not know it 12 At very small coupling g → 0, one might object that the BFKL trajectory is approximately flat J(ν) ≈ 1 along the integration contour ν =real in (3.12). However, the comparatively rapid decayα N =4 weak (ν) ∼ ν −5 still makes the image blurry, and is in any case incompatible with an interpretation in terms of exchange of a finite set of bulk particles, which requiresα ∼ ν −2 . The rapid decay ofα for real ν is only possible because it has complex poles at ∆ − J ≈ 2, 4, 6, . . . and these exist at weak coupling precisely because the trajectory is not flat for real ∆ = d 2 + iν.

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rigorously but we will attempt here a reasonable approximation using the ε-expansion of the Wilson-Fisher theory. There are in fact two nearby trajectories. In the free theory these are the leading-twist trajectory and its shadow, which intersect at ν = 0. At small but finite coupling they mix, and the intersection is resolved, as detailed recently [27]: Here ε = 4 − d and we take ν ∼ ε. Of course, one can cannot really trust this formula when ε = 1. For modelling purposes, we will simply set ε → 0.8 in (3.30), which we find to be reasonably close to the numerical curve in figure 8a.
Ref. [27] gave operator representations which in principle enable to calculate the residue α(ν). We take a shortcut. From the Lorentzian inversion formula, α is the residue of a coefficient function c(∆, J) with poles at (3.30). Using known OPE coefficients along the leading trajectory away from the intercept, we can deduce the fist few orders in the ε-expansion near the intercept, as detailed in (C.16). The Regge limit of correlators may then be obtained by plugging into (3.8). Further simplifications arise because the trajectories have a large slope away from the intercept. The steepest descent method then reduces the total contribution to a discontinuity of α, as discussed in the same appendix: (3.31) where the contour encircles the square-root cut counter-clockwise.
As a nontrivial sanity check, we can study (3.31) as ε → 0. The cut becomes infinitesimal and can be computed exactly. The dependence on cross-ratio is simply proportional to the function P −1 , while the explicit factors (and ρ(ν) ≈ ν 2 π ∼ ε 2 ) control the ε-scaling. We find: This can be compared with the connected diagram at O(λ), which can be calculated exactly as After the analytic continuation to G , the square bracket becomes −2πi log z z in the Regge limit. In the critical dimension ( λ 16π 2 → ε 3 ) this agrees precisely with (3.32). This calculation confirms that the operator L[φ 2 ] in (2.15) is not independent at small but finite coupling and can be re-expressed as a short Regge cut involving double-twist operators.
Our proposed model for the wavepacket-transformed correlator in the 3D Ising model is to take the integral (3.31) at finite ε and multiply by the factors in (3.13) evaluated as ε → 0. This gives the analog of (3.12):  fig. 12 is given. The analytic continuation is obtained once the fit is found. The ✏ and its shadow are represented by the two red dots. We see the prediction for ✏ by looking at the intersection of the curves and the axis.
shadow. As an example the fit of data obtained from ✏ + T at z = 10 4 gives the following estimates : These values are not close to ✏ operator. However the analysis predict their existence as was seen before in perturbation theory. Note that a slight shift in the vertical axis of this curve can land us on a curve which includes ✏. However it should be obvious that using this method to determine quantitative properties of ✏ would be numerically unstable.

Leading Z 2 -odd twist family
In this section we study the low spin operators in [ ✏] 0 family by analysing the correlator h ✏✏ i, which in our notation corresponds to 1 = and 2 = ✏ . According to our previous discussion this correlator leads to data about the [ ✏] 0 family. The operators exchanged in the t-channel, where we fuse with and ✏ with ✏ are ✏ and the families [ ] 0 , [ ] 1 , [✏✏] 0 . The u-channel involves the same fusion as the s-channel, with ✏ so and the [ ✏] 0 family are exchanged. We will use data provided in [5] for the dimensions and OPE coe cients.
There are many interesting physical facts about 3D Ising model that one can understand by studying this family of operators. For instance, the absence of a global conserved current with spin 1. According to the unitarity bound in eq. 3.1, this spin one operator, if existed, must have scaling dimension 2 and thus, must belong to [ ✏] 0 family. Our goal is to verify this fact analytically by extending the inversion formula to conformal dimension of order ⇠ 3 and observe its prediction.
Another compelling question is whether this family of operators contains a spin 0 operator and if it does what is that operator. We try to answer this question by extending C even (z, ) -24 - Note that in the 3D Ising model, the signal's magnitude l gets smaller with increasing energy. In figure 8b we display the numerical integration of (3.34), with ρ and P evaluated at d = 3, and ε = 0.8 in all other factors, as a function of the angle between p and p . We observe a very smooth signal as a function of these angles: the camera shows no hint of bulk point-like particles in the critical Ising model.
While there is admittedly ambiguity in the above model (we only used the first term in the ε-expansion), we believe that the conclusion is robust. The leading Regge trajectory is simply too curved to pump large transverse momenta into putative bulk particles.
It is generally believed that local bulk dynamics (at sub-AdS distances) requires the dual CFT to have a large central charge c T 1 and a large (single-trace) higher-spin gap ∆ higher−spin gap 1 [2]. We see here that these criteria are precisely those needed for the holographic camera to produce sharp images. The large central charge is needed to make the overall coefficient on the right of (3.12) small, so that we can access high energies before reaching saturation (ie. before s bulk approaches the Planck scale). The large gap is synonymous with the leading (single-trace) Regge trajectory being nearly flat.

More images: CFT 2 thermal state and BTZ black hole
Before highlighting the qualities of the camera in a nontrivial state, let us summarize some of its limitations we have seen so far: 1. If pL is not small, the shooting angle will not be precise (∆θ ∼ 1/(pL)) 2. If the wavelength 1/p is not small compared with typical bulk structures, waves of different frequencies will disperse 3. If bulk center-of-mass energies overcome the Planck scale ∼ 1/C T , we can no longer ignore higher-order terms in (2.29) and scrambling may occur -32 -
The first two are standard limits of geometrical optics: these would apply to any camera-like device. They are generally alleviated by using shorter wavelengths. The last two are dynamical limits which prevent access to short wavelengths, thus fundamentally limiting the focusing power of holographic cameras.
Consider a (spatially infinite) thermal state in a two-dimensional CFT. The Euclidean CFT is defined on a spacetime with the topology of a cylinder S 1 × R where the circle has circumference 1/T . We work above the temperature of the Hawking-Page transition so that 1/T is smaller than the circumference of a (possible compact) spacial circle. In two dimensions, this geometry can be conformally mapped to the plane, which allows to write thermal correlators in terms of vacuum ones [56]. The same applies to a general boosted state, which is conventionally described with left-and right-moving temperatures.
If the rest temperature is T = √ T L T R , the conformal transformation is where T = √ T L T R is the temperature in the rest frame. We are interested in the OTOC configuration similar to (3.3): where the cross ratios go to, as δx, δy → 0: , z z → e −ϕx−ϕy T R sinh(πT L (∆t + y)) T L sinh(πT R (∆t − y)) . (4. 3) The correlation functions are then given by the vacuum correlator G (z, z) in (3.8). Since the dependence of cross-ratios on the small vectors δx µ is similar to (3.2), the wavepackettransformed correlator can be obtained as in (3.12). Parametrizing the momenta as p µ i = |p i |(cosh ϕ i , sinh ϕ i ) and specializing the transform to 2d, we have 13 (4.4) withα(ν) and σ o as in (3.12) and )) .
(4.5) These will be interpreted shortly in terms of the bulk geometry of the BTZ black hole dual to the thermal state in the CFT: the intensity I is related to the blueshift factor at the center-of-mass energy at the collision and other factors; the offset ∆ϕ means that a null geodesic fired from the origin with boost angle ϕ x intersects a null geodesic fired backward from (∆t, y) with ϕ y = −(ϕ x + ∆ϕ). 13 The same result may apply to a generic heavy energy eigenstate, following [57].  Figure 9. A particle, initially fired at a right angle from the boundary at t = 0, moves in the geometry of the BTZ black hole. Each slide shows the intensity detected by the active camera as a function of the return angle ϕ y . The shown times range over t = 0.5 . . . 4 in increments of 0.5, with the temperature and other parameters as in (4.6). To add depth information, the images from two cameras displaced by ∆y = 0.1 have been superposed.
To enhance contrast, in all plots of this section we take two derivatives of (4.4) with respect to θ y . This multiplies the integrand by −ν 2 .
In figure 9 we display (4.4) as a function of receiving angle ϕ y for various times ∆t, for a fixed geodesic initially fired at a square angle into the bulk: ϕ x = 0. Note that while the bulk is (2+1)-dimensional, the image on a given time slice is only one-dimensional, similar to how a conventional camera records a codimension-one projection of a scene, resolving the angle-of-arrival of photons but not depth. Depth information can however be recovered in various ways, for example by using two cameras displaced by a small offset ∆y (called "left eye" and "right eye" in the figure). The following parameters were used: , |p x |L x = |p y |L y = 100, ∆y = 0.1. Several features can be observed. First, the image from just the left eye (at y = 0) shows that the particle initially at the origin starts drifting towards the right. This happens because of the plasma's velocity. Second, the angular separation between the two eyes' images decreases with time. Intuitively, this happens as the particle goes deeper into the bulk, the separation being inversely proportional to the radial coordinate, by stereography. The fact that the angular separation stabilizes at a finite value at late times indicates that the radial motion stops. Here this happens as the particle approaches a causal horizon in the bulk. This will be confirmed below by the signal intensity. Finally, sidebands appear at ∆t 3 because, in fact, we defined the theory on a periodic circle y y + 3; past this time, the receiving geodesic Y (v) can travel once around the circle. 14 As a natural alternative to measuring the angle ϕ y (∆t) y=0 , one could orient the receiver at a right angle and scan over lateral displacements, measuring the peak's position y(∆t) ϕy=0 . This data will be somewhat simpler to interpret below. To get a sense of depth, it is natural again to use two nearby receivers. "Parallax" Ξ can be defined as the ratio of 14 In the geometrical optics approximation, we obtain the full signal simply by summing over images. lateral movement to angular change: Thus, using only (near) right-angle geodesics, we can measure three functions of time: ∆y(∆t), Ξ(∆t), and the intensity I(∆t). The former can be determined numerically from (4.5) by solving for ∆ϕ = 0, after which the other two follow readily. This procedure was used to make the plots in figure 10. These plots again show readily interpretable features. The parallax Ξ(∆t) saturates at late times as the particle reaches the horizon, as just described. The linear rise in y(∆t) quantifies the drift velocity at the horizon. At early times, the intensity rises quadratically: I ∝ (∆t) 2 , indicating scale-invariant dynamics, ie. that the bulk geometry is asymptotically AdS. The intensity later becomes exponential I ∼ e 2πT ∆t , which is the smoking gun of a causal horizon [9]. 15 The fact that I grows with time may be counter-intuitive. It is because excitations, both from x and y, blueshift as they go deeper into the bulk. The locally measured center-of-mass energy at the collision thus increases with time. This is a peculiar feature of the active camera. The same would not happen for the radar camera, where the incoming pulse indeed experiences the expected Doppler redshift as it bounces off the receding target.
The three measured functions in figure 10 essentially give the bulk geometry. To see this explicitly, we make a general ansatz for an asymptotically-AdS metric invariant under x and t translations:

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where A(r), B(r) → 1 and C(r) → 0 at the r = 0 boundary, but are otherwise arbitrary functions. Using eq. (2.45) we can predict the signal in terms of the metric and its null geodesics. Integrating the geodesic equation (this is easy since momenta corresponding to spacetime translations are conserved), we find for small angles The first equation determines the collision depth r * (∆t) as a function of time, so the other three equations essentially define A(r * ), B(r * ), C(r * ). (In spacetime dimension d, the intensity is proportional to a factor of Ξ 3−d , see (2.45).) Taking various derivatives yields the following more useful relations: d∆t .
(4.10) For future reference, we have written the last relation in general spacetime dimension. The proportionality constant is fixed by requiring that r * → ∆t 2 at short times; physically, the constant defines the AdS radius. Once r * (∆t) is found, B(r) can be obtained either from ∆t dr * or dΞ dr * . We see that just by taking derivatives of the numerical data in figure 10, we obtain the metric functions A(r), B(r), and C(r)! For the correlator in (4.4), the resulting functions agree precisely with the Bañados-Teitelboim-Zanelli metric [58,59]: where T = T R +T L 2 and = T R −T L T R +T L . We have tested this numerically but did not attempt to prove it analytically (the equations are very implicit).
What have we learnt? Of course, there is nothing particularly remarkable about recovering the BTZ geometry from CFT correlators: the metric is essentially determined by symmetry (it is locally AdS 3 ), just like the correlator (4.4) we started with. The interesting thing about the recipe in (4.10) is that it applies to any situation with translation invariance, and geometrical optics in the bulk (so that we can unambiguously measure y(∆t) and Ξ(∆t) from the positions of sharp peaks). One could write down any metric of the form (4.8) (not necessarily satisfying any bulk equations of motion), compute the OTOC in this metric, and a colleague could recover one's original metric from the OTOC.
Note that we are not claiming that any non-symmetrical metric can be easily reconstructed from 4-point correlators. The general problem may be somewhat analogous to reconstructing the mass distribution of a galaxy from gravitational lensing data or other kinds of images. Getting the images is just one step.

More on translation-invariant states in higher dimensions, and radar
The simple reconstruction (4.10) applies to translation-invariant and rotationally invariant states in any spacetime dimension. It enables to write the bulk Einstein's equations directly -36 -

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Out[]= Figure 11. Geometry of the holographic radar: a pulse sent from a point z reflects off an object (following the thick blue geodesic) at a point P and is later recorded at y. This is shown here in the geometry surrounding a BTZ black hole (the inner cylinder), while all measurements are done within the finite-temperature system at the outer boundary of the cylinder. The yellow surface displays null geodesics from P that reach the boundary. Any points z and y at the intersection of this surface and the boundary would probe the same bulk point P .
in terms of boundary observables. Working in the plasma's rest frame for simplicity, so that y(∆t) = 0 = C(r), we find Einstein's equations: It is very surprising to find the bulk dynamics expressed by a differential equation on boundary observables. This could happen because the parallax Ξ(∆t) is a quantitatively accurate radial coordinate, not simply a qualitative measure of size or renormalization scale. 16 It is an important open question to interpret eqs. (4.12) from the boundary perspective.
In figure 11 we describe the working of the radar camera in the BTZ black hole geometry, with T L = T R = 1.1/(2π) and x-period 2π, ie. just above the Hawking-Page temperature. After Fourier transforming with respect to δx (which singles out a projectile geodesic (x, p x )), the correlator displays a singularity at the conformally transformed version of (3.23).
The bulk picture is that the forward lightcone from the pulse z intersects the geodesic (x, p x ) at a bulk point P . Holding this data fixed, the reflected pulse can be observed at any point y along the future lightcone of P . Conversely, fixing (x, p x ) and y, we would observe a singularity at any z on the past lightcone of P . The microscopic process is graviton exchange in the bulk as discussed below (3.23). (The point z shouldn't be too close to JHEP03(2023)047 x, to avoid signal loss by Doppler redshift.) Note that the projectile does not need to be recovered after the experiment, its position is not observed beyond the point P .
The same construction works in any spacetime dimension and only requires four-point correlators. Any bulk point P whose past and future lightcone reach the boundary in some open sets can be probed in this way. Fourier transforming with respect to small variations δy (or δz) would reveal additional information such as the parallax described above.

Summary and discussion
In this paper we considered four-point correlation functions as a tool to image point-like objects in the putative bulk dual of a field theory. The correlators are decoded by a standard Fourier transform. The images come out blurred in theories that do not satisfy the large N , large ∆ higher−spin gap criterion, as exemplified in figure 6. When geometrical optics applies in the bulk, an arbitrary bulk point P (causally reachable from the boundary) can be labelled by a triplet (x, θ x , y), where x and y are points in the boundary spacetime and θ x is a timelike tangent vector around x (proportional to the momentum injected near x). The corresponding bulk point lies at the intersection of a null geodesic labelled by (x, p x ) and the lightcone from y, see figure 11. More precisely, a bulk point may be viewed as an equivalence class of such triplets.
The construction is technically similar to the bulk point and light-cone cuts [14,15], with two novel ingredients: Fourier transforms and operator orderings. These tweaks enable to single out a unique bulk geodesic (x, θ x ) using just two points, by Fourier transforming the folded OPE limit O(x+δx)(· · · )O(x) with respect to δx. There are two options for using this in a four-point correlator, as depicted in figure 1: one can make a "radar camera" by using two extra points y, z to send and receive a pulse, or one can make an "active camera" by doing the same for a second geodesic (y, θ y ). In both cases, the signal comes from a singularity or peak caused by a bulk scattering process when geodesics nearly intersect.
The fact that the correlator data accurately specifies a bulk point is vividly exemplified by eq. (4.12): in some situations, the bulk's Einstein's equations is precisely equivalent to a simple differential equation on the four-point correlator data. It will be important to understand the meaning of this equation from the boundary perspective.
The active camera may be of mostly theoretical interest since it requires an out-of-timeorder correlator that may be difficult or impossible to measure experimentally, if one had access to a candidate holographic quantum system. The radar camera, on the other hand, is a perfectly conventional expectation value (described by the usual Schwinger-Keldysh two-fold as opposed to a four-fold).
The cameras do not have good analogs in two-dimensional models of gravity: peaks at short impact parameters are a crucial part of the signal. This requires the bulk to have at least 3 spacetime dimensions.
Much, evidently, remains to be understood. A basic question is whether similar correlators can probe other bulk fields beyond the Einstein frame metric. A natural candidate is to generalize the OPE (2.29) to products which vanish at leading power, such as a field O and the stress tensor in the presence of a -38 -

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background φ Ψ (schematically and omitting indices, we have not tested such formulas): (5.1) Again one would Fourier transform in δx to focus on a single geodesic θ x , and try to single out a point on it using small-impact-parameter singularities caused by graviton exchange with a second geodesic from y. Related Regge OPEs with spin have been discussed in [16,36,46]. This would naturally define (if it works) bulk fields in a boundary-anchored coordinate system (x, p x , y). From this perspective, the extrapolate dictionary φ(x, r) ∼ r ∆ O O(x) as r → 0 would be a simple consequence of the usual local OPE since r → 0 can be approached by taking all four points to be nearby. This type of description is highly redundant: a bulk point P can be labelled by many triples (x, θ x , y). It is unclear whether this is related to the highly entangled, quantum error-correcting nature of the vacuum in holographic theories [62]. From that perspective, it is also generally believed that observations in a boundary region R can probe the full entanglement wedge of that region [63]. It is an important question whether, with suitable add-ons, a camera could image this generally larger region.
We assumed that 1/N and 1/λ corrections at small bulk impact parameters are controlled by flat space physics, and effectively treated them as a nuisance. It could be interesting to understand them better.
The camera signals have an interesting interpretation in large-N algebras. Consider a theory with central charge ∼ N 2 c where the OPE can be expanded in single-traces, double-traces, etc. In a classical state |Ψ with large expectation values, O i = O i Ψ + δO i , such that all multi-traces need to be resummed, one finds an OPE for fluctuations δO i that becomes effectively state-dependent. It seems reasonable to expand the commutator between two operators in this regime as integrals over causal diamonds The identity coefficients A ij define a generalized free field algebra which nicely encodes some bulk properties [64,65]. Under suitable conditions, where one can ignore singleand higher-traces, the out-of-time-order correlator (1.4) is essentially the double-trace coefficient C ijkl (x, y; z, z ) in the limit z → x and z → y where the corners of the diamond are approached, up to a convolution against two-point functions. What we saw, thus, is that the double-trace coefficients faithfully images bulk scattering processes at the edge of the bulk causal diamond. This is reminiscent of [66], although we didn't use modular Hamiltonians. The support in the second line of (5.2) has an interesting implication: it makes the Fourier transform analytic in upper-half-plane energy planes for p µ x and p µ y . This is crossing symmetry of the bulk scattering amplitude! Changing the sign of p relates bulk particles to bulk antiparticles. This property is somewhat mysterious from the correlator perspective. Clarifying it could help adapt recent bootstrap constraints in vacuum AdS, which use -39 -

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dispersive sum rules to control bulk equations of motion and bound their deviation from Einstein's theory by powers of 1/∆ higher−spin gap [47,67,68]. Given Einstein's equation around vacuum AdS, a fundamental question in understanding the bulk is why the same local equations of motion should be satisfied around any point in a nontrivial bulk geometry, ie. the equivalence principle. Morally, from the bulk perspective, this is related by consistency of interactions involving massless (at least, compared with the AdS radius) spin-two particle [69]. This suggests that bootstrapping stress tensor correlators could be particularly fruitful.
Four operators is just the right number for bootstrap. It is clearly of interest to nonperturbatively bootstrap camera correlators in non-vacuum states. Note that a bulk metric does not always exist, but correlators always do. Potentially, this could extend or replace Einstein's equations in situations where they do not apply.
These questions have analogs in asymptotically flat spacetimes, where a classical bulk geometry may also be probed using camera-like four-point scattering amplitudes in its background.
Shockwaves at future null infinity (I + ) also deserve further exploration. Matrix elements of the bulk light-ray operators (3.16) at infinity are related to (energy) J−1 flux measured in a region of small angular size θ, around the direction n on the celestial sphere [27] L ∞;n,ν ∝ lim Experimentally, if one could add up the signals with different momenta ν and angles n, in such a way as to compute the integral (3.17), one would be measuring the bulk average null energy (ANEC). In conformal collider physics this could reveal that what appears from energy flux measurements to be a spherical shell in the boundary [3], is in fact a point particle in the bulk. QCD might not be holographic in the sense required here (at least not at high energies), but it could still be interesting to study (energy) J−1 fluxes with complex J in a collider context, perhaps similarly to the nucleon-energy correlators in [70].
with Ω d = 2π . The measure is (note that ρ(ν) here differs from ρ(ν) in [47] by 2 d−2 Ω d−1 ): Integrate over all angles n gives, for x, y two unit vectors, the harmonic functions For integer J, this function is proportional to a spherical harmonic (Gegenbauer polynomial), whence our notation. For more background and spinning generalizations, see [71]. Since the Laplacian on H d−1 is multiplication by minus ν 2 + ( d−2 2 ) 2 , harmonic representations of Green's functions are readily obtained. For example, Poisson's equation is inverted by the following transform The constant is a dimension-shifted version of C ∆ in (2.27). This is used in the main text to write the graviton propagator on the transverse hyperboloid as Π d−1 (η). Finally, the harmonic representation plays nicely with other Lorentz-covariant operations such as ordinary Fourier transforms: the d-dimensional Fourier transform of an eigenfunction is an eigenfunction. Below (3.11) we used the following identity, involving the wavefunction (2.35) in the plane-wave limit ψ p,L=∞ , with the integral running over all d-vector x (which can be either spacelike or timelike). It can be derived by combining the denominators using Schwinger parameters and using (A.3): , (A. 16) with γ in (3.10). As in the main text, we have assumed that p is positive-frequency timelike and defined (−n·x − ) q by analytic continuation from the positive-timelike region, where the parenthesis is positive. Recall that hatted vectors are normalized:p = p µ / −p 2 .
To account for the finite width of wavepackets, we convolve (A.16) with a Gaussian of width ∆θ = 1/(|p|L) in the orientation of p, as mentioned below (2.36). We are only interested in situations where this width is small. The convolution then has essentially no effect unless is large ν ∼ 1/∆θ. In the rest frame of p, say, one can then approximate

C.3 Pomeron exchange in the Wilson-Fisher theory
We discuss here the coefficient functions of the Wilson-Fisher theory in d = 4 − ε dimensions near the intercept. We start with formulas from [74] for the leading trajectory for generic spin. The anomalous dimension takes its most concise form in terms of a variable that we write here as h = ∆+J 2 = 1 + J + O(ε): with H(x) = ψ(1 + x) − ψ(1) the (analytic continuation) of harmonic numbers. Note that even though ∆ appears here on both sides of this equation, the equation can be solved perturbatively in ε to recover the formulas from [74]. The leading-trajectory residue takes a similarly concise form in terms of h, c t (∆, J) = c u (∆, J) = Γ(h) 2 Γ(2h−1) +O(ε 4 ) . (C.12) More orders are available, but this will suffice for our purposes. The above expansions are valid for generic h but they break down when h → 1 (where ∆ → 2 + O(ε) and J → O(ε)). As explained in [27], this singularity is caused by mixing between the trajectory (C.11) and its shadow (defined by the same formula with ∆ → d−∆). We can get a smooth characteristic equation χ = 0 by multiplying (C.11) with its shadow and expanding in ε (see (2.27)