JHEP08(2020)044 Springer June 30, 2020 April 16, 2020 : : August 10, 2020 : of independent states Received Accepted k Published Published for SISSA by https://doi.org/10.1007/JHEP08(2020)044 [email protected] , , consistency arises from an exact degeneracy in the inner prod- . 3 k > Z , whose value can be less than the naive number 2002.08950 Z The Authors. c Models of Quantum Gravity, Black Holes, AdS-CFT Correspondence

In the 1980’s, work by Coleman and by Giddings and Strominger linked the , [email protected] Santa Barbara, CA 93106, U.S.A. E-mail: Department of Physics, University of California, Open Access Article funded by SCOAP ArXiv ePrint: gravitational path integral. Wecomplicated also models, comment and on on otherin possible aspects the implications of individual for extrapolations members the to of the black more above hole ensemble. information problem Keywords: theories. The dimensionrandom of variable the asymptotically AdSin Hilbert the space theory. turns For outuct to defined become by a thediffer gravitational only path by integral, a null so state. that many We argue a that priori a independent similar states property must hold in any consistent such ideas, using features associatedically with AdS a boundaries negative cosmological toconnect constant with strengthen and recent asymptot- the replica results, wormholean emphasis introduce discussions on of a the the role change of Pageical in null curve. models states. perspective, of A We the and explore key bulk this new structure that feature in allow is detail us in to simple compute topolog- the full spectrum of associated boundary Abstract: physics of spacetime wormholes to ‘baby universes’ and an ensemble of theories. We revisit Donald Marolf and Henry Maxfield Transcending the ensemble: babyspacetime universes, wormholes, and theblack order and hole disorder information of JHEP08(2020)044 36 64 54 -sector α 4 61 31 45 49 17 45 41 – 1 – 44 7 24 18 52 32 4 11 ∂ 21 S 62 27 39 14 -eigenstates 41 α n 2 and convergence and 52 n λ A.2 Large 6.1 Summary and6.2 future directions Transcending the ensemble: implications and interpretations for each A.1 Large 5.1 Formulating a5.2 wormhole perturbation theory Perturbation theory in the topological model 4.1 Entropy bounds 4.2 Consequences and interpretations 3.4 End-of-the-world branes 3.5 Baby universe3.6 Hilbert space with Hilbert EOW spaces branes 3.7 with boundaries The boundary3.8 parameter Spacetime ‘D-branes’ 3.1 A theory3.2 of topological surfaces Evaluating the3.3 amplitudes The baby universe Hilbert space 2.1 Path integrals2.2 and ensembles The baby2.3 universe Hilbert space Operators and 2.4 More Hilbert spaces A Limits of moments of the Poisson distribution 5 On third-quantized perturbation theory 6 Discussion 4 Entropy bounds and the Page curve 3 Example: a very simple topological theory 2 The gravitational path integral with spacetime wormholes Contents 1 Introduction JHEP08(2020)044 , ]. G A 1 4 42 = BH ] entropy S 52 1 ] that the above at all times. As 49 BH S using the gravitational ≤ rad rad S S coupled unitarily to the radia- BH S ]. 47 – ] did nothing to explain the dynamics from which the 44 ]) were thus all properly viewed as speculative 41 – 2 – , 39 – 40 , 7 ] and Hubeny-Rangamani-Takayanagi [ , of an appropriate comdimension-2 quantum extremal 3 23 51 , bulk 50 S ] argued that one could obtain this result by computing the generally increases to a maximum, at which time it nearly + G A 43 4 , rad S = 42 gen ] then pointed out that — at least in some contexts — this seemingly- S 49 , is the von Neumann entropy of bulk fields outside the codimension-2 QES. ]). While at this level the physical mechanisms behind such results remain 48 ]. The plethora of proposals for new physics that might be associated with 6 54 – bulk , 2 S , and then decreases monotontically thereafter. The final phase with decreasing 53 BH S of emitted Hawking radiation as a function of time gives an important diagnostic of describes the return of information to the external universe from the black hole. A starting point for such further investigation is the observation of [ Critically, [ Recently, however, it was noted that the ‘unitary’ Page curve, including the turnover of Despite many arguments suggesting that the latter so-called ‘Page curve’ should ac- . Here , could be obtained by combining ideas from holography with effective field theory [ As an example, the firewall proposal of [ 1 ] — or equivalently with quantum field theory in curved space. In particular, under rad rad rad rad replica trick results appeardefined to be theory. inconsistent withfunction-like In one quantities particular, might seem normally rather to call than have a both single taking well- a single meansupposed well-defined firewall value might values, and arise. partition- a non-zero variance. This somewhat mysterious, the derivation fromplies the that gravitational the path explicit integral additionthat nevertheless of fundamental im- lessons novel might physics be is revealedand not by studying required. carefully the dissecting Indeed, the path it relevant integral calculations instead in suggests more detail. See also further explorations of this idea in [ hybrid recipe in factpath follows from integral replica (and trick intum calculations particular of corrected that Ryu-Takayanagi this [ wasformulae [ implicit in earlier derivations of the quan- 43 very general conditions [ generalized entropy surface (QES) , where the surfaceS is chosen so that holography suggests this might represent reviews in [ obtaining this Page curve (includingand [ contained at least some optimistic extrapolation or adS hoc ingredient. equals S curately approximate the resulthow of such black a hole result evaporation, could for be many years obtained it from was a unclear controlled gravitational calculation; see e.g. out the evaporation, even thoughmonotonically the decreases black to hole’s a value Bekenstein-Hawkingis near entropy a zero. standard In quantum contrast,tion system a field model with would in density — which ofa the when states black result, the hole in initial such state models is pure — require The past year has seention. several interesting In developments in particular, the it studyS has of black been hole well-known informa- forwhether some and to time what that degreeFamiliar the information effective von is field Neumann theory preserved or entropy would give lost an in entropy evaporating that black increases holes monotonically [ through- 1 Introduction JHEP08(2020)044 – 55 ]. In than 58 – less 55 for a wide variety , consistency turns n ρ on the future half of a ∗ ) k > Z ∗ ˜ J introduces some simple toy , whose value can be 3 = ( on the past half of the Euclidean Z ˜ J J ] that might be interpreted as com- asymptotically AdS boundaries. Such ,J ] are then functions of these quantities. ∗ , section ˜ J [ 2 49 , Z 44 – 3 – by reviewing the connection between spacetime ], our analysis will show that one may generally 2 disconnected 57 – denotes CPT conjugation. However, the most general 55 ∗ ] argued that spacetime wormholes require the gravitational Hilbert of independent states in the theory. For 57 – k 55 relates this degeneracy to diffeomorphism invariance, black holes, and the 4 After describing this framework in section The output is a description of how (say, partition-function-like) quantities at asymptot- Our goal here is to combine the above ideas to better understand the ensembles asso- This relation will be reviewed below, but is familiar from older discussions [ ]. In doing so, we incorporate features associated with a negative cosmological constant the naive number out to arise frompath an integral. exact This degeneracynull degeneracy in state, means the and that inner many sospace. product a should Section defined priori be by independent regarded the states as gravitational differ linearly by dependent a in the gravitational Hilbert particular ensemble specified by the choice of baby universemodels state. in which the gravitationalsum path over integral possible can topologies. be performed TheHilbert toy exactly models spaces. including are the topological An full and interestingtotically involve finite-dimensional feature AdS of Hilbert the space models becomes is a that random the variable dimension of the asymp- partition-function-like quantities allowed by our formalismCFT include quantities would that describe in matrix aof dual elements density of matrices. operators The entropies asIn R´enyi of well accordwith [ as the e.g. original Tr describe works such quantities [ as bring drawn from an ensemble of their possible values with the ically AdS boundaries have apath spectrum integral. of Below, possibleputing values we determined the focus inner by on product the quantities of gravitational AdS a boundary state created with by anotherEuclidean a state source AdS created boundary, by where a source and asymptotically AdS boundaries.change This in both perspective. strengthens the In particular, resultsory’ we and and avoid allows emphasize the a that use useful of certainWe ‘third results also quantized follow focus perturbation exactly on the- from the any key well-defined role path played integral. by null states. ciated with replica trick computationsof and such to ensembles. extract We implicationswormholes begin for and in particular ensemble-like section members properties, and57 by revisiting the baby universe ideas of [ space to include spacetimes withmoment of compact time Cauchy has no surfaces, asymptotically andspace AdS boundary. thus was for This called part which the of space thewith baby gravitational at universe this Hilbert a sector. sector typically Furthermore,to it led act was the argued as rest that if ofensemble entanglement it the could were be theory part chosen (here of by the an selecting asymptotically ensemble an AdS of appropriate sector) theories. baby universe However, state. a particular member of the volve connected bulk spacetimes with geometries have been termed spacetimeometry wormholes, is or Euclidean. Euclidean wormholes when the ge- particular, refs. [ feature is associated with the fact that dominant saddles in the replica computations in- JHEP08(2020)044 ], n J . M J 57 ∼ (2.1) – 55 . 6 : i J specifying an ] will imply a ]. This is par- J [ M 49 . The notation is , Z n 48 ,...,J , we describe the approx- 1 J 5 [Φ] S − e ]. Φ -dimensional boundary metric on a J d [ D Z J ] where it gives the partition function ∼ -dimensional boundary Φ asymptotics. This is the context in which 60 d Z , . However, there is no harm in letting J +1 , describing the behaviour of the fields Φ 59 := d J includes a E – 4 – ] [Φ]. Each boundary is associated with a set of n J S J [ Z ··· ] 1 J [ connected components, each with an associated Z n D the left hand side as the path integral over all configurations with . We will focus on the case where the boundary metric has Euclidean but the identification of this quantity as a partition function in fact M 2 ], we introduce the following notation for the path integral defined by an ]. In the same way, any entropy R´enyi of any state that can be prepared by sources J [ is usually interpreted as computing a partition function 62 defines Z J ]). Motivated by this interpretation, with an eye toward the ideas of [ typically includes a function on the 61 to be connected, and introduce disconnected boundary manifolds by specifying J ], and emphasise that the appearance of null states is associated with the failure of M Now, the gravitational path integral with asymptotically AdS boundary conditions Our theory will be defined by the path integral over a set of fields (including a metric) We emphasize, however, that we allow very general notions of ‘sources’ and thus very general notions 57 2 [Φ], we then mean the holographically renormalised action with boundary condition chosen to be suggestive of a particular interpretation toof be ‘partition described functions.’ below. Inoperators, particular, so one that may use oneshould sources should as to be some prepare able(and initial to perhaps and represent restricted final to states any any and matrix region) to element should insert of again any be operator some in the dual CFT asymptotic boundary with This equation asymptotic boundaries with boundary conditions specified by ticularly familiar in theof AdS/CFT the context dual [ CFT, dates back to the first(see discussions e.g. of [ Euclideanand approaches following to [ black hole thermodynamics appropriate boundary condition forAdS/CFT context that is field; known e.g.,S as it the will “non-normalisable typically part” specify of the what field. inspecified the by In all cases, by signature, but Lorentzianeach or complex metricsmultiple are such also boundaries, allowed. eachbe with disconnected, We its and will own the generally notationmetric, below take remains consistent. For each field other than the denoted collectively by Φ,admissible with boundary conditions action labelled by near the given boundary.boundary In manifold particular, 2 The gravitational path2.1 integral with spacetime Path wormholes integralsWe and begin by ensembles describing a natural setand of convenience, observables we in any will theory assume ofwe locally gravity. have For AdS the definiteness most control and the clearest interpretation in terms of possible CFT duals. corresponding degeneracy in moreimation general in contexts. which In wormhole section of effects [ are small, analogousthis to approximation. the We close third with quantised some formalism summary and final discussion in section Page curve, arguing in particular that the replica computations of [ JHEP08(2020)044 ). ] for 2.2 (2.2) 56 for any , 2 55 T 1 T associated with

] ] (see [ that would naively 2 , but also contains J

[ ] ] , but also contains 63 2 2 , Z E J J ] ] [ [ 1 2 58 J Z Z J [ [

Z -matrix elements play the ] Z

1 S D as partition functions with J and [ .

Z ] E 1

] ] 2 = and 2 J J [ [ J 󰁈 [ E ) does generally not factorize over ] Z Z ] + Z 2

1

J J 2.1 , [ [ ED

] ] Z Z 1 1 D J 󰁇 J [ [ Z . We use the term spacetime wormhole, or Z D

1 – 5 – not only yields terms of the form 6= E E ] ] 2 2 ]. The natural path integral J J [ [ 2 J Z Z [ ] ] 1 1 Z J J [ [ Z Z D D ] and = 1 = J 󰁈 [ ) must thus differ unless the contributions with extra connections ] 󰁈 Z 1 ] 2.2 2 J . [ J

[ ] Z 2 Z 󰁇 J ] [ 1 Z ] J [ 1 J of terms associated separately with Z [ 󰁇 Z 2 . The gravitational path integral with spacetime wormholes does not factorize. The top

,T 1 T The two sides of ( The presence of spacetime wormholes in the path integral now leads to a phenomenon for AdS black holes,Euclidean which saddle. is We associated thereforemoment with allow assume a terms that change with they inIt extra the lead follows connections, topology to that and of a weproduct at the non-zero least cannot dominant difference for simply between the interpret the two sides of ( exactly cancel amongoption themselves, appears to or require fine unless tuning,that such and the undermine contributions second the the are imposition presumed ofone excluded. non-local local might constraints introduce nature The useful ofof first such the the constraints theory. Euclidean without destroying path It other integral, is apparent such also successes as difficult the to see description how of the Hawking-Page transition component of the bulksometimes manifold; Euclidean see wormhole, figure wormholes to are refer generally to localizedfrom any spatial in such wormholes both like connection. theCauchy space familiar slice and Einstein-Rosen Note of bridge time, the that that maximally and exist spacetime extended on thus Lorentz every signature differ smooth Schwarzschild qualitatively spacetime. The difference between rightEuclidean and path left integral sides for arises becausepair the sum over topologiesadditional contributions in from the terms in which the two boundaries lie in the same connected role of our partitiondisconnected functions). boundaries: The path integral ( additional connected contributions schematically shown as the second term in the bottom line. which is very puzzling fromearlier the discussions standard of AdS/CFT the point asymptotically flat of analogue view in [ which Figure 1 line gives a diagramatic representationdefine of the partition path functions integrals a pair of boundaries yields all terms generated by multiplying JHEP08(2020)044 ], where the will be ded- 2 57 – 55 y b a B ] think of the closed universe t , it is useful to provide a brief n 67 e – r 2.2 64 a P – 6 – given by Euclidean metrics, but our discussion does not i J contain factors associated with compact spacetimes having no

] that they are in fact consistent. The rest of section ], the connection between the above two interpretations is motivated as the expectation value of a product of partition functions in an . We call this the baby universe sector following [ ··· 57 ]

57 – 2 2 – J 55 [ ··· 55 ] Z ] 2 1 J [ J [ Z ] Z 1

]. . Slicing open a spacetime with a boundary and a handle (left) can give a disconnected J [ 57 Z –

The discussion of baby universes is simplest in the context of Euclidean path inte- Before turning to the detailed discussion in section While these two interpretations may at first seem to be in tension, in analogous settings From the bulk point of view, the extra connections appear to describe dynamical in- 55 having been emitted by a (here asymptotically AdS) parentgrals universe. with boundary conditions exclude more general contexts.Lorentzian pieces In of particular, the one metric, can using a choose Schwinger-Keldysh boundary type conditions formalism with in which Eu- boundaries whatsoever. The ideacutting open that the the path Hilbertspacetimes. integral space then of Doing suggests a so that theoryasymptotically identifies we can should a AdS be also new boundaries, slice identified sectoruniverses; but open by not see such which associated figure compact is on insteadname this comes associated slice from with with the any idea spatially of that compact one the can in many cases [ overview. As in [ by realizing that summing overlar arbitrary over topologies manifolds in with ourterms arbitrary path in numbers integrals, of and connected in components, particu- means that generic it was argued by [ icated to providing aasymptotically version AdS of boundaries. this We discussion findconclusions, that that and incorporates using perhaps features these as associated new aof features with [ result allow we strengthened will take a slightly different perspective than that pret ensemble of boundary dualwould describe theories. probabilistic correlations In frominteractions. this the ensemble interpretation, average the rather than connected dynamical contributions boundary (red line) at the moment of time described by the indicated slice. teractions between anaturally priori compatible independent with asymptotic standard regions. AdS/CFT, but it This may point instead of be consistent view to is inter- not Figure 2 geometry on the slice,ent including asymptotically a AdS closed ‘baby universe. universe’ that The has baby become universe detached does from not the intersect par- the asymptotically AdS JHEP08(2020)044 . 6 ) as if for the 2.1 BU H ]. To do this, 48 – 43 -states’ of baby universes in α . – 7 – 4 to be operators in the auxiliary system, and then include a J ). We thus identify the relevant Hilbert space as the space of 2.4 ). In particular, we split each history over which we sum into a ‘past’ ], defined by the absence of additional boundaries besides those required 2.1 68 ] insertions. But this is not the only choice of baby universe state that we can J [ ) includes an implicit choice of the initial and final state of closed baby universes. Z 2.1 indicates that baby universes can affect the physics of universes with asymptotically One might hope to describe elements of the baby universe Hilbert space as wavefunc- One further comment is in order before turning to the details. In the above discussion Note that the ability of Euclidean or complex universes to split and join as shown in fig- Such constructions allow us to describe quite general observables that might be associ- 2 (generalizing in section closed universes in the theory.reasons described We call above. this the ‘baby universe’ Hilbert space tions of all possible spatial metrics (and field configurations on those metrics). A complica- the path integral ( and ‘future’ that meetintermediate on states. some There is sliceasymptotic a where boundaries choice we of are imagine how labelled weconnected summing past cut, component over or constrained of a future. by the complete theof boundary For way set now, either in our of we entirely which cut, will to the the choose so past to that or place our entirely each to intermediate the slice future intersects no asymptotically AdS boundaries Nonetheless, for the remainderit of gives this well-defined section exact we results. will treat the2.2 path integral in The ( babyAs universe described Hilbert above, space one can obtain a natural Hilbert space interpretation by cutting open defined only assome asymptotic small expansions coupling. (perhapstypically with Both fail nonperturbative to loop contributions) converge, expansions andfinite in and there result. may sums The be over distinction no betweenasymptotic nonperturbative obvious, exact expansions sectors natural quantities may or well with will be unique finite important, values way and of to we parameters define will and a return to this issue in section different ensembles. In particular,which we the will factorisation construct property special is ‘ restored, and no ensemblewe is have required. written ourHowever, amplitudes in as all if but the the path very integral gives simplest some contexts, definite, gravitational finite path value. integrals have been gral ( Most naturally, the path integralboundary computes state [ expectation values inby the the Hartle-Hawking no- describe with our gravitational path integral, and other choices will be associated with we can simply allow sources corresponding auxlliary path integral tothis compute construction the in effects more of detail such in operators. section We discuss ure AdS boundaries. In this context, it becomes clear that the definition of our path inte- time evolution. Ininvolving such complex a metrics. case, it is useful toated think with of a putative the dual gravitationalto CFT. describe path Indeed, coupling the integral to set an as of auxiliary observables quantum we system, are as using is is important also in sufficient [ clidean sections of the metric are used to prepare states and Lorentzian sections give real JHEP08(2020)044 (2.4) (2.3) ]. of boundary 57 } m ) as living ‘in the 2.3 ,...,J 1 J . { BU ], in defining the gravitational ∈ H 69 , E BU HH

∈ H E −→ ] m J [ = 0) – 8 – Z m ··· ] 1 J [ Z

) defined by different sources, or even with different numbers 2.3 No boundaries ( ) is not important. Reordering the sources gives equivalent boundary 2.3 ) to be computed with the same specifications. With this understanding, , are generally not mutually orthogonal in any useful sense. Note that the 2.1 ] in ( m i = 0, giving the Hartle-Hawking state with no boundary in the past: J [ m Z States of the form ( We will think of the boundary conditions associated with the state ( However, we can bypass these difficulties entirely by using our asymptotic boundaries compute inner products.path However, as integral explained one in mustthe e.g. make contour [ of a integration choiceor — — instead as in to defines some whethercorrelators a languages, it ( Green’s associated fully function. imposes with the specifying gravitational We constraints simply choose the former, and we take the of sources physical notion of innerbut product is something cannot we must simply compute bepath from the integral. assumed theory. Now, to It some must have readers thusone may follow any from be typically particular an confused uses appropriate form, by first-quantized the path fact that integrals in to quantum compute field theory Green’s functions and not to Here we emphasize thatrepresents this a is state of not the just full a collection state of an on indefinite a number single of universe, baby but universes. that it instead understand that these are simplyof names without the intrinsic meaning. Noteconditions that for the the ordering path integral,case and is so must define the same state. An important special Wick rotation to Euclidean signatureto it Euclidean tends time. to As cause aasymptotically accelerated result, AdS expansion such boundaries closed with at cosmologies respect infinite naturally have Euclidean Euclidean times. signature past.’ They can then be paired with bra-vectors living ‘in the future’ — though one should defined by the specifiednatural for boundary sources conditions defining fora Euclidean the negative signature cosmological path boundary constant. integral. metricsuniverses and While to This in a collapse the is negative in presence cosmological Lorentzian particularly of constant time tends evolution (perhaps to with cause a sinusoidal form), after to define states inconditions, the there baby is universe Hilbert a state space. Given a set of universal time thatmade. might Proceeding be in used this manner specify(the would precisely Wheeler-DeWitt thus require where equation) imposing on the the thechallenging gravitational past/future constraints in resulting cut wavefunctions. the is current This toassociated context is be where splitting made spacetime and particularly joining wormholes of are universes important, should so modify that these the constraints [ tion is that, as usual in a gravitational theory, diffeomorphism invariance forbids a notion JHEP08(2020)044 . ) ) J ): ] as 2.3 2.5 (2.7) (2.8) (2.5) (2.6) J 2.3 [ Z ] for further . 75 – E ] ]. 71 m . 70 J [ E ] i ) reduces to a stan- Z ) corresponds to simply ], and an inner prod- i,m J 2.5 ··· [ 2.5 J ] [ on every boundary, the 1 Z . Z J [ . ∗ [Φ] ∗ ) with boundary conditions Z ··· S ] E ] , then ( ] ∗ n − 1 ], using ( ˜ n E 2.1 e J i, ] [ J [ n J 77 Φ [ Z , J Z [ D Z 76 which is then given by

Z ··· i ··· ) corresponds to taking two arbitrary ‘kine- 3 ] c ] ∗ 1 Z ··· 1 ˜ 2.5 J ] N =1 J [ i X [ 1 Z ], ( J Z [ = no boundary = D , for more complicated systems it would be 70 D ) is Hermitian. If we can interpret Z

) but are easier to check.

3 = D = Ψ – 9 –

2.5 2.8 E E ] HH ]

∗ m n J J [ [ HH Z Z

0 for all ··· ··· = ] ] ≥ 1 ∗

1 is the CPT conjugate operation on boundary conditions ) need not be normalised. In particular, the norm of the J

1 J [ [

) sesquilinearly. This is almost enough to construct a baby uni- ∗ Ψ Z 2.3 | Z

= ] D 2.5 Ψ n

Z ˜ J [ := Z 2 ] for connections to path integrals. As in [

··· ] 78 Ψ 1 – ˜

J ], and where [ 76 ∗ ˜ Z J ), namely reflection positivity. This can be stated as the requirement that ( [ D . This is equivalent to performing the path integral excluding closed components Z Z 2.1 We now have a space of states defined by (finite) linear combinations of the states ( Note that the states ( ] and In the language of Dirac constraint quantization [ 3 J [ , defined by the path integral over all spacetimes without boundary: very useful to find properties that imply ( matic’ states (which may not satisfyconstraints, the and constraints), computing projecting the themcomments, physical onto and the inner space [ product of of statesskipping the satisfying to the resulting the projections. final See answer [ without going through the intermediate steps inherent in [ Thus is clearly required iftheory, our though gravitational it path is integralfor cumbersome is the to to simple verify define toy a directly models standard for studied quantum all in states. section While this can be done uct defined by extending ( verse Hilbert space. The missingintegral ingredient ( is a single property thatdefines we a demand positive of our semidefinite path inner product on finite linear combinations of states ( For most purposes, itdivide by would be sufficientof spacetime to which consider do not normalised connect amplitudes, to where any asymptotic we in boundary. correspondence with formal polynomials of ‘partition functions’ Hartle-Hawking state is givenZ by what one might call the cosmological partition function dard construction in probability theory,variables in defines which an the inner covariance matrix product.expectation of In values pairs particular, of of showing random a thatinner distribution the product with amplitudes is follow nonnegative positive from probabilities semi-definite. would imply that our amplitude is complex conjugated: This guarantees that therandom inner variables with product correlation functions ( Here the right handZ side is just theThis amplitude operation defined should in have ( the property that if we act with the path integral indeed computes the inner product JHEP08(2020)044 ) 0 i} j → 2.3 Φ ). (2.9) 2 {| J , i k j i} i Ψ ) with the Ψ {| i−| one may find 2.3 i i is then a Hilbert Ψ c k | ] or Osterwalder- BU though a standard H 79 BU H are invariant under familiar that in certain circumstances J . , where two sequences N 4 ∈ i . Equivalently, some infinite sums = 0 are infinite sums over states ( i} i BU E and ] Ψ i BU H 3 {| H i,m J [ Z ··· ; i.e., for appropriate coefficients ] – 10 – 1 ]) reconstuction theorems. In this analogy, our ). This may seem like a minor technical point. Of BU i, . Recall that a sequence is Cauchy when J H [ 2.9 81 , Z may appear to consist of formal power series in the ob-

→ ∞ is very similar to the construction of the Hilbert space of the baby universe Hilbert space i Importantly, however, infinite sums with different terms 80 c 4 i, j BU BU ). =1 ∞ of [ H i X H 0 as define 2.5 are defined; i.e., natural sources → 2 J i k j , one expects different slices of the same spacetime to describe gauge has a countable dense subset (assuming that amplitudes are continuous in Φ 3 J ). Roughly speaking, states of ), we now i−| i 2.5 Ψ 2.8 Theorem 3-7 k | ] correspond to (smeared) local operators inserted in the Euclidean past, and J . The inner product between two such sequences is defined by the limit of the inner products [ ] with some convergence property. But it is in fact smaller since the construction is the set of equivalence classes of Cauchy sequences Z J [ → ∞ Z BU The above construction of Naively, the Hilbert space Assuming ( H 4 i, j are equivalent if as of the terms, whichspace, exists so and in is particular theset is same of complete for possible and all sources the members inner of product the equivalence is class. positive definite. It is separable as long as the objects the inner productsthe between (Euclidean) states Wightman with functions.completion finitely construction. many The operator Hilbert space insertions is are again given defined by by the above such modifications to be small,they we lead will to find dramatic sections physical consequences. a quantum field theory fromSchrader its correlation (see functions in the Wightman [ illustrated in figure equivalent states. Butno including longer a be sum relatedthe over by same topologies a number means of diffeomorphism, thatimportant connected to and two compute components such in the for slices factthat effects space may that they of at take this they a the modified need familiar gauge given not form. symmetry time. even rather In contain than particular, It to while will assume one thus might be naively expect the effect of aries where our sources gravitational gauge symmetries.encode As possible a senses result, in one whichof one sources. might may expect However, have one accidentally the shouldgauge introduced expect null invariance an the so states overcomplete sum that set over to it topologies simply no to longer modify corresponds the precisely gravitational to familiar diffeomorphisms. As jects divides out by the setcourse, of from ‘null one states’ perspective ( thenaturally inner leads product to defined a by largeBut any set we gravitational of path usually such integrals expect null that states symmetry due to to the act gravitational trivially gauge at symmetry. the asymptotically AdS bound- construction, as the completion ofinner the product space of ( linearwith combinations finite of norm states defined ( by ( and coefficients may not give risemay to be distinct identified states with in the zero state in JHEP08(2020)044 For 5 (2.10) (2.11) (2.12) . E ] m J [ Z ··· by continuity. ] 1 , ] is J BU [ E J ] [ , H Z ] d m E Z J J [ [ ]. we have two circular universes, ]. As usual, unbounded operators , we may write the action of i Z J . The matrix elements of this Z 2

[ 1 J HH ] [

] BU Z ˜ Z BU m , we now introduce a set of oper- ··· ˜ m Σ J ] H H [ J 1 [ BU J Z [ \ Z H Z ] on ] ··· ··· ] J J [ ] [ 1 ˜ 1 d J Z Z [ J

[ 1 Z [ Z = D ) is dense in Σ – 11 – E = = ] E m E ] ] 2.10 J [ m m J Z J [ [ Z Z ··· ] 1 ··· ··· ] J ) implies that our defining states may be created by acting ] [ 1 1 J Z J [

[ ] -eigenstates 2.11 Z Z

J

] to define an operator [ α ] d J Z J [ [ d Z Z

] ˜ m ˜ J [ ] operators on the Hartle-Hawking no-boundary state, Z J [ d we have only one. These may be thought of as different gauge choices for the same state. ··· Z . In the presence of spacetime wormholes, different spatial slices of a spacetime may have ] 2 1 ˜ J [ Since the labelling of boundaries as past, future, and in between does not affect the Z Strictly speaking, this is the case for bounded functions of the 5 D with the can be defined only on somewhat smaller domains. such operators as extending the action oflater such use, operators we to note the that full Hilbert ( space value of the path integral, the defining relation of the operator Since the span of the bra-vectors in ( Having constructed the baby universeators Hilbert acting space on it. Heretake we any once boundary again find asymptoticoperator boundaries are useful. defined by In a particular,some path we initial integral and over all final configurations states with with boundaries an specified additional by boundary different number of connectedbut components. on Σ Here, on the slice Σ 2.3 Operators and Figure 3 JHEP08(2020)044 , i J are i (2.16) (2.17) (2.18) (2.19) (2.13) (2.15) α | ) that the commuting 2.16 ] is well-defined via J i [ is arbitrary we see d α Z |

complete . E

. Ψ 6 and an arbitrary state

E α α

† ] N HH

n ] HH J [ m

] J \ Z . [ ∗ n \ J Z [ ··· -states for short. The spectrum J. which are simultaneous eigenvectors α = 0 † α ] . ··· in fact defines a )). To show that i Z

1 ] , α J ] (2.14) 1 J | i ∀ [ N ∗ 2.9 α ··· J

= 0 0 [ α is null, and since [ 0 Z ] J | ] is given by taking the CPT conjugate of α

[ ] [ i ∗ 1 Z Ψ

δ ] J ] mutually commute:

0 J \ Z J [

[ [ turns out to have a powerful set of properties. J J d N = α [ Z [ α

HH = = ] as operators on the baby universe pre-Hilbert J d HH [

Z Z Z D Z is determined up to a phase by its eigenvalues † J

D , – 12 – ] α [ ]

i are unchanged by permutations of the sources = d = = 0 J N Z J [ α = ] is normal (that is, it commutes with its Hermitian

i [ are normalisable eigenvectors, writing

α | ] d ) that we may identify our original path integral as E ] d Z J

α E i Z [ | J to write α -eigenstates, or ] m ] [ h d for all possible α

] using an additional assumption about locality of induced cou- α Z J d | ]

m J [ Z 2.11 [ } n

J 57 ] Ψ [ d ) that all Z – J

Z as advertised earlier: Ψ [ J ] Z [

∗ 55 Z d E ··· Z J 2.11 ] [ { ··· ] for all possible 1 ) and ( ] ··· \ Z J ). In particular, for any null state J 1 ] [ HH [ 1

J d = , as the state [ 2.5 Z Z J

[

Z 2.12 0 BU Z D Ψ D H

has a basis of orthonormal states ], we call these ) or ( ] may be either discrete or continuous. In the latter case the 55 J BU [ is also null as desired. d H Z 2.14

] operators: N of

J ] [ α , we must show that it maps null states to null states. But this follows immediately J d } [ Z ] d BU Z ]. To see this, note that we can determine all matrix elements of J It turns out that the set Following [ The set of operators Thus far, we have really defined the [ , we may define A similar result was derived in [ J H [ α 6

α Z Ψ plings. Crucially, this assumption played no role in our argument above. leaving the appropriate modifications for continuous spectrum implicit. set of operators on Z { not normalisable states, but arewe instead use delta notation function in normalized. either However, case for simplicity as if In particular, this implies that each conjugate), so that we may applyHilbert the space spectral theorem. Itfor then all follows from ( Firstly, since the states it follows immediately from ( The last equality follows fromthat the fact that space (before taking the quotienton by null vectors ( from either ( computing correlators in We also see that thethe Hermitian source: conjugate of and thus by combining ( JHEP08(2020)044 ) ]. 2.1 56 , (2.23) (2.20) (2.21) (2.22)

], thus making 55 J by these [ i d HH Z α α ): | | p n ). The only ; we can even α 2.1 basis. Now, by .

] BU 2.16 n 2 α H α J [

). ] α n has vanishing overlap Z 2.7 J ] i [ 1 α Z J | | [ 1 α − Z n α =

i α ···

, ]

2 1 J [ α = 1

[ . ,. ] Z ]

α 2 1 i = 0. Otherwise n p α J i| J i 6 [ -state, this selects a single member of the from the ensemble. The states [

i α α α α Z α HH | | | α X α | ) it was associated with the implicit choice of Z 0

-states, and describes an ensemble with proba- ] α 1 – 13 – HH α , ··· HH h J HH

[ -states is provided by the ‘eigenbranes’ described 0 2.13 ] 0 h |h 1 [ ), tells us that a gravitational path integral ( α Z α J

>

[ = α α α are in one-to-one correspondence with members of the ], and analogous to the random couplings of [

2.21 p α Z -state, 0. p 62 HH . While the Hartle-Hawking state is a particularly simple α = α BU n p

> i label the various theories in the ensemble, the eigenvalues BU H

α α

α H ] α ,...,α X -states, we can compute the general amplitude (

1 2 X α J is the norm of the Hartle-Hawking state ( ,α [ Z 0 [ HH α Z Z -states define a preferred orthonormal basis for ]

= = 1 α . Classical probabilities are sufficient to describe the ensemble, since E J is a superposition of ] [ 2 ), along with ( n i [ i| Z J

Ψ [ Ψ | α | 2.22 Z

α |h ··· ] gives the probability of selecting = 1 J α α [ p p Z ] in the context of JT gravity, which act to constrain the eigenvalues of D ] give definite values for observables in the theory associated with the particular label With our new Hilbert space point of view, it is now clear that the ensemble described Equation ( The above calculation of the matrix elements also shows that the Hartle-Hawking state J 82 [ α , and ensemble so that amplitudes factorize: Any other state bilities above is not unique.the Instead, Hartle-Hawking state through in ( and natural choice, wethe are initial nevertheless state free of to the select baby any universes state is we an like. In particular, if in [ partially diagonalizing thesedistribution operators. because Note the thatproperty relevant required we operators of arrived are the atpositivity, mutually gravitational to a guarantee path commuting classical nonnegative integral ( probabilities. probability (besides its existence) was reflection Specifically, the parameters Z α up our preferred eigenbasis of ensemble. A less extreme example of The normalising factor is quite generally compatibleensemble with dual an to ensemble JT interpretation, gravity exemplified in by [ the matrix where the second followsinserting from complete completeness sets and of orthonormality of the we find fix phases by choosing has non-zero overlap with every with a dense setoverlaps according of to states, and hence must be the zero state. If we define This means that the JHEP08(2020)044 , i = M 1)- ∅ ∂ M . For − (2.24) J Σ= d H ] defining bound- J [ ψ = Σ. As before, the (perhaps with different M m ∂ Σ , . Σ 6 . While this notation is useful, J of closed ‘baby’ universes. We ∈ H ) can be paired with a boundary ], and the Hilbert space of closed i t · · · t on Σ. Note that Σ can have any , except that in addition to closed E J 1 ] [ BU 0 J ), we then have states n BU Z H J = Σ [ i H 2.3 ] to be associated with a connected Z i ˜ J M [ ] we also have objects ··· ∂ ψ J ] [ ) in a natural way if we note that a boundary 0 1 is the solid black semicircle forming the past for any source Z J [ i 2.5 ]. In other words, with respect to the algebra of – 14 – with Z ] J M i [ m , d ˜ Z J M ] are constants [ 2 J ψ [ . -states in the superposition are irrelevant for correlation d to be connected. Note that this does not imply Σ = Z J (where the ordering of the components is meaningful, in α ··· i ] 1 M of an asymptotic boundary with generalizes ( J proceeds much as for may again be disconnected. As a result, one will sometimes [ ) already defines factorising amplitudes, so that our theory of Σ . i ψ Σ M Σ

H 2.1 H H = Σ i with sources ] would define a state on the CFT Hilbert space with spatial geometry M , leaving implicit the other sources by cutting amplitudes in such a way that any given asymptotic boundary J ∂ M [ Σ -states define superselection sectors. ψ is one-dimensional, spanned by the Hartle-Hawking state, which is also the of components Σ H ] in the ‘bra’ (on some BU α i ˜ J H m , and in particular its boundary Σ, is implicitly included in the sources BU [ ) for a given ] is associated with component Σ Σ ψ i H M m -state. We discuss this possibility further in section J [ ], the α ψ J [ of closed baby universes described above. d The inner product on As before, we may choose The construction of If the path integral ( Z t · · · t BU 1 whose boundary need to use a numbervalues of of distinct decompositions Σ = Σ condition case they have the same geometry). Generalizing ( where it is also somewhat awkward if we take a given Σ, as the wavefunction forpath a integral given on CFT field configuration on Σ wouldto be be computed connected. by a Σ When Σ is not, it can be useful to write Σ as the disjoint union Σ = ary conditions on amanifold piece example, in the rightasymptotically panel AdS of boundary figure andinterpretation, Σ consists of the right and left endpoints. In a dual Hilbert space number of connected components, and if ΣH is empty we find again the Hilbert space asymptotic boundary conditions denoted by constructed lies completely on one side ofintersect the one cut. or We more now components generalize of thiscomponents the construction asymptotic into to boundary, allow two thus cuts parts. splitting that suchboundary boundary This gives conditions us at many differentdimensional the Hilbert (perhaps intersection, spaces oriented) depending and spatial on boundary in the geometry particular Σ. on We the thus call choice the of resulting a ( universes unique 2.4 More HilbertThe spaces above discussion concerned the Hilbert space functions of the commuting operators the gravity has a single boundary dual,here. we have In a that trivial special case, case the of operators the formalism described relative phases between different JHEP08(2020)044 . ∗ i i . so ˜ 3 M 2 is an , the (2.25) 2 Σ Σ = Σ , H from the 1 1 i Σ admits isome- constructed by ⊇ i i , and in particular M by such sewings are Σ ∗ i , even if Σ i

H ˜ ] M 1 ) to define a boundary M E i ] i ,J ˜ ∗ 1 ,J M ∗ J [ = Σ ˜ J . This notation is chosen be [ Z i ]  Z ] 2 D M J [ ,J ∂ ∗ 2 = , ψ J ] [ ˜ J Z  E [ ] with

ψ J i [  . While the norms of these states will 6= ) is not invariant under reordering of the Σ

is then constructed as a completion, which ] M

, ψ 1 ] ] by reversing the orientation) and sewing Σ 2 ˜ J J ] = [ 2.24 i [ – 15 – H ,J ψ ˜ ψ ] ,J ∗ 1 M 2 ∗ J ˜ J [ J ] acting on the Hilbert spaces [ D  [ Z J ψ ] . One may also wish to restrict the allowed sources to [ Z

= 1 ) of states in the dual CFT Hilbert space. The distin- d · Z E ,J , , the structure above is properly described as being pre- ,J ] · ∗ so that the sources defined on 6= ∗ 2 ˜ J again denotes CPT conjugation of sources, and the sources J i [ J BU [

∗ ψ ]

H 2 Z ] ],

J ˜ J [ (formed from [ ,J ψ = ψ ] ∗ ∗ i ˜ 1 D

J ˜ ] [ J M [ 2 Z J ψ [

; this will prove to be relevant for the toy model discussed in section ] associated with the closed boundary manifold ψ ] ), 1 . In BU ,J i ] in the ‘ket’ (again on some J ∗ i [ H ˜ is then positive definite as desired. Note that reflection positivity on J J ψ [ [ 2.24

. In particular, the notation in ( Σ ] are given locally by ψ Z 1 H J M [ along Σ M ψ ∗ to ] i 2 ˜ As before, we have operators As in the discussion of We emphasize again that if Σ contains identical connected components Σ We shall write the pairing as It is important that the above sewing is uniquely defined even when Σ ˜ J M M [ M ψ positivity on which preserve the spacebefore. of Again, these null operators states mutuallyoperators commute. in which But can the now map we pre-Hilbert between also Hilbert space have a spaces for plethora with of the different new same boundaries. reason In particular, as includes a quotient with respectthe to the path space of integral null isproduct vectors. it This appropriately procedure defines reflection succeeds on when positive, thethe final pre-Hilbert by space which is we positiveadditional requirement semi-definite. mean we impose The that on inner the the product path inner on integral, not necessarily implied by reflection this pairing makes sense).invariant This under symmetries is of a Σ. special case of the statement thatHilbert states space. need not The be actual Hilbert space components are treatedtation as of distinguished ( agree, and the canonically inner ordered. product of these Thus states in with the generic other no- kets will not (for example, ner product is thenintegral defined as by before: using the above pairing and evaluating the resulting path of suggestive of an innerguishability product of ( points in Σgravitational is boundary motivated either conditions by at a dual asymptotically CFT AdS perspective, boundaries. or from familiar The extended in- sufficiently smooth. tries. In particular,beginning, recall and that at no thepoints point above of was Σ discussion there should a fixed thus be quotient a thought by of manifold diffeomorphisms as Σ of carrying definite Σ. labels, The defining the individual unique sewing condition taking the manifold to on vanish sufficiently quickly at Σ condition JHEP08(2020)044 ) } 0 ] → i ) in J [ ˜ Σ α 6= Σ (2.33) (2.29) (2.30) (2.31) (2.32) (2.27) (2.28) (2.26) ˜ Σ. We ]. H Z 2.25 { ,J .

∗ defined by ] E ˜ ] 2 J 2 [ 2 \ ,n Σ -states. Thus 0 2 ,n Z 0 t ] maps 2 . In particular, α , J 0 1 [ 2 J , one may write E [ J Σ ] corresponding to Σ [ Z 0 n t Z [ H ψ BU 1 J ] ∅ [ α Σ ··· H J ··· ] [ Z , H ] d 0 α Σ ψ 21 into 0 21 J , as we may define ··· 2 [ . J . It follows that all of our ] [ Σ 2 0 i 1 into Z ∈ H BU ] α Z , J α 1 ] 2 2 [ |

H E 1 α by acting with such operators Σ ˜ α Σ . Σ there is an operator Z α δ ⊗ H α 2 ,n ]

] 0 ,m Σ 0 1 ˜ ] Σ 1 J 1 2 [ J t Σ , H m [ J by taking the boundary conditions ⊗ H 1 ˜ [ α,α ˜ ψ ,J Σ J ] H 1 α Σ Z δ [ ∗ ˜ Σ 2 t ψ α Σ J \ ψ J Σ H [ [ H = 0 . ··· H ] commute, as 1 ψ -states of ··· 0 ] α Σ

α ] ··· α to α,α J 0 ] 11 [ Z H . δ 21 → H = 1 Σ J [ ˜ ψ HH Σ J [ ] preserves the null space, and thus is truly J t . One then finds

[ ˜ = , [ Σ E α t 0 ] → ] ˜ J Z that takes the value 1 on arguments Σ ]. α [ M ψ [ Σ ] α

ψ 0 n H ˜ J

2 0 d J 1 2 [ t f H α ψ J [ 0 α Σ 6= – 16 – = d Σ [ α α f ψ d = Σ, then for any ⊗ Σ ]: := . In the special case Σ = ψ 0 f Z ]; from the Σ ,m

J α Σ α ] [ 2 1 E ) and the corresponding property for = 1 H M ⊗ H d α Σ J J H α ψ ··· [ [ ∂ ,n

1 → H -sector, taking ] 0 H ] yields a dense set of states in ]; 0 1 ψ ψ 0 α Σ 1 are disconnected. Note, however, that (when Σ

α ˜ α J 2.25 J Σ m J 2 [ [

H [ J for all d ··· ˜ H α [ Z Z maps from Z ] ⊗ M ] } ψ ]; , ] † ˜

2 21 Υ: i ] J having ··· [ unchanged. For example, evaluating the analogue of ( J α J J J ] [ [ ··· ˜ [ M

[ ψ 0 ]

0 d α ψ ψ 11 α M ψ ] 1 ]

] maps J 1 J J Z [ [ [ ] commutes with J for some function Σ { [ d with the Z ψ J ψ ) are dense in ] d i [

ψ ,m along the manifold Σ. 1 ] i 1 d 0 Z Σ , follows from ( J ∗ J [ ] commute, it is again useful to diagonalize them using HH [ Σ 2.29 | ,m J HH J ψ ) 1 [ | ψ [ H } is one dimensional, consisting of multiples of J d ] [ . As a result, the fact that with Z ) even when i ˜ ··· Σ we define the components of Σ to be ordered before components of α ∅ ψ J ] [ BU t H 11 Z 2.26 ··· H J { ] [ , while ( ˜ Σ = ψ α 11

t f J ] is associated with 0 ∅ [ , each Σ J Finally, note that there is a natural map Υ from The adjoint operator Since the Nevertheless, one can build a dense set of states in [ = 7→ ψ H t

i ψ BU Σ -states we have α but which vanishes on and more generally This maps acts nicely withinsince each acting on | It also follows that concatenation of sources: and, the statesH ( boundary operators leave α One can explicitly build the spaces well-defined on defined by theconjugate state source on which it acts, andthe gluing Hilbert space to splits boundary as conditions of the CPT it does not makeH sense to ask whether on where in Σ may use ( if JHEP08(2020)044 , 0 0 to to S on S α α α = = C in sec- ∂ ∂ S S turns out BU . b H 2 Z α Σ is plausibly the ⊗ H , and demonstrate 1 -states are described 2 α Σ 3.1 α H . We then evaluate its in very simple theories = 0. We will use Υ ) is related to spacetime 2 c , associated with a single 3.7 Z 2.2 ] occurs when there are two- for , and its to denote a Hilbert space with 85 – } c 0 3.4 83 { = H c ], which differs from the partition function , with the path integral defined by a single as the Hilbert space of a dual CFT 85 ], along with the addition of ‘end-of-the-world to be an isomorphism is a precise version of α Σ – 2.3 H 87 α 83 – 17 – , 86 , for non-negative real 62 H . In particular, c ]. We further simplify that class of models by removing any H 49 associated with boundaries, whose preferred value ∂ S and construct the Hilbert space of closed universes . The failure of Υ 3.2 determining the suppression of nontrivial topology, along with a (some- 3.6 0 times that of S c of the class described in section b Z . This theory allows only one boundary condition . The most interesting output of this model is that the spectrum of We begin by presenting the simplest model (without end-of-the-world branes) in sec- It is natural to attempt to interpret 3.3 3.1 amplitudes in section tion to be non-negative and discrete,compatible and with in an fact takes interpretationwith non-negative as end-of-the-world integer branes the values is for dimension then described of in a section dual Hilbert space. The model tion operator bulk parameter what ad hoc) parameter will be determined later by a consistency analysis in section brane’ dynamical boundaries [ notion of a dynamical metricmodels or are tractable dilaton, enough leaving to aThey be theory thus solved of exactly, give topology and a for alone. surprisinglythe many The details clean type to resulting illustration of be results of made to the explicit. which ideas such of ideas section can lead. This section further exploresof the two-dimensional structure gravity. describedsimplest in Indeed, possible section such the theory. model Ourof models described nontrivial are topology in inspired in JT by section gravity recent [ work studying spacetimes version appears in pure JTsided gravity, which Hilbert has space. a two-boundary Wewould Hilbert expect be space that absent but this in no more feature single- realistic is theories. an artefact of3 simple toy models, and Example: a very simple topological theory wormholes. In particular, thesided factorization black problem hole of states [ represented with as a superpositions spatial wormhole ofHilbert (Einstein-Rosen products spaces. bridge) of For example, which ‘microstates’ in cannotparticles, in a be there bulk the theory are corresponding with eternal a one-sided charged standard black Maxwell holes field but but no no charged one-sided counterparts. An extreme be an isomorphism, since this propertythis would is certainly hold not true always inexample a the local in case, dual as theory. section But theanother map potential may ‘factorisation not problem’ [ befactorisation problem surjective; discussed we in will theissue discuss introduction an is and explicit the naturally start associated of this with section. spatial This wormholes new while ( inner product denote the restriction of Υ to diagonal tensor productsΣ; of this the is form thespaces natural in formulation the of context of an ensembles isomorphism and between baby bulk universes. and In this boundary case, Hilbert we would expect Υ Here we have used the notation JHEP08(2020)044 (and ∂ S satisfying ]. The his- ∂ S 88 ) of residual gauge M 8 of end-of-the-world brane . First, we describe how -states. in extracting the correct k , compatible with states in α b Z 3.7 2.9 enumerating the possible numbers of of n α Z in section ∂ S – 18 – of end-of-the-world brane states, the spectrum of as the dimension of a Hilbert space which contains . This remarkable compression of the Hilbert space α k α boundaries, and two such configurations are considered Z Z -states define an inner product on end-of-the-world brane (surface), but the only additional structure we introduce α 7 all labelled 0 S , such boundaries should be thought of as distinguishable even when = ∂ ) on this space. This can arise when a group Γ( S 2.4 for which reflection positivity holds true. For values of ]. Since there is no possibility to add sources in this model, we simply M ] will hold not just for typical members of the ensemble defined by the ( ∂ = 0) the end-of-the-world brane models fail to be reflection positive, and µ S 90 ∂ , 49 , S . Here we find that, no matter how many species 89 48 . Nevertheless, residual effects of diffeomorphism invariance can lead to a non- 3.5 M to denote the boundary condition on any circular boundary. We therefore define our path integral as a sum over such diffeomorphism classes of In this first model, the only boundaries are those fixed by boundary conditions. As We then return to the ad hoc parameter Z For definiteness, we take smooth (not just topological)We manifolds, take and accordingly the use set the of language boundary of conditions to be a vector space, so that a general boundary condition is a subset of the non-negative integers and the rank of the end-of-the-world brane 7 8 b symmetries remains after gauge fixing diffeomorphisms. This naturally leads toequivalence under symmetry diffeomorphisms rather than homeomorphisms. assigns a (perhaps complex)circular boundaries. weight to each non-negative integer equivalent only when they areseparately. related by a diffeomorphism that preserves eachsurface boundary trivial measure use described in section their boundary conditions coincide.set As of a oriented surfaces result, with the space of allowed configurations is the structure that would appeartories in that the can standard appearwith model in boundaries a of path dictated topological integral by gravity(for are the [ each then relevant connected the boundary component) setboundaries conditions. of [ is oriented famously This topological classified set surfaces by is genus discrete and and number of circular 3.1 A theory ofWe now topological consider a surfaces theory ofis purely topological a two-dimensional two-dimensional gravity in manifold is which spacetime an orientation. We thus have neither a spacetime metric nor the conformal or complex reflection positivity for anyZ number Hilbert space is boundedthe as properties required above. to In interpret the particular, end-of-the-world brane the states. reflection positive models have all tations of [ Hartle-Hawking no-boundary state, but in fact for all allowed different choices for thisin parameter particular modify thefind model. the We set find of that for generic states we allow, for states with rank equala to dual or Hilbert less spaceillustrates than of the the dimension eigenvalue importance ofphysics. understanding It the also null shows states in this model that results analogous to the entropy R´enyi compu- in section JHEP08(2020)044 . ), 0 S M (3.2) (3.1) ( ; i.e., χ Z below, 0 with no S g . − 3.7 ˜ Z ) = ]. ) by M ( M . As forewarned in appears completely [ S 3.1 we will encounter a S ∂ M S in ( 3.3 Z , ] M [ S ) from a more complete model. − e M with a unique free parameter ( ) 9 µ M . ( ! µ g . One may therefore expect to write our | m M ) 1 connected components of genus g M 1 g Q Γ( X and replacing each | m of spacetime, Z – 19 – Surfaces , the sum over compact universes, which is an ) = ∂ Z χ ) = has S e := M ( M ( M = µ [Φ] µ S ˜ Z − . These can act only on compact connected components = 0. We discuss how this factor may arise in ) is the most general form for the amplitudes of a two-dimensional e ) = 1 for any connected manifold. It then remains to discuss obeying the appropriate boundary conditions, up to diffeo- χ ∂ 0 M Φ S S M D M ( ) from boundary-preserving diffeomorphisms that interchange µ Z M ( denotes the number of circular boundaries of µ | , we have g ∂M | , where | ) as residual gauge symmetries, and divide by the number of such permutations counterterm. Indeed, as stated above, we expect that the unique local theory of g ∂M ) turns out to have little effect on the physics of interest. It leads only to a change | Despite the apparent uniqueness for the action, we now introduce an additional term Following the principles of effective field theory, we should now write down the most One would ideally like to derive the measure factor Here we take locality to mean invariance under cutting and gluing surfaces. A precise version of the ∂ 9 M local S ( For now we simply note thatit the can parameter be effectively removed just by rescales introducing the definition of above statement is thentopological that quantum exp( field theory (TQFT) with trivial (one-dimensional) Hilbert space on the circle. − the introduction to this section, forad the hoc. moment the In extra particular, parameter a while this is an intrinsicour function form of is asymptotic boundaries, givenperhaps it by most is setting simply not by introducing a new local degree of freedom residing on boundaries. degrees of freedom available toproportional us, to there the is Euler a characteristic This unique is local the such Einstein-Hilbert action action action in in JT two gravity. dimensions, and is the topological term of the boundary for each general action allowed by the degrees of freedom. Fortunately, with only the topological overall normalisation of amplitudessituation (though in at which the our end choicediffeomorphisms of of that section measure permute is compact physically connectedgenus important). components Nevertheless, (necessarily we with regard thein same the measure. This means that, if only contributions to the connected components of (i.e., the onesµ that have no boundary).of the With ‘cosmological this partition understanding, functon’ the detailed form of morphisms acting trivially on the boundaries, weighted by anHere, action we will be contentleads to to define natural the results. modelrelated with Since by a boundary-preserving boundaries well-motivated diffeomorphisms are choice are distinguishable, ofassume already the measure and considered trivial that measure equivalent, since we any will two surfaces path integral in the tentative form where we sum over surfaces factors in the measure, of the form JHEP08(2020)044 , ). 3.7 3.5 (3.7) (3.3) (3.5) (3.6) as a sum

n boundaries). th subset. For Z n i

) that arises when would be naturally . As an a posteriori that does not count M 0 ( S Z χ µ ). (3.4) , , = ) ˜ χ ∂ 0 3.7 M is the number of boundaries S S ( i e n , ! n i ∂ ) n S boundaries in the i M ! ( i gives precisely the number of ways . − n ˜ χ Q ) n ! 0 ) ! i g S ! n 2 is that it precisely cancels boundary , where e m i n M ! ) i ( − 0 Q (until section ! n χ S i n M 0 L ( (2 Q S 0 M µ = , and also where we choose not to label the is the most natural choice to cure this failure. − S n 0 = ∂ then accounts for the fact that the components | ), it is useful to comment further on the inter- = with S – 20 – S ! X boundaries ) = X ∂ X 1 that the end-of-the world brane models fail to be ∂M m 3.5 = M ), and also for the factor of | n S Connected M components ∂ ( = ) nor generates a new partition of the 3.5 S 3.7 S = Ordered lists with ˜ χ 3.5 of connected surfaces

n = Z . As is well known,

L = 0, and

M n ) as connected manifolds ∂ are related by this scaling, it suffices to discuss only a single value Z S ∂

3.5 of S where we choose L are as above. in the action. The amplitudes in our path integral thus take the form i M n , including a factor of χ in terms of a putative dual 0 + 1-dimensional quantum mechanics (which m boundaries into lists of subsets that have Z , and n m , is the usual genus of each connected component that counts handles. th entry of the list n i g Before computing the amplitudes ( It will be useful below to sometimes use an alternate presentation of the sum ( Our action is thus given by Since all values of where pretation of we will sometimes callassociated a with CFT the path in integral analogy of with this AdS/CFT). quantum Each mechanics on the circle, which would Thus we may rewrite ( in the to arrange a list of length are not ordered in thesome original items in sum the ( list bothneither coincide generates and have a no new boundaries term (so that in exchanging ( these items Instead of summing overover surfaces ordered with lists labeledboundaries. boundaries, The we number can of waysa write to separate label factor the of boundaries the is multinomial then coefficient accounted for by including Here which we have writtenboundaries in and terms which of is a given modified simply by Euler characteristic ˜ The practical simplificationcontributions of to choosing in detail, and then to usewe the will above thus scaling to confine understand discussionjustification, all to other we values. the will Until particularly show section simple inreflection case section positive when JHEP08(2020)044 (3.9) (3.8) (3.10) (3.11) ) with . Again, 3.7 u over connected λ = 0, given by the . is a random variable n 0 th power of the sum taking a value in the S Z 2 m 0 Z − S . But since we have no e 2 β e − 1 = ) times the g , 2 ! CFT

− 1 m n H (2 , it is simplest to compute a generating . 0 Z i = 0. This means we have a topological ). This is the case S

λ n e ! e n Z , and invariance under diffeomorphisms of H n 2.7 u h is always the amplitude for any connected β = =0 ∞ g X

=0 ∞ 1 = dim λ 1 X n =

– 21 – contribute for a circle of length χ as the exponential of the sum defined above. We begin by computing the no- = CFT 0 = ) = 1, so we have

S

H Z m e Z ), and in particular excluding boundaries from the n βH M uZ ( − being infinite). In the presence of spacetime wormholes Z e e µ 3.5

? = Tr Z Z as in equation ( Z X Connected compact surfaces compact surfaces, which are classified by genus. The measure is ) as the exponential of a sum over connected geometries. This is := (or perhaps by 3.11 λ ). Indeed, the exponentiation is particularly explicit by using ( N M ( µ connected = 0, in which lists of length over i ). We thus find n λ We now introduce boundaries. To evaluate In the usual way, one may write 3.9 = function and to extract thewe amplitudes wish from to a write power ( series in the ‘chemical potential’ in ( In particular, in our model the path integral defined by the sum over topologies converges. nected topologies. This property determines all amplitudes ofsurfaces. the model. For this,the it measure is important thatn we include symmetry factors in our definition of With our amplitudes definedcount by in ( the Eulercomponent character, of the spacetime value (with of fixed but arbitrary boundaries) after summing over con- sum over arbitrary compact spacetimessum without boundary. Fortrivial this, for we a first connected compute surface, the i.e. model. 3.2 Evaluating the amplitudes We now solve forboundary the partition amplitudes function A unitary dual quantumnatural mechanics numbers is therefore characterisedconnecting by these boundaries, it would thustaking seem nonnegative natural integer to values. find that We will see below that this is precisely the case for our metric, there is nothe notion boundary of boundary implies length aquantum vanishing mechanics Hamiltonian (a one-dimensionalthe TQFT) identity where operator, the which only is observable the is dimension the of trace the Hilbert of space: describe the partition function Tr JHEP08(2020)044 ; ) }

M n ( ˜ χ Z 0 is the

(3.16) (3.15) (3.13) (3.12) (3.14) . Here S

e , . . . , n L that the n 2 . , M Z } 1 u

{ is the number of , . . . , n . 2 ) after dividing by ) , λ 1 boundaries lie in the . We can extract the ( { in Mathematica; also g n n 3.14 , . , . . . m ] B u ) λ while the factor ) in powers of L = λe . u = 1 + ) n ) M ( i p = λe M ˜ χ ( 0 ˜ χ ! S 0 for 1)) as in ( ! e n S ! i n e u i − BellB[n, n i ! n n ( + u n ) may be written as the product n i u =0 i e ∞ n ( X n Q P M λ for the disconnected surface !

u with the sum over 3.13 , and ). We may thus compute the amplitudes χ m L   λ ) 3.9 M X + M ˜ L ( boundaries (Number of subsets in χ M 0 λ n λ Connected – 22 – S } p e from ( + =0 ∞ X n 2 X λ =0 ,...,n ∞ + λ g 2 X , = X 1 is determined entirely by ) to identify   {

+ 3 ! n Ordered lists Partitions of = 3.9 n 3 u uZ of connected surfaces e λ

= = , the double sum in ( = = uZ ). In agreement with our previous result, these polynomials are g

e

log

3

uZ Z n e

log

Z 1

to remove contributions from closed manifolds and thus any mention − 1 th surface in the list. Since ˜ ) which gives Z i − Z by expanding the generating function exp ( Z = 3. There are five distinct ways to divide the three boundaries into 3.7 E n n Z D for the individual components, this disconnected pieces exponentiate, is known as the Bell polynomial of order is the number of surfaces in the list χ n ). What remains is then just to simply count the relevant configurations remaining . m ). Such configurations are classified according to which of the Touchard polynomial B λ M e To illustrate the counting in detail, consider the example of the third moment We pause to note that there is a more direct way to compute the amplitudes ( 3.5 µ = i.e., the case connected components: Here called indeed known to have theZ generating function exp( from a counting ofpartition partitions, defines: graded by the number of subsets of of in ( same connected component oflabelling spacetime, the boundaries. and thus For each byto connected a sum component of over partition spacetime, genus, of it giving then the a remains set factor only of Here the last equality has usedcorrelators ( we first divide by Furthermore, since thedepends factor only on the genus boundaries of the sum of ˜ explicitly from ( where precisely the usual combinatorics familiar from Feynman diagrams, but it can also be seen JHEP08(2020)044 ) ) n ). is of λ . 3! ( 2!1! for L Z λ d 3.15 , ! (3.18) (3.19) (3.17) p N 1 by the n 2! ,

) = uZ d e

with support = ) = (2 ! i Z Z ! n below, a pertur- n Q 5 , ! . , , m ! d L ! d d λ d λ , or more generally by N λ . We may also read this off 1 λ 3! − λ − e 3! e 1!3! ) = ) = ) since the 3 components are all λ λ ( ( d 3! d 1)] is the moment generating function 1!1!1! , − 2 3! 3! 2!2!1! u after summing over genus, independent of , ) would instead list each topologically dis- e , p ( , p + λ ] (based on a Fock space labelled by number ) ud λ 3.7 λ, λ e 57 ( ) ) gives – 23 – d λ + p ( ) = (1 2 ) since all 3 boundaries lie in the single connected d n ! i λ p 3.18 d 3! 3! ! 3 n n , 1 Q =0 =0 ∞ ∞ + 3 3! , d X d X , ! 3 3!(1!) λ = = from ( th moment of the Poisson distribution. The appearance of , m i

! n = = n L n n u ) = (1

uZ ! Z N i 3 e h ! n having manifestly non-negative probabilities Pr(

n 1 Z 1

Q d − 1 is the − and write the result as the Taylor series for the exponential: , Z ! − Z n Z Z as a Poisson random variable with mean ). This gives the identical result B ) only once, but would accompany each term by the number , m ) using the fact that exp [ L Z 3.7 N 3.16 3.14 ]. 91 from ( The alternative counting used in ( ! i n ! 10 n Q ! This is a surprising and remarkable result. As reviewed in section We now interpret the amplitudes in terms of a probability distribution where For indistinguishable boundaries the answer would be multiplied by m 10 bative description of theof theory baby following universes [ and with wormholes treated as a smallboundaries). correction) would have led to the using the fact that the Poisson distribution can beof understood spacetime from contribute the the resultthe same that number all amplitude of connected boundaries. components distribution This (that corresponds is, to the the fact completely that connected the correlation cumulants functions) of are the all Poisson equal to showing that all moments canon nonnegative be integers generated fromWe a thus single identify distribution for directly from ( for a Poisson random variable. Alternatively, one can see this from the amplitudes ( Extracting the coefficient of regarded as a random variable.normalisation To factor do this, we divide the generating function identical but have only one boundarysince each, the the second two components term are has ( notthird homeomorphic but term the has cylinder ( has 2component. boundaries, The and resulting the combinatorics coincideensembles with [ those emerging from random matrix where the first term has ( ponents are counted separately, andabove. there are no explicit symmetrytinct factors term in the in firstdistinct ( line ordered lists that oneof can construct from the connected components and the factor Since the boundaries are distinguishable, the three configurations with two connected com- JHEP08(2020)044 (or , we ∂ C (3.21) (3.22) (3.20) S for any . Every → n defines an BU is discrete, R H is a function f Z : boundaries in that gives rise f + log f 0 m ˜ 0 S from the last line e S i ) = Z , where ( ∂ i ) f S . At the self-dual value | 0 Z we find that the result is ˜ ( S d f | . ) between the ‘bare’ parameter ) . d n ( . 3.9 f n ) ) + λ d ( created by inserting ( m . E + log n g d 0

n + ! 0 ! correspond to complex couplings, with d d S + is a space of functions m d d S λ m λ − m λ Z e is not injective, but is instead two-to-one. B

= , there is a second value Z λ =0 =0 BU ∞ ∞ 0 − λ d d X D e X ∂ S H S e ). Before considering the details of the inner = = = = (or more generally for – 24 – = 1 E E 0 0 ) ) in terms of an ensemble of dual Hilbert spaces. ˜ S m S A.1 . Z − Z ( 3.8

= Z e f form a Cauchy sequence guarantees that n

can then be represented as ∂ ) Z N i S Z D n o ( i g Z n | D takes nonnegative integer values n Z c | , which grow slowly enough for convergence. Demanding that n Z n c =0 c ∞ n =0 that full consistency (in particular full reflection positivity) of the n N P = 4, and smaller values of should have a continuous distribution supported on all real numbers. P ), with argument 3.7 λ ), since Z n C n → . From the point of view of the path integral in a semiclassical expansion it , and hence the same theory. In particular, we find λ R C i : + suppresses connected topologies (and thus describes weakly coupled universes), f ): 1 2 0 S ∈ We can read off the extension of the inner product to states As a final comment, it is interesting that the relation ( Furthermore, for our choice = 2 we have and the physically observable parameter 3.21 0 S 0 0 S − S perhaps in ( A more general state with Taylor coefficients the partial sums entire analytic function (seeproduct, appendix we are thus led to the idea that 3.3 The babyWe universe can now Hilbert give space a complete descriptionstate of can the be Hilbert written space of asthe closed a past, universes linear with combination inner of product but yields the samee theory as ae very small value ofis surprising the that dual such ahence complex to coupling a gives unitary rise Hilbert to space reflection and positive positive amplitudes, probabilities. and to the same This is a strong-weaklarge self-duality of the model in the sense that the semiclassical limit of Although at this stagewill this result see appears in to section model depend in on fact fine favours tuning precisely the the parameter relation e This means that there for a given value of Instead, from our exact nonperturbativeand solution limited we to find nonnegative that values. the support of positive integer compatible with the interpretation ( expectation that JHEP08(2020)044 ). 0 c (3.26) (3.25) (3.23) (3.24) ). For acts as ) where Z 2.9 are repre- ( of natural Z 6= 0) in the , g N n ) BU -states of our for any integer Z α H ( i f (for n c πijZ 2 , and they must form e | states, we may choose . N N . i = below that at least some ∈ n ) = 1, or more generally by i ∈ 4 Z d = 0 eigenstate (given by , = 0 | Z d (

 HH Z f | ) ) ) is that it depends only on the and d +1 . n 2 − 0 πZ 3.7 3.22 has zero norm whenever the function Z dd = 0, so in some sense the space of

Z δ

( sin( )

π +1 ) =

Z n , we can now discuss the , labelled by ) = 0 for all ( 2 Z 2 E d ( / b f π Z d (

1 + 1)! f n 2.3 f ) (for example, − n = of 1)  N (2 ! d πZ – 25 – − E Z (

d λ d 0 ∈ ⇐⇒ d  d we have the otherwise surprising relation sin( =0 = ∞

X n = at will. As a result, the only physical information in is the overlap with the ) the covariance of random variables = N λ Z i n = 0

= e Z E c n on such a function. One of the infinitely many ways to 2 d i D , we must quotient by such null states as in ( ) Z

d | = n

BU c ) πZ Z H we have

Z ( =0 ) = 1 for all ∞ n sin( f f | d

( P . We regard this degeneracy as a gauge equivalence. As described

we can change any finite number of coefficients f i ) vanishes on n i ) Z πZ | evaluated at non-negative integers (also known as the set πZ . When expressed as a sum of the states : g N BU this gauge symmetry is a natural modification of diffeomorphism invariance sin( H but vanishing at other natural numbers, since multiplication by n and Z 2.2 d | f = In parallel with the treatment in section These considerations reveal an enormous degeneracy in how states of Z is Poisson distributed. But the salient feature of ( vanishes on ). To emphasise the impact of the quotient by null states, note that by adding vectors of where the coefficient is chosen to enforce the normalisation a basis for coefficients defining theat Taylor series ofmultiplication any by analytic the function constant represent taking such eigenstates a states non-zero is then value thus to produce thefundamental explantation, degeneracy and observed we above.of will these see conspiracies Such in are sections conspiracies in call fact implied out by for reflectionmodel. a positivity of These more our are path the integral. eigenstates in section associated with allowing topologypower of change this in seemingly thecorrections natural to functional modification diffeomorphism comes integral. invariance are as notAs a But generic, a surprise. but the result, are This enormous instead the indicates highly corrections that correlated. the conspire to enhance the impact of the gauge symmetry, and the form expansion of the state any finite collection of coefficients sented as sums of More generally, fornull any states is theHawking same state can size be as represented theany by function the total constant that space function has beforej the quotient. Similarly, the Hartle- To form the Hilbertexample, since space sin( Z vales of numbers). In particular, we findf that the state This is (up to normalisation factor JHEP08(2020)044 (3.28) (3.29) (3.27) 11 is a harmonic b Z and its eigen- b Z onto the (here, one- , we can impose the

d , in which ), and it is interesting ) 5 Z = ( 2.23 f Z

. We can define the annihilation i ,

connected components. . -state correlation functions. , or equivalently to terms of order -state i 12 = 0 α d d α , which is a natural observable when connected components. We thus find Z + 1) | = 0 is equivalent to restricting the sum on connected components. This does not d d is equivalent to restricting the sum over a , Z ) d Z d † ( † space | d a a † f and (

! λa λ = d √ 1 , √ , e √ N – 26 – Z = = = = 0 =

i i ) b Z d has a natural representation as a harmonic oscillator Z counts the number of connected components of space- ( HH = 0 = f b

Z

BU takes the value Z Z a | | H b a Z , N ) to the given eigenvalue ), note that projecting the states acts as a number operator. b Z 3.22 3.22 ) (and the fact that the analogous equations are identical for any fixed 3.9 ) that arise from spacetimes with . From ( 0 of boundaries on the connected surface), these give precisely the contri-

, is crucial for this simple description of takes values in d Z 3.5 as acting to shift functions of b -states are designed to make amplitudes factorise ( Z = a α n > Z

Since The In other words, the operator Finally, we discuss the spacetime interpretation of our operator We thank Xi Dong forThis discussions is on not this to point. be confused with the free Fock space description of section . But due to ( 11 12 d The distribution of the associatednumber ensemble operator then follows follows a from the Poisson well-known distribution fact in that a the coherent state. oscillator position operator. In this description, the Hartle-Hawkingas state is a coherent state, which can be represented so that we have the relations Hilbert space in which operator exclude wormhole configurations connectingcorrelations multiple between boundaries, disconnected but configurations provides oftion additional boundaries. in It a thus surprising achieveschoice way, factorisa- of which symmetry may factors be onrenormalise instructive spacetimes for without less boundary, which simple otherwise models. only acts Note to that our (unlike the number of connecteduniverses components cannot of split and join, but is not gaugeto invariant note when how they can). ournonlocal model constraint achieves this. that spacetime To work has in exactly an butions in ( that working in theamplitudes eigenspace to with terms eigenvalue where the universe has precisely time! This iswith quite a surprising, Cauchy since slice this if is we not were a to quantity attempt we to would naturally quantise associate by gauge fixing diffeomorphisms states dimensional) subspace where the right-hand side ofλ ( number JHEP08(2020)044 ) i , ψ j (3.30) (3.31) ψ labels an i . ) dictated by i 2.2 , ψ j ψ circular boundaries m . We denote a boundary i ψ . E ) circular boundaries without EOW n i and a future endpoint that destroys m , ψ i n j ) since the bulk path integral with this i ψ ( i j , ψ ). More importantly, the EOW branes allow j ··· -dimensional Hilbert space, so that ’s in the amplitude, and 3) additional circular ψ ) k 3.9 1 Z ) = i – 27 – i by ( , ψ , ψ ); in general, these are CPT conjugate boundary 1 j j j j ψ ψ additional interval boundary segments labelled appro- ( ( , ψ i m n and ψ Z i for an example. D 4 6= ( . Since the interval is oriented, we may refer to it as having a ) j i . We refer to both past and future labels as EOW brane sources. j , ψ j and ψ , the most general amplitude can now be written i i ψ . Equivalently, we can place a topological quantum mechanics on the } ]) we call end-of-the-world (EOW) branes. We choose to include an arbitrary 49 of species of EOW brane, so each of these boundaries is labelled by an index so that it is no longer given by ( , . . . , k k λ 2 , 1 Including the Introducing the EOW branes has two effects. Firstly, they can appear as closed bound- ∈ { corresponds to the preparation of a certain 0+1 dual state brane labels as dictated byboundaries the formed number by of partitioningthe into boundary subsets conditions the and, orientedsegments intervals for using ( each oriented EOW subset, brane forming segmentsat whose a both species circle endpoints. labels by match See connecting the figure source the labels ( The associated boundary conditionswithout EOW for brane the sources path and priately integral with EOW require brane species.is Since then the computed EOW by branesthe summing are following dynamical, over three the all types: path oriented 1) integral species surfaces circular EOW independent whose brane circular of boundaries, boundaries each all labelled are boundary by of an conditions, arbitrary 2) conditions. This coincides with the general notation introduced in section Since the boundariesket-vectors carry so an that orientation, ( the notation distinguishes bra-vectors from In a putative 0+1i dual, the condition thatinterval a between boundary EOW creates branes anboundary EOW condition brane should with compute label the inner product between these states. us to impose awe new have class a of boundaryEOW possible condition brane boundary which species is conditions.past an endpoint Namely, oriented we that interval can creates labelledan specify an at EOW EOW that brane its brane of endpoints of type by type orthonormal basis of stateslocal in data that on Hilbert an space. EOW brane Apart is from an the orientationaries compatible species in with the label, the sum the over spacetimevalue only topologies, it of but bounds. this is largely unimportant, only acting to change the We now extend the(following model [ described abovenumber by introducing dynamical boundaries,i which EOW branes, with zero Hamiltonian and a 3.4 End-of-the-world branes JHEP08(2020)044 ) 3.9 ). In (3.32) . 3.1 that this 3.7 ) in ( 3.7 M ( µ . The solid red lines associated with each

, ∂ Z -boundary). The other is ) S ) but includes additional j ˜ χ Z 0 S , ψ 3.2 i e ) ψ ) for our new model is then )( with the boundary conditions M i ( 3.5 µ , ψ M j ψ M ( X

= E ) is analogous to ( n i µ , ψ n – 28 – j for every circular boundary, no matter how it is ψ ∂ ( . The analogue of ( S g ··· ) 1 i , ψ 1 j ψ ( possible species labels. Since EOW brane boundaries can be m k Z D , resulting in an action which counts only genus and not the number of 0 S = ∂ as the sum over connected surfaces with no asymptotic boundaries in analogy . A spacetime contributing to an amplitude ). However, this sum must now allow for the possibility of circular EOW brane S λ 3.9 We may now proceed to evaluate the above amplitudes. As a first step, we again It remains to specify the symmetry factors that will be the analog of We now know the set of amplitudes to compute and the corresponding configurations define with ( boundaries, each with specified in precisely theby same an way overall for factor counting each the genus, number this of simply possible multiplies such the labelled result boundaries. ( For a fixed where we sum overspecified on diffeomorphism the classes left hand offactors side. associated surface The with measure counting end-of-the-world branes using Bose statistics. are those formed bycompletely circles independent involving of EOW theprecisely branes boundary the alone. same conditions. way as Furthermore, They the such thus genus circles enter are all of our sums in corresponds to a simplethe scaling case of our operators,boundary we components, will and nonetheless comment on once the again extension focus to on otherdoing values so, in it is section useful toable, note they that, will since all not asymptotic contribute boundaries to are symmetry treated as factors. distinguish- The only indistinguishable boundaries degrees of freedom. However,circular we boundary. will We again use includeformed the from a asymptotic same parameter pieces and EOWcan branes. be Again, obtained we by willasymptotic see introducing and in additional section EOW local brane degrees boundaries, of and freedom integrating which them reside out. on While both this no longer form a topological circle. over which we are to sum.before, It the remains Euler only to characteristic specify is the the measure on unique the local configurations. action As without introducing additional Figure 4 indicate asymptotically AdS boundaries,The and spacetime the has dashed twocircle green boundary at lines components, bottom) are each is with EOWformed a by brane the single a topology boundaries. circular pair of asymptotically of a AdS asymptotically circle. AdS boundary segments (a One connected (solid by red a pair of EOW brane segments to JHEP08(2020)044 ), i , , ψ ˜ χ j (3.35) (3.33) (3.34) (3.36) 0 ψ S ways to e ! ! n i n k ! D i D Q ! i i I , th term in the sum I i  i n P ) t Q k shows the new factor to − ! m u n I L e u ) # identity matrix. D n det( ( t k distinguishable boundaries into λ C  × Tr D k 1 n ) represents the number of ways to as before. The from summing over genus and closed i indistinguishable circular boundaries . D ,I k i ( =1 i λ ∞ Z e the I = exp X n D intervals corresponding to some ( L ) boundaries respectively. As we explain C 0 i I S M + + n 2 0 has , ψ  u − S j ) i e 2 " i ψ e , and − X , ψ – 29 – ij j 1 t boundaries ψ ( and ( connected surfaces = exp avoids overcounting equivalent configurations where = ij Ordered lists ) L 1 λ n t Z t ). This sum is itself given as the exponential of a sum of where entry =1 − λ k u X I i,j e distinguishable/indistinguishable . We have I + E D g n species. Summing this factor over ) . Finally, the factor = det( i = n i ) for the i i k uZ I D D i , k i } , ψ  i P n P again counts partitions of the ,I j i for each connected manifold counts the number of ways (including ··· ! ψ D , matrix with entries exp i X ( { ! i ! i I D with k EOW branes. Summing over species of EOW branes results in the matrix * I L, k i D ··· = 1 L n Q × ) i ≤ i 1 k i ≤ ) as a sum over ordered lists of connected manifolds. This is readily obtained 1 , ψ of EOW brane boundaries, including symmetry factors we count 1 = j (with ) by recognizing that the circular EOW brane boundaries enter every sum on the 3.32 n ψ is the ( ij accounts for insertions of circle boundaries t t 3.7 component intervals are cyclically permuted. and we obtain m u For an alternative route to this result where various factors are more explicit, we can Once again, we compute this result by writing it as the exponential of a sum over As before, we can now compute all amplitudes through a generating function, where Z k n e D and where the factor symmetry factors) to assign EOWand brane the labels factor to (labelled) subsets of size the present ( from ( same footing with the genus The comes from boundary components consistingalternating of with product and trace, and the factor of have already been absorbedover into distinct types of boundaries: where connected spacetimes, each weighted byEOW a branes. factor of The connectedinsert contribution on is a a given sum connected over spacetime all (excepting possible circular boundaries EOW we brane could boundaries, which u below, this yields label the boundaries with be we sum over all configurations, with any number of asymptotic boundaries, and fugacities number JHEP08(2020)044 i ψ ) of . d ] with (3.39) (3.38) (3.37) ). Let 3.8 i − ) is the ) for each 92 t , ψ d ‘auxiliary’ j − 3.34 ψ = I d kd = Z   Z labels the EOW a i + and ( ψ ) i ) is clear from the a j = det( Z ¯ ) can now be written ψ d , ψ j . We then recover the = 3.34 d =1 , . . . , k ψ Z Z a X counting the number of ( 3.37 ij ij ! !+ i t t = 1 ) ! D i ), and the defining property i i D =1 =1 k k Q , ψ X X i,j i,j j ) 3.34 ) as i ψ   D ( ( exp ) by conditioning on ij , ψ in ( i t a j C i exp * ! I i ψ ψ i ) , ψ k I =1 a j  k j λ ¯ a i ( X ψ i,j ψ as labels for an orthonormal basis of the d ψ a i p d

=1 ¯ ψ is again Poisson distributed, though with a a X ud − e ) we are computing, we can read off the distri- e Z exp matrix. The index a i – 30 – ) = , . . . , d =0 -dimensional based on our interpretation ( ∞ ¯ * i ψ d X k d 3.37 d λ a , ψ i × e = 1 j ⇒ d = ψ dψ a = ( 1 π +   ) converts the factors (which is ) d i =1 ). ), the relation to the exponential of ( Y a , ψ 3.34 j k CFT =1 i 3.33 Y ψ 3.36 H ( ) and write each term as an average over the Poisson probabilities Z ij variables, our Wishart generating function ( ), the inclusion of factors of t , arranged in a a i = a i 3.34 ψ =1 k d ψ 3.36 X − i,j ) t ) without EOW branes, so given by ( + : ! − λ d ! in ( variables will be interpreted as the components of the EOW brane states = 0 in order to consider the marginal distribution of d λ 3.14 t uZ . To find the corresponding conditional generating functions, we Taylor expand a i λ /L  N − ψ e distinguishable boundaries from the specified boundary conditions (together with det(1 ∈ exp d D ) = In terms of the To make this more transparent, and to simultaneously explain this distribution to the We can now characterise the distribution of ( We now interpret the amplitudes as describing an ensemble, for which ( In comparing with ( * λ ( degrees of freedom. ). The d Comparing with the expectation valuebution ( by identifying the matrix of inner products ( as a Gaussian integral: complex variables brane states, and weboundary will Hilbert interpret space Z in this orthonormal basis. The result is the generatingd function for a standard complex Wishart distribution [ uninitiated reader, we can rewrite the generating function by introducing new value of fixed the exponential in ( p ways to match distinguishable boundaries toweighted boundary sum conditions over into all the possible above-described boundary conditions for each(unnormalised) connected generating component. function forus moments first of set random variables old result ( interpolating EOW brane segments). factor of 1 of generating functions.the generating By functions this ( last feature, we mean the fact that the definition of form JHEP08(2020)044 . - d i, j (3.42) (3.40) (3.41) operators 2 k . to be as large as ], as and discussed k ]. 1 49 -states. These are eigenstates , so we have α α . ) . i

} α

, ψ j α k, Z ) ψ i {

-distribution. In particular, the number α from the 2 , ψ

j χ α and ( ψ BU Z α H ) = min Z = = ( i – 31 –

, even if we include Euclidean wormholes there is , ψ α α j

5 ) b ψ because, along with circular closed universes, we also Z i , ψ 3.3 j , and then multiplied by a random normalization so that \ rank( ψ ( CFT H (where the orientation defines a preferred order). On the other j which for a finite number of amplitudes at any finite order gives no 0 independent standard complex normal random variables , this is a version of the semiclassical Page curve [ S and − 4 e ∼ i a i ψ operator as before, but now are simultaneously eigenstates of the new types of universe whose spatial slice is an interval bounded by EOW branes, b Z , as independent random variables, each chosen from a complex normal (Gaussian) 2 ) as well; note that Hermitian conjugation acts on these operators by swapping EOW brane states appear to be orthogonal, and we can choose i a i k : with probability one we have It is most straightforward to construct At first sight, this appears to require an enormous conspiracy in the nonperturbative This is another surprising and remarkable result from such a simple model, since in In the 0+1 dual interpretation, this means that the wavefunction of each EOW brane k ψ , ψ Z j \ ψ ( We label the corresponding eigenvalues by have say with labels hand, the above-mentioned conspiracies will also imply the existenceof of the new null states. study the Hilbert space interpretation of the model3.5 with EOW branes. Baby universe HilbertWe now space incorporate with the EOW EOWthe branes branes space relative into to the that of baby section universe Hilbert space. This enlarges further in section contributions, which might lead one tosimple suspect models. that it We is an willprimitive artefact show of principal, below studying particularly that namely this reflection is positivity not of the case, the since path it integral. follows from For a this, more we must the we like. As discussedan below expansion in in section obvious sign that apparently distinct EOWNonetheless, brane states in must the in fact completethat be solution linearly the dependent. after number summing of linearly all independent nonperturbative states effects, is we truncated. find As in [ the semiclassical limit (without the exponentially small effects of spacetime wormholes) states is selected independentlydimensional Hilbert and space uniformly atits squared random norm from is drawn theof linearly from unit independent an states, sphere appropriate given by ofby the rank a of the matrix of inner products, is bounded the distribution with unit variance: from the final factor in the integral. The remainder of the integral gives the measure for JHEP08(2020)044 , . a i ). d k ψ a j ≥ ¯ while ψ 3.43 (3.44) (3.45) (3.43) d a i with an =1 ψ d a b Z P ) = i ). Their simulta- i measure in ( , ψ ). 2 j . , ψ j } L ψ directions. For d \ ψ 3.65 ( a . In particular, we write ≤ d k space; this is the overlap M 2 , L ∈ rank ,
α α ) for the eigenvalue of i ) i 0 , ψ α , ψ j Z j -states, since they are eigenstates only α ψ , ψ Z matrices α 0 ( δ positive definite matrices of rank at most k are given by the nonnegative integers αα α δ k Z × b Z f × = k ! α α k i of Z – 32 – Z α λ | α by any of the means discussed in that section, for 0 ) has probability zero in any normalizable state). associated with the choice of Z i s α spaces of square integrable functions on the relevant d d h . k (defined with any convenient smooth measure). For i 2 d, k = d d d M k = L

= ) . We can generate the rest of this eigenspace by acting d = Z M k Z b | Z -states are delta function wavefunctions living in the sub- H HH , this density is given explicitly in ( M Z α

| ( k )-dimensional manifold; we can write ( 2 α Hermitian p.d. =0 ∞ 2

d L { M counts the number of independent real parameters in ≥ d α ) on = = = i − kd Z d d k , ψ = ). Instead they are the projections of the Hartle-Hawking state onto BU are the usual kd j i , the eigenvalues M Z d \ H ψ ( H , ψ = . For j Z 3.3 α ). However, they are now not full \ ψ Z ( H f 3.25 = , supported on some particular matrix ( -states is determined by the allowed sets of eigenvalues, which is constrained is the product of a Kronecker delta forms a (2 d is the probability density function of the complex Wishart distribution with α

) so that the 2 0 = d α k α Z ). Z Z αα M subtracts for the invariance under unitary rotations of the H δ f 3.39 , = 2 and not of k Finally, the wavefunction of the Hartle-Hawking state in this description is given by With this description, the In each such eigenspace, we can now diagonalise the operators As in section 3.41 d degrees of freedom with respect to the measure on our Z b

Z (though any rank other than min( ≤ α α 3.6 Hilbert spacesOur with discussion boundaries of Hilbertof spaces interest is arise not when‘asymptotically yet AdS’ complete. we boundaries. insert In Here complete particular, there sets other are Hilbert of two types spaces states of on boundary, distinguished Cauchy by slices that intersect where Z where appropriate Dirac delta function on the the restriction on rank is vacuous. spaces their inner product as The summands space of restricted rankd matrices as in ( the corresponding eigenspace of with the operators neous eigenvalues correspond to Hermitian d The baby universe Hilbert space therefore decomposes as a direct sum: by ( Indeed, we can still defineexample states by ( of The set of JHEP08(2020)044 . . is 1 , BU 1 BU 1 , H α 0 H H (3.49) (3.47) (3.48) (3.46) H , it thus . States

1 , α 1 maps i HH H

b , for example ψ α -states of , the associated BU L α . The simplest of H L = ,n admits a direct sum 1 R , which is spanned by , ) on , is simply given by the n 1 1 , 1 α , 0 1 H , ) H 0 i α 0 , H . Thus,

H = , . . . k H E , ψ α . By acting on j R ; 1 ψ ,n S = 1 ( HH L 0

, this is degenerate, and i × ∗ n .

αα H } δ , d α < k

(for = , α α = 1 i i , ; b i Z α 0 k, Z . In the same way, this Hilbert space can ψ α E . Recall that the operator 1 by duality) and the two-sided space {

, i H ) , ψ 1 ) ). All of the above states can be produced 0 i , ∗ j n 1 H i α . Reversing the orientation of all boundaries ψ +1 , ψ

M H R j R , ψ = min \ ,n ψ ,n n – 33 – = ( j L 1 L

, 1 n 0 n , ψ α 0 operators right boundaries. We thus denote the associated 0 ( α H H

k H R ··· n = → H ) as 1 dim i -sectors are orthogonal, and i R (related to α on a state of closed baby universes. In particular, we can α ,n 1 . On sectors with ; 2.4 , , ψ i i L is spanned by α 0 1 b n ) ψ ψ j i 1

H , . But we now have an additional possibility where we introduce 0 ψ H i α 1 ( α b , ψ ψ ; H j ∗ m j j circle. We denote the boundary condition by ψ c ψ Z ψ ;

Z i , which we have already discussed. We will be primarily interested in ψ 0 , | from section 0 can be created by acting with separate EOW brane states on left and , and the resulting state by Σ H 1 , α 1 left boundaries and H = d H L are continuous). The inner product on each sector n by acting with one of the BU 1 α , H (or more generally 0 1 , H -states. We have the same direct sum structure as before, 0 Next, we look at the two-boundary sector We begin by considering the single boundary Hilbert space In our model, the most general slice Σ of the asymptotically AdS boundaries will α H -dimensional: α In our case thethink cylinder of degenerates as to half a ofoperator line a by segment (since Σ is a point), which we can right boundaries using a single asymptotic boundarycall that this connects the left cylinder and boundary right.it (with as topology In obtained Σ a by times general cutting ancreates in theory, interval), one and half a one a might state might partition that think function of one on expects Σ to interpret as a ‘thermofield double’ in some CFT dual. be populated by actingon with boundary creatingwithin operators each on states of (where this is to bedefining understood in the appropriate sensematrix given of that eigenvalues some ( ofZ the parameters so in particular,decomposition the different by acting with thespan operator The inner product on such states is these is the one-sided Hilbert space states of the form to two types of boundaries would correspond to CPT conjugateconsist theories. of Hilbert space gives the dual (Hermitian conjugate) Hilbert space, so their orientation; we call them ‘left’ and ‘right’ boundaries of space. In a 0+1 dual, the JHEP08(2020)044 .

∆

(3.52) (3.53) (3.50) (3.51) ∆

, giving

1 , α 1 ; a H , φ ∗ b φ span a subspace

α

; , 0 α to a superposition of α a ; ) i .

2 , φ j α boundary conditions for 0 , ψ ) ∗ b i ∗ i j , ψ , as this minimizes φ 1 ψ ψ

j | , ψ ab 0

j b δ ψ 0

α ( ψ ∗ a , ; α c i α α ; ) ) ab a 1 1 of the , ψ i c i to be ∗ j , φ rank( a -state in question (with eigenvalues , ψ ψ ∗ b , ψ =1 φ ab α 0

1 2 . φ ≥ δ i j ,b

ij the states r α 0 running up to the rank of the matrix of ψ ψ c α α X = ( ( Z ; 0 0 0 α Z , we first change to a more convenient basis . This inclusion could be an exact equality, =1 i a,b,a k 1 ab , αα αα αα c X α 1 i,j ∆ 2 δ δ δ + | ⇒ |

-states:

, . . . , r

− . α ab = = = α ) ; c – 34 – α ab

| ⊆ H

a ; c ab α = 1 α α α 0 δ ) that the rank of the EOW brane inner product is , ; =1 ; ; ; , φ r α 1 =1 a 1 1 ∗ b r X i i = a,b X φ

a,b | 3.41 ab , ψ , ψ α ⊗H +

ab

1 1 c − can be built from a linear combination of factorised states ; 1 0 ∗ ∗ j j c , ( a

α 0 =

aa α ψ ψ c ;

α α =1

0 0 , φ H r ; ; ∗ b α α X ∆ a,b Re φ ; ;

2

i −

r =1 ∗ ab

| ; i.e.,

a X c 1 , ψ α , α 2 2 r

α 1 ; Reflection positivity = ∗ j ; , with the index =1 r ψ H − − X ab

a,b α α δ Z Z | − = , in terms of reflection positivity of the path integral. The same argument = = = = α α )

Z a ∆ ∆ . This suggests that we look for a linear combination

, φ

b ). Specifically, we pick linear combinations ∆ φ α α

) ; i i The above calculation teaches us two things. Firstly, for the norm to be nonnegative Before computing the norm of our ansatz = 0 and providing an identity relating the cylinder state , ψ , ψ i j ∗ j ψ ∆ ψ we have an inequality which applies in all This explains our empirical resultbounded ( by In the last line we have chosen the coefficients inner products. In this basis, we rewrite our candidate null state and compute its norm: one-sided states. diagonalising the EOW brane( inner product inwhich the ( with zero norm.| Such a vector would be projected out of the Hilbert space isomorphic to the tensor productembeds of naturally in two single boundarybut subspaces, only so if this the new tensor state product| From the first of these, we see that for fixed and the inner products of these states are given by JHEP08(2020)044 (3.61) (3.59) (3.60) (3.55) (3.56) (3.57) (3.58) (3.54) , we also 1 , α 1 H = 0, and hence i ∆ | : .

]. . . 1 α

;Ψ Z )] Ψ a

a † as the connected amplitude , φ b , φ ; i.e., when there are enough X , b ∗ a

to diagonalise the expectation k ,

φ φ ) X i Ψ

[( ≤ ψ ab . ;Ψ

δ ij

Ψ a ;Ψ α , b Ψ α Z .

λδ ; , φ ab 1 , φ a ∗ a b Var , X δ . If we now compute the norm of the ∗ b = α 1 (1 + . The variance of the individual terms

φ

1

φ , φ k =1

r = Ψ

∗ a − Ψ

X

) is saturated, we have r φ

λ =1 a,b = ) | ≡ H a X i HH − Ψ r 0

+

, r )

=1 − ) , ψ α 1 3.53 r i a X a j )] =

, such as the Hartle-Hawking state. Specifically, Ψ ; i.e., we take a HH

– 35 – \ ψ i −

, ψ = † , φ ( j

;Ψ b BU

⊗ H Ψ , φ b

| \ X b \ ψ then span the two-sided Hilbert space φ 1 Ψ Ψ ( , -states in this argument, we examine how it fails in a α HH ( φ b

α 0

X ; D α of EOW brane states

b [( Z α Ψ

H

a ; i ∈ H Ψ

HH i

HH φ Ψ = Ψ D |

| , ψ

, and we have ∗ j a = rank Var = ] = ∆ ψ

ψ |

r X 2 [ / ∆ 1 Ψ

− ∆ λ

Var , with = , r , where we also connect it with the Page curve [ a -states, though is generically non-zero. φ α 4 ··· , = 1 , † a, b ) is small, a For example, in the Hartle-Hawking state, the expectation value of the overlaps of To emphasise the importance of Secondly, we find that if the inequality ( XX , φ b φ so we can define ( This vanishes in EOW brane states is already diagonal, Here we have definedfor the variance of boundary condition we find an extra term, coming from the overlaps where state EOW branes to populate a one-sided Hilbert space ofmore general dimension (normalised) state let us choose linear combinations value of the inner product in the state Since the ‘factorized states’ find an equivalence between Hilbert spaces This factorization holds in our model for sectors with energy in section an identity can be used in much more general models, and we repeat it with the inclusion of a conserved JHEP08(2020)044 0 k = or S for − × R 3.2 i λ ∂ with , ,n k S (3.62) L e , which n i 3.1 is d alters the H ) that this eigenvalues 0 Z = S b Z Z 3.38 = | is of order ∂ k S , which is spanned by 1 , 1 H d rescaled by a factor of − ) ij t it . From the boundary perspec- α is a nonnegative integer, so that − Z 0 S = − ) again shows the inner product to be ∂ i S α , as the inner product on EOW brane e , the above result then implies ; ∂ = det(1 k 3.57 S d -states are characterised by from its preferred value ≤ is positive definite. In superselection sectors ). = α ∂ i | Z α R S α E , ψ Z ,n ; ) j L – 36 – \ ψ n tM ( H , and hence a reflection positive inner product on h Tr( is to rescale the quantities and operators associated ∂ i M e S ∂ ): , in which our EOW brane amplitudes are given by the . Note that this is not really an issue of fluctuations in i D , contributing an action proportional to the number of S d Z ∂ , ψ S = j auxiliary Gaussian variables we showed in ( ) = ψ t Z ( , as the same discussion applies to the states dk . In particular, reflection positivity is preserved for all Hilbert α with norm = ( d,k ∂ Z i χ S ), reproduced below with fugacities ij ) gives no meaningful bound relating the rank of the inner product α . We thus see by direct calculation that our path integral satisfies but not those of ; r M b 3.37 , Z 3.58 = eigenvalue M | such terms, so they are collectively important when ˆ Z 2 appears to be rather artificial. > k k and the inner product on . From the gravitational perspective, there is nothing wrong with this model 0 α the higher Hilbert spaces are not tensor products of the lower Hilbert spaces. boundaries. We thus find an ensemble interpretation in which S . In superselection sectors with N R Z i 0 n 1 , , by introducing S ψ = Z ⊗ 0 − N > k ∂ associated with arbitrary numbers of left and right boundaries. But in our model ∂ . There the only effect of S α S ) to the partition function ∈ i ⊗H R e with fixed Z L This changes once we introduce the EOW brane states. The bulk then provides a Let us first consider the model without EOW branes, discussed in sections Returning to the issue of reflection positivity, we should also discuss the Hilbert spaces d ,n 3.3 takes nonnegative integer values which can be interpreted as the dimension of a dual ∈ n 0 , ψ L , j ⊗ BU n 1 α α ψ Hermitian matrix For gives a probability distribution for principled reason to preferstates particular will values otherwise of failH to be positive semidefinite.generating function ( To see this,later we convenience focus and with on the a matrix sector of of EOW brane inner products encoded in a orthogonal states tive, there is a goodZ dual interpretation only when Hilbert space. Nothingchoice from the bulk perspective appears to prefer such values, so our with the times a Poisson randomZ variable, so that the for any positive value of spaces. Complex values are excluded by reflection positivity on boundaries. First we describephysics, how changing and thus inwe particular might explain naturally why incorporate this such value a is parameter in preferred. theand We model. then discuss how positive for reflection positivity. 3.7 The boundaryWe parameter now discuss the parameter H all possible boundary conditions creatingthe such states above can be formed byH combining with But much as above, considering states similar to ( larger. As a result, ( ( the particular parameter fix the eigenvalue of but there are JHEP08(2020)044 13 . is a = 0 d,k (3.63) (3.65) (3.66) (3.64) χ d,k M becomes χ approaches M d 0, the density has a positive . → . As 1, along with the BU , λ d 1)) H to be a nonnegative − − ], to which we refer the d,k k , ( p 1, the probability density ) 93 . ) ), turns out to exhaust the − − d > k M d . ( ∞ to be a Hermitian matrix so that ) , f t 1 Γ( 3.38 ∗ : ) M subspace of ( − M ··· , which is in general a distribution M positive definite k d < d < k ( d,k [ 1) g d,k p 2 ) ) p = − − } ∪ M tM k d 2 Z ( ,M is positive semidefinite: Tr( − d,k M i )Γ( , the inverse Fourier transform of d , so we must find a different way to determine Tr BU N , and here we take − Γ( H dMp dMe e ij – 37 – ∈ 1) k , . . . , k d > k − M defines a probability distribution 2 − d / 2 Z Z k already covered by ( d ( , d k ) Im 1 d = − Hermitian matrices , π d ) i ) = M 0 ij t k it f ( = | M g 1 × − ∈ { h d,k det( = 0, but the probability density piles up near det − d,k k Re d χ d 1 positive eigenvalues and taking the last ,k d,k ) by analytic continuation of the density in of 1 as the characteristic function of the distribution. This is the . N − − × N i k 3.62 | λ M ) is real. . This argument does not apply for 1, the density goes to zero for any fixed positive definite matrix from the zero in A succinct summary answering this question is contained in [ If Our integration measure on Hermitian matrices is defined as the flat measure on independent real tM − BU 13 and we end up with awith probability rank density supported on the submanifold of singular matrices components, Tr( We can intuit this from ( k normalisation factor eigenvalues of goes as smaller nonnegative integer values of values of This is manifestly nonnegativetends and to so definesdegenerate, a but probability is distribution. still This integrable. result This ex- is easiest to see from the density in terms of the characteristic function reader for the resultscontinuous we function of now use. For The distribution For this to define adistribution (that positive is, semidefinite it inner givessuch positive product, values as we when need integrated againstsemidefinite positive test inner functions product is equivalent to the existence of a probability distribution with on a space of functions whether we have a positive semidefinite inner product. bution, ( Fourier transform of the probability densityon function the space of H JHEP08(2020)044 . ) is is is 1 as , d,k 1). 1 | ∂ p ∂ − d,k H 3.65 S H (3.67) χ k for the ∂M | d ≤ +log( log dim 0 . In that case, M ∂ − > S H = 0. = k. ∂ | ∂ S = 0. S ∂M | M = 0) = 1, the integral of + log ∂ t 0 S ( , which is the definition of 0 requires rank . This is nontrivial, because − | d,k S > S 1 , χ 1 ∂ = ∂M S H | ∂ S or = 0. On the other hand, since N ∈ M 0 = 0 does not lead to a reflection positive . This is violated by the distribution ( S d ∂ . We thus find that reflection positivity can − 1 at which the probability density appears to 0 ∂ S can be positive definite only when all sectors = S S − e − , and we can regard – 38 – Z ∂ ∂ BU S , this requirement is most stringent for the smallest e ⇒ H ∂ H has probability density supported on matrices with S d < k , it seems that this degree of freedom should allow = 0 d S M = 0 motivates us to explain the physics that might lead to ∂ ) shows that positivity in is well-defined and equal to unity. The resolution is that is an integer. This is not entirely satisfactory: besides the S is the logarithm of a positive integer, or 0 0 M in each sector , since S 3.53 S e , namely − M α 1. For a given ∂ Z S − . This gives us our final result: k k over all < d < k ≥ = 1 d,k Reflection positivity = d p − M k have N ∈ is not a local action. For example, if we take a cylinder (with two boundaries), we d / | A slightly different possibility is that some local bulk dynamics gives rise to a path However, we can achieve the same effect with a local action by introducing a new The failure of models with As a result, the inner product on We can use the arguments of the last section to slightly strengthen our restrictions on by considering positivity in Hilbert spaces with boundaries, and in particular in ∂ ∂M quantum mechanics, in which case we are again leftintegral with localised the theory at themechanics. boundary, This seems but like one aphenomenon strange which occurs situation for cannot at JT first be gravity. sight, In described but that we by theory, note a any that local precisely quantum bulk this theory gives rise to a degree simply, this can be aeach topological boundary provides quantum a mechanics factor with ofa Hilbert dim nonlocal space effective action fromof integrating our out this theory, dynamics. butsomewhat This only gives artificial if a restriction local definition on for additional boundary conditions that project onto a particular state of this boundary not preserved by this cut and paste. degree of freedom onEOW each brane boundary. boundaries. Thishappens should to Note propagate be that along localised both we at the asymptotic regard boundary, and this and not as part part of the of dual the ‘CFT’ bulk dynamics. dynamics Most that an action counting the number| of boundary components can slice it inwe two form along two its separate length, and cylinders. glue The together resulting the manifold two edges has of four each boundaries, piece so so that For any non-zero numberrequired; of the EOW absence brane of species,theory. a we The boundary find most action that naturalthe choice a theory is non-zero we the used value minimal throughout of the value rest of this section. The discussion leading to ( matrix of inner products in the range full rank, rank with non-zero eigenvalue of hold only when either S be positive, the density isanalytic (so not its integrable Fourier near transform det decaysthe exponentially) distribution and becomes a singular distributionand which which must is be not positive defined definite by on a the principal singular value submanifold prescription, det becomes negative. Even for values of JHEP08(2020)044 ··· (3.69) (3.68) ]. This + E 96 , 3 ) 85 [ gZ ( H 1 3! ) Z ( boundary, so we can f D Z + E 2 ) gZ ( 1 2 ) . for any Hamiltonian Z ( gZ e βH f − D = e + g containing an SD-brane was already present to these boundaries, interpreted as a coupling E . To avoid overcounting different spacetimes . In other words, the SD-brane is not a new E g ˆ Z – 39 – gZ gZ g g ) : e g Z ( f SD-brane . D )). Nonetheless, the gravitational theory (for example, E R + , of baby universes. We may thus identify the corresponding gZ SD-brane e E

) ) Z Z gZ ( ( e

f f D D ], though the framework of the Hilbert space of baby universes = = 82 E , g 62 . In particular, we hope that our model might be obtained as a limit of a the- ∂ ]. The Schwarzian alone is not a consistent quantum mechanics, since the path S SD-brane 95 ) as a coherent state , Z 94 ( BU To study the theory in the presence of an SD-brane, we should introduce a new type f H D is equivalent to insertingobject the at operator all! Instead, a state As before, the notationpath integral. on the But left-hand from side the indicates right-hand the side boundary we conditions learn for that the the insertion of an SD-brane account for them byconnecting inserting to factors the of SD-brane,treating the we new must boundaries as dividewhere indistinguishable by appropriate. and introducing factorials We further thus of symmetry havepresence factors the the of following number an recipe SD-brane of for with boundaries, computing coupling the amplitude in the assign a free (possibly complex)to parameter the SD-brane.allow To for compute any an numberis amplitude (including allowed in zero) to the of end presence theseamplitudes, many additional of times each boundaries; an on SD-brane i.e., the SD-brane, boundary same the we acts SD-brane. spacetime should much But the for same the purposes as of a computing in spacetime in any‘eigenbranes’ way. in This [ will beprovides a similar new in interpretation. spirit We to will the focus discussion on the ofof model D-branes boundary without and EOW of branes. spacetime, interpreted as spacetime ending on the SD-brane. We will 3.8 Spacetime ‘D-branes’ We conclude the discussion ofresults the in model with terms some ofmeans interpretative ‘spacetime remarks an for D-branes,’ object some which on ofbranes the we which are spacetime call seen from can SD-branes the end, below. worldsheet and in An as string such SD-brane theory. is In seen particular, from they are spacetime not as localised D- space interpretation. While wespeculate do that not some have analogous afreedom) dynamics concrete could (or proposal naturally an give to rise appropriate maketive to action accounting at a of this theory residual of time, topology gauge ory we which with includes more a dynamics, boundary and that effec- this construction might offer insight into this possibility. gral [ integral on the circle cannot bepossibility interpreted arises as Tr from aboundary quotient (in by residual that gauge case,the an symmetries Lorentzian SL(2 acting theory nontrivially on on the a spacetime lying between two boundaries, has a good Hilbert of freedom associated with asymptotic boundaries, described by the Schwarzian path inte- in boundary conditions: JHEP08(2020)044 θ g ), 3.70 (3.72) (3.73) (3.70) (3.71) ) tells us is ) introduced ) of the SD- imply that H Z λ . 3.70 3.69 . g − , + . E ) ) N ) 2 Re d . First, we note that N e ∈ θ = 0. In particular, the − πZ ∈ = g 0 Z )) of the single boundary ˜ λ ( sin( , d
π H d, d g

E ( E d E − iθ . In fact, we can move between Z , gZ 1) ) , or Re E λ e g  − +2 Re iθ u u ! d uZ ) in an SD-brane state: d ˜ λ e ) in the Hartle-Hawking state, with = ˜ λe = ( λe 0 Z +2 Re  . A natural basis of states superposing g ∗ u E dd g ( 3.11 π δ SD-brane e e 3.14 iθZ

D D = e

. This has no effect in the amplitude ( = = = exp = exp idθ + g − and the integer spectrum for idθ E ) of ( d e − u – 40 – g π e gZ dθ 2 λe π e = Im dθ 2 π π θ − π π ^ Z − Z SD-brane = exp( :=

SD-brane := 0

E + d + d uZ uZ by adding an appropriate SD-brane. This is a familiar situation e e d

λ D g is somewhat analogous to the determinant det( ^ due to the presence of the SD-brane. The result ( ^ SD-brane is thus defined by the Fourier transformed states, Z ) (single-trace in the dual matrix integral). ^ SD-brane u θ * H

SD-brane

− SD-brane E D 0, this inner product vanishes, indicating that the resulting state is null. ], where it was interpreted as a brane in JT gravity. The determinant is analogous To understand these states better, we may use the representation ( We can also make use of these SD-branes in yet one more way by considering the effect Now, what do the amplitudes actually look like in the presence of an SD-brane? To d < 62 For brane states as an exponential to write them as above basis diagonalizes the inner product: where for simplicity we will now focus on the case different values of of the imaginary part ofand the to coupling see itsis relevance not we fixed must but allowthe instead for representation has a of a different the superpositionshould kind SD-brane be of of understood as different SD-brane to values be state for periodic in with which period 2 Hawking state, but with aany positive different real value values of of thefrom coupling worldsheet string theory, wherethe different coupling values of of an the apparentlyof string free the to parameter same the (e.g. theory Euler (e.g. characteristic) coherent turn states out of to the describe dilaton). different states We here used thea result shifted value of that amplitudes in the presence of an SD-brane are the same as amplitudes in the Hartle- in [ because it can beobject written Tr log( as the exponential exp (Tr log( answer this, we compute the generating function ( This exponential of JHEP08(2020)044 . . 0. E 2 (4.1) -sector β > . For a α H = Σ), the ) 2 . β -sector with

+ M α , so to obtain d 1 ∂ Σ ! Furthermore, β (

= H d − e ]. Z

= (with = 82 Z ,

f SD-brane is not a new H M 1 62 β of arbitrary length − e β which can be diagonalized to -state H I sector arose after constraining 1 α α on a larger boundary manifold . β ) d with a given expected energy a − β φ e . J Σ 2.4 = E is trace-class, this condition ensures ] H βH Z β − associated with a pair of disconnected J ) for the βH [ , e f 1 , this boundary condition describes an SD-brane. This − , b ψ 1

e φ for intervals 2 3.25 H = β ]. As before, we work in some definite (but I E , ] from section 97 × – 41 – J and for which where the [ M α Σ on a boundary manifold . When connected components. However, we see that the ψ H

β J to d I βH 3.3 = Σ − β βH β I e − ) on the rank of the inner product in any e C 6= 0 simply rescales the resulting state g 3.41 simply defines a new source -states is at first sight rather different from the alternative geo- in the Hilbert space α E ] E J we showed this bound to follow from an abstract argument involving [ define a Hermitian matrix ( ψ

defined by sources

α Σ 3.6 that we may call E ] on J α Σ ∈ H [ H constructed by gluing a ψ βH -sector of the given theory and also choose a spatial boundary manifold Σ; i.e.,

φ − α e M β . In section I d = The final property we require of our theory is that the CPT conjugation acting on One property we require of our theory is that there is a notion of time evolution, here This means that we can give a somewhat geometric description of a given = β α f boundary conditions acts trivially on that states operator on given state action of M we consider a particular Hilbert space in Euclidean time.‘cylindrical’ This boundary means manifolds thatAccording the allowed to boundary the conditions general include principles Euclidean of section 4.1 Entropy bounds We now state thisThe form of ideas the are argument closelyarbitrary) using related the to more those general in notation [ from section boundaries. As theargument so reader as may toMore already realistic apply realize, models to it will verya likely is general meaningful have straightforward an bound reflection infinite on to positive the numberthis generalize number of gravitational here of states this path by states in bounding we integrals. any must the impose entropy a of constraint. mixed states We will in achieve 4 Entropy bounds and theA Page remarkable curve property ofand in our particular models theZ above bound ( was thethe strong cylinder role state played by null states, metric interpretation given inthe section path integral totwo spacetimes are with equivalent in theEOW end. branes, and We expect correspondingly a in similar the equivalence JT to gravity contexts arise of in [ the model with it is now clear that taking Re by including a particularfundamental object, (Fourier transformed) but isSD-brane built description from of a coherent state of interacting baby universes. The But this is precisely the expression we gave in ( JHEP08(2020)044 ∈ a ) as φ (4.4) (4.5) (4.6) (4.7) (4.2) (4.3) . a 4.7 -sector. ). Thus βE α β − ( : we have e α β Z ) and ( . by

4.6 α

βH ) − β e ( : of our eigenstates α is defined by boundary a , Z

φ 3.6 . This involves gluing two α BU ⊗

= ; with eigenvalues , ∗ b α must also define an operator associated with two copies of a H

. 2 . φ ; , α Σ / , which can be thought of as α 0 2 ) Σ , φ

a α . Σ Σ t H ) β ∗ a is trace class, a similar argument βH β/ ∗ b ( ≥ t β φ 0 − βE H ∗ α Σ α

( a e a . The norm of our cylinder state is − α Σ 2 δ βH [ Z H Z e / 0 β − βE a

ab ab e = − δ δ ∈ H βE e

− ) acting on = = E α e a β ; α

-state, then implies that the eigenvectors can be ( X 2 ; a α α α [ might fail to be trace-class. Z β β/ − ; ; X of length 2 a ) βH 1 β − – 42 – βH β − β/ , which we can use to define cylinder states by , φ S ( e

−

∗ b

α e α BU φ , spanned by products α ; Z

= ; ) in the 2 α Σ H

α 2 β E = ; α β/ ( 0 in direct analogy with section β/ α ; i a [

Z ⊗H a ∆ ∗ . Together with Hermiticity, it also implies the relation | , φ BU 0 α Σ , φ βH β

∗ b ∗ b ∆ −

H h φ [ φ e =

∈ H so that the eigenvalues must be non-negative. Henceforth, we

E 2 ∆ ) under the assumption that α

), where the operator 2 to create boundary conditions with a circle of length βH/ fails to be trace class cannot occur and we can use ( β 4.6 into the ‘two-sided Hilbert space’ ( − β/ e i [ Z , with a thermal circle βH † α 1 β
, it is important that this computation was performed in a fixed ; − = 2 -states a e S , the overlaps are given by 2 α α Σ to denote such an orthonormal eigenbasis of × , φ 3.6 βH/ β/ ∗ b H ) is the eigenvalue of a − φ β | φ e ( \ on α † Z = 2 βH We now introduce a state We will be interested in forming mixed states on The key fact is then that the boundary conditions The above semi-group property of and their CPT conjugates. This space of density matrices is isometrically embedded β/ − e βH α Σ − \ As in section While we arrived at ( using approximate eigenvectors would in anythe case case bound where the trace of written. and impose that its norm is nonnegative, where The last overlap we requirecylinders is of the length norm of the state conditions Σ then given by of elements of the Hilbert space H via states our spatial boundary Σ. Since these latter states were built from orthonormal eigenstates on the baby universeacting Hilbert on the space and return later to the possibility that chosen to bee independent of thus take yield discrete eigenvalues with finite degeneracy. We will take this to be the case for now JHEP08(2020)044 ) ], ≤ by E ) 61 ( α Σ (4.8) (4.9) E N (4.11) (4.13) (4.10) (4.12) ( as the H N on , by drop- ) operator ) is defined , finding β Gibbs β βE ( H β ( ρ − [ α Z -states (in the e , ), is then given ) Z α a E. β E ( ( βE α − N Z ) = e ≥ ), with the inequality ) subject to the energy a ρH ρ a X ), which is the usual way ( 4.7 ) to be approximately the } β S βE ) ( ) is controlled by the density E − β α ) = ( E ( e . α Z ( ) we can then say that a } S βH ) N with Tr( − P 4.7 β . e Gibbs ( ) α Σ α Z , β ) ( H Z E ( α + log ) = Tr( Gibbs S β + log Z ( ) (4.14) ≤ βE E ) { ( log βE α { E – 43 – β Gibbs ( . This constraint depends on the energy expecta- S ∂ β ∂β α Σ -states, we expect N ≤ α − H ,Z . Specifically, we prove that )) = inf log ) = ) := inf ) to make some more direct statements about the spec- on ab E β ( δ ( E E ρ a ( 4.7 βH α E − any density matrix on S e : ) appears in a stronger bound, constraining the von Neumann . In a thermodynamic limit, we would usually expect this to be Gibbs Gibbs = ρ in the sum. From the result ( ρ Z E E ( α ( ) S ), defined as the Legendre transform of log α Gibbs ≤ a S E Z ) = a > E ), where from our earlier considerations we can define ( β ) for E a α . To bound this quantity, note that ( . Firstly, we can use it to bound the number of orthogonal states , Hφ . The sharpest bound is obtained by minimising over all E S to fix the desired energy, E is precisely the quantity we bounded in ( b ρH E α Σ ( ). Now, we can compute the von Neumann entropy of β φ α β β H ( S Gibbs α ρ ≤ Z = Tr( Gibbs ) ) of any mixed state Z ρ E ρ ≤ ( ( ) for any S ) S β β ( ( α The same quantity We can use the inequality ( ) boundaries are suppressed. Z β ( Gibbs βE Z Legendre transform of where we choose Note that It suffices to showconstraint. this This for is the simply density a matrix Gibbs that state, maximises entropy tion value matrix elements ( computed semiclassically by theassociated on-shell entropy action ofby a the classical Euclidean Bekenstein-Hawking black formula.measure hole. of This the The remains Hartle-Hawkingis accurate ensemble) small. as in This long typical Z is as the the case variance if of connected the wormhole configurations between two asymptotic This quantity is nothing butof the obtaining Legendre the transform canonical ofand log entropy in from the overwhelming aBekenstein-Hawking majority entropy partition of of function. an appropriate blackby In hole. the a This Gibbons-Hawking is semiclassical path because theory, integral with periodic Euclidean boundary conditions [ e where trum of states in with bounded energy dominated by states with energy close toof the states maximum, at so energy ping all states with JHEP08(2020)044 i α Σ ), ∆ β . H | ( 3 [ Z (4.15) on ). This -sector, ρ β α ) is equal ( β α ( , and so Z α β can be made Z Σ H ), which generically ) for all ) and of the radiation ) can be saturated, as 4.9 α Σ 4.7 H 4.10 and take definite values in .

) from ( α BU ; ) of ( . The result that the function E from our models in section H a ( in the Hartle-Hawking state. In ,α E α ( Σ N , φ E α S ) by the same minimisation as used ∗ a ) H is null leads to a gauge equivalence ∈ S φ ) by the larger function ρ

i ( 4.9 ). If this inequality is (approximately) β α 2 ≤ n ( / on ∆ Z | a ) S ρ D 4.10 ( βE βH ). Gibbs S − − e Z e 4.10 a – 44 – X is bounded by = α Σ

H ) is defined in ( = 0 is equivalent to the statement that α prepared with asymptotic sources, the (and R´enyi von i E ; ( 2 ρ ∆ α | again define operators on β/ S ]), which can be populated with a growing number of naively ρ 98 ) show that, for theories defined by reflection positive path

)), after replacing . In particular, we expect this to hold for the old black holes in ) of k ρ E 4.10 ( ( n S Gibbs ρ ( ) and ( S that for any state 4.8 ], the cylinder state is naturally associated with a two-sided black hole with an 2 11 ) can be written as a thermal trace is a strong constraint on the eigenvalues of To connect further with our desire to understand black hole evaporation, we recall In the case of saturation, the statement that We expect that in contexts where the naive number of states in We interpret this result as a semiclassical Page curve. The class of mixed states β ( α -sectors. These entropies are then subject to versions of the above bound in each from section Neumann) entropies α and as a resultthe so context are of their black expectation holes, values any such entropies will then reproduce an appropriate Page Following [ Einstein-Rosen bridge joining the two boundaries.this and We a see superposition the of familiar product equivalence states between emerging as an example of our gauge equivalence. to the actual thermalZ partition function Tr which should be viewed as generalizing the result arbitrarily large, one will findin that our the model bound withthe large discussion above. This requiresbecomes saturation a of the null inequality state. in Note ( that these states mustrelations conspire must to occur after produceby the surprising the Page Bekenstein-Hawking time linear entropy, so to relations thatsaturated, satisfy the the ( between entropy entropy of them. of the black thewill hole black follow (i.e. Such hole the the density is Page matrix curve. bounded on seems that this process canThis produce entropy states comes of a fromwith given the time energy along large with a arbitrarily interior ‘nice slice’ large whichdistinct [ entropy. grows possible with low energy time states.positive (in Our path particular result integral, linearly nonperturbative shows that effects in giving an exponentially alpha small sector overlaps between of a reflection integrals, the density of states inwe any expect to be given by the Bekenstein-Hawking entropythat of we an can appropriate prepare black bycreate hole. pure asymptotic state sources black holes includes by oldHawking collapse, radiation couple black escapes, to holes. and an trace auxiliary For out ‘bath’ example, the system bath. we into which can In the the usual semiclassical description, it here to obtain demonstrates the claimed entropy bound ( 4.2 Consequences andOur interpretations results ( The inequality follows because JHEP08(2020)044 . ] ) ]. is 14 ρ ,α . ( Σ E E have 100 ) ) , 101 H n,α ρ ρ , E ( ( include S ) 57 n n ρ – S S ( 100 n D D , 55 not S D 56 , of such entropies 55 E ) ρ ( ]. This approximation n ] which in our language S 57 ] that information is lost D 99 49 , , for which higher topologies , but the distinction is unimportant 48 E 3 -sectors are superselected from ) n α ρ -sector) defined by some fixed set α Tr( D . In such cases it is natural to use an λ , is the action of a wormhole. This holds -state. This puts additional constraints on 1 α − N G – 45 – , of order S (and not just on a single Σ -states. These are entropies defined separately on each H α , where S − e -states in the Hartle-Hawking ensemble. In particular, while this α ) in the various ρ ( n This is a physically useful notion of entropy as the It is, however, important to note the precise sense in which the entropies More properly, the replica wormholes are saddle points for \ S 14 spacetime wormholes are included assulting perturbative perturbation interactions theory between is universes.was the The also ‘third re- emphasized quantised’ in formalism other of contemporaneous [ literature on wormholes [ as long as the variance of these quantities is small. and for the contributionsa of factor spacetime of wormholes the tofor form be computing nonperturbatively simple suppressed amplitudesare by in suppressed our by models factorsapproximation of of where section the different large universes evolve parameter independently at leading order, and where 5.1 Formulating a wormholeWe perturbation have been theory interested abovenant in effects. contexts But where in spacetime mostconfigurations. wormholes circumstances provide spacetime In the wormholes domi- are such not cases, the minimum it action is natural to expect other configurations to dominate, in black hole evaporation. Thisfrom is long all in ago. directcompanion parallel paper. We with will the conclusions also of discuss [ such connections in5 more detail On in third-quantized a perturbation forthcoming theory principle unobservable. Nevertheless, if onematrix wishes on to the consider full theof entropy space of sources, some entanglement density withthat baby exceed universes the will Bekenstein-Hawking generally entropy andcurve. lead thus to In do this much not larger reproduce more the mathematical entropies expected sense, Page Hawking was correct [ over the computed by replica wormholes (toentanglement with a the first baby approximation), universe it sector. manifestly does the standpoint of asymptotic observers, and entanglement with superselection sectors is in higher moments of the entropy. just been defined. Fromof our perspective, the basicWorking quantities in the are Hartle-Hawking the state then eigenvalues computes the average be much as in theare indeed recent the discussions most of naturalThe replica argument saddle wormholes above points [ shows contributing that toresult similar the of results average any will entropy reflection thenbounds positive hold hold gravitational when not path one just integral. computes on the average, Further, full but it in tells every us that these curve defined by the Bekenstein-Hawking entropy. In particular, the final result will then JHEP08(2020)044 . 6 (5.1) ] acting on the J . In particular, at [ i d ] Z J [ collapses to a single E Z ] h n 1 2.2 J − [ Z ] in place of the low energy Z J D [ d Z yields another state proportional ···

E ] 1 J HH [

The effects on the bulk effective field ] with an operator Z J D [ 15 n ] on Z − J . [ ), at this leading order of approximation we Z d 6 Z ≈ – 46 – 2.11 E ] n J [ Z as in ( ··· ] BU 1 H J [ ] focused on studying microscopic wormholes, with the intent of Z D 57 becomes nontrivial, and takes the form of a Fock space. To see 1 – − ] with a multiple of the identity operator ], the effects of higher topology we study are more closely analogous to ‘wormhole 55 Z J BU [ 102 d Z H , as well as for JT gravity — though it does not hold for all amplitudes, 3 , so the baby universe Hilbert space defined in section

To incorporate nontrivial wormhole physics, we must go to next order in the approxi- Suppose then that, for some theory and amplitude of interest, the contribution from The early works [ We now describe an analogous approximation in our framework. This will serve both to HH In the language of [

15 boundary contributions are analogous tothe quantum Hilbert field space theory propagators. In particular, interactions’, as opposed tointerest the in ‘instanton that interactions’ work. arising from nearby wormhole mouths of primary to dimension. mation, allowing contributions to the pathor integral two from asymptotic spacetimes boundaries, that connect butasymptotic either not one boundary more. are The then contributions analogous from to spacetimes quantum with field one theory tadpoles, while the two Identifying an asymptotically AdSbaby universe boundary Hilbert space can simply replace this level of approximation, acting with any cussed in section as we will discuss below.neglect In the connected a contributions, case obtaining where an it amplitude does, that at approximately zeroth factorizes: order of approximation we may description in place oftheory that an arise effective from integrating Lagrangian. out macroscopic wormholes will betopologies explored in connecting section many boundariesThis is holds suppressed for relative familiar to simple disconnected amplitudes topologies. in theories of interest, including the model dis- contrast, we have wormhole mouths which,classical as or with quantum replica extremal wormholes, surface. aresimilar As to determined a some by black result, a hole our horizon, wormholeswill which will may be typically be have more both a macroscopic natural size andbulk to large. fields. discuss For This us CFT it captures thus boundary much of operators the same physics, and is analogous to using an S-matrix describing physics on distances scales(say, Planck much scale). larger than The theas relevant wormhole’s some scale characteristic length is size scale thebetween associated the ‘width’ spacetime with of regions the associated the with cross-sectionalthat the wormhole context, area. wormhole mouth, it mouths is can thought In be most of contrast, much naturaleffective larger. to the field describe In separation theory, the studying physics using the the effect operators of of the integrating low out energy the microscopic wormholes. In complete the connection to the above literatureinteresting and circumstances to described provide a above better in understandingthis which of section the this represents approximation a fails. distractionreaders from Nevertheless, the may main wish line to of skip inquiry directly presented to here, and section some JHEP08(2020)044 BU H (5.6) (5.7) (5.2) (5.4) (5.5) . E ] 2 which are chosen to be J i [ b , ] in terms of baby universe Z Z , J , i [ . i ED d ··· ] Z ) gives the ‘free propagator’ for ij ∗ 1 HH + δ HH J [ i| 5.5 ∗ as (approximations to) states of a ] i| = J ] Z J i a i [ ] J D [ j 1 J Z + [ Z Z − h h 1 † J Z Z ih ^ 1 | i a − − − Z Z Z h + E 2 , with a more conventional oscillator algebra to give operators ] − 2 i −

] ] i − Z J J – 47 – ··· ] [ J J [ [ J Z [ + ] Z Z i − i ∗ | 1 Z j

| a J 1 [ Z = = 0; (5.3) i − = + Z i i Z Z i we would expect to obtain an overcomplete set of coherent ] is most likely to immediately cap off, failing to create a † i D h ] a 1 J ∼ J J [ HH HH [ = − ] d | | + , where in particular we have Z Z Z i † J J ∗ J ^ i | [ 2 i a a J ] for different d † J Z Z J [ , a h , a ] = 0, d † J 1 Z 2 a J = J a i h , a b Z 1 J a (and by varying J ). As long as these higher topologies are suppressed, we can thus construct a labelled by an orthonormal basis of single-universe states. Repeated applica- are then said to create more universes, which can interact through topologies ij indicates higher order terms. 5.6 † i δ a ··· ] = † j In particular, as noted above, based on the validity of the free approximation One is then tempted to think of the states The resulting Fock space structure can be used to define baby universe creation and , a i a terms in ( useful perturbation theory, where thesingle inner universe product states, in with ( higher topologies contributing vertices. appears to be well described by a Bosonic Fock space built on the single-universe Hilbert We can then write [ tions of connecting three or more boundaries and into which we could incorporate as higher order by the source states). We can diagonaliselinear the inner combinations product of onHermitian the and single-universe Hilbert give amplitudes space, satisfying taking where single closed baby universe, with a wavefunction for the metric and other fields determined One can then writecreation corrections and to annihilation the operators: boundary operators annihilation operators and the algebra [ closed universe. It isto natural create to a subtract single thisjustifying possibility, closed the in name universe which of which case ‘singlewould can we universe require propagate are state’. additional most to Going subtractions likely to another for higher this asymptotic orders description boundary, in to the remain approximation valid. boundary states; i.e., one need only introduce the modified (tilded) states and similarly for states involvingtime larger numbers created of by universes. the Loosely operator speaking, the space- this, we define ‘single universe states’ by subtracting the ‘tadpole contributions’ from one JHEP08(2020)044 , . . i i i Z 62 . In Z i h 17 of JT gravity [ i ) it − β ( ) by which we must quotient , shifted to be centred on Z i , independently for every . As the oscillator vacuum, the ) i Z 2.9 R it Z + β ), a function of the real variables i ( ]. However, higher topologies continue ], are normally distributed at the first Z Z -states. We are led to expect that the h J [ α 49 , Z -states, as we will argue in a moment. The 48 α – 48 – operators. Alternatively, in the free approximation we † i boundaries [ a -universe vertices, so requires their inclusion. n , and hence the n i Z ]. Equivalently, we can describe these amplitudes as the overlaps J [ d Z in terms of the wavefunction Ψ( . Due to the null states, the ‘universe number’ which grades the Fock has continuous support on the whole of ). At each higher order, corrections from interactions would then appear then acts as a position operator (or a free field operator in QFT, where the } BU i i BU 5.5 b Z Z H -boundary amplitudes; the dominant configuration is a ‘replica wormhole’, a H { n . It turns out to be invalid because, while perturbation theory is accurate in ], though it requires summation of a class of ‘tree-level’ diagrams involving vertices of all in ( 49 5.6 16 ], for which the contribution from the disconnected topology decays in time, while ], for example. However, in this respect, we have been misled by the free ‘approx- could be momentum, for example), multiplying by ], and hence the dual ensemble and the i J [ Instead, we are interested in cases when the third quantised perturbation theory fails Before we describe the breakdown of third-quantised perturbation theory, we clarify It is now tempting to use this free Fock space description to describe the spectrum This is equivalent to the statementThis that perturbation the theory is vacuum also state useful of for discussing a the free average field entanglement spectrum theory close is to Gaussian the with d 102 103 Z , 16 17 th entropy of R´enyi an evaporating black hole after the Page time, which can be described corresponding covariance matrix. Page time [ valences. happens to be dominated by entirely, and many topologies must be consideredcompute at amplitudes once. with For example, a thisvery parametrically occurs large when large moments we number of of boundary components, giving Eventually, the connected topology dominates,n giving the ‘ramp’. Aas second example a is sum the of spacetime which connects the to be suppressed in such cases, and a similar perturbation theory remains valid; it simply happen that the mosttopology, but important higher contribution topologies toexamples. remain an negligible. The amplitude This first comes86 occurs is from prominently the a in two spectral nontrivial the recent form connected topology factor gives a contributions that is exponentially suppressed but growing. space is invalid once weby take to into obtain account thespace null is states not ( a diffeomorphism invariant observable. that it is not necessarily signalled by the dominance of spacetime wormhole effects. It may product to contribute only small non-Gaussianin corrections [ to theimation’ measure, the conclusion reached many circumstances, it istrue, not nonperturbative applicable spectrum is in smaller because the Fock space description of the Hilbert of spectrum of the resulting ensemble the nontrivial order described above, with covariance matrix given by the single-universe inner states are built bycan acting think with of The operator label Hartle-Hawking state has a Gaussian wavefunction for each space. The Hartle-Hawking state provides the oscillator ground state, and multi-universe JHEP08(2020)044 for for S S − − e c e − e multiplying the Euler 0 -states and so is vitally ] as a position-like operator, S α J [ d Z , with While any particular process of 0 S − 18 e . We will restrict our considerations 3 ]. The approximation of weakly inter- J [ d Z ] is invisible at any finite order in the third- J [ – 49 – d Z . c . From this point of view, the compression of the Hilbert space 1 , the spacetime wormholes describing third-quantised interactions N − N G G nonperturbative effect, contributing to simple amplitudes as -states are like position eigenstates in the harmonic oscillator. They thus α 19 of order S doubly parameters and create a baby universe wavefunction with a more narrow spread, the The small parameter that suppresses topology is The truncation of the spectrum of In retrospect, it should not be surprising that perturbation theory is of limited use for Crucially, this breakdown of perturbation theory applies to provide a simple example of the latter. Recall that, in terms of the usual bulk per- α This notion is well-defined only in the third quantised perturbationWe theory, mentioned but can above nonetheless the be natural used third quantization interpretation of 3 18 19 surfaces with given boundaries) have already been computed exactly.to diagnose These whether are that all perturbation given theory is self-consistent. but we could equally well have interpreted it as an analogue of free particle momentum. characteristic. It isexpansion in natural that to parameter,the organise with details the higher of such genus third an topologiessurfaces) quantised expansion are appearing (particularly not perturbation necessary as accounting for for theory loops. the diffeomorphismsthus point as of instead we However, wish assume connected an to that illustrate. the To full simplify connected the correlators discussion, (and we thus any sums over connected 5.2 Perturbation theory inTo give the some topological insight into model theapplicability validity in of the third quantised context perturbationto of theory, the the we model discuss model its without of EOW section branes. turbation theory in are already nonperturbative, soan the action relevant expansionis parameter then is a of thesome form (possibly imaginary) constant be quantised. quantised perturbation theory.nonperturbative Thus effect, in or that in an descriptionof exact it solution if could one be turns out seen to only be via available. Our some models At that point, the above approximation is not self-consistentdetermining for the studying spectrum such states. ofperturb observables. around the As minimum a oforder a simple tell potential example in whether of quantum the mechanics, similar configuration we behavior, space cannot if is at any compact, we finite and hence if the momentum should important for understanding theacting spectrum baby of universes isfree thus theory, the not a reliablehave infinite guide expectation to value the forthe details the of number operator. themean spectrum. As universe we occupation In reduce number the the increases, uncertainty and in eventually becomes exponentially large. splitting and joining universesenhanced is by combinatorial suppressed, factors theuniverses counting total (or the joining amplitude number many of of possibleimportant. such processes boundaries). interactions with This is many allows possible higher topologies to become of states with very large universe occupation number. JHEP08(2020)044 (5.9) (5.8) , there n . At the n  λ independent k = 0, and with ∼ i N choices of which ) X λ h ( 1) n − in the expansion by 2 n. n B ( k n − for bounded continuous n = λ fixed i
) , 2 n X ( f , the leading order contribution h → ∞ n . The central limit theorem then 1 2 λ . But this does not apply only for ) and note both when and how that λ = λ is just new encoding of the boundary ( i , There are cylinder components. For 2 2 components, which means either two 2 n ··· λ X k λ B − X 20 + 2 also becomes large. The first sign of trouble h − . Specifically, we may define converges to a normal (and thus Gaussian) 1 n connected components. 1 2 ways to choose pairs of boundaries to join = √ n − / − Z k i 2 X n Z is again a Poisson distribution, with parameter n expansion remains useful because we can explic- 1 disconnected components, which requires one n λ – 50 – = − → ∞ any amplitudes Z λ h − 1) n λ 1 X λ n λ/N − ) approach those computed by integrating against a − 2 λ Z n ( to be large. For fixed n acting to subtract the ‘tadpole’ and set n + λ , when the second term in the above expansion is no longer n λ λ √ the distribution of , this expansion is organised by counting the number of connected ) = . This follows from first observing that a sum of λ 3 , so our perturbation theory will be an expansion in inverse powers of ( λ λ n large with fixed ratio then implies that we can apply the central limit B → ∞ N λ if of order n and (fixed independently of , with the shift by comes from completely disconnected spacetimes, giving λ Z f λ , so this defines the ‘free’ Gaussian approximation mentioned above. 2 2 x However, it turns out that the large Now, let us consider what happens when We will return to the discussion of this wavefunction later. Before doing so, we the large Let us begin by noting a precise sense in which the free Gaussian approximation is − For simplicity of language, we will call this a cylinder even though it packages a sum over surfaces of e expansion to study the moments 20 . Taking . As noted in section same (leading) order. In some sense our freeitly approximation account has for failed. the sum over configurations with all genus with two boundaries. A more precise language might refer to it as a renormalized cylinder. occurs when smaller than the first.by There a are cylinder roughly (neglectingis the sufficiently correction large from toa choosing overcome single the the cylinder; same suppression terms boundary by with twice), any which number of cylinder components again contribute at the At the next order,cylinders, we or have a spacetimes ‘pairspacetime. with of pants’ We connecting canaccounting a continue for trio in possible of this topologies boundaries way with to to any the desired same component order of at large next order, we have‘cylinder’, spacetimes a with component joining twoboundaries boundaries. to join, so we have Gaussian. These are the∝ vacuum amplitudes of a harmonic oscillator, withλ wavefunction expansion fails as we also take condition an additional rescaling toimplies fix that the as variance distribution. Infunctions particular, at large λ theorem to the Poisson distribution as which has mean zero and variance unity. This λ components of spacetime. appropriate at large Poisson distributions with parameter by the same number JHEP08(2020)044 n 1 ). k − λ λ n k ( λ (but n (5.11) (5.10) # in the λ B e i cylinder n 2 λ Z n | . If this is the at large λ as before. Such . At that point . As discussed in 3 n 1 /  2 − Z λ λ n ,... 4 , 3 , . ), but since they are not al- E 2 = 2 λ Z λ ). We may thus approximate is the (normalized) oscillator n λ n k 1 e i is of order

− 0 n | for n ) nλ ) in powers of 1 fixed k ( . Expanding log eλ − n , ( O 1 expansion remains a tractable way to  , so we must have 5.10 λ ' + n X λ ).

2 λ that the free approximation is applicable, λ → ∞ 0 (

nX n n pairs of boundaries to join with a cylinder, q ˆ . This pattern continues, with similar 2 X λ B n 2 3 k λ, n λ 2 λ n q 3 – 51 – 1 + q −  + 2 e the large λ e 2 n n n λ e λ n λ n 2 / λ log = , is dominated by contributions with roughly 1  becomes of order n Z Z i n ), and the middle state lives in the harmonic oscillator ∼ λ n n Z ) ) remains accurate until = ' Z 5.8 λ h

( n ways to select n n and is organized by types of contributing geometries. However, Z 5.10 k i Z B

 n 2 2 Z n . For sufficiently small h   ), writing ) an extra factor of ! 1 k and making its discreteness apparent. th moment of , this perturbation theory breaks down catastrophically, since there is no 5.8 nX n λ 2 λ 5.10 Z ), we see that larger still, ( q , this structure is also apparent from a direct asymptotic expansion of  n ), this gives log 5.10 n A.2 exp n is of order λ n To explain this last statement in more detail, we first describe the state In summary, in the regime Taking From ( ∼ introduced in ( n we therefore have an approximate equivalence between the following states: Here the final equalityHilbert uses space ( of the free approximation. In particular, free approximation. We beginX by translating to the harmonic oscillatorany fixed position variable Z once longer any suppression of connectedin topologies which the with novel many effects boundaries.the of spectrum This null of is states the and regime gauge invariance become relevant, truncating effects multiply ( corrections appearing whenever appendix compute the moments three boundaries each (‘pairs ofof pants’), configurations and of also multiple fromare certain cylinders. those aspects in In of which the two theconfigurations cylinders overcounting for latter end simplicity context, (and on to the the obtainlowed same relevant a we boundary. configurations definite must power now We of previously compensate by included subtracting these off their contributions. Together, these two though it should certainly remaincase, much the less correction than from theterm cylinders when is expressed small as in an the expansion sense of that log itwe is find subleading significant corrections to from the including any number of connected components having In this regime, we cancorrections now can systematically account correct for ( compensating including for the higher overcounting topologies of with cylinder configurations. more boundariescomponents. as This well can as be much greater than one and the analysis will remain applicable, where we have neglectedchosen the more correction than once. fromcorrection Summing ‘interactions’, to over where this the the ‘free same gas of boundary cylinders’ is gives us a multiplicative are approximately JHEP08(2020)044 λ of N . λ  containing N when the free BU . But resolving H λ 2 spectrum is thus − . From the above ) 2 ) this is of order 1, and hence Z  λ control an ensemble n Z πi ∼ . Note that any state N

(∆ BU universe with amplitude λ Z = 2 H πiZ ' 2 α , when the free description e

λ N some ] from the late 1980’s leads to a requires ∆ 57 , this is Z , of order Z 56 variable has an occupation number that N in X Z . and a broader consideration of implications in . After applying the exponential operator, the – 52 – ˆ 2 N 2 itself becomes ill-defined once it becomes of order in the X 6.1 − N X e ∝ . Hence it will connect to 1 ) − of order one (for example, for λ X ( α ] computed at asymptotically AdS boundaries. This version ψ J [ Z as the width goes to zero, where the leading contribution comes from 2 − . But for ) λ X 2 | α (∆ ] and of Giddings and Strominger [ ' ∼ | 55 N N . As a result, and as one might expect, the discreteness of the 6.2 is described in the free approximation by a coherent state with average occupation λ

. This effect becomes of leading order at All these phenomena occur when the state of baby universes has unsuppressed inter- We can also see directly that this regime is connected with the appearance of null states, Now, a wavefunction of width ∆ closed universes and introduce a new boundary, the new boundary will connect to any αZ . At that point, null states appear and, furthermore, the null states are not preserved by e Coleman [ sharp structure in which statesof in a results ‘baby for universe Hilbert quantities of space’ the argument usesassumptions only about manifest locality. properties of the path integral and makes no further into more technical remarkssection in section 6.1 Summary and futureWe directions have seen that combining features of AdS asymptotics with the basic perspective of 6 Discussion As with many works motivated bywish the to black hole focus information on problem, eitherfurther various the readers significance may technical for aspects quantum gravity. of For the this above reason, results we or, separate alternatively, our on discussion their below breaks down. We emphasiseapproximation can that be this used, heuristic butλ is that appropriate for any notion of universe number operator actions with a givenN boundary. Roughly speaking,given if universe with we amplitude have aN/λ state of and thus the appearancethat of between new the gauge Hartle-Hawking equivalences. state and Perhaps the the exponential simplestnumber equivalence is and the free approximation fails. writing occupation number inon terms the of kinetic the term. Harmonicthe oscillator In Hamiltonian natural terms and of integer focusing the discretenessorder width in ∆ the spectumassociated of with the complete breakdown of third quantised perturbation theory. resulting wavefunction is aoscillator shifted with Gaussian, average occupation whichanalysis, number is it (here, follows a ‘universe that coherent number’) the state free of approximation is the validscales harmonic for as universe numbers vacuum with wavefunction JHEP08(2020)044 = , the i, j α of size , where Z a i N 0 ψ S ) for − i fixed can be . The ensem- ∂ ] take definite , we have seen S α 0 J , ψ , for which the e b [ S j Z Z Z − Z ψ one may think of ∈ , the models with ∂ or S k = 1 and restricting 0 α , the probability of e S k Z 0 BU S − − ). In particular, any list ∂ + log ∂ k S S 0 ∈ H e ]. In particular, the full bulk S . In addition, as shown in

0 in which the S Ψ 56

that leave , − ∂ BU S ∂ 55 e S H requires a sizeable compression of the . of are nonnegative integers, and thus only a i a i } and α ψ ψ for some rectangular matrix

a j Z 0 α cannot exceed either ¯ a i ψ

S { ψ α a ) a j . i ¯ P ψ α – 53 – a , ψ Z . = j P ψ α BU ) i and what one may call asymptotically AdS states, and is taken to be larger than , ψ ∈ H . Furthermore, all properties of the full spectrum of j 2 , changes in

BU that are multiples of ∂

ψ

S H d α HH

3.8

Ψ

naturally experiences an ensemble. The superselection sectors are is naturally interpreted as the dimension of the CFT Hilbert space. = ) take the form i b ) are naturally interpreted as the matrix of inner products between BU Z α i p , ψ is an adjustable parameter controlling the spectrum of eigenvalues H . The particular ensemble defined by the Hartle-Hawking no-boundary j , ψ 0 \ j ψ N , this structure ( b S Z ( α ψ ) from section ] is − ∈ ∂ Z J ]), and perhaps also with an extra boundary degree of freedom. Without [ S 0 α e S 3.70  Z − k ∂ 104 . As a result, the rank of any ( to be an integer larger than 1 is equivalent to setting , S k . The ( e α 0 49 S For Models with EOW branes have additional boundary conditions ( While We then explored this construction in detail in simple topological models inspired by Nevertheless, the final result is much the same as in [ Z is a parameter associated with the extra boundary degree of freedom. The potential as being dynamical rather than an external parameter. As a result, our models area − ∂ × 0 ∂ , . . . k S k ble defined by thecomplex Gaussian Hartle-Hawking random no-boundary entries state for each arises of from the choosing independent naive the CFT Hilbert space (which would have had dimension parameters in fundamental theories of physics. 1 EOW brane states ineigenvalues a of dual boundary quantum mechanics. For given (integer) equation ( accomplished simply byto relabelling be states the of ‘newS the Hartle-Hawking theory, state’. choosingsurprisingly In some consistent this SD-brane with sense, state the at idea fixed (exemplified by string theory) that there are no free state gives a Poisson distribution for the that it is constrainede to take valuesto in states the with non-negative integers. Furthermore, taking S eigenstates associated with other potentialeven eigenvalues more turn intriguingly, out unless toEOW be branes are null reflection states. positive only Perhaps when when all Jackiw-Teitelboim gravity with ande.g. without [ end-of-the-world branesEOW (EOW branes, branes, there see isassociated a operator single asymptoticallyThis AdS operator boundary is condition alsopredict present this operator in to the have model a quantized with spectrum EOW with branes. eigenvalues Interestingly, the models values and exhibit factorization.outcome Thus for asuperselection sectors given can state at leastHawking in no-boundary principle state be computed from correlators in the Hartle- theory naturally includes both there is a sensesectors in for which the the algebra two sectors ofwith interact. operators no on access However, the the to associated asymptotically theory with AdS has a states, superselection complete so orthonormal that basis an observer JHEP08(2020)044 - ], α ]— 43 J , [ ]. But ]. But ], since α 42 this pic- 49 Z 109

– 100 , Ψ

57 105 – 55 ] for typical members of 49 ] in reproducing various forms describing the universe in which 49 , α 48 , in every possible baby universe state. Z 43 , 42 using slices at ‘finite time’, or in other words – 54 – BU H ] for more on quantum maximin surfaces. 110 has been measured, no further information is then lost. α ]. Work along these lines is in progress and we hope to report 49 ], and various related suggestions can be found in e.g. [ ] and [ 97 states in the CFT Hilbert space turns out to be linearly dependent due 111 α Z sector -states define superselection sectors for asymptotic observers, any given asymptotic A key lesson from this work appears to be that, at least in sufficiently simple models, At the technical level there remain many interesting generalizations to explore in the In the context of black hole evaporation, for general baby universe states Indeed, in order to explain the computations R´enyi of [ α gravitational path integrals by themselves succeedinformation. in describing In a particular, great dealstruction in of of our the microscopic possible models boundary the theoriesand — also bulk defined of by path simultaneous the integral eigenvalues ensemble leads defined to by the a Hartle-Hawking definite state. con- However, this was possible soon. In the longer term, it is6.2 also clearly of Transcending interest the to study ensemble: more implications realistic and models. We interpretations for now each turn toabove. more speculative comments concerning the implications of our results future. For example, evenif in one the can models formulate discussed thewithout here, Hilbert reference it to spaces asymptotic would boundaries. belike useful Moving to beyond to add the topological understand current matter,Sitter model, and models one also of would to [ explore a similarly topological version of the de server may not be able tothey predict may the exact simply outcome consider ofunknown an the value experiment of experiment involving (in black this to holes, interpretationthey be unique) live. value a of To partial the measurement extent of that the previously ture gives a sense in whichtion interactions during with the baby evaporation universes of formally black leadthe holes. to But loss as of described informa- previouslyobserver in [ can find no operational signs of this information loss. In particular, while the ob- eralized entropy defined by theone expects region this outside. to occur And whenevercan due the exist number to of inside a a a priori maximinthe given independent argument area quantum bulk [ of states domain the that codimension-2 ofof surface dependence dependence where with intersect; the fixed see past also exterior and future [ geometry boundaries exceeds of this domain of the Page curve associatedcan with say the that it black was hole implicit informationhere in problem. we all of see With these that hindsight works, it one and in is fact an moderately exact explicit statement inthe [ at Hartle-Hawking finite ensemble somenumber version of of a this priori compression independent must states inside occur a whenever quantum the extremal surface exceeds the gen- to the presence oflinearly null states. dependent states We also mustdefines argued arise a that in positive a any similar semi-definiterelated theory constraint to physical where on ideas inner the in the product. [ gravitational numberthe of path result Our is integral deeply general related argument to is recent successes closely [ of more than JHEP08(2020)044 is BU -state -state H bound- α α 5 S × 5 . While there is an ) gauge group and . The same doubly 0 S N N − -states, since they are G e ], the gravitational path α would then be associated 87 and the asymptotic dilaton

, 5 α

62 S , of the U( -states in our model, it is also α 49 N nonperturbative in = 4 super Yang-Mills (SYM) is the unique = 4 Yang-Mills is the unique such local the- -states in our model, it remains possible that α N N – 55 – doubly of baby universes. The eigenstates ) do not directly correspond to different λ BU H = 4 Yang-Mills theory as a CFT dual. Note in particular that different N ] operators are then proportional to the identity, so the unique state is trivially an J [ d The more likely scenario may be that If we apply the ideas of this paper to more conventional ‘top-down’ examples of Even in the simple case of JT gravity and its cousins [ Z -state. The theory then exhibits factorization despite the existence of spacetime worm- α holes. Nevertheless, in analogy withsimple typical spacetime wormhole configurationsamplitudes. still Of give course, excellentpossible in that approximations analogy simple to with spacetime certain wormhole highly configurations atypical always receive large corrections. dual. We envisagethere two is simply ways no in ensemble.one-dimensional, which The so baby this the universe Hilbert might Hartle-Hawking space statenonperturbative occur. interpretation is diffeomorphism is the that invariance In unique that stateact the produced in of most null the closed most extreme states universes. emphaticAll version, is The possible then fashion, rendering required every to possible state gauge equivalent. supersymmetric CFTs; in particular, since ory at weak coupling, thesespace new (or CFTs perhaps must are be nonlocal strongly or coupled otherwise throughout violate their SCFT moduli axioms). the ’t Hooft coupling specified by boundary conditions forrespectively), bulk at fields (the least flux if on specifies we a retain family the of usualdetermines theories, holographic partition one dictionary. functions for Instead, for every each all possiblea possible choice nontrivial boundary of ensemble conditions. parameters, would since At thus an least appear at to first require sight, surprising new families of maximally large Hilbert space with a menagerie ofof dual them. CFTs, and However, the thisuniquely Hartle-Hawking is selects state in again tension defines withvalues an the of ensemble established the statement free of parameters the duality, of which the theory (the rank our model, suggesting a tantalising connection to explore in moreAdS/CFT generality. duality, such as type IIBary supergravity conditions, (or there string are theory) severalsimple with possible AdS model outcomes. and The JT first gravity, possibility, is suggested that by a our nonperturbatively complete bulk theory defines a extremely natural completion of theit model remains defined unclear by whether a dual theor double-scaled gravitational how path matrix it integral integral, is uniquely realised selectsin in this the completion, the bulk. sum This over completionnonperturbative is topologies, scale associated was which with associated are nonperturbative with effects truncation of the baby universe Hilbert space in over topologies. In more realistic models, we will surely notintegral be fails so to fortunate. converge.for Though any the given model topologyseries is is with sufficiently zero exactly simple radius computable, that of the the convergence in sum path the over integral expansion topologies parameter is an asymptotic only due to the exact solubility of the model, and in particular the convergence of the sum JHEP08(2020)044 ] are -states. 49 -sector) , α α 48 -sectors for α ] was the idea 57 – = 4 SYM with a 55 N ] focussed on wormholes perturbation theory. We 57 N – -sector transfers information 55 α ] that essentially any effective ], then holding the gauge theory 113 112 expansion, which nevertheless need not N ) expansion, but there is a unique theory at N – 56 – -sector is identical to that of any other, but where the = 4 SYM varies from sector to sector. For some given α N -sector. And since the baby universe state in such sectors does near 1, the action for a replica wormhole whose mouth has area . α n N ], so the amplitude for such processes should be exponentially small 114 [ G = 4 SYM, the bulk path integral generates a non-trivial set of A 4 . This is consistent with the observation [ N N With this in mind, we recall that a key feature of the discussion in [ In any case, the suggestion is that the gravitational path integral should contain the full A final, somewhat intermediate, hypothesis is that the bulk theory leads to an ensemble However, it is also possible in principle to have a unique dual CFT and to nonetheless is of order action. Indeed, theconfiguration specifying couplings the naturally geometry and mediatecan matter transitions be fields replaced arbitrarily in by far which any insideleast other, the any for no black replica given hole matter numbers deep interior theA black holes throat may have become. At in our context, but with onewith important Planck-sized distinction. cross-sections Namely, under [ dominate the in assumption any that physical microscopic process.determined wormholes by But would the the location mouths of of acoincide quantum the with extremal the replica surface. relevant wormholes black As in holeout a [ horizons such result, and wormholes they thus approximately thus are induces macroscopic an in ensemble size. of Integrating highly non-local couplings in the effective that one could integrate outa the modified spacetime wormholes effective and action describe inIn their which other effects the words, in detailed the terms couplings of originalequivalent were theory controlled (from with by the the specified couplingscouplings asymptotic but and where point spacetime spacetime wormholes wormholes of were was forbidden. view) The same to construction will a apply theory with an ensemble of bulk physics in each consistent not change, there isitational no path room integral in should ato teach given us the sector outgoing how for each Hawking information radiation. consistent loss. As a result, the grav- any finite field theory in AdS solvescan the bootstrap thus order emulate by a orderexist in consistent at large CFT any in given a finite large redefinitions is equivalent to averagingsince the dual integrating theory out over spacetime anredefinitions wormholes ensemble might can of be induce couplings. allowed non-local But as operators, well. non-local field interpretation in an asymptotic (say, large description fixed while applying aspace bulk field of redefinition boundary leads conditions. tosome a sense different It function is on thus the which possible the that intrinsic physics while of thedictionary each between dual the theory bulk is and set always of in boundary conditions on bulk fields, averaging an ensemble defined by local bulk field maintain both a non-trivial babyamplitudes. universe Hilbert To space see and this, thecomputes recall associated a again non-factorising that number the foretc). bulk each path As set integral a of (say, result,dictionary in boundary if defined a one conditions by fixed member the on of standard bulk the extrapolate fields bulk map (metric, ensemble [ is dilaton, dual to e.g. JHEP08(2020)044 2 − G ]). is the area 117 0 – A ) of order A 115 ( is the generalised 00 gen S gen ], where S 43 , 42 )[ , we have A G − 0 A close to 1, where n is of order + # log( A G for A 4 ] (and as foreshadowed in [ − ] (though admittedly those arguments are 0 ∼ gen 49 A 78 S ) – − A e ( R – 57 – 76 , gen S , such corrections are associated with extending the 69 4 , there are a large number of strongly correlated small , with ) α This places the QES outside the horizon with order one A Z ( . In addition, since the QES is determined by balancing the gen 21  S . . G − k G G dA e ], for black holes evaporating into a vacuum they should always lie R 45 of order ]. A 30 , 22 ]. Were all of the physics determined by replica wormholes confined far enough 43 , 42 As a result, one may think of the induced nonlocal interactions as modifying the gravi- However, for a full understanding of the physics associated with such interactions it The exact location and nature of the above non-local interactions is clearly of some For example, we can perform the path integral over replicated geometries and matter, while leaving 21 . As a result, the QES tends to be adiabatically close to any horizon, and thus separated unfixed the location ofthe the QES QES where location branchingentropy to between of replicas compute, the occurs. QES which and This istime, we leaves for integrate roughly a over example) its final is location. then integralof The over the integral (stretched) over horizon. the arealeading At of to the the a saddle QES width point, (fixing ∆ where ingoing feature, however, is that theseour modifications models are corresponds highly to non-generic.corrections, where In the the correlations regime conspire thatmake to the in give physical a inner large product number highly of degenerate null so states; that i.e., a they priori independent states are directly implies that the path integralwould computes expect the gauge from invariant physical general product argumentsmost as [ one direct in contexts where it is not obvioustational that constraints; topology i.e., change with should new be terms included). in the Wheeler-DeWitt equation. The interesting ance of the path integral withevolution — topology even change. involving processes Extending that this changequantum to the state arbitrary topology Euclidean — of time the implies slicewords, spacetimes used this to of define is different the a topologies restatementis of to a the be gauge old gauge symmetry maxim unless that related. for it In gravitational involves evolution systems other along time an evolution asymptotic boundary. This then suggested in [ appears one must takeinner into product. account As described the infamiliar corrections section diffeomorphism they invariance of imply gravitational systems for to the a more theory’s general physical slicing invari- quantum effect of evaporating against aA classical effect, rough the estimate saddle-point of is somewhat thethe broad. width area of are the saddle-point alsoamplitude. suggests of that The order the associated typical non-localfrom fluctuations interactions of the will deep then black naturally hole transfer interior information into the outgoing Hawking radiation in much the form inside the horizon, there wouldno be way it no could possibility purify of thefrom affecting emitted the the Hawking horizon exterior, radiation. arises and However, from in anyeffects. separation time particular The of dependence, the backreaction which QES of is suchG typically effects associated on with the quantum spacetimeby is an then amount suppressed only by of a power order of to a large effect as seen directly above andinterest. in [ In particular,hole’s while event horizon quantum [ extremalinside [ surfaces may appear outside the black in this quantity. However, in an old black hole the large number of internal states can lead JHEP08(2020)044 ], we ]) and 97 105 ] in which the disorder ] between two-sided bulk 119 , 121 , 118 120 , 97 , ) should be understood as an example 11 Hermitian inner product of the appropriate rank. 4.15 any – 58 – . Furthermore, following ideas related to [ α Z . 4 Extrapolating this result to more complicated models suggests that 23 that null states must enforce a similar bound in a general reflection 4 ] — which were argued there to be physically distinct — are in fact gauge 122 22 indicates that the long discussed relation [ 4 We now speculate further on the implications of this enhanced gauge invariance for The physics of this enlarged gravitational gauge invariance remains to be understood We believe the explicit demonstration of such a large number of null states to be a lesson It is this bound that leads to the Page curve, and which thus determines the rate at This gauge equivalence resolves a problemIn noted particular, the in spectrum of that possibilities work allows concerning how such superselection 22 23 Alice meeting Bob and Aliceinner product finding to only respect empty Alice’s space, notion there of is particle number no (as reason distinguished, forsectors say, from in the total transform physical under permutations. independent states must inleast fact in to our be model, regardedadditional this as structure. happens physically equivalent. in anone Furthermore, at essentially will random find way manyparticular that in does states which not which infalling respect anevertheless observers any gauge priori equivalent. have seem For vastly example, different just to experiences as have there — can very be but different gauge which equivalence physics between are — and in equivalent. issues involving black hole informationthat and in the sufficiently connectionstates old to other is black works. sufficiently holes It large), (where seems this the clear gauge number of invariance implies a that priori vast independent numbers internal of a priori in detail, especially in thesection context of more realisticblack models. holes and Nevertheless, bulk the thermofield argumentof double of this states ( gaugesectors” equivalence. of [ In particular, we now see that the so-called “superselection cosmological analogues in deHilbert Sitter space space. implicit It inis is random implemented also tensor by much inserting network like randomly modelssee the chosen this [ truncation projections to of into be the the a bulk bulk. direct result However, of we now the gravitational path integral. of fundamental importance.— It the implies gauge that symmetry —had of due been gravitational to previously systems the established.such is above a much gauge mentioned The larger symmetry conspiracies idea date andin back that more at discussions bounds least powerful of to on than black the entropy early hole might 1990’s, complementarity when be proposals such related suggestions (see arose to e.g. comments in [ the above non-local interactionsit are is intimately natural tied toit to think is of this the in change former in termsclean as the of statement secondary of inner the and the product, inner the correlationsinteractions; product latter see and as that again conspiracies primary. section inherent (for in In reflection the particular, positive details path of integrals) the induced we find a physical Hilbert space isargued bounded by in section positive gravitational path integral. which the above interactions transfer information out of the black hole. As a result, while in fact linearly dependent in the physical Hilbert space, and so that the dimension of the JHEP08(2020)044 ], . P to 70 code be a kin bath H H H using an kin for which GF, joint that similarly H H bath on defines a bijection bath ] described a close (even if such small O P ⊗ H 0 ⊗ H 123 R , . Instead, one is free to kin 49 kin kin , it is automatically gauge – ] that black holes may have ⊂ H 44 ⊂ H 41 bath , such that ⊂ H H 0 42 kin code GF H GF, joint H ⊂ H H ), and in particular in which standard vary with the choice of state in 0 GF R from some kinematic Hilbert space kin H P ]. In particular, these works suggested that ⊂ H discussed in that work were constructed without – 59 – 31 0 ] for a variety of more technical treatments. . In this sense, one may think of a general gauge is an operator on O GF 75 kin R – H . Note that there is no requirement for O that recently formed from collapse, it would be natural 71 only on a subspace 0 ⊂ H R bath O for any fixed subspace is considered in the presence of another system with Hilbert phys ⊗ H H bath phys . Within a given such gauge fixing scheme, it may then be that 24 phys H . Now, since H ⊗ H phys ]. There one wishes to reconstruct an operator 0 H 49 bath reconstructs ] as providing a gauge fixed description, where in this case the choice GF H R H and O 123 on , then, even if the bath system by itself has no gauge invariance, one is free R GF 49 – O H bath 44 H , The connection with the above works is particularly clear in the discussion of Petz With this in mind, we recall that the discussions of [ Nevertheless, as with any gauge symmetry, one is free to fix a gauge in order to define A structure of this general sort is inherent in Dirac’s constraint quantization of gauge systems [ 42 24 regard to theconsistent, (random) as physical innerfails product, to be they gauge are invariant. However, not at least gauge to invariant. good approximation we This canthough think the is of interested reader can consult [ effectively let the choice of subspace reconstruction in [ operator invariant. However, since the operators physical Hilbert space space to gauge fix bydefines choosing a a bijection general to lineartensor subspace product that have been evaporatingmay for seem longer than to the bestates Page in inside time. the great black At tension hole. firstof with But sight [ our this such tension bound statements can on beof resolved the gauge by number depends interpreting of the on comments linearly the independent state of the bath. In other words, if the black hole system with effective field theory is a good approximation. parallel between old blackbath’) holes and that the haveinfalling ER=EPR been observers paradigm radiating experience of into only an [ standard external physics system even (‘the at the horizon of black holes the experiences of infallinginterior observers of become a well-defined. black hole Forto of example, radius choose in describing a the gaugeinteriors in are which gauge the equivalentinitially to interior much certain is larger much of black larger size hole interiors comparable decays that to to might size form when an a language (i.e., aas set noted above, of at observables) thewith with what level which one of to may Hilbert roughly describe spacesa call any the physical a gauge Hilbert projection physics. invariance space isfixing In naturally procedure particular, associated as a choicebetween of linear subspace As a result, evenhole for may fundamentally pure fail state to bethis black well-defined as holes, as a a the variant gauge experience of‘no invariant concept. the of interior’, firewall-like One or observers possibility may at inside view described least the in no black [ interior from which familiar physics can be extracted. charges coupled to a gauge field), or even her notion of particle number in a given mode. JHEP08(2020)044 . , to 132 code code is to be H then gives ⊃ H d described in L GF G ], H with no drama 22 ∼ GF,joint A H kin H ] then show the sort , the image defines a | 126 – bath approximately satisfies the ψ h . 124 , i.e., where this dimension is ], the non-violent non-locality , the corresponding proper distance , code R phys BH H 131 S 41 H , e – 40 to define a linear map from  | 127 , is set by the condition ∆ bath code d 29 , d ψ h 24 an associated gauge invariant observable. I.e., . With respect to the definitions of [ ], and perhaps other proposals as well. , p . In this sense – 60 – isomorphic to ` ] to state dependence focus on non-uniqueness of 14 4 19 ψ , 2 − phys d . As a result, with high probability distinct states in 126 H 17  . And for any choice of – define H ] and which is naturally associated with the ER=EPR p R ` phys Furthermore, just as there is a particular gauge (or class 28  bath – 124 H ) for each bath state, it will be possible to choose a partial , ∼ 25 H 25 d 41 phys L , H in spacetime dimension 40 4. d L ]), proposals emphasizing the bulk Wheeler-DeWitt equation [ with at most dimension d < 30 kin , 22 , ⊂ H 20 ψ [ H . On a Killing slice of a static black hole of area-radius 26 21 to make the projection of each via its natural action on On the other hand, the above discussion immediately raises the question of how dif- Nevertheless, the non-uniqueness arguments of [ We note that such a gauge fixed interpretation allows all of the hallmarks of what is will project to distinct states of The non-locality scale If one imposes the constraint that the observer survives (in a recognizable form) for a given proper ], the black hole final state proposal [ code kin ψ 26 25 time behind the blackto hole predict horizon, the then maximum one amount would of expect such a drama genericfootnote consistent gauge with consistent the with observer’s this survival to constraint from that the point. event horizon“non-violent” would physics be for our own universe. While theregauge is fixing surely scheme more can tojust be be as used said one about can to use this(“the Coulomb issue, gauge potential we in in note electromagnetism Coulomb that to gauge”), any define in gauge the invariant operators above scenario one can use any gauge to define proposal 133 ferent experiences of aabove given scenario observer could possibly could be possibly realized be in models gauge that related, are sufficiently and realistic thus to describe how the such leads to othertypes of gauges drama in at which thein horizon, infalling the as interior observers well of experience as theof to black varying hole. gauges gauges) amounts where realizing and the ER=EPR-like observerrealizing scenarios, simply fuzzball it fails scenarios seems to exist (see likely e.g. that [ one can also find gauges state dependence isexpected, simply and a so choice that the of gauge gauge invariant predictions (so are thatof in non states fact identical). that, uniqueness while of related they by appear the at enlarged gauge first symmetry sight described to above. be physically In distinct, particular, must tracing in through fact be gauge fixing of theat form the described horizon. above Inevolution that particular, can one selects be will well-described only by be standard statesthat able local in standard to effective objections field choose theory. [ a Inthe code addition, subspace predicted we note within physics, which and the that such objections are clearly moot in a context where the requirements for a partial gauge fixing;H a complete gauge fixing would result from extending often called state dependence [ paradigm. In particular,black hole in states contexts (states where in one expects to find only a small number of In particular, we may useH any bath bra-state subspace much less than the dimensionH of as defining a partial gauge fixing (meaning that we could choose some JHEP08(2020)044 i λ n Z | (A.1) n c n P with mean Z The scenario described above (in 27 , ) λ ( n B of a Poisson random variable = i i n n – 61 – Z Z h h , and to illustrate the failure of the third quantised 1 − 3 Z in our model. 5 in section BU H The moments are given by the Bell polynomials, Note that if there is a priori no mechanism for selecting one such definition as preferred, then it is natural 27 to adopt a Bayesianmore approach generally and that declare that theygiven all are theorist such studying realized the extensions according system). areblack to hole’ realized The some would question with then probability of equal be ‘whatexperience measure probability an does when describing (or inherently an they the probabilistic observer are priors one, experiencehave decohered somewhat already of when into akin conjectured falling the many above to into Everett that asking a hole branches with ‘what in of high does a probability the an generic the wavefunction observer gauge, observer of simply and the fails that universe?’ to post-selecting exist We only inside on the existence black of the observer would lead to high drama. A Limits of moments ofIn the this appendix, Poisson we distribution studyin the various moments limits. This isconstructing useful states to of ascertain theperturbation convergence properties theory of of sums section KITP for their hospitality duringwas the also final portions supported of inPHY-1748958 this to work. part the As by KITP. a the result, National this research Science Foundation under Grant No. NSF discussing black hole informationWe with also the acknowledge entire interesting UCSBJacobson, High conversations Javier Energy with Mag´an,Juan and Daniel Maldacena, Gravity Harlow,Douglas Xiaoliang group. Gary Qi, Stanford, Horowitz, Steve Herman Ted Shenker,from Verlinde Mark NSF and Srednicki, grant Edward PHY1801805 andsupported Witten. funds in from part We the by are University of a grateful California. DeBenedictis for H.M. Postdoctoral support was Fellowship, also and D.M. thanks UCSB’s Acknowledgments This work was motivated and facilitatedPennington, by second three with specific Xi conversations, Dong, first and with third Geoffrey Steve Giddings, as well as by a long history of level of non-perturbative physics,may and depend sensitively predictions on for thewhich the choice apparently of observer distinct this inside observer extension. old experiencesto are black be gauge holes just related) another maysuch version thus conjectures of be and this considered related idea. physics It in will future likely work. be of great interest to further explore that there are a variety ofto possible coincide gauge (or invariant nearly definitions of coincide)differ the under greatly observer familiar which inside conditions happen outside old old blacksurprising black holes. manner: holes while but One we which may mayleading then well-enough semi-classical rephrase understand level, the how there above to may define statement be an in many observer a possible at less extensions the of this definition at the a notion of observer inside the black hole. The variety of possible gauges would then mean JHEP08(2020)044 . . x λ . In (A.6) (A.7) (A.8) (A.9) (A.2) (A.5) n We will at fixed 28 ) is and fixed . . λ . , ( n n → ∞ B fixed fixed n → ∞ → ∞ in the baby universe 2 n i , n n , the ratio is unity and 1 Z n n | , λ , λ n c ) as ): 0 as λ P n . Substituting this value back ( ( → ∞ → ∞ n )) (A.3) , which we can write as o n . From this, we can estimate the → , n fixed. (A.4) λ 2 n . x ( n log ,

B n n i d n . For large B ∼ p n/d , n ! d 2 2 e Z d ) is a monic polynomial of order | d λ d d λ λ/ n log 2) + λ , n ) + c ( ) − λ =0  ∞ n 1 e → ∞ ∼ − n ( d X ) at large n ( 2 0 n and fixed B n n λ λ o n = = (1 X ( B n −  λ i n ( n + e

1 n – 62 –  λ B − n Z d , we require the moments at large − | , λ to infinity separately at different rates. i n ) = − n n d n ) = 2 λ λ ⇐⇒ Z n ( λ  Z h ( | n , n ∼ d n λ 1 , n p B c +1 ) ) n log log n n λ n ( = ( n o B log log n P + ik − n B n . By definition, the series converges if the partial sums form a n n converges − Z and take the limit of the terms in the sum as n k| 3.3  log i log log n n n n x log √ Z − | e n = n c and convergence ∼ ∼ 1 + ik ) n  ) =0 ∞ n λ X n λ Z n ! ( ( th term in the sum is maximal when n n n k| n d log B B ) = 1, from which we can see that λ log = ( log 0 d B Now we can begin to characterise convergence of sums It may not be that every element of the completion can be represented by such a Cauchy sequence of 28 not characterise such series completely,convergent but series, find and a a sufficient necessary condition condition to to give constrain us them. apartial sums. class It of isdistribution false by for its the Gaussian ‘free’ approximation:wavefunctions. version in where that we case, allow this only class discs of Cauchy and sequences cylinders, yields replacing only the analytic Poisson Cauchy sequence. That is, where in this limit we can take norm of the basis state Hilbert space of section For a more carfulwrite derivation and many moreIn terms this limit, in the the series expansion, becomes it a is Gaussian convenient integral to in where the asymptotic formhence applies the for 1 into the sum, we can find an estimate of For studying convergence of For this, observe that the ratio of consecutive terms in the sum defining and particular this gives us the scaling at large A.1 Large From this, one can check the recurrence relation defined by JHEP08(2020)044 . n | z we R | | such . To n n c f ρ | (A.13) (A.14) (A.10) . From we have a for large n n 0, so . ) log z 1 n − for any R > ) 1, (A.12) . Since this holds Γ(  n . . ) is that the norm a 2 ≤ − ! n n < R (1 | . = z for any | > 1 ) for sufficiently large n n )  | R → ∞ n convergent 1+ convergent c , or more generally a function has order | | n i i i z is entire analytic. (A.11) / | n n n z xZ n Z Z 0 and sufficiently large e n | z | | c 0 as n n n exp( c c c →  > < X n n

| X X X decays faster than i ) n z

) = go to infinity, so for sufficiently large ( i Z ⇒ ⇒ z ) = | n f 1. (

| = z

converges in the disc i f Z | ( . We can strengthen our necessary condition to | n ≤ n z n f

– 63 – c ), for any ) Z z . For the norm of the terms in the series to go | | ⇒ n ) n |

n c c ⇒ A, x | n A.8 n c / ⇒ | | is eventually larger than log / log log n

) := i n log(1 − z 0, we have n convergent = ( log(1 ) | Z f

| n i

for some c →∞ . From ( | n  > n converges = →∞

/ ! n i n Z converges = . In turn, this means that log(1 | n x n n

i | Z n | i n n

c Convergence = Z log | < A | ) for sufficiently large Z | n n ρ | | c ) n n log n  z c X c | | =0 > ∞ − X ), we have the result that n ) = lim sup ), we see that =0 ∞ ) | f X n (1 exp( n A.8 c A.8 | > < / | , and hence lim sup

) , we find that our series defines an entire analytic function, i n z n ( R f Z | | The gap between our sufficient and necessary conditions (order one functions that are Our sufficient condition is absolute convergence, which means that the sum of norms First, a necessary condition for convergence (coming from

In particular, this means thatof any exponential exponential type, function defines a convergent series by itsnot Taylor expansion. of exponential type) is small but nonempty, for example containing Now, from ( this, we can find a simple sufficient bound on the coefficients for convergence, converges, and follows from the triangle inequality for the norm. namely order( to zero, we must havehave log(1 log(1 log enough which means that forTo every show this, we use a result expressing the order in terms of the Taylor coefficients, We can thus characterise convergent series in termsImproving of on the class the of analyticity allowed analyticdo result, functions. this, we we can introduce bound thethat order the of growth an of analytic function, allowed which functions is the infimum over all is bounded, which implies that for all of individual terms go to zero Now, from ( JHEP08(2020)044 ) ∗ u (by ( is of S n . The , with ) by a (A.18) (A.19) (A.15) (A.16) (A.17) n /k becomes 1 ··· − 3.11 n 1 + fixed λ 2 ν λ 1 2 + ν , ··· + + ν 2 λ is of order , which will interpolate 2 n λ n e n log ! n ν = n λ ): ν λ ∼ , ( ∼ − fixed. (A.20) n 1) . ) ··· 2 − B ∗ λ + λ u n u result breaks down when u . ) 2 e λ ( , ( 2 ∗ n λ λ S u + ) e ( log ∗ n 1). 00 . u ν + ) ( +1 → ∞ results. We could proceed from the same n function or product logarithm, n ν πn du ), so  πS − ( λS 2 at fixed ratio u 3 λ 2 e , and the relevant saddle point is the unique log 1 ν ν W √ u W n ( p I − − ∗ O ue = λ – 64 – u u , λ πi 1 fixed e → ∞ + ∗ = 2 2 λ log 2 u ∼ 2 n n n ν n ν e ) = e ) = n , and become relevant when − λ . Substituting this into the steepest descent result, we ! u ) λ = 2 term shown explicitly, first relevant when ( ( (by taking ν ν ∼ n and λ n ! ( S k ∼ λ ,... n B λ ), which permits any use, distribution and reproduction in = n ) 3 ) ∗ ∗ B , λ u ( u and large ) → ∞ ( n ∗ 00 fixed ) = 0 solve = 2 u n B λ ( u n k ( πS 0 λS 2 e S n for CC-BY 4.0 fixed p ∗ 1 1 we have λ u k This article is distributed under the terms of the Creative Commons − k n and λ  λ ν 1) and large . , when it contributes an order one rescaling of  ν λ ). This expression extracts the moments from the generating function ( ν λ √ ( n It is interesting in particular to see how the large B Higher order terms in theat exponential small are given by higher orders in theOpen expansion of Access. Attribution License ( any medium, provided the original author(s) and source are credited. first correction occurs fromorder the where we applied Stirling’sare approximation of to the the form factorial. The terms in the exponential taking large. Taking higher terms all integer powershave of Applying the steepest descent method at this saddle point gives us This result in fact interpolates between our two previous results for large The stationary points positive solution, which defines the Lambert where the contour encirclesfor the stationary origin. points We of can evaluate this by steepest descent, looking Here, we study abetween limit the of large series expression, but we use anof alternative method, starting from an integralcontour representation integral A.2 Large JHEP08(2020)044 ] 88 53 , , in , (2001) (2006) 56 , 74 Phys. Rev. (1994) 6606 ]. , Fortsch. Phys. Rept. Prog. Phys. (2004) 008 gr-qc/0308048 , , Rev. Mod. Phys. 49 , 02 (2017) 092002 (2003) 3587 SPIRE arXiv:0909.1038 IN 80 , Phys. Rev. D 18 (2005) 084013 ][ (2003) 021 , (1993) 1291 JHEP Z. Naturforsch. A , 72 04 , 71 Quantum phase transitions and ]. Phys. Rev. D , JHEP SPIRE , IN ][ ]. hep-th/9306069 ]. Rept. Prog. Phys. Phys. Rev. D [ , , Phys. Rev. Lett. 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