JHEP04(2021)272 Springer April 28, 2021 March 8, 2021 : : December 1, 2020 : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP04(2021)272 [email protected] , . 3 2010.06602 The Authors. c Black Holes, Models of Quantum Gravity

We reformulate recent insights into black hole information in a manner empha- , marolf@physics.ucsb.edu Department of Physics, UniversitySanta of Barbara, California, CA 93106, U.S.A. E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: ied by Polchinski andconcepts Strominger and in techniques, 1994 andreplica provide wormholes. we a The argue simple work them illustrationto aims to of readers to who be be the have self-contained associated a not and, natural yet in mastered late-time particular, earlier to extrapolation formulations be of of accessible the ideas above. sizing operationally-defined notions of entropy, Lorentz-signaturetotically descriptions, flat and asymp- spacetimes.of With asymptotic the observers are help consistentdensity of with of replica black states holes wormholes, given as wecost unitary by of find quantum superselection the that systems, sectors Bekenstein-Hawking associated with experiments formula. with the However, state this of baby comes universes. at Spacetimes the stud- Abstract: Donald Marolf and Henry Maxfield Observations of Hawking radiation:and the baby Page universes curve JHEP04(2021)272 52 26 ? 43 E 17 9 26 50 64 13 45 11 4 4 54 39 36 41 24 41 18 7 – i – 27 55 48 45 29 29 23 13 32 64 60 6 29 57 1 57 7.3.1 AdS/CFT and the factorisation problem 6.3.2 Lessons and comments 6.5.1 Replica6.5.2 wormhole baby universes What is a baby universe? 6.3.1 The PS Fock space of baby universes 4.4.1 What4.4.2 happens at the endpoint Violations4.4.3 of of evaporation BH unitarity Violations of causality 2.2.1 Path2.2.2 integral preliminaries The in-in formulation of Hawking emission 7.1 Summary 7.2 What have7.3 we gained? Further open questions 6.4 Replica wormholes6.5 as baby universe Baby interactions universes with semiclassical control: dropping the PS assumption 6.1 From path6.2 integrals to Hilbert Hilbert spaces 6.3 spaces for Hawking and Baby Polchinski-Strominger universes and ensembles 5.1 Replica wormhole5.2 spacetimes Quantum extremal5.3 surfaces Contributions from5.4 replica wormholes Replica wormholes for other observables 4.4 Challenges for the Polchinski-Strominger proposal 3.1 Incorporating back-reaction 4.1 Polchinki and4.2 Strominger’s proposal Wormholes and4.3 factorization Experiments on part of the radiation 2.2 Path integral version 2.3 Entropies from the Hawking path integral 2.1 Hawking’s Heisenberg picture calculation 7 Discussion 6 The Hilbert space of baby universes 5 Replica wormholes 4 Entropy measurements and potential new saddles 3 Semiclassical path integrals and back-reaction Contents 1 Introduction 2 Hawking radiation and the path integral JHEP04(2021)272 ] built 19 , ] in which 18 28 – 25 ] with the so-called 69 22 Lorentz-signature de- 67 ] for a review and references to 66 ]) that sums over topologies in -states’) for quantities associated intrinsically 23 17 68 α – 1 65 ]. See [ 65 68 70 13 – 11 – 1 – 67 ] which discussed the effect of replica wormholes in Lorentz 24 , 19 ] as to what physics is really being studied. 29 ] to argue that an exchange of dominance between two saddle-point ‘replica 7.3.8 The experience of an infalling observer 7.3.3 Contributions7.3.4 from UV physics Spacetimes7.3.5 with singular causal structure Non-perturbative7.3.6 physics of baby universes More7.3.7 details of unitarity Moving away from asymptotics 7.3.2 Description of superselection sectors 21 , Our goal here is to reformulate the recent progress in a manner that i) focusses on In addition, the recent discussions also rely heavily on AdS/CFT duality or related The bulk of the recent discussions have been couched in terms of Euclidean path in- 20 operationally defined quantities (theobservers), outcomes ii) of can be ‘experiments’ stated performedthat and by analyzed the entirely asymptotic in physics Lorentz describedintegral signature, that and follows sums iii) directly emphasizes over topologies. from While having we a take the low inclusion energy of this gravitational sum path over topologies concepts. Thisarguments was were true made evendifficulties by for as the analogy the physics of with asymptotically spacetimethat wormholes AdS/CFT. flat calls raises into the But analyses question so-called the ‘factorization reliance standard of problem’ been interpretation on of [ raised AdS/CFT. As AdS/CFT [ a presents result, questions have signature did so by studying Euclideanentropies signature as replica wormholes, functions using of them to Euclideanreal compute times. coordinates, and But it analytically isscription, continuing clearly the especially of results since interest topology to to changeLorentz-signature understand is metric. an generally incompatible with having a smooth baby universes and withwith the asymptotic boundaries superselection described sectorsadditional in related (‘ [ work. tegrals. Indeed, even [ hole information and otheron issues [ in gravity andwormhole’ geometries holography. resolves In a longstanding particular, tensionof [ between black the hole perturbative description evaporationentropy and as a the density interpretation of states. of Such the effects black have also hole’s been Bekenstein-Hawking connected [ 1 Introduction Recent work hasthe reinvigorated gravitational the path integral idea provide (see missing ingredients e.g. necessary [ to understand black A Further review of theB Hawking effect Intermediate in states a of fixed baby spacetime universes JHEP04(2021)272 counts BH S (solid curve). u u ) u ( l a m T r e h t S (dashed curve). Consequently, after G A 4 . It will feature many times in our ∼ 1 BH S e g a P ] — it also flies in the face of physics associ- – 2 – u 30 ) u ]). ( 35 H – B S 31 ], it would then be natural to expect the von Neumann entropy 36 S the entropy must decrease, approximately saturating that bound. corrections) as a density of states will be a common touchpoint through- Page u + The Page curve for the entropy of Hawking radiation emitted before time G A 4 = to decrease over time, the above interpretation thus would appear to force the von The resulting ‘Page curve’ is shown in figure In particular, perturbative quantum gravity would suggest that Hawking radiation is Nevertheless, the interpretation of the black hole’s Bekenstein-Hawking entropy BH BH S comes close to saturating this bound, requiring deviations fromdiscussion exact below, thermality. againparticular, as the a downward touchpoint slopingblack to part hole to of be be the returned compared to Page the with curve external various universe. requires calculations. The information literature inside on black In the hole information S Neumann entropy of Hawking radiation toAs become described small by in Page laterof [ stages of radiation the from evaporation. aincrease black while hole thermal radiation that is forms produced, from but rapid then collapse to to ‘turn over’ begin and at decrease a once small it value, ated with quantum field theoryquantum on gravity an (see evaporating e.g. black hole [ background and perturbative essentially thermal, which is inblack tension hole with states. the statisticalis Under interpretation an the that upper standard bound laws on of the quantum entanglement of mechanics, any the system. density Since of Hawking states evaporation causes ( out our discussion.hypothesis We to do be not constantly testedwill take and seem this explored. natural to Indeed, due be while toeven to the a required some success fundamental this by of interpretation assumption, classical this black but success, hole rather see thermodynamics a — e.g. or [ perhaps holography (AdS/CFT), or any other UVwill theory work of entirely gravity. with To asymptotically underlinedirectly flat the to spacetimes last the (though point, asymptotically we analogous AdS statementsbriefly case apply in as tangential well). comments. As a result, AdS/CFT is mentioned only For a while this isit increasing, given is by bounded the by thermalthe the entropy Page (dotted Bekenstein time curve). Hawking But entropy under BH unitarity as a fundamental assumption in this work, there will be no explicit input from string theory, Figure 1. JHEP04(2021)272 1 ] to 19 , 18 measurements through the study of PS with a brief review of the 4 2 ]. ], we will argue them to be a 39 – ], which we dub ‘PS wormholes’. 14 37 14 , 34 . This sets up the standard challenge ] in analogy with the term’s use in biology. We 3 23 in order to describe measurements . This discussion has strong overlap with those – 3 – 1 whose evolution is unitary (up to possible BH S e with density of states interactions with other quantum systems). Bekenstein-Hawking unitarity: of distant observers, black holes can be modelled as a quantum system ]. 14 a directly observable quantity; rather, it can only be inferred indirectly from other – The following sections resolve this problem by identifying new saddles for the gravita- Indeed, a final goal of this work is to make the manuscript below accessible to those We will see below that many of the concepts and techniques related to Lorentz- The same concept was called ‘the central dogma’ by [ 11 1 not wormholes. Although the inclusionbeyond semiclassical of control, PS it wormholes provideslater requires a in assumptions simple about this introduction physics to work. ideas that A will be keydo of such not use follow point this terminology is here, that so that spacetime we might wormholes avoid lead appearing dogmatic. to correlations for BH unitarity associated withis apparent often large deviations called from ‘the the black Page hole curve, information and which problem’ [ tional path integral. Some possibleof effects entropy of by new asymptotic saddles, observers, and especially are on illustrated in section we instead return to theHawking logical effect beginning in and a startapproach fixed in and black section hole the background, in-innone but formalism of emphasizing this that material both will is theliterature. new, be path it We useful integral differs then sufficiently in use fromgravity later this the and most perspective parts perturbative common to of treatments back-reaction discuss in this in the the work. section inclusion of While semiclassical quantum reproduce the Page curve.cations, we Since will this devote extrapolationtreatment significant turns as time explicit out to as to discussing possible. lead PS to wormholes in several effort simplifi- who to have not make yet our mastered the above references. Rather than review those works in detail, signature spacetime wormholes, baby universes,times and described the by like Polchinski are andIndeed, well-illustrated Strominger while by PS in space- wormholes 1994 are [ in not under isolation semi-classical leads control, and tonatural while analyzing apparent late-time them violations extrapolation of of BH the unitary replica [ wormholes that were shown in [ measurements. Looking ahead,BH this unitarity will to be be importantspeaking, satisfied for follow despite our the the Page conclusions fact curveof since in [ the it figure von allows Neumann entropy may not, strictly We emphasize that our definitionIn of contrast, BH as unitarity we isis discuss operational, further referring to below, observations. the von Neumann entropy of Hawking radiation rather more assumptions involvedgravity than system. just In the strict presentunitarity’ unitary work (or we evolution will BH of thus unitarity) instead to the use refer full the to term quantum this ‘Bekenstein-Hawking suite of ideas, which we summarize as follows: often describes this as a result of requiring ‘unitarity’. But, as noted above, there are JHEP04(2021)272 , . ) + + I I ( . Since below. ] of the m − a 40 , I ) 2(a) of operators + i I ψ | ( ) † m + a I ( . Using the Heisenberg . Doing so resolves the |O m ψ 5 h ], or pedagogical introductions 40 . There the spacetime is completely − I . In keeping with the general philosophy of 3 – 4 – ] in taking the quantum fields to be massless so that ], we work in the Heisenberg picture. We thus evolve 40 , the original work [ 40 A ] considered a black hole with a single asymptotic region from creation and annihilation operators + 40 can be described as a Fock space of ‘out’ scattering states, we I of the quantum fields is taken to coincide with the Minkowski + . Following [ i backwards in time to write them in terms of operators at concludes with a summary and discussion of open issues. I + ψ ) | 7 + I ]. I ( 38 . − O by slicing open the above path integrals. We find a ‘baby universe’ Hilbert I ] or [ 6 41 defined at ) + We are interested in the predictions of observations made at future null infinity It then remains to provide a Hilbert space description of the physics of the Page I ( O the operators the Hilbert space at can build all operators at labelled by some complete orthonormal set of modes indexed by And for further simplicity, we followtheir [ initial data isflat, specified at and past the nullvacuum state infinity on In particular, we would like to compute the expectation values The original argumentthat of forms [ fromconsider collapse a of spherically symmetric matter collapseis in of Schwarzschild. an uncharged A matter conformal asymptotically so diagram flat for that such the space. a final spacetime black is hole For shown simplicity in figure we this paper, we willobserver. emphasise the Readers computation seeking of aground observables more accessible should thorough to consult review an appendix ofsuch asymptotic as the [ Hawking effect in a fixed2.1 back- Hawking’s Heisenberg picture calculation production radiation using linearback-reaction or quantum evaporation. fields in Weterms also a of recall fixed a how classical path thisentropies spacetime, calculation integral, of can the without and be Hawking howcal reformulated radiation. the quantum in path This gravity integral discussions review can in lays be the section groundwork used for to the compute semiclassi- the Rényi illustration. Section 2 Hawking radiation andThis the section path contains integral a schematic overview of Hawking’s original calculation [ curve, and to characterisein the section correlations arisingspace from of replica intermediate states wormholes. whichof defines This superselection asymptotic is sectors quantities. associated done withfrom As the the values a viewpoint result, of it asymptotic leads observers. to Again, an the ensemble PS description wormholes of provide the a theory simple iments. We also discussthe challenges stage for to BH introduce unitaryPS and raised challenges include by and PS replica reproduces wormholes the wormholesclassical alone, Page control. in curve setting section We using will calculations alsoof that see replica are that wormholes. fully PS under wormholes are semi- a natural late-time extrapolation between the outcomes of what might at first appear to be completely independent exper- JHEP04(2021)272 , ) − (2.2) (2.1) I ( † n from the a T is a regular , our spacetime = 0 , this rewriting + . Since we took ) r R ) − I . + + I I ( becomes very short in ( I n , † m a , we find T a  + ) T − I I ( † n . a 2 | mn mn . For modes localized at late (retarded) β β (b) An illustrationpropagation of of a the moderetarded-time backwards- localized on at late | R n of a mode at ) + . The vertical line marked X annihilated by all − ) ± + = − 0 I I i in terms of corresponding operators ( i I I n i ) . ψ ( | 0 a – 5 – + + | ) m + . In the shaded region of (b) near a + I I − ) mn ( I + I I α i m ( is shown as a dashed line, and the singularity as a jagged  0

a

I is linear. The Heisenberg evolution is thus given by a |

− m

(

+

〉 n − I . In particular, the wavelength of

N

= − X † m | and a reflected part

| H + + 0 i a I i

will remain in the nearly stationary region and transmitted part ψ

I H T

ψ h . For example, if we compute the expectation value of an 〉

= |

) = R

| ) = ψ + mn + I , β using geometric optics. ( I , thus to be the vacuum ( m − − mn a m , we can write − α I I − N

I

I and those at 0 = r − i + I g , the reflected part n The final event horizon + i s r acting on the Fock space of ‘in’ states, and similarly for I p a t ) a l s l − o For a free quantum field theory, the relationship between creation and annihilation I C will be localized very close to ( asymptotically flat black holetotic with region. a single asymp- (a) The conformal diagramdescribing of a collapse classical of spacetime matter (a ‘star’) to form an n for some coefficients occupation number allows us to compute all observables at operators at Bogoliubov transformation near-horizon region to evolution back to a the initial state at flat asymptotic region is nearly stationary.and There, results the in backwards-propagation a reducestimes transmitted to on part scattering inT a fixed potential the reference frame shown. This allows us to complete the backwards-propagation of Figure 2. line. Future and pastorigin null of infinity are spherical labelled polar by coordinates. The state is chosen as the vacuum of quantum fields in the JHEP04(2021)272 , the + ]. These I 44 – 41 [ − is not pure; that ] or reviews such I + 40 I such that this functional on ], which considered the world- ρ is nonzero, so the outgoing ρ explicitly from expectation 45 ρ mn can be formulated as a path β of that are localized at late retarded 2.1 ij ρ are chosen from a complete basis of m i , the state j . This is equivalent to describing the | − + , i I i I | for readers wishing to review them. Readers . This impurity arises for the simple reason i , we cannot do the reverse, since the operator A ψ − ), it is straightforward to compute the Bogoli- – 6 – | . Equivalently, while we can perform Heisenberg + I of such late-time modes are thermally distributed, I ) . When combined, they establish the familiar result 2(b) , + for any I ], or the use of the adiabatic approximation to evaluate ( | along back to 2(b) 40 2(a) m ψ [ In this description, the actual computation of the effect is . Indeed, one way to define the density matrix of a region u + N + ih , where the states 2 − | I ψ i | I I ih ) using two facts. The first is that, in the region close to j | 2.1 = . We can recover matrix elements ) O . O + ρ I Tr( is not a Cauchy surface, as Cauchy surfaces must reach the regular origin shown ]. O 7→ + 41 , I Famously, despite choosing a pure state on In the above discussion we formulated Hawking’s calculation as the computation of At least for operators associated with field modes This differs from the Hartle-Hawking derivation of Hawking radiation [ 38 2 somewhat more cumbersome.the However, path as integral framework weback-reaction allows will and us see certain to in non-perturbative straightforwardly quantum incorporate the gravity both remaining effects. perturbative sections below, line path integral over trajectories of a particle. resulting from forward evolution will have support on the2.2 black hole interior. Path integral version We now recall howintegral the over computation quantum outlined fields. in section is, it cannot bethat written as as a vertical black lineevolution in of figures operators from is as the linearConnecting functional to that the maps usual Hilbert operators spaceacts on language, as there that is region a to unique values their by expectation choosing values. pure states on seeking a more thoroughas discussion [ should consult the original paper [ expectation values of allstate possible of operators quantum fields on on features are illustrated inthat figure the occupation numbers with grey-body factors appropriate toitative the picture. black hole. The Interactionsbut do details we not of include change a this this brief qual- argument summary are in appendix not relevant to our presentation below, this region thus reduces tois solving a that, Schrödinger-type once problem. the Thelocalized second mode very important close is fact to propagated theuse horizon. backward the In into WKB particular, the as approximation near-horizon aagating to result region, the justify of mode either it the the back second becomes correlators property to use without we of explicitly may geometric completing the optics backwards in propagation further to prop- occupation numbers are positive despite choosing an ingoing vacuum. times (large affine parameter ubov transformation ( spacetime is well-approximated by that of a stationary black hole. Mode propagation in Black holes radiate as a simple consequence of the fact that JHEP04(2021)272 ] : + φ − i [ i , so I + give ) − (2.3) (2.4) φ or on t | φ ( − | i + + H take def- − Σ φ φ h ± | ) + + Σ + φ φ h ( + + between the pair are defined by in- O − ] ˆ i φ (2) (1) − φ [ φ | iI + − ] ˆ O | (1) . φ ] [ (2) − φ [ iI , we can describe its Heisenberg φ e h iI + ) into the path integral. We first − Σ φ ). (2) φ e φ D 2.3 D . Since there are two such overlaps to ± + φ before returning to Hawking emission in (1) to denote the set of local bulk fields over φ = φ ± in the path integral, though these sets must φ D Σ | with a time-dependent Hamiltonian 2) φ , . This leaves us to compute the two overlaps 2.2.1 + ) Z – 7 – φ )) (1 t + = 2) φ , ( ∝ φ + (1 − ( Σ φ − | + i = . We use dtH (2) − O − Z ± φ R φ Σ | . To do this, we can insert a complete sets of states | Σ i = + 2) − + , − φ Σ Σ (1 | h φ + (1) on defined in terms of fields on φ exp ( , and this path integral computes the inner product − ± + = P i before integrating over φ ˆ i O 2) (or more general functionals of i , = to indicate the path integral ( by computing its matrix elements + (1) − (1 − φ φ − φ U φ U | : | | Σ + + are defined as eigenstates of different sets of field operators, on (2) − ˆ Σ O φ | ± . These boundary conditions correspond to eigenstates of the field operators h takes definite values i ± (2) − . ± + φ φ φ above. Both choices will simplify the discussion of various issues in the sec- ˆ | h O and i 2.2.2 2.1 with eigenvalues (1) − ± φ Since Given an operator In our experience, most textbook treatments of path integrals work in the Schrödinger | , they give different bases for the Hilbert space. The inner products Σ + − φ be identified at evolution back to of field eigenstates on which h compute, we have a doubled set of fields Σ the change of basisevolution matrix. operator These may bewe thought will of loosely as use the matrix elements of the time- There is of course a choiceis of associated phases to with be the made(and in choice with defining the of such fact eigenstates, possible that andIn this such boundary choice boundary addition, terms terms it in can can change theshould under be path canonical ultimately difficult transformations). integral consider to normalization-independent action keep ratios. track of normalisations in the path integral, so we inite values on Before turning to expectationinitial values, and we final begin Cauchywhich by surfaces we considering integrate. the path Thesertions corresponding integral of between Heisenberg-picture the operators field consider a path integral with boundary conditions specifying that the fields on tions that follow. Butproceed the slowly departure for from the standardHeisenberg-picture moment. textbook path treatments We integrals will suggests in thus that section section first we review various general features of in-in 2.2.1 Path integral preliminaries picture and emphasize therally co-called associated ‘in-out’ with formulation. computations In ofwill particular, transition continue amplitudes. the to latter However, emphasize since is ourlation expectation natu- discussion of values, path we will integralssection instead below. focus We on will the also ‘in-in’ continue formu- to use the Heisenberg picture as in JHEP04(2021)272 as ) lives + φ ( (2) φ + on which lies in the O + + Σ ˆ O + . We then insert as defined by its ) are typically not 〉 O . The right copy of 1 U ) begins at the ini- i ) − − ( i 1 (1) Σ (1) − + φ − ( − φ φ φ | | φ Σ + | |O (2) − φ h , while the left copy contains fields ] (1) while those for the left copy are defined φ [ i iI e (1) − y , as long as the support of φ | f Σ — or, more generally, by CPT conjugation i ] t n (2) – 8 – e φ [ then the backwards and forwards evolutions will d I I + φ with some choice of weighting. This corresponds to and ] − Σ and is weighted by (1) φ (or more generally by the CPT conjugate of the action on the left [ ] (1) I φ of field configurations on the initial slice (2) φ [ − i iI | − − + ) e φ | and describes a forward time-evolution computing ) ) are defined by the ket-state 2 . . This perspective becomes particularly natural once we incorporate | Σ ( − 2 − i is independent of ( 3 Σ (2) − ) φ before passing to the second branch of the spacetime. The field (1) − φ φ 〈 + φ h + | φ ˆ ( O + A path integral that computes the matrix elements O . This slicing-independence will prove useful in our discussions below. Σ ; see figure + The eigenstates Because our quantum field theory is unitary, if we happen to consider a trivial operator and is weighted by Σ . The combination of these gives the familiar Heisenberg evolution of the operator. The † (2) of direct physical interest.ing But over other field boundary configurationsallowing conditions on a can general be state,wavefunction. written described as by Now, a integrat- since superpositiontheir it of explicit is eigenstates wavefunction, we usually may inconvenient instead choose to to specify describe them states by of introducing further interest through cancel. In thatthe case two the spacetime result branches are isbeing joined. clearly evaluated More independent on generally, of one soto long the of be as choice the glued we of two along interpret branches, slice past an we of arbitrary may Cauchy surface choose the two spacetime branches distinction between forward andby backward the evolution relative is sign implemented between which in may the also act path nontrivially integral on fields. for which the spacetime. The first branchtial (which ‘ket’ provides state a home forthe the operator field on this second branch, andU the associated path integral computes the backward evolution We may equivalently think oftime doubling itself. not the The fields doubledon on spacetime a given then spacetime, has butquantum two gravity the branches effects, space- which since the are geometry glued can to fluctuate each independently other on each branch of the spacetime contains fields φ copy). This conjugation(fixing is the associated field with on theby fact the bra-state that the initial conditions for the right copy Figure 3. JHEP04(2021)272 , + + O I . is inserted . + U + Σ O to a complete + Σ . We are interested ) until . The contour begins i 3 0 | 5(a) , where we also insert a e in the region where + m i ]. t + Σ 48 n ). Finally, it proceeds from the I a – i , so we should take our future 3 z t + 46 n e I r o + L O . For clarity, the various parts of the contour | 0 in the vacuum state h t located on . Since these insertions are restricted to . The contour then proceeds along the Lorentzian + + – 9 – O O = 0 e t m i t n a e d i l by inserting a path integral over semi-infinite flat Euclidean c u − E I at Lorentzian time i 0 + | O . These two copies are identified along + in terms of fields at Σ i 0 | to lie in the far future and to coincide with + An in-in (or Schwinger-Keldysh) contour in the complex time-plane that computes Σ ) is the ‘in-in’ or Schwinger-Keldysh contour, and encodes the natural formulation 4 We can now assemble these ingredients: an initial ‘ket’ state prepared (perhaps) by , whence it returns to the origin (the left spacetime of figure t above guarantees the finalis result then to performed be onthe two independent Cauchy surface copies of of suchweighting the choices. corresponding spacetime, to but our The operatorthe only path identification in integral effectively the performs region a partial to trace the over past the interior of part of Let us now applythe the problem above at general hand. description Thein of resulting the path quantum expectation integral fields is values in shown ofboundary in curved an figure spacetime operator to is supported. Away fromCauchy our surface operator in insertion, any we way our we free to please. extend Furthermore, the slicing independence described the initial ‘bra’ state.figure The resulting spacetime on whichof we dynamics perform when the we path do integral not (see wish to2.2.2 specify a final The state in-in [ formulation of Hawking emission Minkowski vacuum state space and connecting it to the real Lorentz-signature patha integral Euclidean computing path integral,insertion a of Lorentzian the operator path of integral interest, performing backward time forward evolution, and time finally evolution, the preparation of at origin to positive infinite Euclidean timehave to been compute slightly displaced from the axes in the figure. path integrals. For example, in our Hawking effect problem, we might specify the initial Figure 4. the expectation value of at negative infinite Euclideancontour time computes and follows theaxis (this Euclidean part axis of to the the contour origin. corresponding to This the part right of spacetime of the figure JHEP04(2021)272 , + + I 〉 j I | i i were a − | + and (b) i I , which for pure . Along 〉 + 0 + | | + int i will prove h I Σ I I + 5 t n i I Σ + I i j | y f i t , we impose boundary n e + d I I to be covering the interior of the were a Cauchy surface, this + has rank one. If int t n O i + Σ + Σ . This operator insertion corre- I O and to use the results computed | 0 〈 + 5(b) i − 2.1 . If I + | i I 〈 i + . The two branches of the contour remain j is mixed. This joining of the two branches I (b) The path integralelements which of computes the matrix densitythe matrix on two copiesconditions of which weight field configurations ac- cording to the wavefunctions of states is a Cauchy surface for the black hole interior. | would cause the pathdisconnected integral pieces, to indicating fall thatis into two pure. the state Here,two this branches does remain not joined happen along since the + + I I + – 10 – in O , computing expectation values of all operators on + . The must include a piece is not − . I 2.1 + I + 〉 + 0 | I I Σ Σ+ by choosing our operator would then become a product of (conjugate) functions of on + + ij Σ + ρ I O . We depict this in figure + y f , and the state on I i t n e int is supported (denoted by d on I Σ + + O ; in operator terms, this says that our I + i j | . I + , Path integrals computing (a) the expectation value of an operator at Σ , which must coincide with int + + Σ | I 0 Σ i 〈 alone, and hence a rank one matrix describing a pure state. However, this does not i − | j I is equivalent to describing the state there. In particular, we can compute components In practice the simplest way to evaluate the above path integrals may well be to relate As discussed at the end of section of the density matrix on + and ij part of the spacetimegral on is which performed. our path inte- forward and backward time-evolutiontively. They respec- are glued togethersurface along a Cauchy the region where the black blob)trary. but which The is region to otherwise the arbi- future of (a) The path integralpectation which value computes of an the operator ex- right and left copies of the spacetime perform living on it to the Heisenberg-picturethere. computation Nevertheless, of the section useful formulation in in our terms quantum of gravity discussions the below. path integrals of figure i occur because any Cauchy surface black hole as well asconnected a through piece runningis along the path integral implementation of what is often called ‘tracing out’ the interior state sponds to a boundarythe condition Schwinger-Keldysh contour that independently, weights so fieldjoined the configurations along branches are on no the longerCauchy two meaningfully surface, branches then of thistwo boundary disconnected condition pieces. would Our cause the path integral to split into I ρ states Figure 5. components of the density matrix on JHEP04(2021)272 . 1 (2.5) for the case 7 , we fix boundary u (rather than joining and taking the limit I 3 u th Rényi entropy of a in the half-plane) and there copies of the spacetime ). n z at + , quantifying the tension n . This divergence is local . On i + + u 6 I I for all , | z is analytic on the right half-plane, of  | . Specifically, the ‘ket’ boundary τ ) n . To compute matrix elements of f ) ) n n Ce ρ ρ ρ < u . | ≤ ) . We then sew these replicas together 0 Tr( (Tr ) Tr( ) ρ u z (  u ( has not yet been normalised (i.e., that it ( that reaches f S | ρ ρ u ). We will subsequently assume that some log at retarded time u Σ 1 + . For systems with finite-dimensional Hilbert spaces, the 1 – 11 – − y I , but only with the Hawking radiation that reaches n such that + − τ only at its future endpoint I , replicas of th replica becomes identified with the ‘bra’ boundary C + meets r th replica since the insertion of complete sets of states ) = n ρ u for all real I ( | Σ n y | + 1) S c and then summing over a complete set of such wavefunctions. r r from the path integral, we start with on the ( Ce j ) u n i = | ≤ ) r ) j u of points at retarded times | ( iy +1 , studying an appropriate analytic continuation, ( ρ r to construct + on the that reaches 2 f i | | 6 I is defined by ≥ Tr( int +1 ρ n ⊂ r Σ i , the resulting Rényi entropy will be infinite due to high-frequency modes at h u n u I such that which defines the von Neumann entropy . It is often of interest to compute (or to imagine computing) the Rényi entropies c < π 1 For any To compute Next recall that we are interested in Rényi entropies. The First, we must slightly generalize the above discussion to compute the density matrix An analytic function is uniquely determined by its values on nonnegative integers under certain growth associated not with the entirety of 3 , we simply modify the discussion above as depicted in figure = 2 → u u conditions. Specifically, by Carlson’sexponential type theorem (there it are real is constants exists sufficient that Rényis always satisfy such conditions.on infinite-dimensional In Hilbert practice, spaces. the same seems to hold for physically interesting states the ‘entangling surface’ where at the entangling surfaceinterest and (in is particular, state-independent, itregulator so is has it been independent is chosen, of implicitly not for discuss related example the to subtraction resulting of the finite the quantity physics throughout. Minkowski of vacuum result, and The trace completes thisn pattern cyclically. Thefor result all integers is shownn in figure depicted in figure as instructed by the matrixlabelled products by and the trace state in labelled by amounts to setting where we have allowed formay the not possibility have that unitsimpler trace). to work As with noted unnormalized states above, than in to path keep integral track of constructions all it normalizations. is typically them on some density matrix ρ the subset ρ conditions according to the desiredpath matrix integral elements. along We a then partial join Cauchy the surface two branches of the We now conclude our review ofof the entropies. Hawking effect The on mainthe a fixed point Rényi background will entropies with be a ofbetween discussion to subsets the review original of how Hawking the calculation path Hawking and integrals radiation BH may unitarity at be via used the to Page curve study in figure 2.3 Entropies from the Hawking path integral JHEP04(2021)272 . u at large . Two copies u of the density u u u i , which defines a i e . As a result, all | u m u + i Σ t ρ corresponding to the | I d j u e h u 〉 defined by the Bondi d j u r | I a t e r BH S − , after some brief transient I + 〉 0 | I before retarded time u + Σ I will increase linearly with of S u I y f i t n e d decay rapidly when clusters of points are sepa- I – 12 – + I u Σ . We impose boundary conditions on , we perform the path integral on the geometry built from two u ) ) = | 2 0 . I 2 ) 〈 u u ) − ( I ρ u u ( e may be thought of as a tensor product of thermal density matrices Tr( ρ | m i i 〈 t + u d and the von Neumann entropy e I Tr( d n r , identified as shown. a S t 6 e r . u i To compute j The path integral on this geometry computes matrix elements | to be bounded by the Bekenstein-Hawking entropy describing Hawking radiation in the piece , u S u i ρ i | Since Hawking radiation rapidly becomes thermal at (with appropriate grey-body factors) associatedRényi with entropies smaller pieces of As noted in the introduction,require this behavior is inconsistentmass with at BH each unitarity retarded which time would behavior any correlation functions on rated by more thandensity a matrix thermal on retarded time. As a result, over large stretches of time the Figure 7. replicas of figure of the original blackCauchy hole surface spacetime when have joined beenstates to glued together along a surface Figure 6. matrix JHEP04(2021)272 . ] g [ (3.1) eff I , and in ]+ ± g [ I EH I to perform the 2 , we may use boundary 2 . is then a stationary point of the ] g φ,g [ , we treated spacetime as a background matter 2 iI φ e to incorporate a dynamical metric, preparing D – 13 – Z 2.2 := ] g [ eff to form a black hole, and asking for the expectation value iI , we can use ideas reviewed in section e . − g ] g [ I eff . The relevant boundary conditions are similar to the situation I + , with the two branches of the in-in contour joined at a future I ] + g . Alternatively, following the review of section [ ± 5(a) EH I I ]. A saddle-point in the integral over metrics 50 , 49 To describe perturbative quantum effects, it will be convenient for us to treat the We will thus need to specify boundary conditions for the metric. It is natural to pictured in figure boundary. But thus farthe the interior metric we has sum beenthis over means specified that allowed only we possible asymptotically willsaddle at metrics. proceed points by of As studying saddle already points, noted where above, here we in explicitly practice mean 3.1 Incorporating back-reaction We now have everything weWe need first to adapt the begin calculations makingan of predictions initial section using state of semiclassical matter gravity. of at some observable at To incorporate perturbative effects fromhole the evaporation fluctuations by of emission the ofporate metric a gravitons, itself, one-loop such this as effective ‘matter’ black actione.g. effective from [ action integrating should out also linearisedcombined incor- metric gravitational perturbations; (Einstein-Hilbert) action see and matter effective action metric separately from mattergiven fields, spacetime and withmatter begin metric path by integral as ‘integrating a out’effective QFT action: the on the matter. fixed background, For which we a can write as a quantum asymptotic observable using an in-instates formalism. only in In the eitherthe case, asymptotic spacetime we region. specify interior. We the willmatching We metric the place will and specified no thus asymptotics. restrictions include In on contributions particular, the from we metric any allow deep all saddle-point in spacetime metric topologies. impose boundary conditionsweak, in in asymptotic analogy with regions scatteringasymptotically of problems flat spacetime in metrics, quantum where and fieldmatter gravity theory. choose fields) We becomes on in-states will and integrate out-states over conditions for that gravitons do (along not with completely specify a final state, and we may instead compute an sion in a nonlinearconfigurations effective for theory. which the In classical practice,is (or, this stationary perhaps, means the under that quantum-corrected) variationsfluctuations we effective of action around look the these for saddles. metric saddle-point and other fields, and we then integrate over For our review of Hawking’s calculationfield in with section a fixed nondynamicalwish metric, to and we incorporate integrated gravitational onlyside over dynamics of matter by simple fields. integrating toy Wepath now models over integral it metrics. exactly. will Of be Instead, course, difficult we to will out- perform treat (or the even path define) integral the as gravitational a weak-coupling expan- 3 Semiclassical path integrals and back-reaction JHEP04(2021)272 as a over 4 eff I . The most 0 → and also at both ]). In particular, G . − and including per- 51 5 I 2.1 and . So this indeed incorpo- with respect to the metric 4 + I eff , up to effects associated with I − . As is well known, this tensor I − i 0 , the only null geodesics that extend I | i 0 | , so that back-reaction is precisely given 2(a) + I – 14 – . Note that the variation of eff I . In particular, the latter includes a factor of the inverse Newton ] g [ EH I , and is thus very large in the semiclassical gravity limit , rather than imposing two conditions (on fields and on their derivatives) on + /G I 1 On the other hand, back-reaction can be significant when one follows a null geodesic Now, in the original classical solution of figure Let us begin by ignoring post-selection at It is natural to begin by assuming quantum effects to be small and treating Finding saddles can be construed as solving the associated equations of motion. How- And analogous corrections to the equations of motion built from linearized gravitons. 4 that lies just insidegeodesics the can event be horizon studiedwhile of they using begin the the with a original Raychaudhuri slight equation background. negative expansion, (see if Congruences e.g. this initial of [ negative such value is sufficiently the curvatures become largewill (which continue to presumeably call means this Plankspacetime large-curvature a diagrams. scale). singularity This and For is to simplicity, consistentPlanck-scale indicate we with physics, it our by though current a we ignorance jaggedmay and do line become lack on not available of in rule control the over out future. the possibility that a better description to infinite affine parameters towardhorizon. the future As are a thosemust result, that be any lie confined additional entirely to outside nullthere the the geodesic is region event a that close large extends to region insidein the to the original the large black event hole physics, affine horizon. where and parameter perturbative We corrections where thus give the little conclude change spacetime that continues to collapse at least until such time as by the expected stress-energycarries a tensor flux of in positive theThe energy flux to state is infinity and small, a soup flux significant over of changes long negative to energy times, the into or background the occur over black large only hole. when affine they parameters. can build which we have alreadypost-selection integrated in when the the initial staterates state is back-reaction also from (partially) the specifiedsaddle Hawking below, at radiation turning described to earlier. other possible We saddles shall only focus in on sections this constant obvious saddle for our pathblack integrals hole is solution thus that giventurbative was by corrections starting used with as from the a classicalis fixed collapsing precisely background the in section expectation value of the stress tensor of the quantum matter fields path integral, and itof can the be relevant ones. challengingwhat to physics One they determine is entail. whether thusoriginal If one often one calculation has left later to with in finds take the additional simply fact new saddles, searching found one saddles for all will into saddles account. needsmall and to correction correct seeing to the ever, one shouldreasons. realize that The this firstThe is is second not is that that a the wecopies standard impose quantum-corrected of boundary Cauchy effective conditions evolution action ata problem is both single copies for generally Cauchy of slice. non-local. two As a result, there can be multiple saddles that contribute to a given JHEP04(2021)272 at and that u u + . This Σ I E , and in partic- 6 . However, our goal ; that is, the time as a codimension- 4 E , since the phases in E . However, it can be u E Σ u . The one issue we must 6 to construct back-reacted saddles 8 We will idealize 5 that is expected to be under semiclassical . It is not geodesically complete, and does ) at retarded time + have stress-energy fluxed close to the mean, 8 8 in direct analogy with figure I + ) ⊂ u – 15 – I ( u ρ I to each other along a partial Cauchy surface 8 , as this can modify the stress-energy fluxes to u influence other parts of the spacetime. But there is a unique . I E 8 . Instead, it has a future boundary defined by the singularity, on a region + ) . The only difference from working on a fixed background is that u (dotted line in figure I ( 9 to be the retarded time of the past boundary of ρ E . E u 5 from E and refer to it as the ‘endpoint’ of Hawking evaporation in the expectation that N . However, as typical states at E + H as shown in figure We may thus construct a saddle for In particular, let us now use the spacetime of figure Without a better understanding of Planck scale physics, it is impossible to say whether . Taking a one-parameter family of such congruences and using the cuts on which the This expectation will become an explicit assumption for the purposes of section . 5 + + E , and (using our spherical symmetry to rule out caustics and the like) the outgoing null control and makes no reference to strong curvature regions. in this work isthus to that avoid no sensitivity suchbe assumptions to reviewed are effects in needed that section for are the not replica under wormhole semiclassical derivation of control. the A Page curve critical that point will is runs from the regularI origin at the centergravity dynamically of determines the the spacetime collapsing awayof matter from this the to boundaries. saddle retarded The to time contribution the the classical path action integral is fromunitarily independent the on of two a the branches fixed choice of background. of the We contour see cancel, that and the the entire matter calculation is evolves under semiclassical across such effects are typically small.qualitative features And of even when figure they are large, they makeular little by impact sewing on two copies of figure N for the density matrix control. We thus wishconsider to find is a post-selection back-reacted at analogue of figure region of semiclassical control isnot shown contain figure a complete E congruence used to study black hole evaporation so long as we do not ask about what occurs beyond at which Planckian curvatures are first visible asymptotically. and how the singularity and perturbatively-semicalssical evolution in regions ofseparated, spacetime from and which of they are course causally also in the region to the past of the singularity and must shrink. And again, thisPlanck description scale, must at continue to which hold point until thethis the apparent locus curvature horizon becomes is also correspondinglylittle small. more We of denote interest2 can surface, happen after though this it‘evaporation point. time’ reality it describes a region of small but finite size. We define the the incoming fluxto of eventually negative become energyI positive. causes the This expansionexpansions indicates to vanish that to evolve define such through anmust congruences zero apparent begin in horizon, and with the fact a fact escape that more each to and successive more congruence negative expansion means that this apparent horizon small (i.e., for congruences close enough to the event horizon of the original background) JHEP04(2021)272 (3.2) , and we refer E (dashed line) is + , and the outgoing H E u 〉 j | E u = . ) u u − ( I E 〉 0 | N E Hawking ρ is essentially a thermal state with a temperature – 16 – ) ≈ u . ( ) and the black hole forms from collapse of matter. This u u ( ρ 2(a) where the identification occurs is not part of the configuration, Hawking u ρ Σ | i 〈 u ], we refer to the density matrix defined by saddles of the form shown 40 (dotted line). Planck scale physics becomes important at the singularity and E N The saddle-point spacetime for computing the density matrix of an evaporating black The region under semiclassical control in an evaporating black hole spacetime. In the as the Hawking density matrix: 9 Since the predictions for any experiment are encoded in the density matrix, we see For future reference, and because it involves essentially the same physics as Hawking’s , and may influence further evolution of the spacetime. The event horizon Because back-reaction is small, that varies slowly with retarded time that perturbatively-semiclassical gravity suffices to make probabilistic predictions for any original calculation [ in figure Figure 9. hole. The future of theso blue the slice spacetime is weakly curved everywhere, and in particular excludes the singularity. E defined to be the boundaryto of this the past past domain domain of of dependence dependence of as the the singularity black and hole interior. Figure 8. far past, the diagramregion coincides is with bounded bynull congruence the jagged line (called the ‘singularity’), its endpoint JHEP04(2021)272 (4.1) sets of n ] which we will is held fixed, it of our u 53 ) , u ( 52 ) n ( that this crutch can be ρ 6 sets of Hawking radiation, , they violate BH unitarity n ) immediately leads to a saddle u and ( predict physics consistent with 5 ? n ), does ], this conclusion is too hasty. While ⊗ Hawking Hawking ρ )] 14 ρ 3.2 u ≈ ( (1) ρ Hawking – 17 – ρ even when the Bondi mass at [ u ≈ I ) u copies of the state together turns out to allow potential ( ) n n ( ρ will use the crutch of making assumptions about physics that 4 , which in particular violates BH unitarity. Continuing with our u I ), considering 6 identical copies of the same state since they were all prepared in the same 4.1 ? At first sight, this may appear to be a frivolous question; surely it is u n identical copies in hand, it is a straightforward task to test whether a state I n copies of the result already obtained in ( n . 6 Now, what does semiclassical gravity predict for the state We must therefore form several black holes, taking care to prepare them in identical However, with access only to the Hawking radiation produced in a single black hole In this section and the next, we simply observe this phenomenon and study its implications. Inter- 6 section it is true that our saddle-pointthat computation would of give ( new saddle points. pretations of the new saddles and discussions of the underlying physics they represent will be deferred to radiation on trivially However, as observed by Polchinski and Strominger [ initial states, and collect their decaypresumably products. in We end up with way. With is pure or highly mixed.describe below. For example, one may use the swap test of [ imagine performing an experiment towithout directly access verify such to violations. several copiesfamiliar But fact this of that is the a impossible mixed state. stateon Indeed, is a equivalent as to single an an copy ensemble immediate can of consequence pure help states, of us no the to measurement distinguish a mixed state from an unknown pure state. 4 Entropy measurements andOur potential calculation new in the saddles previous sectiona has led highly us mixed to state suspect on thatphilosophy the Hawking of radiation concentrating is in on the predictions for asymptotic observers, we might like to historical literature, section is beyond semiclassical control. Butdiscarded, and we will that semiclassical see gravitational inBH physics sections unitarity. suggests the actual black hole density of states toevaporation, be we infinite. cannot operationally verifycritical that fact the provides state interesting islargely room mixed. devoted for It further to turns physics. this outinto point. The that the this remainder technical In of complications order this of to paper replica describe is wormholes, such and possibilities to without make yet connections delving with the which the black holethe has highly mixed become and Planck quasi-thermaland scale). density indicate matrix the Since blackentropy. these hole Indeed, predictions density by are of the encodedleads states usual to to argument in that arbitrarily be starting large unrelated with entropy to an on the arbitrarily Bekenstein-Hawking large black hole measurement of the Hawking radiation that avoids particularly late retarded times (at JHEP04(2021)272 . ]. 14 4.3 ] turns to all of 14 u ] was largely I -evaporation n 14 to section be resolved just by u . I not 5 copies of the boundary . And the third is that it ] was concerned with the ], recasting the discussion n 5 14 14 , or for the final conclusions of 6 then describes shortcomings of or black holes in a single asymptotic 4.4 5 n or, equivalently, extended from – 18 – , the details of which will be unimportant for our , and section black holes can lead to new saddles, let us first E 1 → ∞ 4.3 u under semiclassical control. It is thus important that n > to reviewing the proposal of [ not 4.1 for all times, deferring discussion of subsets , in the Polchinski-Strominger context this issue will + 4.3 . But replica wormholes will offer a resolution in section in the limit u I I 9 7 are introduced in section . The boundary conditions for computing the components of the , which describes how the black hole information problem is related to the 5 + 8 . 5 4.2 I ] involve physics that is We will treat each black hole as if it is formed and decays in its own separate asymptotic Before diving in, we should remark that the Polchinski-Strominger work [ We thus dedicate section The purpose of this section to describe path integrals that predict experimental mea- ⊂ . Placing each black hole in its own asymptotic region is a convenient abstraction, As described in section This can be thought of as a version of the cluster decomposition principle. 14 7 8 + u region, so longor as space. we prepare black holes whichconsidering are the sufficiently subsets well-separated in time sical control. region. As aconditions result, of our figure boundaryI conditions will be precisely though the conclusions should be equally valid for To understand how considering construct the boundary conditions appropriatethe for purposes such of multi-black-hole the experiments. currentradiation section, For emitted we take to our experimenterThis to will necessarily collect involve all making of assumptions the about Hawking physics that is not under semiclas- theories of gravity inphysics any of dimension. the endpoint In ofconsiderations. particular, evaporation much of [ 4.1 Polchinki and Strominger’s proposal the Polchinski-Strominger proposal, allsection of which will be resolved by replica wormholesdescribed in in termsstring of worldsheet. two-dimensional We models interpret their of proposal gravity more inspired broadly, applying by it analogy to with more general the in terms of experimentalsection measurements at infinity.lack of This factorization is of followedI quantum by gravity amplitudes. a short Experiments aside that involve in only some that it serves as ainto pedagogical the tool technical to complications explainplace the of recent idea replica developments of in wormholes. newout an saddles to The have appropriate without many yet historical second similarities delving tosuggests context, is the some as that replica of wormholes proposal the this of physics of that section helpsof may [ section to in fact lie behind the semiclassical replica wormholes surements of entropy and to connectBefore them doing with the so, potentialin we new saddles [ will discussed admit in [ tothey will the not reader form the thatthis basis work. the of any potential We analyses nevertheless new in review section saddles this advocated proposal here for three other reasons. The first is JHEP04(2021)272 as a int Σ ∪ . We note emerge at + . As noted 10 I not ∞ away from the conjugate copies = int n u Σ is such that , so that this ] is that the black hole . For our purposes, we 8 + to i 6 14 , and 9 i r as a (disconnected) j under the PS assumption. We , and may treat | + u i int Σ separate asymptotic boundaries, ∪ n + is appropriately local: in particular, + 2 9 I I E to larger − I 8 〉 0 | + E i after the black hole becomes Planckian will be – 19 – thus involve + i n I t n specifying a ‘ket’ state i Σ + I , . . . , j 1 j | ) is Cauchy in the black hole interior. n the extension of any evaporating black hole spacetime ( . | ρ | int r . i n Σ h E with Minkowski-like future timelike infinity ] for other scenarios for late-time quantum gravity effects. , . . . , i + 1 57 i – h , we expect a saddle given by extending figure I 54 of the black hole interior, we may treat , An extension of the spacetime of figure = 1 33 int region resembles that of Minkowski space;Σ and (2) for any Cauchy surface Cauchy surface for the full spacetime. PS assumption: beyond the region of semiclassical(1) control the shown spacetime in is figure empty near future timelike infinity n For . Indeed, Polchinski and Strominger describe information reaching the singularity as See e.g. [ 9 with boundary conditions at + that the PS assumption requiresthe that state the physics of of anyindependent radiation of emitted to thestrongly history curved of region the black hole, such as the state on We depict the evaporating black hole spacetime under this assumption in figure not return, a perspectivecan which cleanly we will state the explore required further assumption in as section follows: above, this extension mustgion. involve Roughly assumptions speaking, about our effectsevaporates interpretation completely, of in but the the that assumption strong of informationI curvature [ in re- the black holebeing interior transferred does to a ‘baby universe’ that branches off from the parent universe and does density matrix n specifying a ‘bra’ state Figure 10. have a complete Cauchy surface whenever JHEP04(2021)272 1 to 10 − ! ; see n 10 + 〉 〉 1 2 I i i ) for the | | . But since 4.1 case is shown with the same = 2 (2) n = 2 ρ on all of n ) u for ( ρ and its future (and thus E →∞ 11(a) u PS wormholes are diffeomorphic 1 = lim | | − , they also have precisely the same 1 2 ρ j j ! 〈 〈 copies of the Hawking wormhole with n ; see figure n 11(a) (b) Another contribution to boundary conditions, for whichtions the between identifica- blackswapped. hole interiors have been , . . . n of two identically-prepared black holes. 2 , + – 20 – I = 1 r 〉 〉 1 2 gives a i i | | below. The result satisfies the definition of a spacetime is then simply for 10 ) | n r saddles are diffeomorphic also requires them to contribute ( i ! h ρ 11(a) n bras. We refer to the doubled-spacetimes defined by the . For this reason, and because spacetimes like that of figure and n + i r I j as in figure | 9 replicas, the spacetime which gives rise to the naïve result ( . Note that, although each wormhole involves 1 The Hawking (a) and Polchinski-Strominger (b) wormholes contributing to the density describing the decay products at | | n > 2 1 11(b) j j (2) 〈 〈 ρ kets relative to the -copies of the Hawking wormhole of figure Indeed, the fact that all For The PS assumption immediately allows us to sew together two copies of figure n n distinct wormholes over which our path integral must sum, one for each permutation of ! -evaporation density matrix hole are shown. (a) Extending figure Hawking wormhole. Two copies of this worm- precisely the same weight to the path integral. We therefore find the components of our in figure leaves the domain of semiclassicalto control), since all validity as the Hawking wormhole to be interpreted potential saddles. it is clear thatby we simply can pairing then ‘bra’n build and further ‘ket’ boundaries wormholes with inthe different identical ways. boundary conditions Thisnon-trivial pairings construction as defines PS wormholes. The single PS wormhole for the disconnected copies of were often championed by Hawking, we refer to this spacetimen as the Hawking wormhole. boundary conditions the boundary conditions are invariant under independent permutations of bras and kets, define a back-reacted saddleeither top for or the bottom of densitywormhole figure matrix given in the introduction, since the boundary consists of two complete and Figure 11. matrix JHEP04(2021)272 ) ) . ), n 4 4.2 = 2 4.2 (4.3) (4.4) (4.2) n . Such , Sym( in ( copies of ··· n ··· + Sym(2) i that would de- act to permute ) , where n π ) ( π ) indicates various ) admits a natural n U j | 4.2 4.2 n that she would collect ⊗ Hawking Sym( . , ρ i ) ∈ ) in ( n Hawking ( ρ π | π i n ··· n i Hawking , where the ⊗ Hawking ) + ρ ρ -evaporation Hawking radiation n n i · · · h Sym P (1) Sym( i ⊗ · · · ⊗ | π ∝ j of the | black holes and allow them to evaporate, (1) ) π n i n | ( ρ n ), it is useful to introduce a unitary operator is the nontrivial permutation in ⊗ Hawking Hawking ) = – 21 – ρ i ρ ]. This means that we simply measure the swap π 4.2 | n π 1 i i U 53 h ) , ) n n indices: 52 where as a geometric symmetry operator acting on n X X π Sym( Sym( π ∈ ∈ U π U π i ⊗ · · · ⊗ | 1 = i is a projection onto the completely symmetric subspace that is = in the symmetric group = | ( ) S i π π n interiors. But once evaporation has proceeded to completion, we n π ( U ), the rules for our semiclassical path integral, while treating PS ρ U n n ). We will explore the Hilbert space interpretation in more detail in S that the interiors are no longer attached to a corresponding external 4.2 ∈ collections of Hawking radiation: 4.2 π , . . . , j n 1 P j | ! ) 1 n 11(b) n ( interiors as indistinguishable objects obeying Bose statistics. We could say ρ = | n , which acts to exchange the two copies of the radiation. In terms of our previous n identical independent copies of the mixed state S . Sym 6 n P , . . . , i by the diffeomorphism which permutes them. Momentarily dropping the We can now ask what our experimentalist should expect when she tries to verify To understand the implications of ( Since this may at first seem surprising, it is useful to note that ( As a result of ( 1 for each permutation i + h π that the radiation iscopies mixed. and perform For theoperator a swap simple test concrete [ example,notation, we this operator take is the case of we find where invariant under all permutations. states among the We can equivalently think of I state means that we must includeThis a then symmetrisation leads as is to familiar ( section from Bosonic Fock spaces. U spacetime. As a result,treats there the is no longerthat anything each to black distinguish hole them.freedom interior The to is sum describe like in a ( theHawking Bosonic partners. state particle, When of carrying we the trace many these matter internal out, that degrees having formed several of interiors the in black the same hole quantum and the ingoing scribe from a single evaporation. Hilbert space interpretation. Afterwe we collapse must trace out the see from figure potential corrections, including anybeen that identified. from We possible will assume further such saddles corrections that to have bewormholes not negligible as for yet saddles, the imply restcollected of that by section the our state experimenter differs significantly from the state denotes the symmetric group on and where we have not normalised the state. The ellipsis ( density matrix to be given by a sum over all permutations JHEP04(2021)272 11 . it is (4.7) (4.8) (4.5) (4.6) . For ρ of our S Hawking ρ 1 Hawking obtain the . We thus ρ − S copies of the n n ) to summarize always 4.5 th swap Rényi entropy, n , so the outcome of such a τ . , U . 6  ) ) ρ ( n , . ( 2 . )  S ρ n τ − 31 (2) e U . On the other hand, for pure ρ  1 S , and the last equality relates it to the − Sym(  . ) = ρ 2 10 ∈ corresponding to the eigenvalues of ρ as ). For a highly mixed state such as ) = 0 log Tr is invariant under the action of 1 n 1 ± log Tr +1 2.5 (2) – 22 – is invariant under 1 − swap − n ··· copies, ρ ) ) = Tr( n S n n implicit, since it will be clear from context. Once ρ ( := − ρ τ ⊗ ρ := = (1 2 swap 2 . In other words, if we are inspired by ( S τ S S swap n Tr( , and we predict that our experimenter will as defined in ( S ) ) the expectation value is very close to zero. It is thus essentially 1 ρ . We can therefore perform only a handful of measurements and ( − N 2 ) = 1 G S +1 (2) ρ ) it is manifest that -dependence of the state on each copy could potentially be removed by an alternative ) is known as the ‘purity’ of n is not Hermitian, but it can still be measured since it is normal (commutes with its 4.4 in the definition of ) it is manifest that 4.5 π of order n U 4.4 ) = Tr( 2 , S (2) from measurement of ρ 6= 2 is a cyclic permutation of the , the expectation value of such outcomes is S ρ n τ +1 Tr( More generally, any measurement (of more complicated permutations for example, or Nonetheless, the predictions for experiments on any one of the black holes are essen- Now, from ( For There are corrections from nontrivial PS wormholes as described here, but they are exponentially small copies, the density matrix on the remaining radiation is much the same as 10 11 adjoint). This is equivalent to measuringHermitian both operators. its Hermitian and anti-Hermitian parts, which are commuting in the entropy. This prescription for normalising the path integral. See also footnote tially unchanged by then inclusion of theThe primary PS effect wormholes. ofparticular the If black new we hole, contributions trace is butof not over radiation. rather to to alter introduce the correlations state between of several radiation different for sets any again, from ( measurement will always be unity, and even complete tomography tofrom obtain a the pure density state, matrix) as will will reproduce be the made expectations more manifest in section where We leave the then this swap entropydefined vanishes. through This the can expectation value beradiation of as generalised a to permutation an operator acting on find result her observations by defining the ‘swap (Rényi) entropy’ second Rényi entropy (which has equally likely that theguaranteed measurement to gives obtain distinguish reliably between the two cases. swap measurements performed on twomatrix uncorrelated copies of a single (normalised) density The quantity ( measurements have two possible outcomes JHEP04(2021)272 11(b) , which does not , where the iden- int 11 Σ for radiation collected S , and thus two disconnected − . We have attempted to swap I ∪ + . 11(a) I + I (b) A saddle inary which is the effectively swap cancelled on by thenamical an additional bound- swap dy- in the bulk. on – 23 – being joined along the slice Hawking ρ 10 . ) (2) ρ S Tr( is diffeomorphic to the spacetime defined by taking the trace of is obtained from a spacetime wormhole, in the sense that disconnected involves two disconnected copies of 12(a) + The two PS wormholes contributing to the path integral computing the expectation . These are essentially the same saddles pictured in figure I is diffeomorphic to that for the trace of since 12 Hawking ρ 11 For future reference, we note that the expectation value of 12(b) (a) A swap testin saddle the with bulk. naïve connections this argument as a ‘two-replica’ calculation.result Furthermore, as discussed above, theboundaries Hawking become connected through thehappens dynamical due bulk. the In two thereach copies Hawking the of wormhole asymptotic this figure boundary. Bothunitarity features due are to closely the associated with large the entropy failure of of BH 4.2 Wormholes andAs factorization noted above, thematrix path on integral boundaryboundaries. conditions So required despite the to fact compute that it the involves only density one density matrix, we may characterize summing these saddles gives preciselyfigure the same result asand taking the trace of thethe saddles radiation in to check whetherdynamically the hidden this state from is us mixed, by but performing the a gravitational matching path swap integral of has black hole interiors. from two identically-prepared evaporating black holesitational can path be integral. directly formulated Two as saddlein a points figure grav- satisfying these boundarytification conditions of are the shown black holeboundary interiors conditions can be appropriate either to ‘unswapped’ our or ‘swapped’, swap but test now expectation with value. The point is that Figure 12. value of the swap operator, JHEP04(2021)272 . − sets I (4.9) ]. We n ∪ 58 . + I 7.3 acting on ) n , since the current , and then perform ( , O . For this, we would 10  11 − I n ⊗ Hawking ρ ) . We might attempt to proceed + E I ( to factorize into a product of ‘one- π . We postpone interpretation of the U ρ u ) n ( . O 5  Tr ) – 24 – n ). First, we do not expect that our low-energy for a given initial state at X Sym( 10 ∈ + π I =  ), the expectation value of an operator ) n ( 4.4 ρ when studying the path integral of figure ) n E ( , where (along with other difficulties) we will discover them to be O  ]. We shall return to this issue in the discussion of section Tr 61 4.4.2 – to the past of some retarded time 59 + I discussed predictions for the swap test as applied to the entirety of radiation of 4.1 u , and found that they are consistent with a pure state. Here, we will generalise this + I From the PS proposal ( On the other hand, we have seen above that the Polchinski-Strominger proposal gives However, there is no clear way to compute this path integral semiclassically, even after If one believes in BH unitarity, it might thus seem natural to seek a one-replica calcu- I for the replica wormholes introduced in section of Hawking radiation is given by on to ask for thepart predictions of the PSresults proposal to when section we measureinconsistent only with the BH radiation unitarity. on Nevertheless, the this calculation will be a helpful warm-up replica’ S-matrices. Aficionados ofthe the black hole AdS/CFT information correspondence problem willof is thus AdS/CFT a recognize [ special that case of the so-called ‘factorization4.3 problem’ Experiments onSection part of the radiation operationally-defined entropies indicatingnatural the to final expect state whateverculations to physics of lies be the behind above pure. this S-matrixthe operational components. ‘two-replica’ As purity Hawking At to a least wormhole also in calculation result, enable some of it cal- sense, is it should then cause problem affects all of theblack interior hole’s Hawking initial partners size. and scales Resolvingwith with this a a by boundary choosing positive a power condition of prescriptionwill is the to instead equivalent replace take to the the the more singularity not conservative black point offer hole of an final view answer state that to semiclassical proposal this gravity of simply question. [ does there is no obviousapply prescription when we for integrate the overchosen quantum boundary may fields not at conditions be the a or stationary singularity,by and point measure the of the that boundary the spacetime action conditions. we we depending have This on should endpoint is what variations of a are more evaporation allowed severe problem than the one encountered at the making assumptions about theby endpoint using of a single evaporation copy ofthe the evaporating path black hole integral geometry of offinal figure the boundary quantum conditions. fieldssingularity on But (the this we jagged background run with lineeffective into appropriate theory in difficulty initial will figure due and be to valid the in presence the of high-curvature the regions future near the singularity. Second, lation that describes black holemight evaporation. try Rather to than concentrating compute ontion components observables, of we of the the Hawking S-matrixlike radiation directly, to or at compute equivalently the the path wavefunc- integral with boundary conditions on a single copy of JHEP04(2021)272 , u on I + is the (4.12) (4.13) (4.10) (4.11) I ) ) comes u ( terms, but 4.12 ! of the cyclic n , in which case 1 ) . − u . As before, we  ( τ Hawking 2 u . (or more generally S ) of the radiation on . As a result, the u ) ) u + 2 u I in the tensor product Hawking n ( ( I ⊗ Hawking ¯ i I S ) 1 ( ) , ρ ), the two terms give u − ) u S , τ ( o ≈ . in I +  ) , where o U ( ) ) ) ) u n 4.12 u 23 to include the region u I u S ( ( u Hawking ( 2 ( ( ( ρ ) I ¯ S ) ) = S Hawking 2 + ) and the inverse u ¯ . The minimum in ( )), but now acting only on + S u I I − 1 I ( ( I e Hawking 2 Hawking 4.8 n 13 Hawking ( n ( τ π S S 1 ¯ Hawking 2 S is of precisely the same form as the S + U U − − ¯ , S  )  e τ ) ,  u ) u ( U sets of radiation, there are ( ) u (2) ( u n ρ + Tr ) I log Tr u (  In analogy with ( τ Hawking 2 1 I S ( U – 25 – Hawking n 12 1 − − Hawking S 2 S e ) are shown in  n S 2 h n from equation ( ) that is swapped. To understand the contribution n ⊗ Hawking ), this yields . However, except for the term defined by the identity − u is the complement of Tr ρ τ 4.12 1 log ) I min u u 4.5 , where there are two terms: min = I I ) := ∼ ) ( u ∼ − ∼ n ) ( = 2 ( S ) u  ( O n u ( replaced with the Rényi entropy swap n are both very large. The two geometries of the path integral ) . From ( S swap n , where ) = Tr u swap 2 S ) ( u S  u ( I (2) , all terms in this normalization factor are exponentially small. Thus ( ρ Hawking 1 S ) ρ u ). Hawking 2 = Hawking 2 ⊗ S I ¯ ( 5.1 S π ) = S , +  ) u I for the cyclic permutation ( Tr ( Hawking ) S ρ ) u u Generalising this to cyclic permutation on We begin with the case We will ask for predictions when we measure a swap operator associated with the Hawking state. Thus we find I There is an exception when both terms are comparable ( I u 12 τ ( Hawking 2 with the second coming from the relation additional permutations give further interesting corrections: see footnote from choosing only the dominant saddle. only two terms are important,permutation the we identity are permutation measuring. Otherto PS at wormholes least are one exponentially of suppressed the relative two included terms. where we mayS approximate thecorresponding function to the as computation ( a minimum of the two terms because contribution of the secondfirst, term but to with I The first term isstate the expectation value ofsecond the Rényi swap entropy operator ofof the the part second ( term, noteS that the product of two swap operators is again a swap operator: generalising ( U capturing the Hawking radiationencode that the emerges result before in the a retarded ‘swap time Rényi entropy’ which the permutation operator acts.factor Strictly determined speaking by we setting should divide bypermutation a normalisation the resulting corrections are negligible. where we have here used the more explicit notation JHEP04(2021)272 once E , the expec- ).
(2) 4.12 ρ ) u These difficulties I ( ] in making the PS 13 S ) and ( 14 Tr 4.11 ? E (b) The geometry with swappedthe interiors second gives term in ( for two sets of radiation. – 26 – u I may suggest that the semiclassical gravity predictions for by appealing to the recently-discovered replica wormholes 5 4.1 ] concluded in their context that black holes in fact violate BH 14 ,[ ). 4.4.2 4.12 The two PS wormholes contributing to the computation of ]. Nonetheless, we will first discuss the issues in more detail so we can better ) and ( 19 , 4.11 These problems are described below. Using arguments related to the problem that will Here the term ‘remnant’ means an object with unbounded entropy (that is, infinitely many internal 18 13 (a) The geometry givingin rise ( to the first term assumption, but it wouldconclusions be without a such assumptions, greatthroughout and improvement the with if calculation. the we semiclassical were approximation able justified to arrive at thestates) same below a fixed mass. 4.4.1 What happens atAs the observed above, endpoint we of losethe evaporation semiclassical black control hole near the is endpoint of of Planckian evaporation size. We have thus far followed [ be described in unitarity and instead describedwill black all be holes resolved asof in ‘long-lived section [ remnants’. appreciate this resolution. The observations of section an asymptotic observer alwaysHowever, conspire if the to only relevant produce contributions fromwith results the further path consistent integral consideration with are one those BH discussed still above, unitarity. finds serious problems. Figure 13. tation value of a swap operator acting on 4.4 Challenges for the Polchinski-Strominger proposal JHEP04(2021)272 is are , it thus ) M 1 (4.14) u ) mono- ( . Since u ) ( u decreases u ( ) swap u ( Hawking S ¯ S swap Hawking gives the total en- S S ) and u obtained by formally ) ( ) u ( u . monotonically decreases ( . From this, we can solve o ) 2 ) ) is dominated by the first u − u ( Hawking swap ( D ¯ S S before retarded time Hawking 4.14 R S 1 + )+ − u . The swap entropy in the Hawking saddle exceeds the I . In particular, G Hawking ( Hawking , so ( u ¯ ) S ¯ + S . The emitted power (energy per unit I u , ( ) I R u 13(a) ( Hawking and on S ) becomes dominant, and u , while ) Hawking I ∞ spacetime dimensions the black hole mass ¯ ( S ∞ – 27 – Hawking ( , at which point there is a first order phase tran- D , where we computed the expectation value of a in the Hawking saddle, up to order one corrections S < ) 13(b) n u + ) ( , and thermal entropy is produced at a rate per unit u 4.3 2 is proportional to ( I Hawking − min of the initial black hole by a factor of order one. This Hawking S R BH S ∼ S Hawking ), BH ) ¯ Hawking S S u ( S 4.13 , and ) = . Meanwhile, in 3 swap u 1 − ( S − D R R 1 of all radiation at limit of ( − ) Hawking G 1 ∞ S ( . → u n respectively. In particular, the sum as a prediction for the von Neumann entropy that an asymptotic observer would u ) Hawking u S ( We can be very explicit for Schwarzschild black holes evaporating by production of At early times we have To understand these quantities, we must simply note that Hawking’s state does not swap dimensional analysis. In particular,the the geometry, production which of provides Hawking the radiationtime) only is length is determined scale therefore by proportional to time proportional to proportional to discrepancy occurs because blackhence hole produces thermal evaporation entropy; is the generalized thermodynamicallyin second irreversible the law is Hawking and not saddle. saturated by evaporation massless particles, since the various entropies are determined as a function of time by sition, the second saddle-pointback in to figure zero.disagrees While quantitatively this and isThe we key qualitatively find point very a is much thatBekenstein-Hawking result like the entropy which entropy the is Page incompatible curve with in BH figure unitarity. tonically increases from zerobetween the to same values. term, corresponding to theincreases saddle-point until in figure contain significant long-range correlations, sothermal can states be emitted regarded at as different a times.well-approximated product by This of the means uncorrelated that thermal entropythe of time Hawking radiation emittedtropy before and after from the vicinity of the boundary between S deduce by performing measurementsbefore time on many copies of the Hawking radiation emitted However, the same considerations apply directly to Rényi entropies as well. We interpret We now discuss thecyclic result permutation of acting section onvon the Neumann radiation entropies arriving are at the more calculation familiar in and terms more oftaking the physical the ‘swap than von Rényis, Neumann we entropy’ will phrase 4.4.2 Violations of BH unitarity JHEP04(2021)272 , ]; E ) > 0 u 14 was u 63 ( u , (0) (4.15) BH 62 S for which 0 at time Hawking u S + I exceeds , ) 1 0 u u ( , > , we have ) # 1 2 ∞ − − (0) 14 ( D D T Hawking , (0) BH ¯  S 2 1 E BH − − u u S > S D D , Hawking ) = )  − 0 S 2 1 u E 1 ∞ u ( − − ( u  = D D (increasing black curve) to agreement at

 − ) − . For four-dimensional black holes, Page u r E 1 1 ( u BH depends on details of the endpoint of evaporation u "  Hawking du e Hawking ) ) ) g S dS is the time taken for complete evaporation. S a − ∞

( P ∞ ∞ 1 E ( ( u . u Hawking  – 28 – S (0) Hawking S du Hawking Hawking BH with Hawking S S S u dS ) = ) = ) = = . In particular, u u u u ( ( ( . However, because BH ) 0 BH S (decreasing black curve). The transition occurs at the p u S as the Bekenstein-Hawking of a black hole with mass given by the Bondi mass ( a Hawking ) 15 . Hawking Hawking . ¯ w ) u 1 S ¯ s ( S S u − N ( 48 S . G = 1 BH -dependent swap entropy computed from PS saddles (solid black curve) rapidly r Hawking S u ≈ ¯ S Hawking r ¯ S The ) = 0 is the spacetime dimension and u . Equivalently, this is entropy of the black hole when the radiation arriving at ( with and in fact also at all u D u (0) We can define The total entropy of Hawking radiation 15 14 Hawking BH at time emitted, where the preciseis definition a of positive emission power time of is not important sincebeyond the semiclassical evaporation physics. timescale We(by can our PS safely assumption) ignore scale these with details, the since original the size effect of on the black the hole. entropy does not one finds a violation of BHhas unitarity computed by a the factor corresponding of ratiofor for example, von for Neumann Schwarzschild entropies black inhe holes various computed radiating cases [ by emission of gravitons and photons which depends in detailpoint that on is the important dynamics for through us is greybody that factors. it is However, greater the than one. only Indeed, as shown in figure where The only undetermined parameter relevant for us is the (constant) ratio everything up to a few unknown dimensionless constants to find large S S violating BH unitarity. Figure 14. transitions from agreement at small JHEP04(2021)272 , ) ]). u ( , so 16 , 64 4 10 , 5 Hawking 2 , ¯ S 4 -evaporation n and ) u ( Hawking 2 S . . Recall that this is an expression for 5.4 4.3 copies of a subset of Hawking radiation. As n is also very large. But the Bekenstein-Hawking – 29 – ) u ( swap 2 S acting on ) u I , it predicts the swap entropy measured by our experimenter at above, in our context giving the expectation value of the cyclic ( u τ I U 4.3 , as discussed in section u on , which we explore in section ) ) u n ( ( ρ to depend on the entire future of the black hole! This is particularly sharp is fixed by the current mass of the black hole. So from this analysis it would u ) entails a possible violation of causality. In particular, since it involves the Hawking BH 2 ¯ S S 4.12 The above paragraphs describe a problematic violation of entropy bounds, but only ]. We are now able to make this more concrete, since in the past year a new class of ; such further corrections will be explored in the next section. 5.1 Replica wormhole spacetimes In a sense, replica wormholeswe are first a revisit generalisation these of inwe PS a will wormholes way studied that reconsider in is the section suggestivefinite swap of entropies retarded the of required time the generalisation. Hawking Specifically, radiation that emerges before some quantitatively, via a path integralcan over be spacetimes trusted where everywhere. thecontribute semiclassical This to approximation implies other that observables,density the and matrix replica in wormhole general geometries to must also the components of the saddles have been argued towill exist. shortly These become are clear. knownform as They studied replica in were wormholes section discovered for aspermutation reasons contributions operators that to pathwe integrals review of below, the the replica wormholes reproduce the expectations from the Page curve 5 Replica wormholes It is natural to ask ifSimilar the ideas above have challenges been might be investigated17 resolved in by various finding forms further for new many saddles. years; see e.g. [ depending on the swap entropyto obtained. even Such throw violations of thethat causality consistency a appear consistent of large framework enough above will require calculationsu additional into corrections doubt. to the swap We entropy take at finite this to suggest Perhaps an eventhat greater ( problem thanentropy the failurea of finite BH time unitarityif is we imagine the first observation (or performing at this an measurement AdS at boundary), a whence finite we distance can from subsequently throw the matter black into hole the black hole entropy appear that black holesmass, a have serious an failure unbounded of BH number unitarity. of internal4.4.3 states below Violations any of given causality by an order one ratio.problem by However, refusing as to let is theso familiar black that hole from it evaporate other remains freely discussions, and at insteadIf a we feeding given can this it size magnify with for time matter the as iswill long as very both we long, become desire (perhaps very then even large, in eternally so as the in middle [ of this period JHEP04(2021)272 . is int ext Σ Σ Σ to be 4.4.1 u Σ . Indeed, . The ex- I which acts ∂ ext τ to match the Σ = of the Cauchy ) are built from γ u int Σ Σ copies of Hawking = 2 if we like). In this ] for further details . These spacetimes n n + 66 I for that is expected to emerge , . . . , n = . The resulting spacetime ], and [ = 1 ext = 0 (the ‘island’) generalising 13(b) Σ r r 7.3.4 65 applied to , where the boundary conditions , I u ) ; this is also fixed by the boundary 15 u r ¯ I . Next, we have an exterior piece ( M ) is fixed by the boundary conditions, so τ which we continue to call r ( with any choice of permutation we desire. U u τ u I ¯ Σ int M I of the evaporating black hole spacetime and , the most interesting possibility again arises to match the cyclic permutation joins to ) r which stays far from regions of strong curvature, r u I – 30 – where they are sewn together. The surface u I M M at retarded time ( Σ along Σ to the regular origin τ joins to 0 + r U + r ¯ I M I M . to where we sew the replicas, and to also allow a more general on r . Here, the boundary conditions do not uniquely specify any , labelled by a replica index meet at a common codimension-2 boundary , so u Σ r = 2 begins. Finally, we have a Cauchy surface for the black hole E with a nontrivial permutation. For the boundary conditions τ , where the boundary conditions require us to join spacetimes ) arose from choosing the permutation on M ¯ n ext ‘ket’ replicas + M I ext Σ n I will have several past light cones, one for each bra spacetime that Σ 4.13 for γ imposed by the boundary conditions. on and u 15 u I extends to meet and the resulting loss of semiclassical control discussed in section I , which require such an identification in a neighborhood to the future of I + E . , where ext on u u I Σ τ I , reaching from the original regular origin (before the black hole evaporates) (and also one for each ket spacetime). This is an important feature, but we int γ Σ . The novelty of the replica wormholes is that we take the Cauchy surface u We can already see why such spacetimes might avoid the PS-wormhole’s dependence This description also applies to replica wormholes, but generalised to allow a more gen- The PS wormholes for this amplitude (pictured in figure pieces, consisting of I conjugate ‘bra’ replicas n and deferring discussion ofis further depicted implications in to figure section on physics near By joining replicas along a Cauchy slice on connected, so that for Lorentz-signature spacetimes of this form,singularity: the points causal on structure must havemeets an interesting at will treat it only briefly below, referring the reader to [ terior surface specify that bra andthe ket interior spacetimes island pieces arecomputing connected the in expectation the valuewhen trivial of we way, take but the sewing we permutation sew on along eral choice of Cauchy surface splitting of this surface intomust pieces. remain The unchanged, region butsurface we is are split free into toin two choose the pieces: discussion how a above, the partial and remainder its Cauchy complement surface in The path integral includes agiven sum calculation over is all dynamically possibilities, determined. and Into the particular, dominant the the permutation interesting swap for new contribution entropies a permutation ( conditions on retarded time interior to the evaporation endpoint sewing rule, and we can join divided into three pieces,we with have a a different region rulewith for the sewing cyclic replicas permutation alongstretching each from retarded piece. time First, after the final evaporationregion, of we the sew without black permutation, hole so (we can take radiation on 2 n terminate at a future Cauchy surface the expectation value of the cyclic permutation JHEP04(2021)272 along replica . ext n Σ = 2 t x e n Σ and I γ imposed by the acting on the is- I ) u π I ( τ . We will see that such U 9 of the replica number into two pieces 1 u Σ — is excluded from the spacetime → n E are the same as in (a), but are swapped . I ext Σ (b) A replicasame boundary wormhole conditions, geometry obtainedby from with changing (a) the the land’ identifications along the ‘is- lying to the past of the future lightcone of in the resulting geometry will be determined E , and then perform the integral over metrics. γ I – 31 – , and incorporating such back-reaction will make the γ , as well as the operator I u < u . 9 u Σ . The connections along γ in a given single-sheeted spacetime result in different geometries -sheeted spacetime which includes the insertion of a permutation for two sets of Hawking radiation. The right spacetime is an n γ u I acting on the island . The configuration shown is not a saddle as it does not incorporate back-reaction (and we might also sum over nontrivial permutations I ) . This arises from two copies of I u int ( Two geometries in the path integral contributing to the expectation value of the swap τ I Σ U -sheeted whole, so our gravitational path integral integrates over all inequivalent n The matter path integral in this replica wormhole spacetime is a Schwinger-Keldysh (after the black hole forms), and thus remove the dependence on UV physics until the (a) The ‘trivial’ geometryconditions with appropriate the for boundary acting the on swap operator the Hawking wormhole in figure Different choices of for the choices of land). If saddle-points exist, thedynamically location by of extremizing an appropriate action. path integral on an operator boundary conditions. In principle, wespacetime, should in compute this particular for for every such all replica choices wormhole of the entire singularity — and inunder particular consideration, the endpoint just asreplica for wormholes the exist Hawking wormhole forE in all figure times black hole reaches Planckian dimensions. from the structure nearspacetime the metric special complex. surface Thatrequires deforming is, it passing away the fromcoincide real contour Lorentzian with of metrics. the integration (real) However, through the Hawking the replica saddle wormhole desired in saddle saddle the will formal limit Figure 15. operator acting on wormhole. To obtain this, wethe divide codimension-2 the partial surface Cauchy surface along the island JHEP04(2021)272 . τ n u = 1 . To (5.2) (5.1) ] ]. See n ]. The γ . This [ 73 A 1) 72 , − acting on 65 n . The general τ ( n need not be an n is not infinitesimal. at retarded time . These data are equiva- 1 + γ However, we are able ], the condition for a − ] for saddles with Lorentz- we introduce nontrivial I n 16 . 19 66 , is trivial and there is only ], building on [ .  by the same permutation . We will reformulate the ) ) (up to equivalence under changes 18 n u = 1 71 I ( I int ρ I n to first order in ) and by = 1) Σ u γ limit defines the ‘swap (von Neu- n + I I ∪ 0 ( ( 1 , encoded in the ‘swap entropy’ 17 τ u U → → Sym(  1 n matter S − + n log Tr is a quantum extremal surface (QES) [ . ] bounded by ) 1 I N I u – 32 – ∂ ( ∂ 1 [ G − ext 4 is still under exploration. A Σ n = swap 1 n γ − S , studying the path integrals with boundary conditions ) = 1 and taking the u → ; n > ) := n n 4.3 I , but can still state the calculation in terms of the u was found some time ago: see [ that computes the normalization of the state. Nonetheless, ( ( I ) 1) u gen ( swap S − n ρ S n ) := lim ( Tr by a general partial Cauchy surface u as a functional of the partial Cauchy surface ( int exactly the permutation group gen . Our strategy will be to emulate this for replica wormholes as described S swap Σ , we found a new interesting contribution to this path integral arising when u , studying the problem for S ] for related constructions in Euclidean signature and [ n I = 1 70 4.3 – n 67 , 18 ] for discussion of saddles for real-time path integrals when 66 Before reviewing the argument, we recall the definition of a QES. This is a ‘quantum In section See [ We have written 16 17 signature boundary conditions analogous to those considered here. that leave the domain oflent dependence unless invariant), our rather than spacetime its includesboundary). bounding a surface closed universe component (that is, a partial Cauchy slice with empty a quantum corrected version, the generalised entropy: Since the matter fields are pureentropy on on a the full partial Cauchy surface, Cauchy the surface second term is also the matter condition is that the splitting surface also [ version’ of an extremal surface,go which from is ‘classical’ a to stationary ‘quantum’ point extremal of surface, the we simply area replace functional the area function with the original saddle for by continuing the problemdependence to on study the a choicegeometry of neighbourhood and of associatedsaddle matter to state. exist at order As pointed out in [ we chose to join theas replicas we along apply the on blackabove, hole interiors replacing calculation of the pathinteger. integral For on such geometries in such a way that Continuing this to non-integer mann) entropy’ will not only beof convenient, but Rényi also entropies physically givesthe interesting, same the since observables von the as section corresponding Neumannappropriate limit for entropy. computing Specifically, thecopies we of expectation the will value radiation first of emitted compute before a retarded cyclic time permutation The interesting question nowsemiclassical is saddle whether for this given boundary replicacase conditions wormhole for at topology integer some can replica replicato yield number number make a more new progressto by non-integer considering a formal analytic continuation of the calculation 5.2 Quantum extremal surfaces JHEP04(2021)272 , , ) ] u N u u γ [ G I I A 5.2 4 (5.4) (5.3) times ∪ n I ∪ ext ]. These : that is, produced ] (see also Σ 80 u using some – 76 I – I ∪ 78 ∪ 74 : matter + ext replicas of a single S Σ I n . 2 ) . u ) portions of this Cauchy (using the ‘bare’ value I ∪ I I ∪ u u 1 I I∪ − N replica symmetric I ( . This is much the same as Σ = n G u S I I ∪ between replicas. It suffices to 1) and ( n − I∪ τ I n ρ S ( . We can express this in terms of − 1) − e − n I  n is stationary to first order variations of ( is finite and not UV sensitive, since matter − gen (1) matter S gen Z  S is the von Neumann entropy of matter fields on These spacetimes are – 33 – (1) matter -replica matter path integral as ) Z u . n 18 ) = . γ u is that I  log I ) gives the normalisation of the state on the unreplicated I . n n . The matter effective action is thus given by ( n I∪ ∂ + n I ∪ ρ γ )). Relatedly, if the theory has higher derivative terms ρ ) ( ) and a finite subtracted expression for = u I = u I 5.2 N I γ ( ) I∪ G , except we now are computing the Rényi entropy on τ -fold cyclic symmetry possessed by the boundary conditions, as matter = Tr( n ρ ( matter n U S Z  2.3 ), ) n Tr ( log matter = Tr( 5.2 Z n ]) that the combination ) 77 . We can thus write the for the matter state on the final Cauchy surface u (1) matter of them after CPT conjugation) to the past of a Cauchy surface ) Z I n ( effective action, plus a term from the Rényi entropy on ∪ I u in the state under consideration. For a matter QFT, this entropy is divergent and reminds the reader that this matter entropy term depends on where we choose this at the EFT cutoff in ( I = 1 u ( N First, we compute the matter path integral on such a geometry. As noted above, the We now sketch how the generalised entropy functional and the QES prescription arise The definition of a QES τ gen I n Here we depart from the historical presentation of the arguments in order to simplify the discussion. G S U 18 . In the definition ( The factor geometry, and is independentthe of the Rényi entropy of thethe matter discussion reduced in density section matrix not just on This will enable us to describe the problem inreplica terms wormhole geometry of is a the singleof Schwinger-Keldysh copy. contour giving theby expectation unitary value evolution from the initial conditions on spacetime ( and gluing them along that Cauchy surface,surface though we on perform the this gluingcheck using that a we cyclic have permutation rations, a since saddle-point the varying symmetry only ensures amongst stationarity such to replica variations which symmetric break configu- this symmetry. from the path integral, byditions looking for computing for replica wormholethe geometry saddle-points respects with the boundary con- well as the two-fold symmetrycontour. swapping ‘bra’ We are and thus ‘ket’ considering branches metrics of that the are Schwinger-Keldysh obtained by taking features are not specialto to black replica hole wormholes, thermodynamics, butusing are the for finite familiar example. IR fromconvenient value quantum regulator Operationally, of corrections which it is suffices local at to evaluate ( depends on the choice ofthe UV appendix cutoff. of Nevertheless, [ there isfields strong give evidence an [ equalof and opposite infiniteor renormalisation non-minimal to couplings toterm gravity should (perhaps be induced replaced by by quantum the effects) corresponding then notion the of geometric entropy [ in partial Cauchy surface to meet γ I ∪ This is the more standard way of describing a generalized entropy. The final argument JHEP04(2021)272 ) I = 2 -copy n n appearing , which lies for γ ) + with opening 16 γ I I ∪ ( . In particular, since , leaving a cut in the cone = 1 ]) there is an additional matter ext n 69 S Σ replicas along the surface n ]. 86 – 83 , we have succeeded in describing the matter 19 1 . – 34 – γ , one cone corresponds to the top two spacetimes (one ≥ with a swap (or more generally, a cyclic permutation) as -sheeted geometry (sketched in figure n However, we can find a saddle-point by solving the n I 15(b) . To understand this, it is helpful to imagine deforming 20 γ so that we satisfy the equations of motion there, but this replica wormhole geometry near the splitting surface ) when we continue this close to is not restricted to be an integer. We now do the same for γ n 5.2 ] suggesting that quantum back-reaction should be large at singularities in = 2 times the action on a single copy. This is almost true, but as n 82 n , 81 , and joining them back together with cyclic identifications. Our -dependent way. γ n ] (following similar Euclidean observations in [ ), here computing the entropy of gravitons. In particular, the path integral thus incorpo- ). In this vicinity, the 65 γ 3.1 and so do not accumulate to become significant. A sketch of the γ . In particular, this requires back-reaction that will modify the geometry on each π . The two cones are joined along n replicas is simply 2 I Since the Einstein-Hilbert action is local, it is tempting to say that the action on n Smoothness of the Euclidean geometry implies that such back-reaction can be small and that a one- What we have called the matter path integral should include linearised metric fluctuations as explained 20 19 rates small deviations fromare replica-symmetric local metrics. at In practice, this is ratherloop subtle, perturbative but treatment the is subtleties self-consistent.relates It to would classic be analyses interesting [ tothe understand causal in structure more such detail asof how that this such associated singularities with may our be splitting related surface, to and a to conjectures Morse that index the [ presence angle replica in some after equation ( time direction, so thatvery we close can to think ofis the obtained geometry from as the beingemanating metric Euclidean from on (at a least singlegeometry locally, copy must by be slicing smooth the implies at that each single geometry is not smooth: it has a conical defect at our described in [ local contribution at the surface the Schwinger-Keldysh contour to pass through our final Cauchy surface in an imaginary This extra term will becomein the the matter generalised von entropy Neumann ( the entropy Rényi entropy isintegral defined in for such any athe way integral that over replica-symmetric metrics. ‘ket’ and one ‘bra’) andalong the other to the bottomindicated two, by identified the along colours andreplica arrows. wormhole spacetime The conical to defect be in smooth each at replica is required for the resulting Figure 16. at the tip of the cones. Comparing to JHEP04(2021)272 1 → (5.6) (5.7) (5.5) ) and 22 defect n . There π 5.5 n 2 ] by 1) − in ( 69 . , n γ ( times what we (1) EH 65 S n ) and the matter 1) , and in the n -copy metric, which 5.5  n But this contribution within that geometry. 1 γ ] for details of this procedure − 21 . n 66 γ ( ]. . ] 66 , N ) γ ] and [ [ u ) ; G u 15 A ; I 4 ( I ) ( into an integral in the directions parallel to 1 gen ] for a full discussion of the Euclidean case. − S gen R n S 87 1) R 1) − − N n − ( n 1 (1 ( πG − e − 16 ). Together, these just give − – 35 – γ e 5.3 (1) QES EH D I X ∂ Z niS = γ = from ( , namely the local gravitational action ) , the second class of terms give a small correction so we ∼ −→ γ n n 1 , and which we can study for small values of ( ) EH  — plus matter effective action). Secondly, we have the two n -replica manifold. See [ ) n iS n  ( ρ , the weighting is provided by the second set of terms: 1 6= 1 ) (1) matter γ u Z n − , which is a problem which can be continued in ) defines a good variational principle on the singular single copy defined by ( I using the Gauss-Bonnet theorem. See [ γ ( n 5.5 γ π U  , and not by a phase. The same basic phenomenon was observed long ] N γ Tr [ ). For from curvature with delta-function support at G limit, the area term above gives the geometric term in the generalised en- A 4 ] -fold replicated geometry. ) 1 N 5.3 γ 1 [ n G − -fold quotient of the A 4 ). Note that this is a real contribution when the action is evaluated in Euclidean ]. n n → -replica action and the singular Einstein-Hilbert action, we must subtract this ‘by ) − 1 n 15 n 5.2 (1 − − n For the integral over The above considerations fix a saddle-point geometry on a single replica. However, We now have a description for the path integral over replica symmetric configurations e For one way to see this, we can split the action It is also true that ( − 21 22 , which gives the area, and a transverse two-dimensional integral. We can evaluate the latter integral on γ a small circle centred on in Lorentz signature near singularities in the causaltaking structure the of the formThe associated Lorentz with signature case follows by analytic continuation; see also [ such contributions cancel completely in the final expression. there remains a residualNote integral that over we codimension requirefor two a the surfaces saddle full point for this integral as well if we are to specify a saddle from the singularity for terms making up theentropy generalised in entropy, ( namely thecan ignore area them term at in first,state. obtaining ( simply Since the we path integral will that also computes need the norm to of divide the by this result to get our final expectation value, that we can nicely continue in are two types ofterms term which appearing are in independent the of the action which normalising weights factor thiswill integral. call First, the single-copy we action have (that is, the gravitational action — including a contribution tropy ( signature, so despite the Lorentziantial setting it weightsago the in path [ integral by the exponen- is smooth and socorrect has no such singularhand’. piece. The gravitational To action make on up the for replica this manifold difference is therefore between given the [ In the boundary condition at limit we return togravitational the action original on this smooth single-copy geometry. singular(1 configuration, Now, we would if find we ashould were contribution not simply be to there, evaluate since the we are really evaluating the action on the equations of motion from varying the metric on a single copy while imposing the JHEP04(2021)272 ]. 96 and ] for (5.8) ] and I to the 66 95 and can , while the = 1 1) . But since 1 are ensured − N n − (1) EH G gen n S ] first stated by ( S is finite and not saddle. Recall that 94 , gen will back-react on the S 93 = 1 γ n the dominant saddle is given . 1 ) such that it is stationary to first u Expressing this in terms of the ; γ is naturally of order I ( near . These features of 23 ) γ u n gen ], other saddles can sometimes play important I S . But their inclusion appears to only strengthen in saddles for real-time path integrals. ∪ . A trivial case is when the island the singularity at 89 1 , gen 9 1 int S QES − , this effect is quadratic in 88 , I (Σ n min ∂ (1) 18 EH n > = – 36 – , it has become known as the ‘island formula’ [ ext S γ I S ∼ ] for related comments on corrections near transitions in which + ) ] ]. In this context where the quantum extremal surface 92 γ u G [ – ( 4 72 A 90 swap ) = as a limit of partition functions over replica manifolds which S reduced the study of replica wormholes near ) are precisely the quantum extremal surfaces, since these are u takes the smallest value. ; gen γ 5.6 ( 5.2 for island bounded by surfaces is stationary. We may attempt to approximate this by including S ], following generalisations to time-dependent situations [ gen S gen 73 gen gen S S S ) is a version of the Ryu-Takayanagi formula [ ), we can summarise the resulting replica wormhole contribution by the 5.8 5.1 we are at stationary point for = 1 The first term in Incidentally, the argument reveals why we should expect that The result ( The saddle-points of ( This is not always sufficient. As described in [ n 23 is compact and hence bounds an island study of quantum extremal surfacesfor in us the this original is semiclassical the ‘Hawking wormhole’ in figure roles — especially when twothe QESs arguments has presented similar here. values of Seesaddles also [ exchange dominance. long times involved there is nonetheless a QES. 5.3 Contributions from replicaThe wormholes discussion of section if we have a sensible effective theory. second matter entropy term willthat typically in be most circumstances, athis a small QES correction in will of not be order alwaysnontrivial one. close classical the to extremal This case. a surface means classical but, as In extremal we surface. particular, will see However, for presently, evaporating due to black the holes parametrically there may be no any such dual descriptionoutcome so of our ‘swap’ interpretation experiments is performed rather on multiple different, sets instead ofUV predicting Hawking sensitive. the radiation. We obtain are smooth, with no singular features at the surface Engelhardt and Wall [ inclusions of quantum corrections [ γ These were all originallyterpreted stated as in a the von context Neumann of entropy of holographic a duality, dual with quantum the system. result in- Here, we do not assume That is, we evaluate order variations, and if there are multiple such surfaces we choose the smallest result. the points at which only the dominant saddle-point andby noting the that term for inswap entropy which ( simple formula in principle we should alsosingle-replica realize metric that and for thus changeat the value of the single-replicathus action be ignored for thefurther purpose discussion of of computing back-reaction our at swap finite von Neumann entropy; see [ where the last step indicates a saddle-point evaluation of the integral over surfaces. Now, JHEP04(2021)272 for γ (5.9) . The E u is close to ) u is well-defined ; I ) ]. Those works ( u ( 21 gen , S app v 20 . 17 . o ) u ) ( u ( BH p exists very close to the event horizon , the apparent horizon of the black p u u ) a ,S ) v u ( u ( is defined as the ingoing time which intersects ). To locate the QES, we first define the swap ) S N u ( G Hawking – 37 – S app . This QES thus becomes dominant after the Page n ) v γ u , as shown by the dashed blue null lines. The QES . Given our spherical symmetry, we may define the ( . This is sketched in figure ) u β min u exists soon after the black hole forms (after roughly a BH ( is dominated by the area term, so , just behind the event horizon. S ∼ , with the corrections to this advanced time being of order ) γ γ ) u app ( u v ( swap app for the swap entropy. It remains to ask whether there might = v app S to follow the Page curve: ) v v are empty, in which case we obtain the original Hawking result u ) u I I ( ( ∂ = swap so that for a given outgoing time S matter γ ) S u ( ] showed that a QES computing A sketch of the location of the nontrivial QES in an evaporating black hole. The soon after formation of the black hole, up until the evaporation time ) = app 21 u u , v lies at ingoing time ( ) 20 u ( The generalised entropy of This is precisely the question that was studied in references [ Hawking swap the Bekenstein-Hawking entropy time and causes stationary under variations in thethe event outgoing horizon null since direction. the blackfor This hole times is surface evaporating, is so slightly theworks outside [ function at advanced time close to the inverse black hole temperature showed that a non-trivialscrambling QES time, whichfunction is logarithmic in hole lies at ingoingapparent time horizon as the (spherical) surface on which the area of the transverse sphere is hence the QES S also be a nontrivial QES in this spacetime. Figure 17. red curve isthe the outward location null of direction.the the apparent The apparent horizon function horizon, atS outgoing where time the area is stationary to variations in JHEP04(2021)272 4.4.3 . The in- γ . We have u copies of the to be zero, the n E ]). We emphasise ] with AdS asymp- on 28 τ . Indeed, this is what – 21 , term would be station- E ] 24 20 γ G [ , giving a third term over 4 A ) is located near the past light u ( γ before retarded time ) as a reasonable extrapolation of to approach Hawking + E , we can think of the PS wormholes ¯ S γ 4 I at which the by at least a factor of order one. Since ⊂ γ ) = can vanish on a nearby surface ) u u > u , the replica wormhole generalized entropy u . This provides some justification for using u ; ( E E I gen u int S – 38 – (Σ → u < u . Furthermore, since u Hawking gen . If we choose the area term from -replica system. Indeed, distinguishing between two possible ¯ S S n E 4.4.1 is stationary to variations in ingoing and outgoing null = ). Since this term arises from saddle-points which are not identically-prepared black holes in the same universe to γ gen n , we also avoid the causality issues described in section 5.9 S + in the limit I γ , so that is smaller than int ) u who allows ( = Σ , with a focus on the region . This models the actual results of measurements made by a sophisticated 24 BH u + I S I I ∼ under semiclassical control, where we take ) u ; are I In summary, we have considered a context where an asymptotically flat black hole We thus no longer require any input beyond semiclassical physics or assumptions about To make contact with the discussion of section The physics that allows such a QES to exist is rather generic, and in particular is ( For a black hole above the Planck scale, the experimentalist must be very sophisticated indeed, since the 24 , resolving the problem of section gen requires exponentially many copies of the then studied the expectationradiation value in of cyclicexperimentalist permutation operators relevant expectation values are exponentially small. As a result, distinguishing between the two branches which will happen to thethe QES PS wormholes aftercontrolled calculations evaporation at (i.e., earlier for times. radiates to that we ignore. E cone of the relevant cutfor of the PS proposal. However, we can think of the PS wormholes as a limit of saddle-points in the presence ofis the in new any QES case andS irrelevant. the For accompanying any replica time wormholes,these we quantities are see both thatnever large, it the dominate, difference and is in also fact large. can At only follows provide that PS at wormholes most an exponentially small correction as spacetimes of the abovehole form interior, in which wecorresponding simply generalised take entropy the is islandwhich to we should be minimise the in entireunder ( black semiclassical control, it is unclear whether or not it should really be allowed. But large gradients in entropyand arising asymptotic from region. the large Asto relative a any boost result, saddle between the ofeffective the corresponding the action near-horizon replica classical as wormholes discussed action, are above. but not only related exist as saddles of the quantum-corrected area terms. This isto possible the due event horizon, to producing thedetailed a arguments, exponential logarithmically we divergence growing refer of contribution the to outgoingtotics reader the geodesics to (though entropy. the near for For original similar referencesthat [ there calculations is with no classical flat extremalary surface asymptotics close on see to its [ own. The entropy term is thus critically important for the extremisation, with independent of the dimension orit asymptotics is of sufficient the to spacetime.directions. argue Using that spherical The symmetry, outgoingunsurprising variation that of the the outgoing area variationgoing of vanishes variation on is more the subtle, apparent requiring horizon, a so balance it between quantum is entropy and classical JHEP04(2021)272 that (5.10) ) u ( ) n , and give , and only ( ) is simply 1 ) ρ u ; we interpret ( ) u , 5.10 n i ( constructions in n > I n limit. It remains ρ 1 = 2 → , . . . , i n 2 n , i . One should thus expect 1 ). It is now natural to ask ) i | u ) . The result ( ( 4.7 ) has a sum over exponentially ) u u ( n ) ( I n ρ ( 5.10 ρ | 1 − n evaporating black holes. This gives , . . . , i to be modified by replica wormholes as well. 1 n u , i – 39 – n I i to compute matrix elements of ] for analogous numerical h n from u 70 u I ,...,i X 1 and thus swap entropies ( I i ] for discussions and explicit examples of classical real- ) copies of = u 66  n I ) ( n τ ( U ρ ) copies of u as a sum over matrix elements of the density matrix n ) I ( u τ I U ( -evaporation radiation density matrix  τ n U ], which inspired many of the considerations in this section. ]. replica saddles (without back-reaction from quantum fields). Tr 97 98 n limit of the swap calculation as a prediction for the result. With the new QES, 1 labels a complete set of boundary conditions on i → However, this argument leaves open whether the required contributions to matrix From one perspective, such contributions seem inevitable. We can write the expec- The expectation is that the replica wormholes exist also for integer n individual terms. Nonetheless, wereplica argue wormholes may below well that give the saddle-pointsvalues conclusion of for is (Rényi) matrix entropy plausible, elements forin and an or the that unknown for smaller state the candidate generallyover requires entropy. expectation that a associated More number sophisticated with ofcopies; copies methods the see which e.g. improve simple is [ the swap exponential coefficient test, of but the the exponential best algorithm still requires exponentially many elements or other observablesthus are also whether large replica wormholes enough needsemiclassical to to computation. make appear an Since explicit at the appearance right asmany the hand saddles side terms, semiclassical in of their a level, ( and semiclassical description of the sum need not tell us anything about the then integrate over allidentifications to possible perform such the boundaryreceives sum replica values shown wormhole on of contributions, the matterand this right-hand-side. fields thus must Since of with be the the true appropriate left-hand-side generic of observables involving the right-hand-side as well, where a reorganisation of theintegral with path fixed integral matter studied fields above on in which we first perform the path discussed in [ tation value of describes radiation on 5.4 Replica wormholes forSo far, other we observables have consideredof the permutation contribution of operators replica wormholeswhether to such the topologies expectation can values also contribute to other observables. This question was also similar results for theabove expectation experimentalist value to of avoid permutationto the operators. establish awkward this step This in of wouldEuclidean full, allow taking signature, the though the and see seetime [ integer [ measures the action of the correspondingher permutation. measurements to The deduce experimentalist the might vonthe then Neumann use entropy of the radiationfollowing on from the replicapredictions wormholes, of the BH result unitarity. reproduces the Page curve, affirming the evaporate, captures the radiation emitted up to corresponding retarded times, and then JHEP04(2021)272 ) ), at γ , as The + + 5.11 (5.11) I I 25 . ), as the on BH S u 5.10 : + I ) than for expectation ). 5.11 5.11 ] goes a long way towards 98 ). . ) (2) 5.10 ρ ) u ( 2 O ) u – 40 – ( 1 ) for the replica wormhole. But there is no such O γ ) should run over exponentially many terms so that ) may receive its leading contributions from replica wormholes if Tr( vanishes due to a symmetry. ) 5.10 5.11 Hawking approaches null separation from the retarded time ρ ) γ ), and no corresponding late-time suppression of contributions u ( denotes which ‘replica’ of the Hawking radiation on which it acts. In O 5.11 r Tr( ), since we now compute the expectation value of a product of operators r evaluated in the replica wormhole geometry. But at the qualitative level i , and 2 u O is a simple local operator such as the value of particular field mode on 1 ) u hO ( r O However, the effect of this saddle should be much smaller for ( The reason for this is in fact much the same as for expectation values of the swap To illustrate the point, we first consider a very simple observable, namely the product An exception to this would occur if the Hawking saddle gives an extremely small answer (or exactly zero) 25 small corrections ofleading-order this corrections order on the in left-hand-side each of off-diagonal ( matrixfor element some other lead reason. tothe For one-point example, the function ( desired action (associated withextensive the effect area for of from ( the Hawking saddle. Sosaddles, one and expects thus replica to wormholessuppression to give by contribute corrections exponential as factors subdominant which agreessums are with on suppressed our the exponentially heuristic right-hand-side in argument of from ( ( values of the swap operator.answer because In the the matter lattersense case, entropy that replica in it wormholes the dominate grows Hawkingoperator the saddle linearly the late-time is with one-loop-corrected naturally time. action ‘extensive’ in of As the the a Hawking result, saddle for becomes expectation larger values of than the the swap singularity as the surface occurs near the apparent horizon ofbetween the the black hole classical sufficiently area farfor in and the the such existence past. a of The a logarithmic interplay may nontrivial similarly singularity QES exist was above. a precisely It semiclassical what is replica therefore allowed wormhole reasonable saddle to for expect ( that there the gravitational action (throughand contributions the associated matter with effective action, the inthe area that role case of the of the matter the Rényi surface entropy.function matter In effective computing action ( isthis played behaves instead much the by same the as logarithm the matter of Rényi the entropy. two-point In particular, it has a logarithmic integral do not include anyas connections between explained the below two asymptotic wedynamically regions. connect anticipate Nevertheless, the saddle-points boundaries in via replica the wormholes. gravitational pathoperator. integral There, which the existence of a replica wormhole saddle relied on an interplay between Here retarded time contrast to our studies of thedifferent swap values operator of above (which mixesthat boundaries associated each with act on a single boundary the corresponding boundary conditions for the path more detailed study isthis required aim. to establish this carefully; [ of expectation values of simple operators inserted on different copies of values of simple operators. Our arguments will be rather heuristic and suggestive, so a JHEP04(2021)272 are limit 5 1 → n copies of radiation is not n on ) , and which still comes just from u ( 3 ) n ( ρ . But this means that the results of inde- n . How are these results to be reconciled? ⊗ ) ) u u ( ( – 41 – ]. The intermediate states in question are states 14 , Hawking . In particular, the swap entropies obtained in Hawking ρ ρ 12 9 , 11 as computed as in section ) u ( ), since it would seem to require finding a back-reacted (and presumably Hawking ρ 5.11 ]), or by studying some appropriate family of quantities with an 98 In this section we answer these questions by cutting open the path integrals described The simple answer is that the density matrix However, it also raises many questions. While we now have a path-integral derivation In practice it may be rather challenging to check for a replica wormhole saddle-point for features relevant to gravitational theories. ensemble. That ensemble depends on a choice of6.1 the initial state of closed From (baby) path universes. To integrals set to the Hilbert stage, spaces wecomputations begin of by quantum briefly reviewing amplitudes the and relationship their between the Hilbert path space integral formulation, emphasizing distribution. The reasontually is commuting that set such ofcan asymptotic operators be observables acting simultaneously can diagonalised on besemiclassical into the regarded gravity superselection Hilbert predicts as sectors. space a resultsBH of It mu- for unitarity, though baby therefore asymptotic the universes, appears precise observers which that dynamics which is are not consistent uniquely determined with but chosen from an from a sum over intermediatewhich states. are Before much diving the in sameof we as closed briefly in preview ‘baby’ [ the universes centralthe which ideas, resulting propagate correlations are between restricted distincttation to asymptotic values be of purely boundaries. asymptotic classical, observables so But as that random we variables may selected regard from expec- some probability of cluster decomposition. Howuation? are we What to form interpretHilbert can predictions space of these interpretation the correlations of theory these between in results? experiments such take? a sit- so And far, what to is the obtain a Hilbert space interpretation of the correlations between boundaries radiation is still the saddle-point pictured innot figure equal to the Rényi entropies of simply equal to the tensorpendent product and widely separated experiments are correlated, and thus give rise to a violation Moreover, it does soin by principle. computing a quantity that is experimentallyof accessible, the Page at curve, least ourof new observables ingredients do for not the affect Hawking the radiation computation from of expectation a values single black hole. The density matrix of 6 The Hilbert spaceThe result of reviewed above, baby showing universes that replicaof wormholes suffice Hawking to radiation make the follow swapgives entropy a a completely Page semiclassical curve, computation that is supports satisfying Bekenstein-Hawking in unitarity. many ways. In particular, it quantities like ( complex) solution to thejust gravitational as equations for sourced integer by Rényi the(as entropies. quantum in It effective may [ action, be directlyanalogous tractable to in the simple von models Neumann of entropy, gravity to evade the complications of back-reaction. JHEP04(2021)272 , and cut by setting elsewhere. M φ Σ at which the . However, as E + u I , formally spanned Σ H . In a semiclassical ap- − becomes a Fock space for . , where the inner product φ Σ Σ Σ in the sum over intermediate in order to explain the main = H we will describe the modifica- Σ on 4 + φ Σ below, the simplest way to pass respectively. We can now perform 6.5 φ Σ 6.2 before some retarded time + (for example), and integrating over I of our choice. This cut manifold has two new – 42 – , we ensure continuity of the fields at ± . We have a Hilbert space | Σ Σ M Σ φ which, as part of the boundary conditions, is required ih u Σ φ Σ | Σ , and include the geometry of φ Σ D labelled by field configurations on R i , the past and future sides of . This would describe a system with a boundary. However, it will = Σ + u φ on the boundaries | 1 Σ ± φ and , and then integrate over all field values − at time Σ φ Σ this comes at the cost of requiring some assumptions about the evaporation. + 4 = is given by a functional delta-function setting I boundary. As will be described in section approach that all lead to the same Hilbert space as long as they agree asymptotically (where − i φ − φ Σ φ In the context of evaporating black holes, we saw that the semiclassical path integral This is somewhat complicated by the inclusion of dynamical gravity, when we also sum This cutting and gluing has a Hilbert space interpretation as the insertion of a resolu- A Hilbert space appears when we cut a path integral into pieces, writing the integral | = (now including the induced metric) need not define orthogonal states. But something + + φ semiclassical physics. in section We will thus at firstideas revive involving the in ‘PS passingassumption assumption’ to for of a the section description next intions few required terms sections, to though of avoid in closed it, section and universes. in fact We to will avoid use using this any assumption outside the domain of situation we are most naturallystates led on to a describe a partial gravitationalto Cauchy Hilbert slice meet space describing the be conceptually simpler and cleanerwithout to consider instead afrom Hilbert the space of former closed to universes the latter is by making complete measurements on see below that the effect ofof replica the wormholes inner can product be described on as the a Hilbert dynamical space modification atwas helpful the cut. for computing observablesblack on hole becomes Planckian and the semiclassical treatment is no longer valid. For this Hamiltonian constraint or Wheeler-DeWitt equation.here, We will but not need we the do technical notewith details that the this cut. modifiesΣ the Due inner to product gauge on invariance,stronger the it is Hilbert is true space here, natural associated as that the distinct inner field product is configurations determined on by the dynamics. Indeed, we will over the topology andalong geometry some Cauchy of surface spacetime.states. As But in diffeomorphism QFT invarianceslice makes we this can more cut subtle.the the geometry We path is have integral fixed many by choices boundary of conditions). These different choices are related by the h proximation, where the pathwith integral approximately is Gaussian computed weighting,linearised this by fluctuations Hilbert fluctuations about space around the a saddle. saddle-point fields To obtain the original path integralφ on tion of the identity, by field eigenstates over the cut aslet a us sum discuss over this intermediatethe for states. geometry a QFT along Before path aboundaries incorporating integral Cauchy dynamical on surface gravity, a fixedthe path background integral spacetime on this manifold with boundary, imposing boundary conditions that the JHEP04(2021)272 + O : I i + i (6.2) (6.1) | . The I int computes . The path ⊗ H int + for this space. 18 Σ ij I i } H O + with orthonormal and I set by the state i + int i , where we obtain the + + I H + {| I 〉 I i | I . int + I . ) | ⊗ H j ∗ int + ih 〉 of a pure state in a i | I | ( ai ⊗ H , we sum over all states of the interior ψ ) to obtain a Hilbert space interpretation. + int ∈ H + i int ∗ I a I Σ | i H , we have decomposed the geometry into int b j i . h . The Hawking wormhole computing this : a int ij + ai |O| Σ int O i ψ | I h ⊗ | a ai bj 〈 + – 43 – + ¯ ψ ψ I ⊗ H on and I aj , obtaining a Hilbert space i ¯ = + ψ i + | O int X I ij i,j,a,b ai Σ I O H i,j,a ψ is not sufficient to write the expectation value as a sum | = . We choose an orthonormal basis j 〈 P + + + i,a , a Cauchy surface for the black hole interior. To obtain a X I = I I int = H . In particular, for now we will make use of the ‘PS assumption’ Σ i hOi Hawking , computes the wavefunction 4 ψ i ρ | a | to simplify the discussion. and X i,j,a of a state in and 4 + 3 ai = I ψ . (with matrix elements a Cutting the Hawking wormhole computing the expectation value of an operator hOi + } by the state I int int i a Σ {| To glue these spacetimes back together along Once we have cut along both We begin with Hawking’s calculation using a single black hole and computing the The path integralwavefunction, on which the is a conjugate state bra in the spacetime dual gives space and the take complex the inner conjugate product, of obtaining the this Hawking density matrix for the state on disconnected bra and ket copiesintegral of on the the Hawking ket spacetime spacetime as withthe shown boundary wavefunction in conditions figure imposed on However, cutting only along over states, since the geometrythus is also still slice connected the throughbasis the geometry black along hole interior. We must expectation value consists ofsome bra final and Cauchy ket slice. copiesslice of to Using the consist The black of PS holeHilbert assumption, space spacetime interpretation, we joined we may can along Hilbert first choose space cut this this of geometry ‘out joining along states’ Cauchy Rather than going directlydiscussing to the the Hilbert replica space wormholes of interpretationculations of of most sections the interest, Hawking we and willintroduced Polchinski-Strominger warm in cal- up section by expectation value of some operator and on left spacetime computes theover conjugate all wavefunction, intermediate and states, to obtain the expectation value we6.2 sum Hilbert spaces for Hawking and Polchinski-Strominger Figure 18. defined on The path integral on the right spacetime, with boundary conditions on JHEP04(2021)272 ) ) n n n 6.4 6.5 ‘bra’ (6.5) (6.6) (6.3) (6.4) n . ) would BU i 6.5 n , and along in ( BU + ) n i with the n n ⊗ ) the resulting I ( , . . . , a ρ int copies of the black H 1 ) 4.9 Σ a n n | ( n π , . . . , a b 1 n a a , it is tempting to identify δ ‘baby’ universe, born from int , . . . , b ⊗ | involves a sphere of zero size. We 1 ··· Σ + 26 b n E h (1) I n i π . at j b n that involve n ‘ket’ copies of 1 ai i b a int 4 ¯ ψ n ψ δ . We must also include information is thus insufficient to give a complete Σ ) n aj − i must be obtained by comparing ( n + copies of ¯ n ψ a I n I ψ a , and we have BU X ), we will use the term ‘PS baby universe’ Sym( X , and the density matrix ⊗ · · · ⊗ | ab ∈ H n ··· δ π + = 1 ⊗ 1 6.5 i I i j = = . In that case its inner product would factorize i 1 j ψ b | 1 | – 44 – i ¯ int ψ | int . But if we made this identification, the state ( BU 1 i n i H i i a int 1 | n n a b a H h ψ ψ Hawking . n n ρ | 7 by cutting along i copies of the asymptotic Hilbert space ··· h , . . . , a ,...,a ,...,b 1 X 1 gives i n 1 1 . In particular, we would not find the sum over permutations BU 1 a b a n | a is the Hilbert space of closed (baby) universes. We obtain the H n int ⊗ ψ = ) Σ n n i BU by tracing out the baby universes, (1) n , which we may think of as a closed H ρ possible ways, this inner product involves a sum over permutations: + ,...,i ,...,a ( X n -fold tensor product int , . . . , b ! 1 1 i ⊗ I 1 n a n Σ b h H , . . . , j = 1 , but instead by computing the inner product induced by PS wormholes. i j -fold tensor product of ) | , where is entangled. The Hilbert space on ) n n BU ( n . After cutting them open in this way, for each term in ( . , where we have ( + BU ψ H + ρ | int | I n 6.4 I Σ ). Since the PS wormholes involve pairing the ⊗ H ). We will resolve this tension below by not making assumptions about the inner copies of the inner product on + n 4.9 with the ⊗ I n Specifically, the correct inner product on Since we have obtained So far, this is a fairly conventional description of information loss. But we will go , . . . , i 4.4 We say that the baby universe is closed as the boundary of 1 H i BU 26 h copies in any of the will comment further on this in section product on Replica wormholes lead toin similar section modifications to the inner product that wewith will ( discuss H into would be simply the be the tensor product in ( density matrix on copies of ‘ket’ spacetimes are identical. In particular, they compute the wavefunction of the state in beyond this by considering thehole. computations of section Thewormhole, Polchinski-Strominger so wormholes to consist obtaincopies of of a multiple Hilbert copies space of interpretation the we Hawking can again slice them along about the state on the black hole formedthe in reader the that ‘parent’ there universe.When are required In two for slightly introducing clarity different this (mostlyto notions terminology, in distinguish of we section the warn baby above universe notion in from the others that literature. may arise. This is a mixedmatter state on because we havedescription traced of out the the original black state hole of interior, the with system which on the An orthonormal basis on JHEP04(2021)272 ) is 27 α 6.6 (6.8) (6.9) (6.7) 28 . In place of , so that . As a result, int int int . ] Σ Σ H ¯ α n on is mathematically anal- α, [ . We then compute i α φ a . F , | Sym a would be a quantum field (a ¯ int α a α BU H to be a Fock space built on α E a n n b BU P ¯ α − with this weighting. H Sym e ] ··· a φ 1 =0 [ ∞ ) defined by the inner product ( b ¯ for matter fields on M α n ) as a Gaussian integral: α ¯ α d φ n a 6.5 ⊃ a 6.6 ) by diagonalising the physical ‘single-universe’ α : dα . The integration variables are complex BU 6.9 a int ··· H Y 1 H = 1 a Z α -fold symmetric product ] for comments and warnings about using this analogy to – 45 – 1 BU Z D n

99 1 = :=

) gives the states of a single anti-universe. would label points in spacetime, int BU BU φ i H E n ] ¯ α ) here, since this is not in fact quite the full baby universe α, [ ⊃ , . . . , a F 1 D a labelled by an orthonormal basis of states | n a α Polchinski-Strominger: is chosen so that Z where by integrating over all functionals , . . . , b , where we integrate over all field configurations ] 1 b φ (the dual space of BU [ h

∗ · ∗ int α we could instead use field configurations

] , with both baby universes and time-reversed ‘anti baby universes’. The second H φ a [ ∗ int φ α D ⊕ H We use complex fields so that we onlyPassing from count the contractions gravitational between path ‘kets’ integral and to ‘bras’, this integral not over between functionals two R int 27 28 ogous to the passageover field from configurations. particle In dynamics thisfunction to of analogy, spacetime), a and description the Gaussianin of weighting the (for quantum QFT a field free action QFT) theorylines, can is then which as given by be we a the understood have action. pathinner implemented as The integral product. rather arising kinetic term due However, implicitly the to ininterpret reader the ( the should Hamiltonian see physics. constraint [ on particle world- of amplitudes kets, for example. This distinguishes baby universes from ‘anti’ baby universes. The normalisation ‘baby universe fields’ the labels a functional of these fields. Then the Gaussian weighting can be written as an exponential familiar representation of the Bose inner product ( proposal (and leave behind itstrue challenges), for as replica many wormholes, of andcertain the also weak lessons for assumptions). learned gravitational here path will integrals remain more generally6.3.1 (under The PSThe Fock predictions space of of the baby Polchinski-Strominger universes proposal can be made manifest by using the The physical predictions that followmay from not the immediately state be ( clear.of We will the describe fact this in that somesults. the detail The below, Polchinski-Strominger result taking proposal advantage will is remain simple useful enough when to we later allow move explicit beyond re- the Polchinski-Strominger We have written ‘contains’ ( Hilbert space. As we willH see below, it is naturalsummand to extend 6.3 Baby universes and ensembles in the Polchinski-Strominger proposal, theFock baby space universe built Hilbert on space the contains ‘one-universe a states’ Bosonic Note that this is the inner product on the JHEP04(2021)272 : n + I (6.14) (6.15) (6.10) (6.11) (6.12) (6.13) copies, )). n i ∈ H α 6.9 Ψ are not fixed, | a and use the same α . BU , . ai H n -evaporation density α b ψ BU can also be written as n a ¯ E Ψ . n α , n i α i int α Ψ a Ψ , we may write this result H X ) and instead simply fix each n α ... ··· = ( ⊗ Hawking 1 1 is now to be regarded as a fixed BU , i ρ α i α b

dµ n α i Ψ ¯ · Ψ = ⊗ n as a subspace of 1

Ψ of the Hawking radiation states on j α i ) 30 n | = Ψ . ¯ . This factorisation means that, for Ψ Ψ ⊗ α int 29 i ai a  α H integral over Ψ α ψ | ... a α Ψ ih | 1 α α α i j Ψ h ¯ Ψ Ψ a ih | fix D X single α , )( Ψ −−−−−−→ n = = | α ) ⊗ – 46 – i i ( i ) n α n | Ψ associated with ( dµ α . But the theory does not give a specific prediction to emphasise that α Ψ i dµ Z ), we may regard α α ih for each evaporation: , . . . , j Z , . . . , j = α Ψ 6.7 1 | 1  α ) j Ψ j as | | where | n ) ( ) ( = n n ρ + ( ( is obtained by a ρ ρ | I | n term in ( + n α ) (which in the PS paradigm is given by the Gaussian ( I n fix ( Hawking ) = 1 ρ α n −−−−−−→ ( with some weighting. In other words, the parameters , . . . , i , . . . , i 1 1 ) , the Hawking radiation can be described by a pure state i i α n copies of dµ h a h ( ρ α n gives the copies are uncorrelated. Since the inner product on ) on n int H 6.5 . Instead, it gives a probabilistic one. For an asymptotic observer, the black hole copies of some pure state i α n parameter pertains to all asymptotic observers at all boundaries. In particular, for from multiple black holes are correlated in such a way that they are always compatible This should be contrasted with the result obtained in the absence of PS wormholes for As a result, any given set of actual measurements Now, the above potential factorisation property is spoiled by the fact that we must With this representation, we can write the components of the Ψ | We remind the reader that quantum mechanics measurements are associated with projection operators Since to some specific value, then the expression completely factorises between the α + 29 30 a and that, while expectationwith values projections, can a be direct inferred measurement of fromof quantum the mechanical quantum expectation relative mechanics. values frequencies would violate of the the linearity outcomes associated inner product. which the an integral with respectusing to independent the integrals same over Gaussian measure with for formation and evaporation can thus beto described pure in states, terms of but an with S-matrix an taking unknown pure states S-matrix selected from an ensemble. for some measure I still integrate over but instead selected fromof a probability distribution. Note,black however, holes that the the state same at choice Here we have included a superscript complex number depending ona our given choice value for of each If we for now ignoreα the integral over and also between ‘bra’ and ‘ket’ indices: matrix ( JHEP04(2021)272 , ) ). of n ( − i and ρ ) for I Ψ b (6.18) (6.19) (6.20) (6.21) (6.16) (6.17) | Tr i B h 6.15 = i ], though those Ψ 13 : – a ] = 0 α 11 b , . Similarly i a ,B with classical matter † a HH A | + m † I b B ] = [ b ,

··· ,B 1 a † b to the normalising denominator HH .

]. But with PS wormholes we find A ) ] B n † . n ( 31 ˆ † a α : ρ = 0 . One might also consider the possibility of i ) BU A ] = [ . i i α, b α a [ˆ ( α HH B | ··· F | copies of a pure state as described above. ,B HH dµ

a 1 | a + b † a α n B ) in terms of a Fock space generated by baby -matrix [ † a B A HH $ = A

6.6 = = – 47 – -fold experiment, as the predictions of ( ] = [ = = i i 31 BU n b . i a m α b ¯ ˆ α BU | ,A HH ) is nothing but the expectation value of (complex) | dependence. It would be worthwhile to understand this issue in a a E ] a ˆ α n A ¯ arising from the Polchinski-Strominger proposal gives a α 6.9 [ A -dependence for Hawking ,..., ) n labelled by a set of complex eigenvalues ρ 1 α, α [ b ¯ , ( to denote the oscillator vacuum in order to think of it as a , F BU ab n i mutually commute, so we can write the Hilbert space in terms i D δ dµ α † a | ˆ α HH | ] = in the oscillator vacuum † b , . . . , a a 1 and ˆ ,B a α are given by a Hawking | a a annihilate and create a baby universe in the state ˆ + α B Although anti-BUs did not appear in our discussion above, they naturally † a I A 32 ] = [ ), which is a classical mixture of † b and ,A 6.14 a ), we have used a A A [ ]. Now, the Gaussian integral ( To understand this from the perspective of the baby universe Hilbert space, we can Note that the measure 6.16 These are much the same as the baby universe creation/annihilation operators of [ This gives non-normalised wavefunctions. While the normalisations can be absorbed into the measure, annihilate and create a time-reversed object that may be called an ‘anti’ baby universe 99 † 32 31 b doing so appears toadditional introduce involving wormholes a connecting the mild pathwhich integral should for be expected todetail, remove but this such a treatment is beyond thereferences scope worked of in this a work. realin basis. [ There may also be minor differences associated with subtleties discussed All the operators of the position eigenbasis where In ( Hartle-Hawking state for reasons that we will explain momentarily. ‘position operators’ B (an anti-BU). form the intermediate states if(associated we with considered a the white time-reverseand of hole quantum fields our that in boundary explodes the conditions vacuum to state but form with a time-reversed Hawking smooth radiation at Here, by the Hawking density matrix instead represent the Bose inneruniverse product creation ( and annihilation operators, experiments at instead ( complex multivariate Gaussian probability distributionthe Hawking for radiation the wavefunction. components The mean is zero, and the covariance matrix is given We dub this the Hawking result for the JHEP04(2021)272 . j j ¯ ψ ¯ Ψ , create i (6.24) (6.22) (6.23) Ψ ai ¯ ψ † a ˆ α a . Had we used j ˆ ¯ P Ψ = ). i ˆ Ψ 34 ) that the operators 6.24 of Hawking radiation + 6.24 . † . The latter statement is I and not that i † a i i ˆ ,B | α † ˆ Ψ commute. j and and its adjoint ) follows from the fact that the ˆ ¯ Ψ , and in particular that creation a a ˆ . A, B, A , Ψ ˆ i α ) as an amplitude computed by the 2 α | inner product to derive this result, | 6.14 ˆ a Ψ α α | 6.10 ai ih a BU ψ α | P a H ˆ 1 2 ¯ α α specifying both an initial state of matter on It is manifest from ( by acting with the − i a e i X Ψ 33 dαd ) from this Hilbert space point of view, let us – 48 – HH = | Z i ∝ i 6.19 ˆ = Ψ ) by inserting the completeness relation bulk spacetimes compatible with the stated boundary HH | , which thus also commute with ( 1 † a α 6.9 h all -evaporation state ( α ), and using the Gaussian wavefunction of the oscillator n 6.19 as a ‘boundary-inserting operator’. Indeed, it is tempting to refer to these as in the ˆ Ψ α of closed universes. ] it in fact follows from fundamental properties of the gravitational path ) describes the inner product of two states that involve only baby universes and not BU never appear alone. Similarly, the time-reversed boundary conditions H 22 6.10 † a ) simultaneously diagonalize the operators A either creates a baby universe in some internal state or annihilates a time-reversed 6.21 . The corresponding operator + BU which will collapse to a black hole as well as a final state Although we used explicit results for the Let us first illustrate this rather abstract-sounding statement by recalling that, in the One may thus refer to Equation ( I H − 33 34 ‘boundary-creating’ operators. Butboundaries one should in realize this that sense.differ both from These the baby are universe thus creation not and standard annihilation operators creation-annihilationanti-BUs, or operators, in and other in words states particular created from integral. The point isquantum simply that gravity we path may integral regardSince with ( the the path integral specified sums boundary over conditions, conditions the built result from ordered. is As independent a of consequence, the how associated the operators boundary conditions might have been are all built fromoperators the (complex) ‘position’would operators define operators involving as argued in [ on baby universe. Theoperators, path one for integral each computes separate boundary. an expectation value of a product of such present case, we have boundaryI conditions on a single integral over states ( a consequence of aproposal, more in primitive that fact the that boundaryobservables will conditions will for remain continue computing true to expectationHilbert when define values space we simultaneously-diagonalizable of go asymptotic operators beyond acting the on PS the 6.3.2 Lessons andHaving comments completed our derivationnow of pause to ( extract some useful lessons. The first lesson is that the appearance of only into the right handvacuum side of ( We obtain the Gaussian integral ( JHEP04(2021)272 i on Ψ i Ψ to be ) n ( ρ , so that the i ˆ ¯ Ψ , expectation i . Note that we in terms of the i on the right-hand- ] by the absence i j HH | ¯ i HH Ψ | 100 HH ˆ Ψ | i is defined by the new through the action of in the Hartle-Hawking ˆ Ψ b † a ¯ α ˆ α BU , , a ). a H sets of boundary conditions α ˆ α to be the adjoint of n conjugate boundary conditions i 6.23 ˆ Ψ n of additional asymptotic boundaries. -evaporation density matrix described above. However, we will find n i α absence Ψ | – 49 – . Any such cut splits asymptotic boundaries into two BU H ]. 99 ). Indeed, the correlations between different copies of Hawking , which will generally differ from ( i (from the bra). Since we can again expand i 6.23 ] and [ HH ¯ Ψ | i 22 ˆ Ψ | -states, we would again find the . In the previous sections we thus have implicitly chosen some initial state α α † j h ˆ Ψ (or in another state). Since these operators all commute, we can move any i = j HH ˆ ¯ Ψ | The different interpretations are readily be described in the operator language by Finally, before proceeding to replica wormholes, we pause to note that the Hilbert Had we instead chosen the baby universe initial state to be e.g. Another lesson that becomes clear from the perspective of the baby universe Hilbert or i ˆ the latter in theside. ket-state But as if well, wethe there had right. would used have In the beenCPT-conjugates general, latter additional of in the entries the the rule of ket bra-state,above is boundary the we mutual-commutativity that conditions. means would the that This instead amplitude the requiresfurther find operators contains discussion the additional can both in copies be [ the of simultaneously bra the diagonalized boundary as claimed; conditions see and the intermediate baby universe states joining the two sets. writing an amplitude as the expectationstate values of products of subset of them to the right where they act on the ‘ket’ state and move the remainder to forming the ‘bra’ state.giving The rise same to path different integralbaby Hilbert can universe space also interpretations Hilbert be — space sets, cut though in depending always several involving on different the whichother ways, same side defines of the the ‘bra’, cut with they the lie. overlap between One the set two defines being a obtained ‘ket’ by state a sum and over the wavefunction space interpretation on which weone have particular concentrated way thus of faras cutting is forming the not a unique. path ‘ket’ integral, state, It regarding and arises taking from an overlap with the values in this state(from would the be ket) adding and additionalsame boundaries basis with of boundarya conditions classical mixture ofa the different same mixture pure in states which the probability distribution for include any number ofthe universes (connected universe components number of isappear space) not from including even nothing. zero; diffeomorphism indeed, Instead,of invariant it boundaries if is in universes defined the can according path split, to integral the join, which spirit determines or of the wavefunction. [ for closed universes.conditions But with recall that the our experimentsnothing amplitudes more to was were be added entirely to performeduniverse specified adjust state by by must the have our baby boundary been universe specified asymptoticIt state. by observer, is the As for and a this result, reason that do that our we not choice call of use it baby this a term Hartle-Hawking for no boundary a state state of a single connected closed universe, but a state that can space is a sense inuniverses’ which our given predictions by depend ( whatradiation we may are call mediated the ‘initial byset state the of of baby exchange asymptotic of boundaryΨ baby conditions universes, modifies and the we state have of seen that each JHEP04(2021)272 , i

b ¯ a, | HH

(6.25) (6.26) 2 j ˆ Ψ 1 j ˆ Ψ (corresponding i and

HH | HH into our discussion of amplitude,

. 2 i 5

. for the black hole interior ˆ ) Ψ i = 2 1 i b ¯ HH by taking a partial sum over

int n ˆ Ψ 2 a, i + Σ | ˆ Ψ I + 1 i i ˆ Ψ is a Bosonic Fock space. The replica 2 . This Hilbert space description will † j HH | ˆ by incorporating ‘interactions’ between Ψ BU BU 1 ab † j δ H H ˆ ( BU Ψ

bj H to write the amplitude as the overlap between ¯ ψ † HH ai ˆ

Ψ – 50 – ψ , = ˆ Ψ a,b i X 2 = , j . The intermediate states are then 1

j | HH HH (2)

ρ 2 † j ) is not the most natural one if we wish to describe an interme- | † j 2 ˆ Ψ ˆ Ψ i . This is a useful crutch to simplify the exposition but, as we will , i 2 i ˆ 1 6.26 Ψ ] (though this does not directly apply to replica wormholes for E i . ˆ Ψ h I 77 ∂ and =

runs from the regular origin below the black hole singularity out to the as a Cauchy surface for an evaporating black hole, where we remind the γ to trace out the unobserved piece of the state. HH int int

Σ 1 Σ † j i, j ˆ Ψ 1 i and ˆ Ψ ). In such a case, we can choose our Cauchy surface + 1 For now, we will continue to make the PS assumption that allows us to treat the union This interpretation ( We illustrate this with a simple example computing the I on which we join the replicas lies inside the event horizon. This is well-motivated, since I a QES is guaranteed tofocussing lie behind conjecture the [ event horizonn under > the assumption ofto the pass quantum through of reader that endpoint of evaporation explain later, we will beaddition, able for to upgrade simplicity the here argument we so will as only to remove consider this replica assumption. wormholes In such that the island We now incorporate thebaby replica universes. wormholes We introduced can intheory think section of of ‘free’ the baby Polchinski-Strominger universes,wormholes proposal in then discussed the modify above sense the that as innerbaby a product universes. on prove useful in theto context adapt of this replica description wormholes. tothe incomplete In indices measurements particular, at it will be straightforward 6.4 Replica wormholes as baby universe interactions exchanged in the T-channel ofciated some with QFT a scattering process, Hilbertinto which space would two for naively parts the be (as QFT asso- surfaces). associated opposed to to However, a the timelike in usualto surface the Hilbert lie that operator spaces in splits associated the space description same with above spacelike baby the Cauchy universe intermediate Hilbert states space continue diate state in real time. Indeed, it is somewhat analogous to describing intermediate states so that the intermediate statesorder consisted the of boundary-inserting two operators baby universes.states Alternatively, we could re- to the trivial contribution whereof the one black baby hole universe interiors and are one not anti-universe (corresponding swapped) and to the states nontrivial PS wormhole): between them. We interpreted this earlier as the overlap between the states the left to act on the ‘bra.’ And we can finally insert a complete set of baby universe states JHEP04(2021)272 , int ) so n (6.29) (6.27) (6.28) copies where, for the = Σ do not n ] i I 0 I [ 5 a | , QF T ) n i will be a sum S n − i a e . a | 00 n ). Thus we should a 00 n b . If we take δ I 6.28 0 ]. ). acts as the permutation ) i ⊗ · · · | i n 1 ) ( 00 66 a π I | a ( , a 0 0 n ) or ( )( π b 00 2 a I | δ U ,a ( 0 1 π . To include a replica wormhole, 6.27 a ··· U α ¯ α ) 00 1 | 00 2 a in a natural way as a sum over real n ,a 00 1 b b i 0 2 δ a ). Summing over all possible combi- a 0 | α 00 1 (1) 6.9 in the split π ,a 0 2 a γ 0 1 a b ¯ . The operator α δ n 00 1 | ⊗ · · · ⊗ h – 51 – 1 ⊗ int ,a ⊃ b 0 1 associated with the island fields, summing over indices to induce the required for its complement: h H a and another Hilbert space for its complement. ( However, as we discovered by computing amplitudes i ¯ α α BU 00 int I i , making gravity dynamical naturally leads to finite we insert a term 00 2 ⊃ a n n 35 | H ,a ) to the baby universe inner product as an interaction, . Such contributions are then exponentially suppressed 0 2 BU 5.2 I ,a = 2 i 00 1 X n 6.27 ,a n 0 1 , . . . , a a 1 ‘scattering’ of baby universes. Indeed, we can borrow standard ). We can be a little more explicit by choosing a basis of states a copies of a given state they would compute | n fields and copies of n , . . . , a 6.6 n n 1 a as encoding not only the matter state, but also geometrical degrees means ‘contains terms of the following form’. The states and inner which factorises between an orthonormal basis of states n α | −→ i n ⊃ n a , . . . , b , for | int 1 + b H h in units of the cutoff. I , . . . , b γ 1 for b ) by pairing baby universes with permutations, the replica wormholes introduce h i 00 6.6 is the Rényi entropy of and a corresponding basis , a 0 ] a I ) for the inner product in terms of integrals over I | [ Using language analogous to that of the Feynman diagrams of perturbative QFT, we Note that if we interpret the sum over states Note that adding analogous terms to the inner product in a continuum quantum field Now, just as the Polchinski-Strominger wormholes induced extra terms in the inner 6.9 = This fact is deeply related to the fact that the Hilbert space of quantum field theory does not factorize on those parts of the i 35 QF T n a into the integrandnations on of the replica right-hand-side wormholes of exponentiates ( this factorinto a (and tensor similar product of terms a Hilbert for space all for in ( we insert a product of connections. For example, for evaluated by deforming the contour. For further discussion seecan [ think of thegiving contribution a ( ‘vertex’ for techniques from perturbative QFT to compute the associated effect on the expression think of the states of freedom (perhaps including the location of Lorentzian geometries, the saddle-pointappear replica directly wormholes since discussedof they in highly are section oscillatory complex. phases, which The (as is direct familiar sum from over steepest states descent integrals) can be discussions on S by the area of with the path integralcontributions in from section the terms on the right-hand-side of ( island theory would naturally given a vanishing contribution. Indeed, in direct parallel with our products on the right handof side the of black this hole equation interior areπ Hilbert taken in space, the tensor productwe of recover the terms in| ( product ( new terms with a permutation acting only on the associated island. We thus write where the notation JHEP04(2021)272 , n u at BU H Σ + H using (6.30) I BU does not H . We thus . However, u of radiation I i BU α 5.1 H Ψ | so that , we may cut the path E u to be well to the past of to a rather complicated I copies of some given u a n Σ ¯ α a u < u ) remains true, with the mod- of boundaries on which these α n a n ⊗ 6.14 ) P | − α e Ψ ih α , choosing u Ψ from is determined by the path integral, and iii) | I ) )( for justification). α BU α . In particular, the expression ( ( – 52 – H ], the Hilbert spaces are in fact labelled by the asymptotic dµ 7.1 dµ 6.3 22 Z . We will then sum over all boundary condition on the = which extend to meet the asymptotic boundary ) E u n ( Σ . We may then further choose ρ u . only on the subset E + I . For simplicity we restrict to the case where the asymptotic regions are defined by described above. The details of this measure are not of primary im- u ) Σ α ( dµ Although in the Hawking saddle it arises from , ii) that the inner product on . We may then expect the relevant saddles to remain under semiclassical control 36 + . Indeed, we will obtain a Hilbert space interpretation by slicing open the path -evaporation density matrix given in equation ( BU n 5 I H In particular, since we impose boundary conditions only on In a replica setting, we require several such cuts. The resulting Hilbert spaces As we now show, all of these results can be derived using physics that remainsTo fully proceed, we follow the same basic strategy as in our study of Rényi entropies in On the other hand, aside from this modification of the inner product (and the cor- As described for the Euclidean context in [ 36 spheres. We also mention that, in the current context, there is a notion of ‘anti-boundary’ or ‘conjugate associated with suchcuts cuts end. are labelled by the number geometry of the slices n integral along Cauchy surfaces the associated retarded time both the singularity and section integrals and the associatedspecify replica the wormhole state saddles on intersect discussed the in future section light conerest of of as desired. under semiclassical control.those However, using the the arguments PS areon assumption necessarily a above. first more Some reading abstract readers of than may this thus paper. choose to skip this section it turns out thatrely the on most the important PS lessons assumption.space from These lessons the include Hilbert i) space thethe existence interpretation of fact do a that not baby asymptotic universe quantities Hilbert and define the simultaneously diagonalizable associated operators existence on of superselection sectors. 6.5 Baby universesThe with above sections semiclassical developed control: the notion droppingthe of PS PS the baby assumption. universes PS and assumption the Thisgeometries’, associated allowed the us associated to amplitudes, give and a the very resulting explicit inner product treatment on of the ‘saddle-point for the ified measure portance for us, except thatwhich the follow modified the measure Page is curve dominated (see by section states non-Gaussian measure. responding changes to theuniverse wavefunction creation of and the annihilationarguments Hartle-Hawking operators), or state the there and conclusions are of the no section algebra further of changes to either the that it modifies the original measure JHEP04(2021)272 . . , u n − is u u in to +1 \I Σ I u I a u n u Σ ⊗ H Σ (6.31) (6.32) (6.33) ): Σ 1 are not and on i H → H 1 u 6.28 i n 00 I meeting the , a 2 H ), ( u , a as linear com- a . But now this 0 ) | , where Σ i n a ): n | n u H 6.27 a H \I ( i rather than on the = and u anti-boundaries. (There ˆ Ψ i i u Σ without boundaries is 2 ¯ n a for a state in of closed universes. | I . 0 as part of the boundary . , a u which choose a different 00 n i 1 u i ⊗ · · · | H a ) I BU 1 n a 00 n Σ | b u a | H ∪ δ ( 0 i ) u )( ˆ n Ψ I indices. Σ ( ( π , . . . , a 0 π a 1 a 0 n on due to contributions from replica U .) Finally, while one might at first expect b boundaries and ) i δ | on the Hilbert space of state on i n a, a n ¯ n,n | n . Thus we write ⊗ 1 ) i b just as in equations ( on the Hilbert spaces H n ··· i u | H i associated with the location of these spheres on ( 00 1 I i ⊗ | on its complement to a ˆ Ψ u ai a 00 1 ¯ | n 00 -boundary Hilbert space b in two, so the state ψ ) a 0 n, δ n u 0 a ( H a allows one to canonically identify Hilbert spaces with X (1) ai acting on the – 53 – + | ⊗ · · · ⊗ h π and ψ 1 a = important. The states π I b 0 1 b i h I i,a to label a basis of states on ( δ is n i i P i ⊃ a | ). i ⊃ u n . The notation here is similar to that used above for to be a basis of states on a Cauchy surface , . . . , a i indices must be paired without permutation, while replica I 1 i n a a 00 | | a ) to define an operator , . . . , a u to denote the boundary condition that fixes both matter at 1 ( i i i is not just the product on the island a , . . . , a | Ψ ˆ Ψ 0 Ψ 1 n n a a | H . If we consider the path integral for any single copy of the spacetime, does not exhibit Bosonic statistics. This is associated with the fact that u , and impose boundary conditions for the quantum fields on . n , . . . , b u u 1 , and we take H b Σ h + I the Hilbert space on for the case of sphere boundaries is associated with ¯ n u n, H H Again we may use Nonetheless, we still find contributions to the inner product from replica wormholes. Furthermore, we can write states on the With this small change in boundary conditions, most of the considerations above will , choosing a notion of time-translation on + boundary’ (associated with thespace anti-baby universes discussedis below), a such natural that linear isomorphism thethe from Hilbert most the space general dual to Hilbert space I also depend on thedifferent advanced values times of Using our bases, the action of this operator takes the form Because boundaries are distinguishable, itthe is first important slot that (we we could, added if the extra desired, label define other versions of wormholes give rise to the permutation that collapses to formAnd the again black we may hole take andoperator a adds state a boundary, increasing the value of In the second linelabelled we by have the split state now the extends index to infinity.conditions Translating require the that discussion the above to this notation, the boundary their entirety. Such contributions again permute island regions baby universes, but thereas is distinguishable, a the crucial order difference. ofthe the Because same, we and treat asymptoticwe boundaries specify the asymptotic identifications betweenconditions, Cauchy so slices there can be no terms in the inner product that permute copies of boundary at time truncate it at the result computes a wavefunction (with binations of a basis wormholes. In particular, as wenot discuss the below, trivial the Hilbert Hilbert space, space but should instead becontinue the to space hold. Weentirety simply of take we emphasize that JHEP04(2021)272 . ) i in u j | ( i , i . We ˆ i Ψ | = 0 u n and gives u I 〉 after time , and for j | ) we defined n + . We have flat asymp- . The compositions ) I n u 6.33 ( . This is the boundary . ij ˆ ρ BU 19 → H specified by the states H u +1 = I n carry information not just about 0 H n H to itself for any H n ): H u acts by inserting a conjugate boundary ( † i ) ˆ Ψ u ( † j thus acts by inserting a boundary condition ˆ Ψ ) – 54 – must be identified as shown in the asymptotic re- u ( i u | ˆ ) and gluing it to an asymptotic boundary associated i Ψ 〈 . j ) i u ( † j HH involve two copies of such boundary conditions, computing | , so we have ˆ Ψ ) i at time u ˆ Ψ ( 15 + ji ρ ) = I )ˆ u −→ u ( ( . By definition, the adjoint operators map between Hilbert spaces ij ) ij ) ˆ ρ ˆ ρ are then operators that map u u | ( ( ) † i u ˆ HH Ψ ( ij h i ˆ ρ , which justifies the choice of notation. ij ˆ Ψ u ) P u I ( The boundary conditions corresponding to the operator † j . But again we allow all possible gluings deeper in the bulk, and in particular ) = ˆ Ψ u 2 ) Σ u ( ) := The composition More concretely, the adjoint operator (2) u ρ ( ij ˆ we allow nontrivial replica wormholes. corresponding to a completecondition one in-in would choose contour, for as computingradiation the shown components on of in the figure density matrix of Hawking (with boundary condition labelled by with the state onto which add it a acts boundary (theboundaries in first requires the such a first boundary, corresponding slot). since gluingpart As in of of in the ( our respective discussion spacetimes above, on this the gluing asymptotic of asymptotic A convenient but abstractadjoint method operators of avoidingin the the extra opposite informationρ direction involves using to the particular act within the closed universe Hilbert space would thus like totruly ‘trace associated out’ with baby this universesrise extra and to information, which the mediates leaving Page the curve. only correlations the on piece6.5.1 of the state Replica wormhole baby universes ordering). In particular,case it we means cannot that talk thesespaces. about operators Intuitively, this diagonalizing no is them longer becausethe commute. since the closed Hilbert they baby And spaces universes, map in but between any also different about Hilbert the state that escapes to gion. We do notthe specify dashed what curved. happensand to the In the future particular, spacetime awayample, Cauchy this from slices the allows the need spacetimes the asymptoticTr( not region, in spacetime inside be figure to identified connect in with the other same boundaries, way in their entirety. For ex- Figure 19. totic regions as pictured,The with Cauchy matter slices boundary meeting conditions at JHEP04(2021)272 ) u ( ij ˆ ρ (6.34) . . However, γ of the E that mediate (which in the ) ), the resulting u . BU ( ) BU α ij u 6.34 H ( ρ were amplitudes of ij ). ˆ ) and, as explained in 5 i ρ above, it follows that i ∈ H in ( ), joined at ˆ ρ . ∗ HH (left), where the RWBU i HH 6.34 | | I ) 6.3.2 u HH ( | ) ij 20(a) u ˆ ρ ( 1 . We could thus perhaps refer to i n I i ∂ ˆ ρ . = ··· ) γ 6.4 . . We will refer to the result as a replica . Furthermore, they can be simultaneously – u u -state), the eigenvalues ( 3 ) above. We give some interpretation of the BU α i BU ). Since this is the natural object in a replica 2 between insertions of 6.2 i H H ρ – 55 – ] reviewed in section )ˆ 6.26 BU 6.34 u 22 ( . H and 2 i 1 n B i ˆ ρ ) and the teal slice (labelled H | I HH h n -states, giving rise to superselection sectors and ensembles as α ,...,i X 1 i -states of the specified baby universe state from mutually commute on α ) u ( , where using the PS assumption we interpreted them containing both a ij ˆ ρ , they should also contribute to more general amplitudes 6.3.2 5.4 . The replica wormholes gave particular contributions to ( ) In particular, let us consider the intermediate states If we insert intermediate states of As before, the superselection sectors mean that Hawking radiation emerging from In this language, the swap Rényi entropies computed in section Using the general argument from [ u , while the second part is a partial anti-baby universe. The two much match at the ( u ij ˆ part may be calledΣ a ‘partial baby universe’,boundary consisting of of only the the island,this island and RWBU region as they on a are some consists ‘BU joined of — at the anti-BU red bound slice state’. (labelled See figure wormhole discussion, we willof call section it an RWBU.PS Similar baby states universe were and considered awhen at PS we the anti-baby consider end universe. replica Thiswish wormholes conclusion instead to must of avoid be universes PS slightlywe with wormholes, modified can and Planckian in curvature still such particular think as when of those we our that intermediate end state at as naturally containing two parts. The first replica wormhole construction of wormhole baby universe (RWBU), tobaby contrast universe it (PSBU) with discussed the in Polchinski-Strominger sections notion of the correlations between boundaries in ( between consecutive experiments on differentthat black achieves holes. this aim, For see a appendix Hilbert space6.5.2 description What isWe a now baby pause universe? to more carefully explore the notion of baby universe associated with this section Hilbert space interpretation generalisesintermediate ( states duenatural to way replica to wormholes describing in intermediate a states moment. of baby But universes this in is a not real-time the process, most the form That is, they wereρ correlation functions in the Hartle-Hawking state of products of the one black holeclassical evaporation correlations, is described by correlated adecomposition with classical into a that probability emerging distributioncases determined from discussed by above the another. is the Hartle-Hawking These no-boundary are state the operators diagonalised by a basisbefore. of In aare given interpreted superselection as sector the (an componentssuperselection of sector the that density emerges matrix before for time the Hawking radiation in that JHEP04(2021)272 (for a 20(a) 1 and ∗ − I I D I S . Figure 2 − , right). D 3 S ≥ n times an interval, with a corresponding to ‘ket’ and , which we can think of as 1 B 3 − . , has the topology D 3 , S B = 2 . Such a state is somewhat unusual in n = 2 BU n make up the Northern and Southern ∗ I at the other. A Euclidean continuation for propagating between the two boundaries. i ∈ H γ −→ from ‘ket’ and ‘bra’ spacetimes respectively) B and HH ∗ of the constituents above together along | I ) I – 56 – 20(b) n u t ( x e ij Σ and ˆ ρ I boundary γ 2 , left), or an interaction of RWBUs ( − I , of which D 1 the topology of spacetime is S = 2 − ), this notion of ‘intermediate state’ may seem unnatural in D n S is a spacetime which asymptotes to a closed Euclidean manifold = 2 ) 6.26 u ( n ij ∗ ˆ I ρ replicas (as shown on the left) making up a replica wormhole has topology n ] constructed replica wormholes in a two-dimensional spacetime for a 19 We may describe the correlations between sets of Hawking radiation arising from replica times an interval, with the asymptotic boundary at one end of the interval, and 1 − . We picture the resulting Euclidean spacetimes for such boundaries as shown in figure D ∗ n (b) To build a replicaI wormhole, we sew propagation of a RWBU ( a RWBU with topology hemispheres respectively. (a) Each of the S two conjugate copies of thejoined island ( along their common resembles the geometry on the right, which could be described in terms of propagation of For example, for As described after ( -dimensional spacetime), with the two hemispheres of . For our case of a black hole formed from collapse For example, [ (right) shows the resulting Euclideanjoin continuation of each replica. A replica wormhole will boundary lying at each endin of the terms interval. of It a is then baby very universe natural to with describe topology this cylinder Lorentz signature. Nevertheless,Euclidean notion it of is intermediateboundary the state. condition correct To see Lorentzian this,B continuation note of that a aD natural Euclidean version of the ‘bra’ segments of the boundary, joined along an asymptotic spatial wormhole spacetimes asappearing mediated in by intermediate states exchangeLorentzian such of signature, as but ‘replica has wormhole a natural baby Euclidean universes’ continuation. (RWBUs) Figure 20. JHEP04(2021)272 , of 9 ) = 3 u ( n ) n ( ρ early subsets, but n identical black holes, n , a pair of pants for = 2 n for the Hawking radiation arriving ) is sufficiently to the past of the future u ( u ρ , the geometry of the Hawking saddle is weakly – 57 – E in an asymptotically flat spacetime. To formulate + , and hence expectation values of operators acting on I u ). From the Euclidean perspective, it is natural to think black holes are well separated, we may approximate each n sets of early Hawking radiation, so that the state n 20(b) before some retarded time + In the limit where the No part of our later discussion caused any direct modification of the above conclusions. , with the other half being the island from a bra part of the replica wormhole. I B separate asymptotic boundaries. But in performing the sum over all geometries with of spacetime. Then boundary conditions onsuch boundary our conditions gravitational we path allowedas for integral geometries so-called then ‘spacetime which involve wormholes’, connectcorrelations which distinct we between define asymptotic the boundaries. Such geometries introduce to measure a swapwhich operator leaves that the remaining acts late as subsetsof a fixed. identical-but-independent permutation quantum Such among observations states performed give these we on a many accordingly direct copies refer way of to measuring the entropies, associated and expectation values asblack ‘swap hole entropies’. formation and evaporation as occurring in a separate asymptotically flat region However, we notedexperimentally that measure observers its who entropy.violation possess of We only BH thus a unitarity imaginedcollecting that single experiments the involved decay to copy forming products verify of andthat of the a was evaporating each, above emitted and system before identifying cannot the some ‘early’ particular subset retarded of time. each We collection then asked our experimenter curved, and all perturbativeresulting corrections entropies are of small. theof Hawking Of the radiation course, remaining far at blackHawking exceed late hole. unitarity the retarded (BH Bekenstein-Hawking times We unitarity). entropy refer the to this phenomenon as a violation of Bekenstein- interest. We thus began withwhich the familiar can Hawking be saddle described usedat in to the compute form of a figure that density radiation matrix or associatedlightcone (Rényi) of entropies. the If endpoint of evaporation on Hawking radiation collectedand at perform the relevant computations,integral, we which used in the the Lorentz-signaturein semiclassical gravitational path- limit field involves theory, a classifyingone sum all works over possible to saddle-points. identify saddles In interesting tends gravity, saddles as to and be hopes that rather they difficult, dominate so the in amplitudes of practice of 7 Discussion 7.1 Summary This work has focussed on describing semiclassical expectations for experiments performed and so forth (asof in such figure wormholesanalytically as continue describing to interactions Lorentzianmatrix signature between in boundary closed the conditions, universes. correct‘island’ we sense from arrive to And any at describe ket when precisely our part we density the of the situation Lorentzian described replica above. wormhole spacetime The is then just half two-sided black hole, with topology of a cylinder for JHEP04(2021)272 , as E , where 11 limit where 1 → ]. They considered n 14 to introduce the idea , defined by inserting 4 BU H . By this mechanism, one n ⊗ ) u ( ρ are ‘swapped’ in all possible ways ) acting on u ( ρ ]. A particular goal was to reconcile the 13 ], all such operators can be simultaneously – 22 operators 11 component of the density matrix of Hawking – 58 – ij ], which are closely associated with the quantum that there are other saddles which resolve these 5 19 , -Rényi entropies — at least in the ]. We thus briefly reviewed results and arguments from n 18 21 , for each 20 BU H . But as argued in [ on u . The correlations between multiple sets of Hawking radiation can incorporated these ideas into a conceptual framework for the grav- ) 6 u BU ( H ij ˆ ρ sets is not in fact equal to the tensor product n It is crucial that the correlations between sets of Hawking radiation can be described are Finally, section Nevertheless, we saw in section Such an approach was advocated by Polchinski and Strominger in [ the relevant boundary conditionsis in an the operator gravitationalradiation path before integral. time Fordiagonalized. example, there In particular,defined the it path is integral to easy sum over to all show topologies with that the they required boundary mutually conditions, commute. Since we both strict and classical; i.e.,will find any identical observer sets who of Hawking forms radiation,be and but thought the evaporates as particular identical radiation of black state asfeature, obtained holes may we being considered chosen the from expectation ain values particular or classical states matrix probability and elements distribution. other of asymptotic asymptoticWe To quantities observables found explain that that one this might such expect quantities to in be fact c-numbers. yield the path integral. Thisnot led meet us asymptotic to boundaries, slice‘baby’ and open universes which spacetime we wormholes associated alongthen with surfaces be understood which a do as Hilbert arising space from of a sum closed over intermediate states of these baby universes. itational path integral inof the Coleman and presence Giddings ofoperational and spacetime verification Strominger of wormholes, [ the building Pageunitarity on curve associated described the with above Rényi insights with entropiesso, the of we apparent the sought failure Hawking-saddle a of Hawking BH Hilbert radiaton. space To interpretation do by cutting open the relevant contributions to The result is thencalculations that are our more swap tractable —unitary. perfectly This reproduce is powerful the evidence Pagewhich in curve the favor associated Bekenstein-Hawking of entropy with the is BH idea indeed that the there black is hole an density operational of sense states. in issues. In particular, ouranalogous swap to experiments those receive describedextremal contributions surfaces in from studied [ replica in wormholes [ those references, translating them to our Lorentz-signature asymtptotically-flat setting. relative to the Hawking saddle.without We reviewed the this proposal technicalities in of section features replica with wormholes, finding BH-unitarity-compatible swap Pagemodel entropies continues curves. that to share violate However, certain BHus on to unitarity (and regard closer the perhaps inspection PS alsosaddle-points. wormholes, this causality), which as contain well a as singular requiring and strongly-curved region might hope that observables suchwith as BH the unitarity. swap entropies will be nevertheless be compatible including in the path integral athe class various of interior ‘PS wormhole’ connections spacetimes between shown copies in figure of these JHEP04(2021)272 ;

) n ) (7.1) (7.2) (7.3) splits u ( ρ BU H Tr(

-values will be α selected from the ]. ) ], the situation is more -sector will be expo- u ( 22 is defined so that the α 13 ij –

ρ ) 11 dα u HH (

n and fails to follow the Page ], replica wormholes do not ] does not apply. There is no ) j -function measure. . u n u 29 ( α i δ 101 E n ρ ) j (where u n i ( ··· ˆ ρ n ) j dα n u 2 i ( ··· ρ 1 i| ) j for all such operators. The correlations 1 u α i ( ··· in almost any given computed by replica wormholes. The Page ρ HH 1 | ) BU j -state giving ) )

1 u α i ) α α n H ( ˆ ρ ( ) n , the state of Hawking radiation from any given |h 1

) i j u of 1 – 59 – ( u dµ i ( ρ

ρ . A different choice of state results in a different HH ρ HH ) = ) holds). This defines the notation of the final line, | Z D α

α α Tr( . And while the set of all possible ( ( 1 = = Tr( µ states, we write this as an average over superselection

) = dµ dµ . Its entropy grows with = u

( . Thus as emphasized in [ | for the argument that these operators are normal, in the ) α ) n

α u n (

34 ...j ih ρ ) 1 ] as the baby universes which provide decoherence carry no energy or

α j u | n ( 102 ρ -copy density matrix as an expectation value in the ‘no-boundary’ ) ...i

dα n 1 n ) = In particular, the swap test provides a measurement of ( i R u ρ ( Tr( 37 of baby universes, which was an implicit choice in our earlier calculations. BU Hawking through the measure ρ ∈ H BU

HH

, the formulae above make manifest that our results depend on the choice of state It is important that the value of This framework finally allows us to reconcile the entropy results described above. So Explicitly, applying this to calculations of the density matrix of Hawking radiation we i ∈ H For much the same reason, the energy conservation critique [ BU 37 HH nentially close to the average value curve is therefore a robust prediction, accurate up toexperimentally-accessible exponentially ‘dollar small matrix’. corrections.similar Indeed, to We as described thatmomentum. long discussed ago in in [ [ of replica wormholes. it does not measure compute the ‘true’ vonthe Neumann entropy entropy of of the the state state projected of to a the typical radiation; superselection instead sector they [ give black hole is curve due to entanglement with baby universes.sectors, But, this as is entanglement always thefurther is case entanglement for unobservable. superselection with the Evaporatingso same that additional baby measurements designed black universe to states, holes deduce the correlating induces entropy the produce the decay Page products curve with the help H | measure, with an extreme example being an long as the initial baby universe state is sectors, with probability measure completeness relation where we write thisensemble as defined an by expectation the valuedetermined of measure by random state-independent variables considerations involving the algebra of our operators on The first line writes our state By inserting a complete set of from a probability distribution of superselection sectors. can write boundaries. See also footnote sense that they alsointo mutually superselection commute sectors with for the theiris algebra adjoints. a of asymptotic basis This observables. of meansbetween In simultaneous that multiple other eigenstates sets words, there of Hawking radiation can then be described as classical correlations the output of the path integral cannot depend on any ordering of the multiple disconnected JHEP04(2021)272 ] 22 – (7.4) 18 bound- is pure, ¯ + X I ]. 19 , 18 boundaries to the . Since the variance is the 2 X

connected copies of the single-copy state.

¯ n X X

X

 ] in which black hole states tunnel into =

) 57 ¯ , X X 56

X Var(

– 60 – −

¯ X X

) = X over spacetimes which connect the ¯ X Var( X (or the entropy) is ‘self-averaging,’ meaning that any given sample from ) n ) u ( ρ is self-averaging if these connected contributions are dominated by the discon- Tr( (which we take to be complex in general) is self-averaging if its ensemble variance X X We have focussed here on black holes in asymptotically flat spacetimes. But analogous Interestingly, despite the fact that the replica wormholes computations rely only on Now, without any obvious reason to exclude replica wormholes from the gravitational separate black holes must then be a tensor product of In particular, it is natural to ask for a complete description of the physics in a particular the action of variousprecisely swap to operators the replica on wormholes the and non-gravitating associated7.2 auxiliary entropies studied system in then What [ leads haveWe we now gained? we reflect on the position in which we are left after drawing such conclusions. boundary conditions, one can alsosystem couple the that AdS can system to absorbspacetime. an Hawking auxiliary We non-gravitational radiation may and viewthe remove this current it as paper, an from withinto analogue the the which of asymptotically auxiliary our the AdS system experimenter experimental playing uploads processes the the described role Hawking in of radiation’s the quantum ‘quantum state. memory’ Studying n But this in incompatible with the swap entropies computedcomments by can replica be wormholes. made inreviewed above many (and other on contexts which much as ofically well. this AdS work Indeed, boundary was based) the conditions. were original performed While works with it asymptot- [ is common to study AdS settings with reflecting to constrain certain aspectsrule of out the scenarios final such stages aswhite of those hole evaporation. described states in For at [ example, lateof it times stored appears (after information to reaching at Planck thisthen size) late there and stage. can thus In release be particular, large no if amounts superselection the full sectors. final state The on state resulting from the evaporation of servables acting on the Hawking radiationprediction or for entropies, measurements but of it fine-grainedthe declines quantities density to such provide matrix as a of the definite Hawking off-diagonal radiation. elements of the part of the evaporation that remains under semiclassical control, they can also be used classical or invoking new physics),semiclassical we gravity. are The scenario compelled is towith that consider unitarity, it the and predicts following observations with scenario toHawking the for always entropy. be density But compatible of itclassical black does gravity hole not gives states definite predict being predictions the given for detailed coarse-grained by unitary questions, the dynamics. such Bekenstein- as Instead, simple semi- ob- boundary conditions aries. nected spacetimes. path integral, and in the absence of as-yet-unknown additional contributions (either semi- the ensemble is parametrically likely totity be parametrically close to the mean. A given quan- is much smallerconnected than two-point its correlator, mean it squared, is computed gravitationally from a path integral with say that JHEP04(2021)272 states provide α not ]. The earlier works ]. 22 104 -sector in which we live. , α ], the correlations between 12 , 103 11 So in our context the effect of 39 – 61 – To explore this question and the associated physics of 38 -states is describable as providing a distribution of random α -state as determined by the initial conditions of baby universes as described α ] for a recent review. Each member of the ensemble is thus a local theory on the 105 With the above as prologue, we now point out an important difference between the It is important to emphasize that measuring the actual state of Hawking radiation is First, however, we should briefly discuss the predictive status of a framework involving As has often been stressed to us by SteveOne Giddings, could no think clear of answer it was as provided defining by the a discussions local of effective theory on scales larger than the black hole, but then 38 39 -states as minimal Bayesian priors, assigning credences to different possible superselection of the black hole undergoingsuch evaporation, a integrating local out effective such theory wormholes on will black hole scales. wormholes and baby universes from the 1980’s andthe early black 1990’s. hole itself would simply be treated as a particle with a large but finite number of internal states. smaller than any macroscopic scaleand the of resulting interest. ensemble In of couplings that for case, they terms can (suchsee be as [ ‘integrated the out’, cosmologicalscale constant of interest. term) But in since the a typical scale local of effective replica wormholes action; is that of the event horizon a situation which haswords been will much necessarily discussed apply. in cosmology and to whichwormholes many studied of here and the those same studiedprimarily in studied the the effect late of 1980’s microscopic [ wormholes, and in particular of wormholes much The situation is thusparameters much (and indeed, the we same could identify asaction these when parameters giving with working the coefficients with in S-matrixcan the a effective for view theory the black with hole unknownabove. formation free It and thus evaporation). has the Alternatively, same we logical status as any other measurement of initial conditions, this is also thestandard only systems, sense though in inparameter which controlling that frequentist the context probability width makes one of definite may the predictions distribution. use for thetantamount number to of experimentally experiments determining, at as least a in part, the α sectors and thus touniverse particular initial states states for thiscertain the features perspective are Hawking allows common radiation. to usaveraging all to observables allowed Now, superselection with make for sectors, parametrically definite or general when sharply predictions baby we peaked only consider self- probability when distributions. But superselection sectors. One issuesuccessive is experiments that, mean as that pointed we‘probability’ out of cannot by getting use [ a a particular strict stateonto frequentist of a the interpretation branch radiation. for of the Any thesome given wavefunction particular observer with superselection will sector. a decohere baby But universe it is state natural tightly to concentrated instead around interpret ‘probabilities’ of a density matrix thatso reproduces without the the Pagefrom help curve, the of and above replica for considerations? superselection a wormholes. sectors, single we evaporation We willour must might introduce considerations do then so some far, ask: ideas but that which what are were studied have not in we directly more apparent gained detail from in [ superselection sector. Even for a single evaporation event, such a description must yield JHEP04(2021)272 ] 108 , -states -states . This 18 α α BH . S e # e 7.3.2 ]. 40 107 , 106 ]. In the exact solution of these 22 bounded in number by the Bekenstein- 41 which suppresses more complicated spacetime – 62 – has any useful semiclassical description, since it BH S − BU e nonperturbative effects. The effects are not merely H From these considerations it would appear that -state for the baby universes. The answer to this question 42 doubly α -states lie in the regime of semiclassical validity. α as for subleading saddle-point geometries, but are of order BH S # e . BH S e This issue is exemplified by toy models of black holes which are so simple that we This seems unlikely, and semiclassical physics seems similarly unlikely to determine It thus appears that we will not obtain a local semiclassical description of superselection Moreover, these effects may not be determined uniquely from the semiclassical expansion since (as is the Unless, perhaps, we introduceWhile new there objects to are resum no certain propagating contributions: degrees see of section freedom in these theories, we may nonetheless model the 42 40 41 -states in the occupation number basis leads to large weights for terms involving very than case in JT gravity) theFor sum over JT, topologies there describes onlyselected is an from asymptotic an expansion an extremely thatstructure does ensemble natural required not by of completion converge. such random of ato completion. matrices, the identify For an sum more since obvious realistic completion. over the models topologies it topological is defined expansion unlikely that by precisely we Hamiltonians will fits be the so lucky rigid as strong suppression is associatedsuppressed with geometric their saddles. arising from an infinite sum of exponentially- black hole interior by ‘end-of-the-world branes’ with a large number of internal states, perhaps much greater expand in a small parametertopologies. of order If wesuperselection truncate sectors with that the expansionturns above out at features. to any be Inof finite fact, order sensitive order, the to we precise see spectrum no of restriction to and the even simplermodels, topological model the introduced superselection inbut sectors [ which have are features remarkablea expected when discrete appearing of spectrum from unitary ofHawking a quantum entropy. black gravitational But systems, hole path this is microstates, integral: not they manifest from have the semiclassical approximation, where we particular, if we trycorrections to to sum the semiclassical over approximation thecorrections in to large each the number term final of may answer. accumulate semiclassical to terms yield involved, large small may perform the path integral exactly, namely Jackiw-Teitelboim (JT) gravity [ universes) become important atparticular leading interaction order is because compensated the byregime, exponential the there suppression number of is of any no possibleis guarantee such no interactions. that longer In even this good approximately approximation a by Fock truncating space the of path single integral universes, to and any finite we number do of not topologies. obtain In a will hinge on whether the precise spectrum ofα possible superselection sectors.large The numbers basic of reason baby‘interactions’ is of universes. that baby universes writing But (i.e., the for topology exponentially changing large processes occupation which split numbers, and the join just the sort of non-locality required for the scenariossectors discussed by in integrating [ out topologyask changing if processes. one can Onexplicitly the find other a includes hand, local an we semiclassical initial might description still that retains such processes, but which cannot be absorbed into a shift of local coupling constants in a useful way. Indeed, this is JHEP04(2021)272 ], we interpreted 112 are well-defined, and 6 ]), such a bound requires 35 ], following [ 22 superselection sector. More precisely, ], what is the ‘niceness condition’ which 37 ] show that a single topology produces the ‘ramp’ every 109 , -states. In [ – 63 – α 108 ] with periodic Euclidean boundary conditions). Since old ]. These relations rely on the same doubly-nonperturbative 110 111 Nonetheless, we can make much stronger statements by taking an states. For example, [ 43 α ] for a more detailed discussion. ] showed that the number of linearly independent pure states below a given 22 22 ] and generalisations [ However, the attentive reader will still want to be assured that we have not thrown As a result of these considerations, we have no reason to expect semiclassical physics It would thus appear that we can say little about individual superselection sectors By focusing on clever averaged quantities, semiclassical calculations can nevertheless give more indirect 22 43 hints at the structure of in the spectral form factordiscrete that spectrum is with characteristic statistics of long-rangeform resembling eigenvalue that factor repulsion of which and more a hence directly randomof indicative signifies of matrix. all a a topologies However, discrete the or spectrum feature going (the of beyond ‘plateau’) the a appears spectral geometric to description. require summation weakly gravitating regimes? In theensures language that of we [ mayinteractions neglect with topology large changing baby processes universesThe in key involving observation contexts replica in where wormholes this BHthe or regard unitarity matter is was that entropy not replica is in wormholes so danger? become large important only that when the sum over internal states can compensate for the improved because the semiclassical approxiationits itself suggests validity. The principled reasons approximation to predicts doubt its own break-down asout it should. the baby withblack the holes bathwater. in a If given semiclassical superselection physics is sector, inadequate what to ensures describe that old we may still trust it in physics as discussed abovesuch in relations relation as to novel nonperturbative manifestations of diffeomorphism invariance. to be a goodlection approximation sector. in While the this interiors has of of old course black been holes suggested for before an the individual situation superse- is now much than allowed by thesurprising Bekenstein-Hawking linear entropy relations between (seestates such e.g. must states [ be (equivalently, some unexpectedlyand linear ‘null’, so combinations must with of be vanishing setof inner equal [ product to the with zero every state). other This state, was seen very explicitly in the toy model energy (say, prepared by forming andHawking partially radiation evaporating onto a various black possibleby hole and states) the projecting is thermodynamic the bounded entropyHawking in (defined type every path by superselection integral the sector [ black inverse holes Laplace-transform have of large a interiors Gibbons- and thus naively give rise to many more internal states that the Hilbert spaces ofthat intermediate they states each considered have in awormholes section discussed positive above semidefinite showed that inner theBH product. entropy of unitarity Hawking For radiation on example, is average, while consistentrequiring with general the consistency replica consistency with arguments BHsection show 4 unitarity something of for much [ stronger, section 5 of [ using only semiclassicalstatistical physics, properties. and thataxiomatic we approach can and only making access use of averaged consistency or conditions. other Specifically, simple let us assume involve a regime where quantum fluctuations of spacetime topology are untamed. See JHEP04(2021)272 BU H ]. The 61 – , factoriza- super Yang 59 6 ]. = 4 115 N has a one-dimensional 5 S × 5 44 super Yang Mills and type IIb string super Yang Mills is the unique dual, but that = 4 = 4 N N – 64 – . G A 4 ], the more well-established examples of gauge/gravity duality & S is one-dimensional, so that all states of baby universes are somehow 113 , ) involve a unique dual theory. ], there is a possibility that BU , which has become known as the factorization problem [ 5 108 ]. 22 H S 4.2 × 114 5 , 22 -sectors defining different quantum error correcting codes in the sense of [ α There has not been any entirely satisfactory resolution to this puzzle. It thus remains This tension is not entirely new; rather, it brings to the fore an old puzzle, touched To be specific, in the asymptotically AdS context, our considerations point to the idea As described in [ 44 field theories with onlyMills one using the member standard of bulk-to-boundary the dictionary. ensemble being givendifferent by bulk superselection sectorsdictionaries would map then to be thisthis related as dual by the (perhaps using non-local) distinct bulk dictionaries. field redefinitions. In One effect, might the also think different of equivalent [ to be seen whether(perhaps e.g. due to type the IIb propersemi-classical string inclusion supergravity), of theory various or in stringy whether objects AdS this and bulk features that theory go is beyond in fact dual to an ensemble of surprising from the gravitationalconnect point different of boundary view: componentsbut contributions appear it from to bulk seems spoil spacetimes the arbitraryFrom the that above to point factorization of exclude property, view suchtion of requires the spacetimes that baby from universe Hilbert the space gravitational discussed in path section integral. AdS/CFT correspondence equates gravitational amplitudesboundary with conditions fixed to asymptotically the AdS partitionmined function by of the a conformal CFT, boundaryis with of background disconnected, geometry the locality deter- gravitational of ‘bulk’as the spacetime. the CFT If product immediately that implies of boundary that partition the functions result on should each factorize connected component. But this result is models of gravity [ (such as the paradigmatictheory duality in between AdS upon in section complete description in termssign of of the a superselection dual sectors conformal that we field inferred theory, from butthat the semiclassical in existence gravity of should which be replica there dual wormholes. dual not is to to no an a ensemble single unitary ofWhile CFT, such examples but theories, should of with instead such a be dualities different have theory been for recently each discovered superselection for sector. simple two-dimensional 7.3.1 AdS/CFT andA the potential factorisation concern problem withunderstanding the of above conclusions the is AdS/CFTprovides the correspondence. us strong tension with The with examples the point of traditional is theories that of this quantum correspondence gravity with a nonperturbative, UV effects only when wePlanck units; have i.e., a when region with entropy7.3 exceeding the area Further of open its questions We now perimeter close in with some open questions and further comments. usual exponential suppression of topology change. We therefore need to consider these JHEP04(2021)272 , 108 , 22 a mixed state of 4.2 ]. Perhaps these ideas can be 58 -states. Certainly, it remains an α – 65 – ]. But the more pertinent question for us is whether 118 ], in a situation like the standard AdS/CFT setting having – 98 116 -state. While these appear to be new objects in the theory, they α ]. Somewhat less concretely, as pointed out in section , we were rather pessimistic about describing individual superselection sectors 98 7.2 ]. These are dynamical boundaries for spacetime (analogous to D-branes providing In the context of evaporating black holes, the idea of providing boundary conditions Now, it may well be that physics similar to replica wormholes appears naturally for In the light of replica wormholes, the factorisation problem is directly related to the 120 , 7.3.3 Contributions from UVWe have physics been careful totested, make use and only in of regimes low-energyHowever, where we physics cannot which there rule is out is well-established the no and possibility that reason the to quantities we expect have studied that are sensitive it fails to be trustworthy. is not new:revisited this as is an essentially effectiveoutstanding the open description final problem of to state find babyof proposal a more the universe [ complete, transfer and of perhaps informationsuperselection more from physical, sector. a description black hole to the outgoing Hawking radiation in each universes (much like regarding D-branes asnew a fundamental coherent objects). state of Does closedto strings, something theories as similar which opposed apply are to going rich enough beyond to these include toy evaporating models, for black holes? spacetime in the black hole interior to produce a pure state of Hawking radiation appeared recently in toy119 models is that ofboundaries ‘spacetime on D-branes’ or which ‘eigenbranes’partially) the [ fixing string an worldsheetcan can also end) be which thought have of the as an effect emergent, of collective (perhaps description of a coherent state of baby 7.3.2 Description of superselectionIn section sectors directly in terms of standardscope semiclassical for a gravitational relatively physics. simple description Nonetheless, using there a is different language. still One such idea which has spoils factorization: see e.g.replica [ wormholes are relevantcoarse-graining. in Paraphrasing [ a situationfactorization where and we without have performed superselectionwormholes as no sectors, the such can first explicit term we in nonetheless a understand systematically improvable replica expansion? torization [ Hawking radiation represents a failureare of computed factorization: by a componentsexactly of product when the of the density two amplitude matrix ‘S-matrix’ similarly boundary factorizes. conditions, andtheories the with state a is single pure unitary dual after some appropriate coarse-graining which explicitly black hole information problem,necting since multiple the boundaries. entropy We computations dosince not involved immediately wormholes the require con- boundary factorisation for conditions thewhich for entropies, explicitly separate spoils boundary factorisation. componentstities are But which correlated if do in we require a decompose factorization, way the it Rényi appears entropies that into the quan- wormholes remain and spoil fac- JHEP04(2021)272 , 84 , with 15 + I has multiple γ ]), which we high- 122 , )? If so, how are we to + 121 I , where interactions with baby universes 7.2 – 66 – , the past light cone of any splitting surface 5.1 -state. But since this selection depends on fine details of the UV α (and similarly one such part for each of the ket-spacetimes). ] for an example of timelike separated islands in a cosmological context. It may γ 123 Conversely, our work above took as a fundamental assumption that the low-energy This idea that such causal singularities should be included is not new (see e.g. [ One such set of ideas is the fuzzball proposal (reviewed in [ ]), though its implications remain to be fully explored. One would like to understand (‘bra’), we add furtherreplica-wormhole-like forward identifications. and backward Such branches, time-foldsthe with might past the be of possibility used a of splitting to nontrivial surface connect (where the physicsgravitational is path understood). integral sums over topologies. While this is a common discussion in understand their effects onare measurements there performed saddles by with asymptoticSee multiple observers? [ splitting Similarly, surfaces thatbe are possible causally to related probe tothe the each study physics of of other? such out-of-time-order settings correlationconstructed using functions. from time-folds, That one as is, may branch instead be of of familiar forward each from replica evolution being (‘ket’) and one of backward evolution 86 just how general such causalin singularities saddle-point can geometries. be, and Forreplica in example, particular wormholes can what lie one singularities outside find arise horizons saddles (and where thus splitting in surfaces the for past of The fact that replicaspacetimes wormholes with can singular causal provide structures gravitationalintegral. play saddles an As important strongly noted role in suggests in section disconnected that the parts. gravitational path In particular,join it at has one such part for each of the bra-spacetimes that solve the information problem.when It the suffices if parametrically this largestate physics interior of kicks can in Hawking play radiation only a are after role, required. the and Page time, 7.3.4 when large corrections to Spacetimes the with singular causal structure completion, with extra dimensions,ignorant strings, of branes and the so details:to forth, average it the over low-energy does the gravity thewith is possibilities. one best comment: job In while it the thebefore fuzzball can hope the literature in horizon suggests of forms, that the making the from face tunnelling such our event considerations of a happens we its connection, see ignorance, that we this which conclude is is in fact unnecessary to horizonless configuration. The amplitude tothis tunnel is to compensated any for given by configurationsituation the is for large small, superselection number but sectors of described possiblewere in states. similarly suppressed We can individually, compare butuniverses. this compensated to for One the by might aselecting speculate large a population that distinguished of the baby fuzzballs replace the baby universes, effectively to solve the factorization problem in the AdS/CFT context. light due to some conceptualdiscussed similarity above. with physics Specifically, of onelapse an piece does individual of not superselection lead the sector to fuzzball formation proposal of is a that horizon, gravitational but col- instead there is a tunnelling event to a to more exotic physics from the UV completion of the theory. Indeed, this may be required JHEP04(2021)272 + 45 I , the Polchinski- of Hawking radiation 6.5.2 , we found this proposal Hawking ρ should evolve to orthogonal 4.1 − evolving to a pure state on I − I . In section 4 . – 67 – . But did the PS-assumption lead to the correct 2 Hawking + ? And on a similar note, does the non-perturbative ρ I , perhaps by throwing a particle with two possible Tr 7.2 − ] and defining the Hartle-Hawking no-boundary wave- I 124 for these two black holes is zero. But this is not the case for + I . We can check this using a swap test, except that we now prepare two black ], some readers will ask if there might be formulations of quantum gravity in + 100 I , then the swap expectation value is If we remain within the semiclassical regime, considering only experiments on the ra- As an illustration that challenges may lie ahead, we give an example in the context + If the particle’s internal state transmits perfectly into the black hole interior in the semiclassical ap- I 45 diation before the black hole becomes too small, then weproximation, do so not that have both suchat initial a states sharp give contra- rise to the same density matrix holes with orthogonalinternal states states at into theswap black operator hole. acting on Unitaritythe demands PS that proposal: the theThus expectation expectation the value value Polchinski-Strominger of is proposal the exponentially does small, not but result nonetheless in positive. a unitary S-matrix. of the Polchinski-Strominger proposalto in section give predictions consistent(for with example, a as pure probedinner state by product on the is swap conserved,states test). so on two But orthogonal unitary states evolution on also requires that the Our work above focussedtarity, but on it the is Page notsemiclassical curve. it gravity make itself This predictions enough that is tomore are a guarantee detail? in unitarity prominent line for with signature asymptotic unitarity of observers. in BH other Does ways, uni- and in does each PS-baby necessarily come attachedthe to RW an baby anti-baby universes? so that Orof the null is result states this is as question more described like fundamentallytheory in ill-defined have section a due meaningful to distinction the between presence universes and anti-universes? 7.3.6 More details of unitarity The difference between the two wasus in to part discuss due the tocomplete path the fact integral projective that associated measurement the with at PSconclusion? forming assumption a allowed We black presume hole thetions, and but full the what non-perturbative performing are theory a the results? to allow Do the such resulting boundary baby universes condi- resemble the PS-babies, or There also remain certain questionsdiscussion about of how non-perturbative baby corrections universes. willStrominger affect assumption For our led example, to asreplica a described wormholes certain in led notion to section ofcan a PS different roughly baby notion be universe, of while thought RW our baby of analysis universe. as of In a particular, bound the state latter of a PS-baby and a PS-anti-baby universe. Page transition in AdSfunction space [ [ which it fails to hold. This important issue7.3.5 also deserves further attention Non-perturbative in physics the of future. baby universes treatments of gravitational path integrals, and despite its utility in describing the Hawking- JHEP04(2021)272 . 7.2 ]), this remains to 28 – -parameters, thus deco- 25 α – 68 – ] to effectively ‘measure’ the 21 ] is evaded because, in a technical sense, information is lost: 127 , 126 ], the most interesting question would appear to be what form of conceptual 125 However, the situation is less clear for multiple identically prepared black holes or more If the baby universe state is simple (as for the Hartle-Hawking state), our path integrals It is clearly of interest to explore this further, not least in the context of cosmology. of an auxiliary bath systemhering as the in different [ superselection sectorsservers have of no the access to gravitating the spacetime.dividual bath, superselection one Since sectors. might infalling expect The ob- their firewall experiences problemwe then to must arises be deal with described full with by force. the in- In vast addition, number of null states required by the discussion in section conventional description that an infalling observerThe will firewall experience paradox no [ drama atthe the late horizon. radiation is not required to be entangledcomplicated with baby the universe early states. radiation. In particular, in the AdS context one could make use quantum cosmology). We will notfew say comments anything below. definitive on this question, but we makedescribing any a one black hole are dominatedwith by a the usual smooth semiclassical interior black hole until spacetime, the singularity. This gives us no obvious reason to doubt the Our main focus inhole, this paper in has asymptotic beenquestion regions. to of compute predictions for observables We the defined observers havesince far who it not enter from is the the far directly black from black hole. commentedare obvious This how upon inevitably is to more give the part challenging, a of gauge more invariant the description difficult of quantum such observers, system who of the black hole (a situation familiar from Indeed, in analogyproblem’ with [ Everett’s treatmentframework of (if the any) quantum wouldand allow mechanical not a ‘measurement just sharp the discussion intermediate of steps — experiments7.3.8 are whose subject final to records The quantum — gravity experience effects. of an infalling observer ditions imposed on finitebe ‘cutoff’ shown in surfaces detail. asexperiments In in on particular, implicit multiple our in black conceptthe holes e.g. of separation will [ between cluster approach them decomposition, is our in taken ‘separate the to universe’ sense infinity, is idealisation that as as yet only an expectation. (like commutativity of operators acting onbe the an baby universe approximation Hilbert to space), morementers but realistic subject it settings. to can only gravitational Any physics, actualthat even experiment if such will only real-world involve weakly. experi- experiments Whileabove it would (or is be involving natural an well-modeled to auxiliary assume by system the coupled to idealized AdS, ones or described involving sharp boundary con- motivates a careful study of the situation7.3.7 when we consider several Moving different away initialWe from studied states. asymptotics black hole formation andprepared collapse and in an measured idealised at setting,black using holes asymptotic states placed boundaries, that in were and separate using spacetimes. experiments This allowed with us multiple to make very clean statements diction with unitarity. Nonetheless, it provides a warning that more must be checked, and JHEP04(2021)272 . ]. u ω of − 41 i I ψ along | on L ) ] and [ − 40 I backwards in ) ω, as does + ( , the state a − I , − ) ] in working in the I ω, I − ( 40 a I , ) means that the spacetime . ω, ( + along − u † using the number operators a I ω I + ω, ( (right). (left). On † I a . We follow [ 2(a) 2(a) + I . The argument below follows [ region where the spacetime remains nearly 2.1 along I – 69 – u be can be found by studying the behavior of the at late advanced times (i.e., large affine parameter ) − ), the localization at large + ; see again figure I + associated with modes of definite positive frequency I T I ) ω, localized at late retarded times (large affine parameter + ( . a I . + L , has the same positive frequency reaches ) I ω, ( + R R a 2(a) ) I + ω, ( I † a ω, ( † a ) without leaving the Phase − ), the desired backwards propagation can then be broken into two phases; see amounts to solving a Schrödinger-type scattering problem, resulting in a reflected ) = I and a transmitted mode + + ω I R I ; The reflected mode In the first phase (closest to Now, linearity of the quantum fields also means that the desired backwards evolution We wish to characterize the state of the field on ω along ( quency mode v stationary. As a result, the right panel of figure is very close tothe that collapse of will have aits either gravitational stationary effect dispersed black is to holeapproximation minimal) that distant as or the parts any region will of is transient have exactly the fallen effects stationary, the asymptotic into associated evolution the region of with nascent a (where mode black of hole. definite fre- In the of the operators corresponding field modes.wave And equation. the For latteralong modes behavior is obtained by solving the classical with respect toHeisenberg some picture, affine so we parameter needtime to to express evolve them the in operators This terms of will the give corresponding a operators Bogoliubovof transformation occupation that numbers at will allow us to compute the distribution Recall that we consider a freeuncharged massless collapsing quantum field black on hole thethe spacetime classical quantum spherically of field symmetric figure coincides with the Minkowski vacuum on N A Further review of theThis appendix Hawking completes effect our in briefcurved review a spacetime of that fixed the was spacetime derivation begun of in the section Hawking effect in a fixed are grateful for supportCalifornia. from H.M. NSF was grantand also PHY1801805 D.M. supported and thanks in funds UCSB’s partresult, from KITP by this the for a research University was their DeBenedictis of alsoGrant hospitality Postdoctoral supported No. during Fellowship, in NSF portions part PHY-1748958 by of to the this the National work. KITP. Science Foundation As under a resolutions remains a fascinating topic for both discussion and furtherAcknowledgments investigation. We thank Steve Giddings andwork. Douglas We Stanford also for thank conversations Mukund motivating much Rangamani of and this Robert Wald for comments on a draft. We What it means to discuss physics in this context, and how it relates to previous proposed JHEP04(2021)272 T T back T . The fact . Bogoliubov T α BU H , the transmitted mode u begins in some initial state before the black hole becomes BU . It is then natural to ask for u u H I I travels through the region where the T unobserved, we have necessarily lost some u is localized at large – 70 – L is naturally interpreted as the coefficient for emission of T contributes only to what are usually called the Bogoliubov parameters associated with mixing between creation R β required for predicting subsequent similar measurements. Since we are ] by noting that in the overlap of the regions corresponding to phases 1 BU ), and an asymptotic observer creates a black hole before making a complete mode is localized in a region close to the horizon that is well approximated 41 i , we discussed a Hilbert space interpretation of replica wormhole calculations, H T HH 6.5 | and completing the calculation. This is the 2nd phase of the backwards evolution − modes are thermally distributed. Consider in particular a real-time process in which The Hawking effect is thus associated primarily with the transmitted mode On the other hand, the transmitted mode . This means that I T + the state of leaving the radiation which emerges afterinformation. time The state of babythe universes unobserved will thus part become of mixed thea due asymptotic map to entanglement state. from with density As matrices a to result, density this matrices process (a is quantum best channel) described on as consecutive measurements on differentgive sets an of alternative Hawking Hilbert radiation, space so interpretation. in this appendix(perhaps we projective measurement for the Hawking radiation on B Intermediate states of babyIn universes section in particular allowing onlytoo measurements small. on However, this a description region was not particularly natural from the point of view of fall into the black holecoefficients and must be agree absorbed, in and thermalmode also equilibrium. to that Thus the remains the fact present that (squared)the in absorption fraction original and of mode emission the by original a radiating black hole. that it corresponds only topotential the barrier part into of the the original regionthe mode near famous that the “grey-body” was horizon transmitted factorsfact during through into that the the when the phase evolving Hawking modes 1 toward effect. evolution the introduces future Here the corresponding the transmitted name part would comes from the by Rindler space. Furthermore, sincewith the the corresponding (approximate) Rindler time time-translation translationis coincides symmetry purely outside positive-frequency the with respect black hole,approximate to Rindler the in time. this Minkowski vacuum region But in anyof smooth this state Rindler will region. locally Thus the occupation numbers has high frequency withapproximation respect may to be natural freely-falling usedto observers. to justify As a the result, usethat the was of foreshadowed WKB geometric above. optics Butcan in rather than be propagating complete seen the full [ and calculation, 2, the end the result parameters (which map annihilationthe operators more to interesting annihilationand annihilation operators) operators. as opposed to spacetime is dynamical. However, since I JHEP04(2021)272 . . u ). is to , Σ ) BU ) to BU u ρ (B.1) ( ) BU . The whose H † j ) gives u B.1 H ( ˆ u Ψ BU ij I B.1 ρ H BU ), the map is chosen to (1981) 2640 | ρ is built from on = = ˆ ) α ) u 0 N 22 ih ( u BU i BU ( H α , one on the ‘ket’ | ρ ρ ˆ α Ψ + ) ρ u = , and I ( u i ˆ I Ψ BU , ) | ρ . Thus u j ( † j ih BU i ˆ J. Math. Phys. Ψ meeting . The point is simply that we , ) simplifies if u ⊗ | BU → H Σ ρ )) )) = 1 ) B.1 is an operator on u u u ( H ( ( † j † j . BU ij ˆ Ψ ˆ ) Ψ ˆ ρ identifies these two branches along ρ u -state (so that ( u BU BU maps . It is not meaningful to specify a priori α to itself. Its matrix elements are computed ρ ij ρ ˆ ]. † Tr ρ ) 1 ) ) BU u u u (to be defined below), which enacts the partial H ( ( ( H i i u j – 71 – ˆ Ψ ˆ Ψ ˆ Ψ ( SPIRE ( Tr u u IN that describes the state of both the baby universes [ Tr Tr u i,j Third quantization formalism for Hamiltonian cosmologies X . However, it should be borne in mind that other states on the radiation Hilbert space on ) , and so ), which permits any use, distribution and reproduction in ⊗ H 1 | u -eigenvalues of ( j α ij ih BU 7→ N (1984) 233 i ρ | → H H 23 BU ρ ) = ˆ BU u ( H CC-BY 4.0 past these obervables and then use the cyclic property of the trace in i ), recall that a density matrix ˆ ) Ψ This article is distributed under the terms of the Creative Commons ) u ( General relativity: dynamics without symmetry B.1 u † j ( maps † j ˆ Ψ ) ˆ Ψ u ( i ˆ Ψ , the second factor Int. J. Theor. Phys. BU K. Kuchar, N. Caderni and M. Martellini, We can also use this same simplification more generally if we only wish to use ( As is the case for all our discussions, the formula ( To explain ( More generally, we can write down the map from an initial density matrix on H [1] [2] References observer falling into a black hole. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. By essentially the samethe argument same as result above, as tracing tracingcan the commute such observables observables against againstorder ( to use and operators exist and may be of interest, for example to describe the experience of an In particular, ifleaves the the baby state of universes baby are universescomponents unchanged, are in while given producing an by the the state compute expectation values of asymptotic observables (thought of as operators on over all such choices; replicaidentifications persist wormholes will to the lead edge to of semiclassical the contributions associated where island. asymptotic the boundary conditions (so thatobservables). is part In of that the case superselected one algebra of finds asymptotic operator a map from the one-boundaryby Hilbert a space path integral boundedbranch by and a one pair of onasymptotically, Cauchy the producing surfaces ‘bra’. an operator Thehow on far operation this identification persists into the interior. The gravitational path integral sums where we have introduced the operation trace over the unobservedon radiation. The firstnormalise term the in trace the to tensor unity. product is an operator a final density matrixand on the state ofexplicitly the as Hawking radiation to which we have access. We can write this map JHEP04(2021)272 , ]. 09 214 309 38 SPIRE IN Pisma Zh. Class. JHEP , ][ ]. , (2020) 044 (1987) 337 Replica wormholes SPIRE 08 Phys. Lett. B IN , 195 Nucl. Phys. B [ ]. Phys. Rev. D (1987) 167 [ , , (2019) 063 46 JHEP (2003) 021 (2005) 084013 12 , SPIRE 04 IN 72 arXiv:1911.12333 [ (1988) 904 [ JHEP ]. The entropy of bulk quantum fields Phys. Lett. B , 37 , ]. JHEP Replica wormholes and the black hole Disruption of quantum coherence JETP Lett. ]. , , SPIRE ]. Field theory, quantum gravity and (1988) 854 ]. IN SPIRE (2020) 013 Phys. Rev. D SPIRE ][ IN , , in IN 307 05 ][ SPIRE [ SPIRE IN Phys. Rev. D ]. IN ]. 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