Bulk reconstruction and the Hartle-Hawking wavefunction
BBulk reconstruction and the Hartle-Hawking wavefunction
Daniel Louis Jafferis
Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, USA
Abstract
In this work, a relation is found between state dependence of bulk observables inthe gauge/gravity correspondence and nonperturbative diffeomorphism invariance.Certain bulk constraints, such as the black hole information paradox, appear toobstruct the existence of a linear map from bulk operators to exact CFT operatorsthat is valid over the entire expected range of validity of the bulk effective theory.By formulating the bulk gravitational physics in the Hartle-Hawking frameworkto address these nonperturbative IR questions, I will demonstrate, in the contextof eternal AdS-Schwarzschild, that the problematic operators fail to satisfy theHamiltonian constraints nonperturbatively. In this way, the map between bulkeffective theory Hartle-Hawking wavefunctions and exact CFT states can be linearon the full Hilbert space. a r X i v : . [ h e p - t h ] M a r ontents The gauge/gravity correspondence provides the best understood example of a completetheory of quantum gravity, that is, a precisely defined quantum system which is approx-imated by general relativity in the appropriate limit [1–3]. An aspect of this is thatgravity is a non-renormalizable effective theory, whose ultraviolet completion is providedby string theory. However for the purposes of this note, it will be surprising infrared,albeit nonperturbative, constraints that will play the central role.In recent years, there has been considerable progress in elucidating the constructionof bulk gravity observables in the dual CFT language. Some surprising phenomena havebeen found, particularly in situations involving causal horizons in the bulk. In particular,contradictions appear to arise between the existence of linear operators associated to bulkobservables behind horizons and the semiclassical analysis around the expected smoothspacetimes.A paradigmatic example is the information paradox of black hole evaporation [4]. Inthe asymptotically AdS context, the evolution is obviously unitary by the duality with aunitary conformal field theory. Thus the paradox is recast as a problem in the existenceof CFT operators that obey the expected bulk evolution equations and commutationrelations in the weakly curved region behind the horizon that is relevant to the Hawkingprocess.These arguments were sharpened in the work of [5–8], which demonstrated that ex-ponentially small corrections in the Planck expansion cannot resolve the paradox. Inparticular, there cannot exist CFT operators that approximate the naively expected be-havior of local bulk fields behind the horizon to within exponential accuracy in typicalstates of the black hole. It was suggested in [6] that this implies a sharp breakdown ofthe bulk effective theory even in certain regimes with low curvatures - that most blackhole microstates have firewalls near the horizon.The fascinating work of Papadodimas and Raju [10–12], and related work of [13–16] and [17], showed how to explicitly construct bulk operators behind horizons in 1 /N perturbation theory around a given approximately thermal pure state, using an analogof the KMS relation. The striking feature of their construction is that it depends onthe background state, so that it is impossible to obtain a linear operator associated tothe bulk observable that is valid on all microstates. This can then evade the AMPS(S)paradoxes. In the present work, I will focus for simplicity on the situation of a two sidedblack hole, where similar issues also arise [11, 18].Given that these interesting puzzles only appear when considering semiclassically dis-tinct configurations, rather than in perturbation theory around a fixed background, it isappropriate to use a non-perturbative framework to describe the effective bulk physics.To that end, in this work, I will investigate the relation between the bulk Hartle-Hawkingwavefunction [9] and the dual QFT states. The asymptotically AdS arena is furthermorea promising one to make the Hartle-Hawking formalism more precise.The main new feature that can be seen in such an analysis is that the observableswhose existence as linear operators on the full Hilbert space would lead to inconsistenciesare already ill-defined as operators on the wavefunction of the bulk effective theory. Inparticular, they are not non-perturbatively invariant under spacetime diffeomorphisms. Iwill show that they fail to obey certain non-perturbative consistency conditions analogousto the Wheeler-DeWitt equation [21] that implements the Hamiltonian constraints inperturbation theory.One may define more refined, gauge fixed versions of these bulk operators that arenon-perturbatively diffeomorphism invariant. These involve additional choices, and onlyagree with the naive observables on a small subspace of the full bulk effective theoryHilbert space. For these operators, it appears that no paradoxes arise, and they shouldbe given by linear operators on the exact Hilbert space.This paper is organized as follows. In section two, I sketch the basic logic of this work,and review the apparent obstruction to a linear map between certain bulk observablesand CFT operators, in the context of eternal AdS-Schwarzschild. In section three, I argueusing the Hartle-Hawking formalism that such observables fail to be nonperturbativelydiffeomorphism invariant. In section four, I describe appropriately gauge fixed versionsof bulk operators defined relationally to the boundary. Finally, in section five, I discusspossible implications for the information paradox and speculate about the appropriateframework for describing bulk measurements. The relation between the Hilbert spaces of UV theories and their IR effective theories is,in general, one of projection. Thus one expects that the states of the IR Hilbert space canbe mapped to states of the UV theory. They only span a subspace, since short distance2odes remain in an adiabatic vacuum.Consider the Wilsonian renormalization group flow in a quantum field theory on a d dimensional spatial lattice, obtained by passing to an effective description with twice thelattice spacing. The Hilbert space of a 2 ⊗ d block must be replaced by the Hilbert spaceof a single lattice site, of dimension M [22, 23]. This is done by a projection map, ⊗ i ,...i d =1 H i ,...i d → H block , from the M d dimensional space to an M dimensional space. The cokernel of this mapshould be interpreted as states in the microscopic theory in which no short distance modesare excited. Clearly information is lost during each step of the rg block transformation,and the projection must be chosen judiciously to avoid projecting out physically importantstates.The Hilbert space of the long distance effective theory, H EFT , is the subspace of the fullHilbert space of the microscopic theory in which only long wavelength degrees of freedomare excited. This should be sharply contrasted with the subspace of states with totalenergy below some cutoff (in an IR finite system with a discrete spectrum). The latterwould contain states with small numbers of ultra high energy quanta, which are not partof the effective theory, and would fail to contain coherent states of long wavelength modeswith large total energy. For the present purpose of examining nonperturbative constraintsin the bulk, it is important that H EFT does contain such coherent states, the quantumanalogs of low curvature solutions of the non-linear classical equations of motion.In standard constructions, there is a fixed projection in the above definition of H EFT ,which, for example, might select states of lowest energy density. An alternative possibilityis the use of state dependent projections, in other words, a non-linear map. For example,in the MERA block transformation, the Hilbert space is projected onto the subspacewith maximal entanglement with the rest of the system [24]. This type of rg flow forstates has proven to have significant computational utility in constructing ground statesof certain many body systems. However the physical meaning of the state dependence ofthe resulting effective theory observables is unclear. In particular, if it were significant insome situations, it would seem to violate the linearity of quantum mechanics, which wecertainly do not expect in long distance effective observables in condensed matter systems.It is therefore important in the gravitational context to ask what a fundamental statedependence of bulk observables would imply for the Hilbert space of the effective bulktheory. In particular, if the theory in AdS is dual to a large N conformal field theory,what is the relation between the exact CFT Hilbert space and the bulk effective theoryHilbert space?It is essential to describe the bulk theory in a non-perturbative framework, since theconstraints and paradoxes under discussion that motivate state dependent constructions3nly appear when comparing states that differ non-perturbatively. Perturbative treat-ments only refer to the behavior of observables in the subspace that can be obtainedby acting with parametrically less than N single trace operators on a given state with aknown gravity dual. The CFT expressions for bulks fields around the AdS vacuum [25,26]and in other situations where they can be constructed using the bulk evolution equationsfall into this category. Even if no causal reconstruction is possible, all constructions clearlylead to linear operators [27] in such code subspaces. The question at hand is how these canbe patched together over the full space of configurations described by the bulk effectivetheory.State dependence of bulk observables in the emergent bulk gravity theory would seemto imply that there does not exist a linear map between the full bulk low energy effectiveHilbert space and the exact Hilbert space. Such maps would then only exist for subspacesof the bulk Hilbert space defined perturbatively around a given state. A scenario of thistype would be that a MERA construction of the CFT states determines the bulk state.Since the projection in MERA is done on maximally entangled subspaces, the result wouldbe a non-linear map on the Hilbert space.I will argue instead that the observables in question fail to be non-perturbativelydiffeomorphism invariant. In other words, such observables do not exist as linear operatorseven in the bulk gravity effective theory. Thus there is no contradiction with the existenceof a linear map, H bulk (cid:44) → H CFT , between all physical, gauge invariant states of the lowenergy bulk theory and states of the CFT. The existence of such a linear map is completelynatural, given the AdS/CFT duality of the quantum systems. Furthermore, it appearsthat there is no need to restrict H bulk to be smaller than the expected domain of validityof the bulk effective theory.These situations must be examined in a non-perturbative long wavelength bulk effec-tive description that appropriately imposes the constraints of temporal diffeomorphisminvariance, such as the Wheeler-DeWitt wavefunction. That formalism can be mademore precise in the asymptotically AdS context, as I will discuss in section 3. It is non-perturbative because it allows discussion of configurations that are significantly differentthan the vacuum (for example, states related to solutions of the classical non-linear equa-tions). It is a low energy effective formalism in the bulk because short distance modes arenot excited.The Hartle-Hawking wavefunction is defined by a euclidean path integral up to a slicewith on which the spatial metric and other bulks fields, collectively denoted by h , arefixed. This data should be understood as defined up to spatial diffeomorphisms. Thetemporal components of the metric at the slice, the lapse and the shift, are integratedover. This is because they appear in the action without any time derivatives. Given astate | ψ (cid:105) , this path integral defines the wavefunction, Ψ( h ), giving a linear functional onstates of the bulk Hilbert space, h : H bulk → C .4n important feature of the AdS arena is that adjusting asymptotic sources in theeuclidean path integral produces many states, as opposed to the situation in cosmolog-ical global de Sitter spacetime, where only the Hartle-Hawking no boundary state [9] isnaturally defined by a euclidean path integral. The states obtained by euclidean pathintegrals with sources for single trace operators generate the bulk long distance effectiveHilbert space, H bulk .By the standard relationship between the path integral and the Hilbert space in quan-tum mechanics, the inner product between two states | ψ (cid:105) and | ψ (cid:48) (cid:105) is obtained by perform-ing the euclidean gravity path integral with the asymptotic sources that defined the twostates. Similarly, the overlap (cid:104) h | h (cid:48) (cid:105) between two of the Hartle-Hawking kets is given bythe euclidean path integral with intrinsic metric h and h (cid:48) on two spatial slice boundaries.In the asymptotically AdS context, the two noncompact slices must asymptotically joinat the AdS boundary.A crucial feature is that because no gauge for temporal diffeomorphisms has beenfixed, the overlaps between the kets, M hh (cid:48) = (cid:104) h | h (cid:48) (cid:105) , are generally nonzero, and the ketsdo not form a basis: they are overcomplete. This simply reflects the fact that there aremany spatial slices of the same spacetime geometry.The ket states | h (cid:105) are thus not linearly independent. For this reason, the matrixelements of any gauge invariant operator A hh (cid:48) = (cid:104) h |O| h (cid:48) (cid:105) must obey certain constraints.In the perturbative analysis, for h (cid:48) = h + δh , Hartle and Hawking showed that theseconditions imply that the A hh (cid:48) satisfy the Wheeler-DeWitt equation, in other words, A commutes with the Hamiltonian constraints.The euclidean path integral additionally allows one in principle to determine the over-lap between arbitrarily different kets, which would be challenging to understand by inte-grating the Wheeler-DeWitt equation. As I will explain in more detail in section 3, theresulting conditions are nonperturbative analogs of the Hamiltonian constraints. Thesewill be the key to the present analysis.One way of understanding the state dependence required in the constructions of bulkoperators is that naively distinct bulk states are not linearly independent. In other words,the code subspaces based on different background states have unexpected overlaps. Forexample, using the duality with the exact CFT, [11, 12] demonstrated that very long timeevolution of the two sided AdS black hole results in states whose overlap does not decayto 0, but rather remains finite, albeit exponentially small in N .The main point of this note is that this overlap of seemingly distinct states has thesame origin as the lack of independence of the Hartle-Hawking kets. For this reason it isalready a feature of the bulk effective theory Hilbert space, and the associated constraintson operators are imposing non-perturbative diffeomorphism invariance.To illustrate the idea, consider the superselection sector of quantum gravity with twoasymptotically AdS d +1 boundaries. The spacetime geometry may be connected or not,5epending on the state. However, the full quantum system consists of two decoupledcopies of a large N CFT. The CFTs are non interacting because the Hamiltonian densityin gravity is a total derivative, hence the total Hamiltonian consists of two decoupledboundary terms, H = H L + H R , even when the spacetime is connected [29].A particularly straightforward example of a non-perturbative IR paradox appears inthis context, as pointed out by Marolf and Wall [18]. Consider a right framed bulkobservable, such as a field operator at a position defined by a fixed proper distance,relative to empty AdS, along a geodesic that extends from a point on the right boundaryat a given direction.In any factorized state, this observable should be represented by a purely right CFToperator. After all in that case the left CFT is neither interacting nor entangled with theright one, so it seems that from the right point of view it cannot have any effect.On the other hand, the thermofield double state is dual to the eternal AdS-Schwarzschildgeometry [29], described by the metric ds = − f ( r ) dt + dr f ( r ) + r d Ω d − , where f ( r ) = r + 1 − G N M Γ( d/ d − π ( d − / r d − . The most salient feature of that spacetime is the Einstein-Rosen bridge connecting thetwo sides. Thus the right framed bulk observable can enter the causal domain of the leftboundary, and the observable cannot commute with all left operators in that state.This contradicts linearity of the bulk observable, because the thermofield double stateis a linear combination of factorized states, as we know from the microscopic CFT de-scription, | tfd (cid:105) = 1 (cid:112) Z thrm ( β ) (cid:88) E e − βE/ | E (cid:105) L | E (cid:105) R . In [18], it is suggested that the AdS/CFT duality is incomplete in this situation and mustbe supplemented by a choice of connected or disconnected dictionary. In this work, I willargue instead that the bulk observables do not define non-perturbatively diffeomorphisminvariant operators on the bulk effective Hilbert space. One must make additional choicesin specifying the observable to make it gauge invariant, which determine whether it actsnon-trivially on the left CFT. These play the role of the choice of dictionary, but we willsee that they are already required to have a well-defined action on the Hartle-Hawkingstates in the bulk effective theory.It is important to note that small corrections to the bulk gravity behavior of theseoperators cannot resolve this paradox. In particular, using arguments similar to thosein [5–7], the works [11, 12] demonstrated that any linear operator must have order 1disagreement with the expected properties of a right framed bulk field behind the horizonon most states that are dual to time shifted versions of the eternal black hole.The evolution of the thermofield double state by the left Hamiltonian is simply the6pplication of a large diffeomorphism that acts nontrivially at the left boundary. Thismakes it clear that such states, | ψ T (cid:105) = e iH L T | tfd (cid:105) , should remain in the domain of validityof the bulk effective theory, even for long time scales T = O ( e S ). This time shifted stateis geometrically the same spacetime manifold as the eternal black hole, merely with adifferent origin of time on the left boundary. Thus any observables that are truly definedonly relative to the right boundary should be unaffected, and the region near the horizonhas low curvature for all t .A very simple way to see that exponentially small corrections to the observables can’thelp is to consider the bulk observable, C , that counts the number of connected com-ponents of space. Suppose it has an expectation value of 1 + O ( e − N ) on the states | ψ T (cid:105) in which the two boundaries are connected by an Einstein-Rosen bridge, and anexpectation value of 2 + O ( e − N ) on factorized states, in which the spacetime consistsof disconnected left and right pieces. Using the microscopic CFT description of | ψ T (cid:105) as √ Z β (cid:80) e ( − β + iT ) E | E (cid:105) L | E (cid:105) R , one obtains a contradiction, because averaging the resultfor | ψ T (cid:105) over a sufficiently long interval of the parameter T approximately projects ontoindividual energy eigenstates.In more detail, if (cid:104) E, E | C | E (cid:48) , E (cid:48) (cid:105) = 2 δ E − E (cid:48) + (cid:15) ( E, E (cid:48) ), where (cid:15) is exponentially smallin the black hole entropy, then (cid:104) ψ T | C | ψ T (cid:105) = 2 + 1 Z thrm ( β ) (cid:88) E,E (cid:48) e − β ( E + E (cid:48) )+ iT ( E (cid:48) − E ) (cid:15) ( E, E (cid:48) ) . It is impossible for the phases in the second term to add coherently for all T over a intervalof order e S in such a way that the double sum over the exponentially large number ofstates of the exponentially small quantities (cid:15) ( E, E (cid:48) ) /Z thrm ( β ) is of order 1. Therefore, formost T , the result will be the average value of 2 up to exponentially small corrections, incontradiction with its expected value of 1 + O ( e − S ).A sketch of the argument in [11, 12] for the more interesting operators that measure afield at bulk points defined relationally to the right boundary is as follows. Consider suchan observable, for example the particle number operator, N , constructed out of the rightframed field operator, which has an O (1 /S ) expectation value in the thermofield doublestate. It will then also have an O (1 /S ) expectation value in the time shifted states. Thenone can average over exponentially long times to obtain12 T (cid:90) T − T (cid:104) ψ t | N | ψ t (cid:105) dt = (cid:88) E e − βE Z ( β ) (cid:104) E, E | N | E, E (cid:105) + (cid:88) E (cid:54) = E (cid:48) e − β ( E + E (cid:48) ) / Z ( β ) sin(( E (cid:48) − E ) T )( E (cid:48) − E ) T (cid:104) E, E | N | E (cid:48) , E (cid:48) (cid:105) . The second term becomes small when T is large enough, which proves that the firstterm is also of order 1 /S . This is true for all temperatures β . Performing a Legendretransform, one can approximately project on to factorized states. It is then easy to show7hat this implies that the right framed operator must have a small expectation value inall factorized states of the same energy. This is inconsistent with the bulk predictions.In the next sections, I will analyze this situation in the Hartle-Hawking formalism. Itis applicable here because all of the time shifted states are well-described by wavefunc-tionals in the low curvature regime. In particular, one can find nice slices even in theseexponentially time shifted states which have small intrinsic and extrinsic curvatures, ifthe black hole is large in Planck units.Before turning to the more detailed analysis of these observables in the bulk Hartle-Hawking framework, it is interesting, although simple, to see that the nonzero overlapbetween a factorized like such as the vacuum | (cid:105) L | (cid:105) R and the thermofield double stateis already calculable in the gravity path integral. It is precisely the lack of orthogonalitybetween these states with connected versus disconnected semiclassical descriptions whichled to the paradox.The euclidean gravity saddle that contributes to (cid:104) , | tfd (cid:105) = Z S d (cid:113) Z S d (cid:112) Z S d − × S = 1 √ Z therm , is just euclidean AdS d +1 . Note that in the CFT language, the vacuum caps that producethe state | , (cid:105) and the cylinder that produces | tfd (cid:105) are conformally flat, so the overlappartition function is not only topologically but in fact conformally equivalent to S d . Thedenominator is just the usual normalization of factor for the states produced by thesepath integrals.Figure 1: The factorized vacuum state is produced by a pair of hemisphere asymptoticboundaries, while the black hole state is produced by an annulus asymptotic boundary.The metric is not the induced one associated to the figure. The left and right figures showthe two gluings described in the text, with a disconnected versus a connected slice.8he inner product between the vacuum and the thermofield double state in the grav-itational path integral is shown in on the left in Figure 1. It can be interpreted as anamplitude that the thermofield double state is a disconnected geometry. That this isnon-vanishing is not surprising, given that the disconnected configuration can even bedominant over the black hole (below the Hawking-Page phase transition). On the otherhand, one can equally well glue together the bulk configurations shown on the right inFigure 1. Here it appears to be an amplitude that the vacuum is a connected spacetime.A important aspect is that it would be completely wrong to add together these “two”possibilities. They are just different depictions of the same saddle, as can be seen in Figure2, which is an alternative, but topologically equivalent, depiction of the two slicings ofFigure 1. This is precisely because C , the number of connected components of space, isnot a nonperturbatively diffeomorphism invariant operator. It depends on the choice ofslice, and so fails to obey the Hamiltonian constraints non-perturbatively.As explained above, a paradox very similar to that described in [18] appeared for theputative observable C . By the AdS/CFT duality, it should be 2 on factorized states, and1 on the thermally entangled states, which contradicts linearity. What we see now is that C is simply not gauge invariant, so this indicates no obstruction to a linear map between H bulk and H CFT .Figure 2: Here it is clear that these are simply two different bulk slices of the sameconfiguration, described by the euclidean AdS saddle. The left shows the disconnectedslice and the right shows the connected slice. The slices are at t = 0. The black hole stateis produced by the annulus asymptotic boundary in the middle of the spheres, while thevacuum is produced by the cap boundaries at the top and bottom of the spheres.The Hartle-Hawking kets with different topologies are not independent, and no com-plete linear operator C can act on them with different eigenvalues. The nonzero overlapbetween these kets implies topology ambiguity, not dynamical change of the topology. Astate that “definitely” has a given topology still has a nonzero probability to have a differ-ent one. This is particularly clear in the AdS context, since the global Hamiltonian makes9o appearance in this discussion. If a related non-perturbatively gauge fixed operator wasdefined, then its dynamical evolution could be discussed.This fact is an important point of consistency in the proposal that entanglementencodes spatial connection [30, 31] by nontraversable wormholes [32], since entanglementis not measurable by any operator. The nonexistence of a linear operator that measuresthe spatial topology has been pointed out by [33–35] and in related work of [36], based onthe duality with the CFT. This is true even without the presence of horizons, as shownthe examples of [34, 35]. Now we see that this is already a feature of the bulk effectivetheory.It is more subtle to see that the geodesic defined operators are not gauge invariant be-yond perturbation theory. As explained in section 4, it requires extra data to define them,so that the geodesic dressing lies in a space-like slice. This is necessary in order to havea straightforward action on the bulk Hilbert space. The end result is that appropriatelydiffeomorphism invariant versions of these observables do not lead to paradoxes. Consider the collection of states that can be produced by performing a CFT path integralwith sources for single trace operators. This class encompasses all states that are accessiblein the bulk effective theory. Given any such state, we want to find its description as awavefunctional Ψ( h ). Here h represents the data of the bulk metric and fields on a spatialslice. They are required to obey AdS asymptotic conditions that are described below,and we should consider h to label equivalence classes with respect to the redundanciesgenerated by spatial gauge transformations obeying the appropriate boundary fall offconditions.The Hartle-Hawking prescription [9] is to perform the euclidean path integral withsources on the AdS boundary, up to a slice with data h ,Ψ( h ) = (cid:90) g | ∂M = h, g | ∞ = J Dg e − S ( g ) , (3.1)up to a normalization constant, where the time cut boundary of space is ∂M , and theAdS asymptotics, g | ∞ , are schematically given by the sources J that define the state. Inother words, one only integrates over metrics and bulk fields that obey the appropriateFefferman-Graham fall-off conditions. Writing the metric near the slice as ds = ( N − N i N i ) dt + 2 N i dx i dt + h ij dx i dx j , (3.2)where x i are coordinates on the slice, one integrates over the lapse, N , and shift, N i , atthe slice, keeping only h ij fixed. One similarly integrates over all over bulk fields, with10dS asymptotics given by sources and fixed on the slice as ϕ | ∂M = ϕ ( x ) to obtain thewavefunctional Ψ( h, ϕ ).This formalism is general, but for concreteness, one may consider gravity describedby the Einstein-Hilbert action, S = (cid:82) M √ g ( R − (cid:82) ∂M √ hK , where in AdS Λ < K ij = N (cid:16) − ∂h ij ∂t + D ( i N j ) (cid:17) appears in the boundaryterm. More precisely, the AdS asymptotic boundary should be cut off, and the actionobtained as a limit after subtraction of the appropriate counter terms, as discussed in thiscontext of producing Lorentzian states from euclidean sources on a cap in [19, 20]In AdS, the spacetime metric is required to have the asymptotic form ds = L z dz + η µν dy µ dy ν z + . . . , where y µ are the boundary spacetime coordinates, L is the AdS radius, and the subleadingterms vanish in z → h ij dx i dx j = L z dz + . . . . Moreover, the z shift mustobey N z → z → h . The asymptotic vanishing of N z implies thatthe ADM Hamiltonian is a nontrivial boundary operator, rather than a constraint.The path integral above should be understood as being evaluated in bulk perturbationtheory around its saddles. It is still non-perturbative in that one may consider states and h that differ classically, in other words, at leading order. One should only allow the metrics, h , on the slice which have no Planck scale features. More precisely, one must perform thepath integral in a bulk effective theory with cutoff Λ (cid:28) M Pl , and the slice metric mustlive in the cutoff configuration space, with no wavelengths smaller than Λ − . If there isno bulk saddle with small curvatures that contributes, the result will be close to 0. SomeCFT states may have small amplitude on all h , since they contain Planckian or stringscale objects in the bulk. As discussed above, contributions from subleading euclideansaddles will lead to exponentially small corrections to the Hartle-Hawking kets, which willnot significantly affect the conditions of non-perturbative diffeomorphism invariance.The bulk effective theory Hilbert space is spanned by states that can be producedby performing such euclidean path integrals with sources, as can be seen by analyticcontinuation of Lorentzian bulk configurations (in general the sources will be complex).The Hartle-Hawking path integral then defines a map from H bulk −→ C . Using theHilbert space inner product defined on the states | ψ (cid:105) ∈ H bulk , one identifies | h (cid:105) with astate (strictly speaking, a limit of states since these may have divergent normalizations).As usual, the relation between the exact Hilbert space and a low energy effective Hilbertspace is by projection, H CF T → H bulk . Thus we can also think of | h (cid:105) as living in H CFT ,by applying the adjoint of the projection map.11his set of kets is overcomplete, and the matrix of their inner products, M ( h, h (cid:48) ) = (cid:104) h | h (cid:48) (cid:105) , given by the euclidean path integral between slices with intrinsic data h and h (cid:48) , aregenerally nonzero. For this reason is not entirely trivial to determine the inner producton the wavefunctionals Ψ( h ) = (cid:104) h | ψ (cid:105) , in other words to find a K hh (cid:48) such that (cid:104) ψ | ψ (cid:105) = (cid:82) dh dh (cid:48) Ψ ( h ) ∗ K hh (cid:48) Ψ ( h (cid:48) ). The kernel K would simply be the inverse of M if the set waslinear independent. Instead, there are many matrices K that satisfy M KM = M .One such kernel is a differential operator in the space of h , representing the innerproduct as (cid:104) ψ | ψ (cid:105) = (cid:90) (cid:89) x dh ij ( x ) h ik h jl + h il h jk − h ij h kl √ h (cid:34) Ψ ( h ) ∗ (cid:32) ←− δδh kl − −→ δδh kl (cid:33) Ψ ( h ) (cid:35) , where the product is over all points in the spatial slice [21]. This results in an integrationover spatial metrics that excludes the local Weyl factor, which is a timelike direction inthe superspace of metrics. In this work, I will not need to use the inner product on thewavefunctionals.An important caveat to this path integral based approach to the bulk effective theoryis that the full rules for summing over different spacetime topologies in the gravity par-tition function are not known. For example, in a calculation in AdS d +1 , with asymptoticboundary S d , one might including a sum over topologies with extra handles. Cuttingthese with a time slice could include a disconnected compact spatial region.One of the puzzles is that in the system with two asymptotic boundaries, the CFTimplies that correlation functions in the vacuum, (cid:104) , |O L O R | , (cid:105) , factorize between theleft and right. However, if connected spacetime saddles contributed, this factorizationwould appear mysterious on the gravity side. This issue arose in [28] in the attemptto define the partition function of pure gravity in three dimensions and find its CFTdual. Note that in the example of the vacuum correlators for a 2d CFT on a pair ofcircles, there is no such saddle, since no smooth hyperbolic 3-manifold may have twodisconnected S boundaries. However, as discussed in [28], such euclidean saddles doexist with disconnected higher genus boundaries. In that work, it was proposed that thegravity path integral simply does not include them.In spite of this gap in understanding, the discussion of the present paper is unaffected.The reason is that the issue at hand is about the different topologies of slices that cuta given spacetime saddle, and their ket overlaps. Thus the question of whether otherspacetime saddles should be summed over is not important for the characterization ofnon-perturbative diffeomorphism constraints.The Wheeler-DeWitt equation (cid:18) − h − / ( h ik h jl + h il h jk − h ij h kl δ δh ij δh kl − R ( h ) h / + 2Λ h / (cid:19) Ψ( h ) = 0 , R is the intrinsic curvature of the slice can be derived [9] from the lack of inde-pendence of the kets | h (cid:105) , which implies that Ψ( h ) = (cid:104) h | ψ (cid:105) is a redundant data. Thedifferential equation results from expansion of the kets under small variation, | h + δh (cid:105) .There are also non-perturbative constraints that would be hard to see by integratingthe Wheeler-DeWitt equation. In particular, there is a conceptually identical lack ofindependence between the data with different topologies of the spatial metric, h . TheHartle-Hawking procedure defines wavefunctions Ψ( h ), Ψ( h , h ), ... which each possiblespatial topology, including the number of connected components. However the Hilbertspace does not consist of separate factors, since the kets are not linearly independent.Unlike in gauge theory, in gravity it is impossible to fix the gauge even in perturbationtheory by imposing a condition on the spatial h . This is because the Wheeler-DeWittequation is second order in δ/δh . Furthermore, standard gauges do not actually fix thegauge nonperturbatively.In particular, the diffeomorphism group for different topologies is different. In the twosided system, the difference in the group of diffeomorphisms is associated to the existenceof the gravitational analog of the Wilson line operators discussed in [37, 38].Consider the example of spacetimes with a pair of asymptotically AdS boundaries.The AdS/CFT duality implies that the kets with disconnected topologies | h , h (cid:105) in factspan the entire Hilbert space. This is hard to see explicitly in the bulk effective theoryanalysis, since obtaining the formula for connected kets in terms of disconnected ketsinvolves partially diagonalizing the matrix of overlaps, M hh (cid:48) = (cid:104) h | h (cid:48) (cid:105) . The result willdepend on the details of the cutoff scheme Λ.For a different class of supersymmetric horizonless geometries [39] in global AdS , itis possible to see the linear relations explicitly, using the explicit description of the dualCFT microstates [34, 35]. Without this exact dictionary protected by supersymmetry, thebulk effective theory analysis would in general lead to a linear relation that was cutoffdependent. Nevertheless, the matrix, M hh (cid:48) , of kets overlaps is a UV safe quantity.The states | top , h (cid:105) are overcomplete (here h represents a metric on the topologicalspace labelled by the first argument), so it is mathematically inconsistent to define alinear operator by independently specifying its action on each of these states. But given alinear operator O , one can compute its matrix elements A ( h , h ) = (cid:104) top1 , h |O| top2 , h (cid:105) .Then these must obey some relations. By considering small changes in h with a fixedtopology, one can show that A must commute with the Hamiltonian constraints. Similarly,by considering different topologies, one can show that A ( h , h ) obeys further conditions.These are nonperturbative constraint equations. A collection of such matrix elements thatdon’t satisfy these constraints do not give a mathematically consistent linear operator; itis not gauge invariant under temporal diffeomorphisms.The AdS/CFT duality implies that disconnected topology states span the Hilbertspace of the two sided AdS system. Therefore defining the action of an operator O on13hose states determines its action on kets with other topologies. It will be perfectly welldefined on the thermofield double state.What is disallowed is to say that one defines an operator like C , the number of con-nected components of space, which obeys C | h , h (cid:105) = 2 | h , h (cid:105) , C | h (cid:105) = 1 | h (cid:105) . This isjust as bad as writing a set of matrix elements that fail to obey the Hamiltonian con-straints. Therefore there is no candidate operator in the bulk effective theory that countsthe number of disconnected components of the spacetime.The euclidean gravity path integral implies that naive basis states associated to dif-ferent spatial topologies are not linearly independent - they are related by the constraintequations. Therefore the number of spatial components is not non-perturbatively diffeo-morphism invariant. There is no associated operator on the full bulk effective Hilbertspace that satisfies the non-perturbative constraint equations.It is interesting to contrast this fact with the structure of the classical phase spaceof general relativity. The latter has distinct components with different spatial topology,since there are no non-singular topology changing solutions. Thus naive quantization ofthis phase space would seem to imply that they give rise to orthogonal subspaces of theHilbert space. Moreover, the topology of space at a given asymptotic boundary timeappears to be a diffeomorphism invariant property of nonsingular spacetimes.Similarly, there is no straightforward Lorentzian interpretation of the euclidean in-stanton that describes the overlap of the factorized vacuum with the eternal two sidedblack hole. One perspective here is that semiclassical quantization and the path integralformulations of the bulk effective gravity theory disagree, and comparison with the exactCFT result is what that tells us that the path integral method is the correct procedurein this situation. Of course, the disagreement is non-perturbative in the G N expansion.However, it is more precise to say that Lorentzian semiclassical quantization is agnosticabout these issues, since the relevant solutions always involve singularities, which areoutside the domain of validity of the bulk effective theory. Analogously, the infinitesimalversion of the Hartle-Hawking ket overlaps (cid:104) h | h + δh (cid:105) are described by the Hamiltonianconstraint equations with a straightforward Lorentzian interpretation, but it not easy tointegrate up to see the overlap between kets with different topologies. This is becausethe slice has to become singular somewhere along any deformation which changes itstopology, and again one exits the regime of validity of the long distance theory. The factthat using the euclidean path integral one may directly define and calculate the overlapof non-perturbatively distinct kets is a main virtue of the Hartle-Hawking formalism. In classical gravity in asymptotically AdS spacetimes, one may define relational observ-ables, for example the value of a bulk field at some regularized distance, z , along a geodesic14hat extends transversely into the bulk from a fixed boundary point, x . These are fullygauge invariant, since AdS boundary conditions require that diffeomorphisms vanish atthe boundary.In perturbation theory around empty AdS in Poincare patch, one may think of suchan observable ϕ ( x, z ) as the bulk field in Fefferman-Graham coordinates. In general, thisis not a good description since Fefferman-Graham is not a good gauge around generalbackgrounds. The geodesics from different boundary points can intersect, causing thiscoordinate system to break down.In the quantum path integral, one can insert such an observable, since it is well-defined for any off-shell metric configuration. However, it is a novel feature of gravitythat this does not automatically imply the existence of a well-defined operator on thephysical Hilbert space. In quantum field theory, one can perform the path integral up tosome time, act with an operator, and then continue the path integral, so there is a directrelation between insertions and operators. In gravity, there is no canonical choice of timeslice, so this procedure does not work. For example, insertion of ϕ ( x, z ) into the pathintegral defining the wavefunctional Ψ( h ) is not well-defined, since for some metrics, thegeodesic will exit the slice h . One needs to appropriately gauge fix the bulk operatorsnonperturbatively, so that they are compatible with the slices.If it was possible to pick a non-perturbatively good gauge for diffeomorphisms, thenthere would be a simple relationship between insertions in the path integral in that gaugeand diffeomorphism invariant operators. One of the essential points is that no such gaugeexists, as can be seen by the fact that the causal relations between observables depend onthe spacetime.Another way of describing the same feature is that there are multiple ways of preparingthe same state from a euclidean path integral. For example, the thermofield double statecan be produced by a path integral with cylinder boundary or via the microscopic relation | tfd (cid:105) = √ Z ( β ) (cid:80) e − βE/ | E (cid:105) L | E (cid:105) R . In general, an insertion of a geodesically definedobservable into a path integral involving one or the other of these need not give the sameresult. Therefore there is no well-defined action on states.It is possible to define closely related operators that are non-perturbatively diffeomor-phism invariant. But additional choices are required, and the resulting operators do nothave the expected action of all states. The following is one such prescription.Consider kets | h, GD (cid:105) defined by doing the path integral only over metrics that have theFefferman-Graham form along a single geodesic originating at the boundary at a point x ∗ and terminating at a point in the interior at a fixed regulated distance, z , definedwith respect to empty AdS. In other words, one restricts the integration over the radialcomponent of the shift to vanish along this geodesic between the AdS boundary andthe point defined by z , in addition to fixing the spatial gauge on the slice to have theFefferman-Graham form along that geodesic. The path integral is thus only over d + 115imensional metrics whose restriction to the t = 0 slice is given by ds d +1 = h ij dx i dx j + N i dtdx i + Sdt , where h zz ( x ∗ , z ) = Lz , h za = 0 , N z = 0 for 0 < z < z ∗ , where the index i runs over all spatial directions, z and the boundary spatial directions, a . This gives a well-defined linear functional on the Hilbert space, h GD : H bulk → C .Note that there is no integration over metrics in which this geodesic is chronal (ie. no twopoints on this space-like geodesic may time-like related), so these kets will span a smallersubspace than the Hartle-Hawking kets.These kets are still not all linearly independent, but ones with different values of thefields along the special geodesic will be orthogonal. This is because there is no integrationover the shift along the geodesic, thus in the path integral that computes (cid:104) h , GD | h , GD (cid:105) will have the marked geodesic in the two slices identified. Therefore one can define a bulkfield operator along the geodesic using these kets asΦ GD ( x ∗ , z ∗ ) | h, GD (cid:105) = ϕ ( x ∗ , z ∗ ) | h, GD (cid:105) . This is well-defined because the definition of the restricted kets implies that they areorthogonal if the values of ϕ at the point ( t = 0 , x ∗ , z ∗ ) are different.In this way, one can define a nonperturbatively gauge fixed version of the bulk field.The action of this operator on other states is determined by the above definition, supple-mented by the condition that it annihilates the orthogonal complement of the full Hilbertspace.Note that in the eternal black hole, this particular observable never enters the regionbehind the horizon (in the classical approximation), since the geodesic is exactly radialand, if long enough, would pass through the bifurcation surface directly into the left causalwedge. That is sufficient to phrase the paradox [18] reviewed in section 2.To reach points in the upper quadrant of the Penrose diagram, one needs a geodesicthat begins with a general slope in the z − t plane at its origin point on the right boundary.This is easily obtained, since the action of the special conformal generator, K that has x ∗ as a fixed point changes this slope. Therefore, one may simply consider the operator e iaK Φ GD ( x ∗ , z ∗ ) e − iaK .The crucial fact is that the geodesic gauge fixed kets only span the Hilbert space ofstates in which the marked geodesic lies along a spatial slice containing the entire t = 0slice of the boundary. In the two sided asymptotically AdS context, that includes theleft boundary opposite to the right one where the point x ∗ lies. Since the bulk operatoris defined in terms of these states, it includes the projection operator onto the subspacewhich they span. Such an operator is explicitly left time dependent, so it will fail toexactly commute with H L .Consider the commutator of left and right boundary framed geodesic observables.16his would seem to vanish in all factorized states, since the operators would only act onthe left and right Hilbert spaces respectively. However, the appropriately gauged fixedoperator acts on both boundaries. One can compute (cid:104) , | [Φ GD, L (0 , z ∗ ) , Φ GD, R ( x, z (cid:48)∗ )] | tfd (cid:105) by finding the non-analytic part of the euclidean path integral with a pair of geodesicinsertions, as a function of x . This corresponds to the insertion of the geodesics inthe geometry shown in Figure 2, and it is clear that they may cross in the interior forsufficiently large z ∗ , z (cid:48)∗ . Therefore the commutator of the gauge fixed operators is nonzeroeven when acting on the factorized vacuum. It is exponentially small, since it is suppressedby the exponential of the action of the saddle shown in Figure 2. Note that the slicecondition for the geodesics is obeyed in the euclidean saddle in this situation, so that theaction of the gauge fixed operators is indeed given by a straightforward insertion here.Furthermore, the action of Φ GD on the time shifted states, | ψ T (cid:105) , is very different thanthe naive ungauge fixed observable. The time shifted states are described by the samegeometry, but with a shifted origin of time on the left boundary. Therefore for sufficientlylarge T , the left t = 0 slice will no longer lie on a spatial slice with entirety of the rightframed marked geodesic. Thus the operator Φ GD as defined above will have a completelydifferent action on those states. In the classical limit, it would simply annihilate them.This resolves the paradox. To consistently define a bulk effective theory operator,one must refine the standard description to make the operator obey the Hamiltonianconstraints nonperturbatively. This involves making various choices; the above is oneexample. The gauge fixed operators are perfectly well-defined, and no paradox arises forthem. On the other hand, their action on most of the exponentially time shifted statesdiffers significantly from the naive description. In this work, I have argued that bulk observables that have been shown to lack reconstruc-tions as linear CFT operators over the entire range of the bulk effective theory also fail tobe nonperturbatively diffeomorphism invariant in the bulk description. This resolves thediscrepancy with the gauge/gravity duality, and implies that there is no obstruction toa linear map from the Hilbert space of the bulk effective theory over its entire expectedrange of validity to the boundary conformal field theory Hilbert space.A simple example is the connectedness of space in the two sided asymptotically AdSsystem. It is a well-defined observable in classical general relativity. However, usingthe description of connected spacetimes as entangled combinations of microstates of thedual CFT, one can show that there cannot exist a linear operator C that measures theconnectedness [33–35]. The solution of the puzzle is that the connectedness of space doesnot satisfy the Hamiltonian constraints nonperturbatively, and so does not exist as anoperator even in the bulk gravity theory. 17he bulk physics is often described in terms of boundary relational operators, like afield value at a location determined by some proper distance along a geodesic extendingtransversally from a given point on the boundary. Such geodesically specified observ-ables are well-defined in classical gravity. Furthermore, in perturbation theory aroundany given configuration, there is a well-defined quantum operator, which can be repre-sented by a boundary CFT operator in the associated code subspace of states that areperturbative excitations around that fixed state. For example, such bulk observables canbe constructed in terms of CFT operators in perturbation theory around exponentiallytime shifted versions of the eternal AdS black hole in the two sided AdS system [11, 12].These states are all described by the same spacetime manifold, with a shift of the originof time on the left boundary, and thus are in the domain of validity of the long distancebulk effective theory.However, from the microscopic description of the eternal black hole state as the ther-mofield double entangled sum of factorized CFT microstates | E L (cid:105)| E R (cid:105) , one can show thatno linear CFT operator can agree with the expected bulk matrix elements in all of thecode subspaces [18].The resolution is similarly that the usual spacetime path integral formulation of thegeodesic defined operators is ambiguous. Moreover, if one defined an operator usingbulk perturbation theory around different configurations whose wavefunction is peakedon a ket | h (cid:105) , then the total object would fail to satisfy the Hamiltonian constraints non-perturbatively.There exist better ways to define such an operator in the bulk theory, that involvea partially gauge fixed set of kets. These make it well-defined and gauge invariant, butthen it does not agree with the results of bulk perturbation theory around all states ofthe bulk effective theory. In particular, in the operators defined in section 4, there is aprojection on to states in which the geodesically specified bulk point is on a spatial slicewith the opposite boundary. This results in an order 1 disagreement with the matrixelements of the naive observable in perturbation theory around most time shifted states.In this way, after making such gauge choices to properly define the bulk operator, thereis no obstruction to its action as a linear operator in the CFT.It would be extremely interesting to explore the role of similar constraints in the blackhole information paradox, for black holes given by a pure state in global AdS. It is inprinciple straight forward to express any perturbative bulk calculation in the language ofHartle-Hawking wavefunctionals, as relations acting on the collection of kets on which theconfiguration is peaked. Presumably, repeating the Hawking argument in this frameworksimply will not result in a contradiction. But that in itself gives little information, sincethe expansion of the bulk kets in terms of states corresponding to boundary CFT localoperators is very complicated, and cannot be computed in the bulk effective theory.One surprising fact is that the entire black hole formation and collapse process in AdS18an be captured by spatial slices from a fixed time, t , on the boundary, by taking anyboundary spatial slice which is spacelike to the end of the horizon where the black holefully evaporates. This description of the Hawking process is complimentary to the morestandard picture in which one takes gauge fixed spatial slices whose time evolution is tiedto the boundary time. Then the process looks like the formation and evaporation of alocally thermalized configuration in the dual CFT, such as a region of deconfined phasein a large N gauge theory. In the CFT description, it is clear that the time evolution isunitary. But the bulk description must already encode the evaporation at a fixed timein the boundary (for which unitarity is tautological, since there is no evolution under theglobal Hamiltonian).From that bulk perspective, one has pure constraint equation evolution, and in prin-ciple there is a euclidean path integral calculation of the overlaps between the initial ketsand the final ones. If the black hole is formed by the collapse of low energy density matter,then the initial wavefunctional is peaked around kets that are in the domain of validityof the bulk effective theory. Similarly, the outgoing Hawking quanta are individually oflow energy, and that state will also be in the domain of validity of the effective theory.The amplitude between such kets at early and late slices (ending at the same boundarytime t ) will be exponentially small, since they are connected by some nontrivial euclideansaddle. However, there is an exponentially large number of relevant late time kets, de-scribing the outgoing Hawking quanta, and one expects that the early and late sets of ketsare linearly dependent; that is the pure constraint equation evolution. Finding the linearrelations again involves partly diagonalizing a matrix of overlaps, which will probably exitthe domain of validity of the effective theory. So, unsurprisingly, one does not expect tobe able to calculate the exact unitary map from the infalling matter state to the finaloutgoing state of Hawking quanta from the bulk effective theory alone.An intriguing possibility is that by taking into account more Planckian configurations,associated to ER=EPR wormholes connecting the black hole interior to the outgoingHawking quanta [32], one can use the linear dependence between kets of the resultingdifferent topologies to obtain a more detailed understanding.The discussions in this work about the restrictions of non-perturbative gauge invari-ance on bulk operators still leaves unresolved the physical question of measurements inthe gravitational bulk. It is well known that there are no local diffeomorphism invariantoperators. This certainly does not imply that what we actually measure in gravity arenonlocal observables, in the sense that there is a projection onto eigenspaces of nonlocaloperators. The Hamiltonian that describes the measurement process is always the integralof a local density, and the physical measurement process cannot, for example, change theADM energy encoded by gravitational fields at spatial infinity.The quantum description of a measurement involves coupling the system that onewishes to measure to an apparatus. This interaction can be written schematically as19 int = O app O sys . Decoherence processes in the apparatus result in a density matrixthat is very well approximated by one that is diagonal in the eigenbasis of O sys . Thisapproximation can be made parametrically good in the limit of a large apparatus, so theresult is equivalent to projection on to an eigenstate of O sys , with probabilities given bythe Born rule.In gravity, there is no canonical way of separating the interaction Hamiltonian intodiffeomorphism invariant system and apparatus operators, since for a local measurement,it must be defined relationally between them. A useful analogy is an interaction of the form ψ † app W ψ sys in electrodynamics, where W is a Wilson line connecting the apparatus andsystem such that the expression is gauge invariant. There is no canonical way to separateout a gauge invariant O sys - one must add a Wilson line to infinity or otherwise pick agauge. Moreover, even if one makes such a choice, the dynamics is never well approximatedby projection on to eigenstates of O sys , since that would affect the electromagnetic fieldfar away. In electrodynamics, the resolution is that one doesn’t call such interactionsmeasurements - rather it is an exchange of an electron between the system and apparatus.However in the bulk gravitational theory, all observations are of this form, and we doconsider them to be measurements in practice. Therefore it is necessary to examine thephysical measurement and decoherence process directly; there is a irreducible obstructionto replacing it with projection on to eigenstates of an operator.From this perspective, the problem of finding CFT descriptions of bulk observations isa dynamical one. At large N in certain quantum field theories, one should find that extranon-local operators decohere, ie. that in appropriate states which include a measuringapparatus, there is decoherence in the eigenbasis of such an operator. The only coarse-graining that is intrinsically defined by the system is that given by time averaging. Givena short time scale, τ , one can construct the time dependent density matrix, ρ ( t ) = (cid:90) t + τt dt (cid:48) U ( t (cid:48) ) | ψ (cid:105)(cid:104) ψ | U ( t (cid:48) ) † , where U ( t ) is the time evolution operator. Then an operator is decohered in the state | ψ (cid:105) if it approximately commutes with ρ ( t ) for a long time T (cid:29) τ . This, a priori , dependsnon-linearly on the state.The novelty in strongly interacting theories with holographic duals is that, at leastaround certain states, additional non-local operators can be made to decohere. Thesecorrespond to approximately local bulk operators.For this Lorentzian question, it is possible that MERA-like constructions of the states,and the resulting state dependent operators will play an important role. But it is notclear what is the correct abstract framework to discuss such observables. The fact that the QFT Hamiltonian is local explains why the usual local operators can be made todecohere in this sense.
Acknowledgements
I would like to thank Xi Dong, Monica Guica, Daniel Hawlow, Anton Kapustin, AitorLewkowycz, Don Marolf, Kyriakos Papadodimas, Eric Perlmutter, Suvrat Raju, SteveShenker, Herman Verlinde, Aron Wall, Xi Yin, and especially Juan Maldacena for stim-ulating and helpful discussions. I also thank Noah Jafferis for creating the figures. Thiswork was supported in part by NSFCAREER grant PHY-1352084 and by a Sloan Fellow-ship.
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