JHEP12(2015)099 Springer August 16, 2015 : December 16, 2015 November 18, 2015 November 30, 2015 : : : Received 10.1007/JHEP12(2015)099 Revised Accepted Published doi: Published for SISSA by . 3 1503.08207 The Authors. c Models of Quantum Gravity, Space-Time Symmetries

If quantum gravity respects the principles of quantum mechanics, suitably , giddings@physics.ucsb.edu E-mail: Department of Physics, UniversitySanta of Barbara, California, CA 93106, U.S.A. Open Access Article funded by SCOAP ArXiv ePrint: to identifying commuting algebras oftant localized properties operators. of In the addition algebraicproposals to structure, of suggesting this a impor- and fundamental related role observations for pose entanglement. challenges to Keywords: on the structure ofsponding the to algebra of regions. observables,gravity and This gives in paper an particular suggests important ondepartures that perspective its from study on subalgebras the of the subalgebra corre- analogous structure naturein of algebraic the of local correspondence structure the limit quantum quantum in of field long-distances/low-energies. theory. theory are Particularly, there Significant found, are working obstacles relevant quantum structures rather than byis quantizing supported classical by spacetime. difficulties of Thisrole such viewpoint quantization, for and by locality. the apparentby In lack tensor of finite factorizations a or of fundamental theproperties discrete Hilbert quantum of space. systems, the However, important generic evenindicate in that structure type local factorization is III of quantum provided the von field Hilbert theory Neumann space algebras is and problematic. of Instead it long is range better gauge to focus fields Abstract: generalized, it may be that a more viable approach to the theory is through identifying the Steven B. Giddings Hilbert space structure inalgebraic quantum perspective gravity: an JHEP12(2015)099 14 3 15 16 14 13 6 14 8 – 1 – 12 1 16 3.3.1 Unitarity3.3.2 of black hole evolution Entanglement, and comments on spacetime from entanglement 3.2.1 Asymptotically Minkowski states A possible primacy of quantum mechanics over spacetime suggests that spacetime Lorentz invariance is also well-tested and hard to modify, suggestingOn a the possible other funda- hand, locality is difficult to even carefully formulate in a quantum theory With these general principles in conflict, a central question becomes what to trust. 3.4 Other perspectives on the algebra of quantum gravity 3.1 Algebraic structure3.2 and (non)locality Further structural aspects 3.3 Possible implications of algebraic structure in turn indicatesmodification, that since the diffeomorphisms local assume local invariance manifold (general structure. covariance) isshould also emerge, likely in subjectmechanical. an If to this approximation, is the from case, this structures also suggests that that are one should intrinsically take a quantum- different approach of physics, at leastfundamental role when for sufficiently space generalized and time. to accommodate themental possible role for lack it of when a describing states corresponding toof flat gravity. or This nearly-flat spacetime. is associated with the lack of a precise role for classical spacetime, and basic principles. Quantum mechanics has a theoreticallygies, rigid and structure, experience forbidding has many shownThis, possible it patholo- together is hard with to its modify many without introducing experimental these tests, pathologies. suggests that it may be a cornerstone Physics currently faces a foundationalgravitational crisis, phenomena: encountered the when principles we ofand attempt quantum mechanics, to the the understand principle principles of ofby relativity, locality study are of in black basic hole conflict. evolution, which This has is no particularly consistent sharply description revealed respecting these 5 Conclusions 1 Introduction 4 Algebraic aspects and puzzles of AdS/CFT 3 Algebraic approach to quantum gravity Contents 1 Introduction 2 Synopsis: algebraic structure of local quantum field theory JHEP12(2015)099 ]. In particular, 1 ], parameterizing breakdown 4 – 2 – 2 – Section three then turns to algebraic aspects of quantum gravity, introducing essential This paper will focus on an initial investigation of the idea of approaching quantum Such a “quantum mechanics first” approach, together with the need for a theory of What kind of additional structure can a quantum-mechanical theory have, beyond the hole evolution, and commentsment, both and on in the challenges particularfour to to discusses a the how fundamental idea algebraic rolecorrespondence, structure that for if could spacetime entangle- this be emerges correspondence a can fromquantum ultimately pertinent gravity entanglement. be in focal shown anti Section point to de reproduce for Sitter the the space. details AdS/CFT of Section five closes with conclusions. spacelike separations. This significantly differsone from no the longer algebraic finds structure commutingvides of subalgebras LQFT additional associated since with justification spacetime forof regions. the LQFT. It Section locality also three pro- bounds thenevolution, of and discusses extra other [ structure, aspects suchstates. of as the structure, S-matrix, namely It relevant symmetries to also and asymptotically-Minkowski discusses a possible connection to the question of unitarization of black Properties of this algebranamely that can the be fundamental theory inferredgeneral matches based relativity onto in on an effective the an theoryreveals long-distance/low-energy assumption description novel limit. given of algebraic by Even “correspondence,” structure, thisSpecifically, departing amount gravitationally-dressed, from diffeomorphism-invariant of operators the information fail to local commute algebraic at structures of LQFT. emergent geometry.” The next sectionture will give and a algebraic brief structure overviewfactorizations. of of the LQFT, Hilbert in space struc- particular discussing the problems withpostulates and tensor beginning an investigation of the structure of the relevant operator algebra. gravity. This, combined with importantgravity, aspects and of in the particular long-distancethe properties behavior S-matrix, of of likely quantum provides black important holes constraints and on cosmology, and and clues toward constraintsgravity the from from basic theory. such an algebraic perspective, and on a possible role for such a “quantum in LQFT, subalgebras ofthe the resulting observables subalgebra may structure be mirrors associated the topology to of regions the of spacetimequantum spacetime, gravity manifold. to and be approximated bygests LQFT, that in we should the likewise low-energy/long-distance focus limit, on sug- algebraic structure in seeking a fundamental theory of are “locally finite,” with finitecally density has of in degrees addition of a freedom,However, tensor in such factor the as structure, case on which a offactor gives lattice, local structures a one quantum quantum cannot typi- field notion be theory ofby (LQFT), precisely “locality.” the as defined. algebraic we will Instead, structure review, a of such more the tensor fundamental operators role acting is on played the Hilbert space [ and attempting to quantizephysics, it. and Plausibly spacetime, quantum locality, mechanicsto and plays more a the basic fundamental equivalence quantum role principle structures. in emerge as approximations existence of a Hilbert space? For finite quantum systems relevant to physics, or those that than the program of starting with a classical description of spacetime, via general relativity, JHEP12(2015)099 . The H ]. 9 – 5 , where one considers, correspondence limit – 3 – into corresponding tensor factors on which the Local Quantum Physics nicely describes both the 1 H , specifically that this spectrum is restricted to the closed µ P ]. 1 , and observables which are self-adjoint operators acting on H However, it turns out that such a tensor factor structure is problematic even in LQFT; For finite or locally finite quantum systems, one often goes a step farther, and describes This net of subalgebras, with inclusions of subalgebras corresponding to smaller sub- Locality itself is encoded in the algebra of observables, whose basic building blocks A well-known “folk theorem” holds that the assumptions of locality, Poincar´einvari- For a basic reference, see [ 1 been used, for examplehole explicitly and its in surroundings discussions might of be black thought hole of as evolution,as separate we where tensor will the factors discuss, black [ quantum gravity appears to present even greater challenges. of subalgebras. a factorization ofdifferent the subalgebras act. Hilbert Thus, these space tensorto factors subregions. also can In be models thought for of the as structure corresponding of quantum gravity, such constructions have also operators associated to subregions thatoperators, are anticommute). spacelike separated commute (or, for fermionic regions, therefore mirrors theproperty basic of topology the LQFT of on the it. underlying The manifold underlying and manifold the can locality be reconstructed from this net of the momentum operators forward light cone, and commonly, that there exists aare unique the ground state. field operators.into subregions, and Specifically, to the each classicalthought such subregion, of spacetime one as geometry can fields can associate an smeared be operator with divided subalgebra, functions e.g. up with support only in that subregion. Then, ifold, commonly Minkowski space.of It a moreover Hilbert assumes the space Poincar´egroup describes standard the quantum symmetries structures of Minkowskithese space, symmetries and (and any as internal discussed symmetries)of by are Wigner, implemented the by relevant a groups. “ray representation” Additional assumptions are commonly made about the spectrum quantum-mechanical properties of LQFT, and theirof interrelation with spacetime. the locality structure ance, and quantum mechanicsQuantum imply Physics the approach begins basic with structure a of notion LQFT. of an Specifically, underlying the classical Local spacetime man- general theory. Second, oneshould expects match that onto a the moree.g., LQFT low fundamental description energies quantum in and theory a weakpropriate of gravitational quantum gravity fields. description of As LQFTalso we is known will as provided describe Local by in Quantum Algebraic this Physics. Quantum section, Field an Theory, ap- To gain understanding of how oneemerges might in formulate an a a approximation, quantum iting theory spacetime is in structure first which is useful spacetime instead tointrinsically assumed review quantum-mechanical as how fashion. a LQFT, foundational intion The ingredient, which is of reason an described LQFT is underly- in serves two-fold. an as First, a such possible a model descrip- for the kind of structure we require in a more 2 Synopsis: algebraic structure of local quantum field theory JHEP12(2015)099 : i in ˆ a ω | , or i (2.3) (2.4) (2.5) (2.1) (2.2) 1 not | and , in other is a factor i H are a A | i . It has been ψ R i| ˆ ψ corresponding to | ⊗H H L Type III subalgebras H 4 2 = isometrically onto A type III factor H ] for modes of frequency ]. H , 11 E 14 , , o i 13 factor. ω ! b i ω i 1 can be written as a von Neumann algebra b ˆ , 2 0 i| 2 − H i| ˆ 1 that maps | ˆ 0 βω/ † ω | 2 dx ) b − is called a + √ e † ω ω + ∈ A i b ˆ ( A ]. 0 2  S W i| 10 ) dt ) = ˆ 0 | ω ω – 4 – ω − Y ( ( , then

z 0 z = = n are complementary Rindler wedges, with which we ): unentangled states of the form 2 =1 ∞ i Y | M t i ds tanh 2.5 A ∨ A = 0 | −| ]. For related comments see [ M ), ( 10 ) = , there exists a i ) by assuming that we are treating fermions, and taking the infinite ) = exp 0 H ω | ( 2.2 x < ( 2.2 ∈ A B S E . For more details see [ and E | where one thinks of each Rindler mode as either occupied, t | = 0. 3 ∗ labels the different modes. The problem is the infinite entanglement → ]”, i x > β 6 WW , and a toy model for the Minkowski vacuum is , i 5 0 | . Indeed, an even simpler model for such a state is given in the “qubit ap- = 1, π ) is a unitary operator generating squeezed states [ ], and other references in [ ω W ( ∗ 12 yield tensor product factorizations of the Hilbert space. The basic problem can be = 2 S W β The type-III property refers to classification of von Neumann algebras, where the In short, there are two problems with describing factorizations of For completeness: if the space of bounded operators on This may be obtained from ( See [ not 3 4 2 together with its commutant, A such that for everywords projection temperature limit, manifest in the expressionsthe ( Hilbert space; they have infinitefield excitation, theory and, Hamiltonian, for example, they if haveentanglement. one infinite This introduces energy a means due typical there to isshown no the that well-defined infinite factorization this amount type-III of behavior “broken” is the generic behavior of LQFT. where the index with and proximation [ unoccupied, where could associate subalgebras of thecorrespond field to algebras of operators a acting LQFTspace on on does this the not space. left correspondingly Thethat or subalgebras factorize. describe right Rindler Rindler Formally, excitations one wedge.respectively, of can and the However, introduce formally Rindler the one states vacua Hilbert can for write the the right Minkowski or vacuum in left the Rindler form wedge, do illustrated by discussing two-dimensional Minkowskihas space in metric a Rindler description. If this the two regions subregions in LQFT: thelong-range first gauge fields. is One the might type-III hopesuch to as property, avoid these and the problems lattice, the by introducing but second a we regulator, is will the return existence toalgebras of problematic associated aspects with of LQFT that always momentarily. lead to type III subfactors. JHEP12(2015)099 ]. 19 , . 18 2 ⊗ H 1 H ]. = 16 H A problem with such a lattice 5 ], which e.g. can effectively “screen” the as a tensor product 15 H – 5 – ] and is a significant part of the apparent misconceptions that lead 6 , 17 The basic structure in which a physical system has different subsystems is particularly In short, in LQFT there is apparently no natural way to decompose the Hilbert space Breaking the symmetry seems particularly problematic in the context of describing the Of course, a regulator that makes the number of degrees of freedom finite seemingly The second problem is associated with long-range gauge fields. For example, in a gauge For a discussion of factorization and entanglement in lattice gauge theory, see [ 5 important in physics, anddom appears here to appear be independentindependence manifest of of in subsystems those the is on centralfor physical Proxima to world; example discussions Centauri. degrees of subsystems measurement, of Moreover, corresponding where free- such the to statistical world the has physical system being measured, and the the recent literature. Asfined we relative will to describe a furtherthe subsystem below, Hilbert structure entanglement space, given is and bya most the challenge such absence clearly to of a de- a such particular fundamental a tensor role natural factorization for factorization such of structure entanglement. appears to be to the proposal that physical black holes have such singular, or “firewall,”into behavior tensor [ factors, without drasticallytry modifying of the the short-distance theory, structure and andThis this symme- is problem particularly becomes pertinent even more to acute discussions in of theories entanglement, with which gauge have fields. been popular in evolution near a blackstates hole with horizon, where very the largemodes dynamics that relative naturally become boosts, forces Hawking like one“disentangled” radiation. at to those a Indeed, cutoff consider between scale the such infalling as assumptionto the singular that matter Planck horizons scale such [ can and states play a outgoing that role in are the dynamics leads number of the states ofhighly-boosted the theory particles, that whose otherwiseof would presence be such is legitimate a states implied procedure (forentropy, by as example, shows the Lorentz up regulator invariance). in is divergent removed. The contributions to result quantities like entanglement has the potential tothe overcome decomposition the of type-III a problem, HilbertA space and specific for also a example simplifies gauge is the theoryor regulating by description other by introduction of of introducing regulator surface a isinvariance. charges. lattice. that This it generically indicates badly that, breaks e.g. the in symmetry, the specifically Rindler Lorentz case, one has truncated an infinite freedom on the boundary between the subregionscharges. [ However, that changes the basicconditions content on of these the theory. surface One charges,reintroduces may roughly impose a that additional non-trivial those from entanglement thehas structure two no between sides way the cancel, to two but write subregions. that the Thus, original one Hilbert space theory such aswithout quantum creating electrodynamics, its itof long-range is space field. impossible into to twosubregions So, will subregions, create have if an generic a effect we on excitationsthe charged for the subregions. other of particle example subregion: One charged approach field consider to lines particles this cross dividing problem in the is a boundary to one between introduce region additional, of charged, degrees the of two JHEP12(2015)099 , in order to H of a i ], which do not rely ψ | 21 ]. Specifically, we will 22 ]. 20 where operators corresponding to 6 7 exhibits this statistical independence and is a H – 6 – has an inner product, which is (anti)linear H . structure we have noted, . H i 8 ψ | 0 ψ h The space Physical configurations will be described by rays algebraic (Q1) Hilbert space (Q2) under addition and multiplication;denoted the product of two states is to exist, due to the type-III property and the existence of gauge fields, unless we A key question is what additional structure needs to be provided on LQFT, via the Local Quantum Physics description, thus provides an example of a quan- Often one wants toA restrict to natural the temptation algebras of is bounded to operators. referUsually to expected to the be approach separable, e.g. advocated to in permit this thermal ensembles. paper as “Algebraic Quantum H 6 7 8 introduce additional structure that does violence to the physics. Fortunately, theGravity,” net but of this name is already taken by the different approach of [ independence of different subsystems of ainteraction, physical system, measurement, which is etc. important forphysics, describing In a tensor finite product or decompositionusual of locally prerequisite for finite describing quantum interaction and systemsHowever, measurement, as as of well discussed interest as entanglement, above, to etc. of in LQFT, we do not expect such a product decomposition describe physics. A critical ingredient for physics, generalizing locality, is that of statistical on assuming an underlyingassume space, the time, following or postulates notion to of begin “histories” to [ lay a quantum foundation for physics: In confronting the puzzles offor quantum gravity, physics we is assume quantum that the mechanics,duction fundamental suitably of framework generalized. space and In timequantum particular, concepts postulates, since is such a not as priori necessarily those warranted, intro- of one universal needs quantum a mechanics general [ set of when we turn to gravityan we underlying do classical not geometry, however so expect this the structure quantum needs structure to to be be generalized. based3 on Algebraic approach to quantum gravity tum theory with subsystem structure whichphysics matches and well also our our observations need of fortions the independence in locality of simple of physical terms, systems and inalgebraic for order a structure to basic describe can description interac- of bethus measurement. furnishes used Moreover, us to as with noted, reconstruct an this the example of underlying the spacetime type manifold. of quantum This structure needed to do physics; ized in terms of the different subsystems commute, andHilbert apparently space. doesn’t This require is tensordescribe a factorization the rationale of for centrally the focusing importantof on subsystem the this structure, spacetime. algebraic and structure: the the net subalgebras captures the topology observer. But, this locality and independence of subsystems can fortunately be character- JHEP12(2015)099 ]) — but then one is no longer 23 of such a more basic theory with general – 7 – , of localized Wilson lines, or even of operators that create correspondence µν F the underlying analog of spacetime, we expect a very different fundamental structure One focus of this paper is such preliminary exploration of the possible role of this Our general approach to a quantum description indicates we should focus on the algebra But a particle is inseparable from its gravitational field, and screening of gravity is Gravity introduces further puzzles. First, we don’t expect there to be a notion of In treating gravity, let us assume that the basic long-distance/low-energy properties define to ultimately emerge. algebraic structure, and tothis algebraic investigate structure the that novel can be features, inferred in from comparison the long-distance/low-energy with behavior LQFT, of of quantum mechanics andand Hilbert the equivalence space principle emerge now fromlimit. in this Second, more a if basic we structure, do fundamentalof in assume the role, that the theory general correspondence approximately locality, relativity “right,” gets spacetime, then theof long-distance we local do gauge structure commuting not expect operators toof — have the even the theory same in is kind the different of correspondence from notion limit, that of the LQFT. algebraic If, structure in turn, the algebra of operators is used to of observables, and lookcharacterizes the for physics. interesting refinement Atapproximate long of those distances/low the of energies structurealgebraic LQFT, we structure of expect but is this these expected gravity algebra to observablesit has be to that to new different incorporate from properties an that of that underlying LQFT. spacetime tell First, structure; we us do instead that not we expect the might full expect that, with create a zero-energy operator byover integrating all a spacetime, field and operator,necessarily or might true, product hope if of gauge this field invariance avoids is operators,considering also gravitational a required coupling localized — (though see operator. [ this is not a particle-antiparticle pair togetherwith with no a net charge, localized are field not configuration: required to these modifyproblematic. operators, the Any asymptotic localized field. excitationation is of expected such to an have excitation some therefore non-zero energy, modifies and the cre- long-range gravitational field. We can an underlying classical spacetimerelated on problem which regards to locality.observables “anchor” In which the have a an net gauge effect of theory onlygauge on invariant such operator field a as strength algebras. localized QED, subsystem we by A considering can action formulate of local the we can take asrelativity a in the key long-distance/low-energy guide limit. the as In a that limit, field gravity theory is withthat, of a course in gauge described introducing symmetry, that aencounter of structure similar diffeomorphism challenges such invariance. as as This in suggests with that LQFT, long-range either of due gauge statistically-independent to fields. the subsystems, type-III we behavior, or associated to describe statistically independent, or local, subsystems. of the theory arejob indeed is approximately to well-described find byalso a general long modification relativity, distances) but so of that that the we our structure have at a fully short quantum-mechanical distances theory. and This high means energies that (thus, operator algebras of the theory comes to the rescue, and describes the necessary structure JHEP12(2015)099 A change In fact, even quan- 9 , creation of a quantum q ]. to another, or if there is unitary 24 H with charge φ . of “observables;” these are self- H – 8 – A apparently contains no operators that are local in the LQFT A There is an algebra (A) adjoint operators that act on This can be made rather precise in the weak-field limit of gravity. As we have described, A lot is clearly unknown about the algebraic structure of quantum gravity, but if we do As we have reviewed, in LQFT/Local Quantum Physics, the analogous algebra is fur- Some comments are in order. First, we assume that the operators in this algebra For a more detailed version of the following discussion, see [ 9 tum electrodynamics (QED) exhibits some similarLet behavior us — begin though with with keyof QED. differences. this If field we will have be a accompanied scalar by field creation of its long-range electromagnetic field. There sense. Consider excitations built onspacetime, a quantum such state corresponding as to Minkowskihave a large, some space. semiclassical energy, and As thusthe to noted, full alter any operator the localized long-distance creatingis gravitational that excitation not field. excitation, local. is For together expected that with reason, to its inevitable gravitational field, or a perturbatively quantized versionmore of basic it structure. in For thateven example, limit, in if that the we is “field discover an theory”fundamental that important behavior limit, certain guide of we operators to the expect the do theory. that not lack commute of commutativity to mirror the more locality in gravity isbelow. more But problematic, nonetheless, asture, we we and would have also like described how to the abovestructure subalgebra know in and structure to the of will what correspondence LQFT see emerges extent limit. further in there an is approximation to analogousassume that struc- that it has a long-distance/low-energy limit that matches onto general relativity, ther refined by a subalgebradence) structure, is and encoded locality in (or theregions. more commutativity of generally Therefore, subalgebras statistical associated it indepen- relevant with to is spacelike a separated clearly theory of of quantum great gravity. Of interest course, to a simple identify subalgebra properties structure encoding of the algebra to characterize states, forwhich example may correspond by to findingstates, definite eigenvectors attributes for and of example eigenvalues a ofevolution by state. operators, generated mapping They by from can a“interaction” one also terms. hamiltonian, state be in used of which to such operators can serve as elementary are gauge invariant, that issupplants they it. are Second, left such unchanged operators by can local play different gauge roles symmetry in or a whatever theory. They can be used these long-distance/low-energy properties is a viableabout, one, and these should moreover, serve constraints as on, critical clues such a fundamental3.1 theory. Algebraic structureWe and begin (non)locality with another postulate for the quantum theory of gravity: gravity. If the approach of trying to find a fundamentally quantum theory that matches JHEP12(2015)099 . y (3.4) (3.5) (3.1) (3.2) (3.3) and ; (3.7) x  ) 0 ] (for a related , z ˆ ν 24 x ( zz h 0 all along the line, and . ]. These are most eas- dz z . µ , ξ E ) ∞ ν Λ 24 z y , ∂ µ ( Z . ∂ ∗ ˆ µ 29 ) by defining [ − φ – ∂  + ν . z quickly decays — the field relaxes 2 1 ξ  3.1 µ 27 z µ ν z A ∂ + )) (3.6) dzA dx x ˆ µ − ( z µ dzA → µ ∞ x h y µ z µν z x dx V  Z z ) h A ]; a more general dressing is relevant for non-static Z + µν iq 25 dz → µ iq h  ∞ x x  ( z + µν – 9 – ); Z φ x µν ( ) exp − η φ ) exp x , h ) = ) ( x x x ( φ ] by averaging over directions of the Wilson line, to = ( ) = φ Λ( µ x 2 Φ( 24 ξ ( iq ˆ µ e ) = ds + ) = x ], which for static charges would be the combined Coulomb µ ) we can in some cases arrange their electric strings not to However, this makes the situation worse: while for a collection → x Φ( ) 26 ,V x, y 3.1 10 x ( ( → D ]) φ µ zz x 28 , dzh 27 ∞ x z Z 2 1 ) commutes with observables spacelike-separated from this neighborhood. Of − x, y ( ) = D x ( Such constructions can be generalized to gravity [ Of course, in QED, or more general non-abelian gauge theories, we can, as noted above, z More precisely, this produces the dressing of Dirac [ V 10 -direction: where charged particles. One can construct an analogconstruction, of the see Wilson-line [ dressing ( ily explored inMinkowski the space, linearized approximation. Consider, specifically, linearization about The linearized gauge symmetry is course note that once created, theto “electric string” a along different electricfields field (dipole [ field), andeven extends local) to gauge infinity. invariant operators Nonetheless, in this QED is existence one of of localized the (and key differences with gravity. which only creates aThus, non-trivial field in a neighborhood of the string connecting produce the Coulomb field. of operators of theintersect, form so ( that theare operators always contributions commute, to averaging commutators. over all directions meansstill that formulate observables there that are strictly localized to a region. An example is of the form This operator however failsalso to two commute such with operators thesymmetric do electric dressing not field is always commute obtained at [ spacelike separations. Another more constraint. One such configurationz is created by a simple Wilson line, say running in the This is gauge invariant under are many such possible field configurations that can be created, satisfying the Gauss’ law JHEP12(2015)099 (3.8) , into , but if R z direction. ]. They in z 24 )[ 3.1 and moreover, even 11 ). This absence of any at which the particle is ), but at different x 3.5 ˆ µ 3.3 , and thus with curvatures, x µz axis, so that they don’t meet, h z 3) − D ( / 1 ) at the same ) 3.6 GE – 10 – ) = ( directions along the E ), are not invariant under ( ( in the region is such that the radius is less than the . This dressing can be shown to be gauge invariant x R z ( E R ), with ]. E opposite ( 24 R ) are unstable; nonlinearities are also expected to contribute to ≤ ) have analogous properties to those for QED, ( 3.6 R 3.6 of gauge-invariant operators for gravity, as compared to the algebras for LQFT. ), and creates a “gravitational string” extending to infinity along the A denotes indices excluding 3.5 µ There is a natural conjecture about when such gravitational strings become too dense However, we do not expect this construction to hold in the non-linear theory, where For example, if we have two operators ( Since locality seems quite important, at least in low-energy physics, and since we have Since all localized excitations are expected to carry energy, and gravity can’t be In fact, one can average such Wilson line operators to find an analog of the Dirac The operators ( Though, one may create a particle at one point and annihilate one at another. 11 commutativity when the energy Schwarzschild radius, created by operators ( this. The gravitationalget field too lines close. will spread out, andto in commute the in non-linearwhich theory, the such cannot full gravitational lines theory. enter from Specifically, different angular if directions, we we expect consider failure a of region of radius the linearized approximation, and thus apparently build up commutingthe subalgebras. gravitational strings, eventhem if has “thickened,” a have large energy,Moreover, amount even and of in where the energy. linear a Ultimately theory, large as we we number expect have of noted, commutativity the to gravitational break field down. configurations we run their Wilson linesthen in these commute inact the with linearized multiple copies theory. ofconsider Moreover, these such in lines operators, terminating the maintaining in commutativity. linearized spacelike Indeed, theory separated we we regions, could can still with commutativity in local or even localizedalgebra operators appears to be a veryseen important that an difference approach to between describingbras, the locality let is us through further the explore existence the of extent commuting to subalge- which we might find such commuting subalgebras. Sexl solution. Properties of these operatorswill and be their further extension to described those in for moving [ particles screened, there is noscalars, clear such analog as of the the Ricci uncharged scalar operator ( the resulting Coulomb-like operator willsuch generally operator. have nonzero commutators with another dressing of electromagnetism; this symmetricalSchwarzschild solution, dressing creates at a spacelike linearizedcreated, separations version and from of a the the boosted point version is expected to create a linearized version of the Aichelberg- under ( general do not commuteand with also the canonical they conjugates dopected to not to in create general a commute state with that each decays other. to a Like “more in natural” QED, field they configuration. are ex- However, here ˆ JHEP12(2015)099 ), E ( R ]. This 24 particle when , the nonzero r m (3.9) ), with gravitational strings ), we expect the commutant , 3.6 3.6 that, in the weak- I LQFT 12 O i 0 | I O . We expect the same behavior in gravity, I Y | /r 2 0 q – 11 – i ≈ h ]. There, for similar reasons, we don’t expect to find 0 | (again working in general dimension) for two massive I 23 3 contains operators O − D I A Y , and this is found from the gravitational dressing [ | 0 h M are operators of LQFT. I GmM/r and O m ],” which states that an N-particle state fails to be well-described by The algebra 4 algebraic structure, by introducing a correspondence postulate: – 2 (C) field limit of low-energies/longduce distances, correlation functions approximately computed repro- in LQFT, e.g. where the = 1. We also expect two highly-boosted particles, that source Aichelburg-Sexl 3 − now the center of mass energy, we expect ultimate failure of commutativity. D spacetime dimensions. This would offer further explanation of the “multi-particle E approximate An important question is to better characterize the full algebraic structure, extending Similar behavior is expected to appear for other gauge-invariant operators, such as the So already, at the linear level, or in its extension into the nonlinear theory, we appear So, it appears that even a pair of operators of the form ( Indeed, this is supported by a related argument. If we consider the electromagnetic It also appears that such commutativity is state-dependent, in the sense that two operators may appear D 12 of this structure, such asthe subalgebras, local exist. algebras Another of important LQFT goal emerge is from to it understand in how suchto a have correspondence larger limit. commutatorsof when two the operators commutators may are only evaluated be in small certain when states. evaluated on That certain is, states, commutators e.g. “near” to the vacuum. it consistently into the full quantum theory, and to understand what further refinements this to be discovering something verygravity important is about quite how the different algebraicfind from structure an that of analogous quantum of subalgebra LQFT.tations structure Specifically, in to we different that don’t regions of appear ofapproximately LQFT, in spacetime. to corresponding limits, necessarily Such where to gravity a localized is structure exci- behaving appears weakly. to We can only attempt obviously to arise characterize to shrink with increasing powers of the single operator. relational observables considered in [ precise subalgebras corresponding to some notion of “regions” of spacetime. operators. However, oncewith they reach the region where their separationrunning in is opposite of directions, size dowe act not multiple provide times true on the subalgebras vacuum in with the a single nonlinear operator theory. ( If commutator of the two operatorswith commutator is of of size size particles with masses is expected to be anGM/r important modification to thedressings, dynamics and of the collide mass from opposite directions, to initially be described by commuting locality bound [ LQFT when the greatestradius separation corresponding between to the the particles total center-of-mass is energy less of than thedressing the particles. of Schwarzschild two Coulomb-dressed charged particles separated by a radius in JHEP12(2015)099 . A and 1 (3.10) O . A where 2 O . 1 O H = ]. Here we don’t have the H 31 , . For example, if subalgebras 30 λ , 23 iλH − e and parameter = – 12 – U ∈ A H As described by Wigner, a symmetry is implemented by (Sy) a representation up to phase of the symmetry group on On the other hand, we can also consider states which are expected to have a no- The preceding comments seem particularly pertinent in cases of closed cosmologies, An example of the possibleWhat role other of structure such symmetry is will necessary be for discussed physics? momentarily. Of course, typically, as we have em- Often in physics symmetry plays an important role. The symmetries that are present lie in commuting subalgebras, corresponding to different degrees of freedom. Such 2 tion of spatial infinity, suchMinkowski as boundary those conditions. which Here correspondhave we to extensions make asymptotically to some anti the comments dealso former. on Sitter be the or (Further made latter, comments below.) on which also the AdS/CFT correspondence will important way to make progress in“platform” describing of evolution [ an asymptoticstatements spatial must infinity be with made respect withof to reference which scalar to we fluctuations other can looks features localize,inflaton of and field.” like the so X state, at e.g. “the the spectrum reheating time determined by the value of the context. Note inexpected particular to that, match of thatonly course, governed be since by apparent the the evolution,So, weak-gravity Wheeler-de as once limit again, Witt described of the equation, through key evolution any relational questions is seem evolution observables to contained may return in such to as the de structure Sitter of space, the where algebra identification of the correct relational observables seems an can be identified, aO basic interaction termevolution may can take be the viewed as form a producing prototype correlation for between interaction, theseclearer and degrees understanding measurement. of of freedom, But, how and this any is such is evolution far is too precisely general described and in we the need gravitational a some contexts this evolution may be described by considering unitary transformations for certain self-adjoint operators phasized, one begins with asubalgebras. subsystem Then structure, one or in can LQFT, describe a evolution, corresponding e.g. network through of interaction of subsystems. In will depend on the vacuumanti on de which Sitter we space, work,group e.g. de one is Sitter corresponding present, space, we to or postulate Minkowski space, one with less symmetry. When such a symmetry While the algebraic structure, orgravity approximate is, algebraic structure, as ofrequires we a further have theory study, of argued, we quantum turnsuch very to a important a theory. brief in discussion better of other understanding structure its useful in nature, describing and 3.2 Further structural aspects JHEP12(2015)099 . | ~p ≥ | 0 p excitation of the ground state any in the physical Hilbert µ P except in the sense that for any finite boost, 13 – 13 – ], and differing from standard LQFT. Systematic 34 (defined in the asymptotic metric). Note, however, – r 32 at a distance is confined to the closed forward light cone, . i 3 0 The spectrum of the momenta | H − D = 4, one must still exercise care in treating massless photons and gravitons, and a true D (Sp) space Typically, we alsostate, assume the existence of a unique ground 14 GE/r These asymptotic states are expected to be characterized by their momenta and other In particular, matching onto the known long-distance structure of quantum physics, Nonetheless, we might expect that the states of the theory do fall into representations There is a further subtlety here. As described above, One may define asymptotically Minkowski states asNote that those for states which are well-described by LQFT 13 14 important clues to the more fundamental behavior of quantum gravity. near infinity. S-matrix does not exist.electromagnetic dressings To described avoid above, this, or formulate one appropriate may inclusive work scattering at probabilities. higher D, properly treat the gravitational and quantum numbers, and the S-matrixappropriate is expected infrared to behavior have analytic ofquantum structure field associated the with theory. theory, In approximatelythe gravity, it long-range matching behavior is onto of also that gravity expectedstudy [ of of to the local have analytic other behavior of new and features constraints on capturing the gravitational S-matrix may provide include states where weso have two their or interactions more can such beparticle single states neglected. is particles This the that traditional identification are assumptionof of needed widely the asymptotic for separated, theory. states formulating an in S-matrix terms description of we expect that thereof is the a Poincar´egroup. subspace Moreover, of we single-particle expect states that that the furnishes “multi-particle” a states representation of the theory of the Poincar´egroup. We might also make the following assumption about the spectrum: that some care is neededand in increases its orders energy of and limits. thusgroup asymptotic A on field. large asymptotically So, boost Minkowski there may is states, one be not has an applied action to to of work such the atboost Poincar´e a sufficiently state, produces large a radius strong sothe field the theory extending field also to remains serves infinity, weak. as and The an in limit ultraviolet this of cutoff. sense infinite an infrared cutoff on (We may also seek to describe these states ascorresponding evolving to with the respect Minkowski tofield vacuum time is at at expected infinity.) infinity. totypical Far modify size from the a effective gravitational localized finite energy state, this field will be small, e.g. of If quantum gravitygeometries has with asymptotically a Minkowski boundary sector conditions,structure we of expect in significant states extra this that sectorof that can the may theory. be be Specifically, thought usefulunder according in of the to further postulate Poincar´egroup. as elucidating Sy, If corresponding we fundamental so, expect to properties states these states may to be transform assigned definite momenta and energies. 3.2.1 Asymptotically Minkowski states JHEP12(2015)099 – – 2 6 Ψ, and , 2, within √ ] considered / = Φ ) i χ 18 , 10 36 i  | , with i 01 | λ to sharply define entan- i| = ( χ | i  = Ψ i | λ χ | necessary 2 and √ / ) i 11 – 14 – i  | 00 | ]. 42 = ( – i  39 ]. What is apparently needed is evolution that, with re- , Φ | 8 35 , ], consider a Hilbert space spanned by the states of two qubits, 18 , 43 7 , 6 , 38 – 36 . With respect to this subsystem decomposition, the Bell states are unentan- ] instead propose a much “softer” form of effective transfer, where information is − , 42 – Entanglement entropy is usually defined in finite or locally finite quantum systems Interestingly, we have found that in gravity locality is difficult to formulate to begin = + 39 , ]. This ultimately appears to conflict with a statement such as the equivalence principle, λ Indeed, specification of suchglement. a For subsystem example structure [ and is consider the Bell states, it. If asked whetherchoose these another states are basis, entangled, where many these would can first be react rewritten “yes.” But, as we can Study of entanglement has beenitself au courant, emerges and from it entanglement. hasgiven even the been It discussion proposed is of that this worthwhile spacetime paper. to reassess thewhich role possess of a subsystem entanglement, structure defined by a tensor factorization of the Hilbert space. ticular, such effects may beof “small” parameterized deviations in from an LQFTsuch effective evolution. deviations field It to theory would the be framework fundamental satisfying in obstacles to terms to more closely LQFT connect 3.3.2 locality described in this Entanglement, paper. and comments on spacetime from entanglement corrections sufficient to unitarize blacka hole “harder” formation form and decay. ofmation Refs. such [ dramatically transfer, modify where the the8 physics new in effects the responsible vicinity fortransferred of transferring from the infor- a would-be black horizon. holehole interior Refs. size, to [ in its a atmosphere, fashion on that scales an determined infalling by observer might the characterize black as “nonviolent.” In par- information [ with; this certainly provides encouragement thatof lack of the LQFT story locality of is an blackterize important hole evolution, part evolution. and see An how important it question approximately is matches to onto more that completely of charac- LQFT, together with hole evolution canmatrix be describing unitary; formation andapparently unitarity, disintegration utterly here, of fails can a tosive black capture e.g. violation hole. the be of correctspect For unitarity defined dynamics, black [ to and as holes, appears the that LQFT to semiclassical of predict description a the of mas- S- the black hole geometry, nonlocally transfers ings, and, if smallthis enough, section behaves will discuss like some a other system possible in implications3.3.1 empty of flat the space. preceding discussion. Unitarity The of remainderAn black of hole underlying motivation evolution of part of the preceding discussion is to understand how black The preceding discussion has reviewed howtems, even since in precise LQFT one tensor isdepartures factors limited from can’t in traditional defining be local subsys- defined;4 structures we are then even have more foundwhich profound, that assumes as that with anticipated a gravity, in physical [ subsystem may be isolated from its gravitational surround- 3.3 Possible implications of algebraic structure JHEP12(2015)099 ] is even more puzzling. It 45 , 44 – 15 – elements of different subalgebras. Such interactions 2 O and 1 O , with 2 This is directly related to the type-III property that we have described, O 1 15 O = i H Indeed, the notion of “building up” spacetime from entanglement is puzzling from The proposal that entanglement builds up spacetime [ The situation seems even more problematic in quantum gravity, where we appear to Already in LQFT one encounters difficulties in quantifying entanglement by entangle- Though, other entanglement measures such as relative entropy of entanglement may be well defined. 15 proximal nor connected. 3.4 Other perspectives onIt is the also algebra worth of notingto that quantum quantum there gravity are gravity. other For perspectives example, on the the algebraic proposed structure covariant relevant entropy bound of Fischler and structure! Again, spacetimecorrespondingly structure encoded appears in the more hamiltonian primitive thatentanglement. can than govern Put entanglement, the nature differently: and and is evolution oneby of a may the spacetime propose bridge. that But,that an there encodes ordinary is interactions EPR nothing between pair in the the is partners hamiltonian connected of of the a pair. widely-separated pair In this sense, they are neither in governing thesubalgebras, structure then of interactions theterms between hamiltonian. “nearby” subsystems If arecan a alter encoded subsystem entanglement in between structure interaction arise different is from subsystems, the defined but entanglement, by it if is that not was clear what how was they responsible would for defining the spacetime short, it would appearor that subsystem subalgebras, structure, comes whether first,it defined and is by to with tensor the respect factorizations extent to to this which structure. entanglement isanother precisely defined, perspective. Another place that the spacetime structure appears in LQFT is vacuum and finite energy excitations of itthis are is infinitely far entangled, as from we saying havehave described. that just entanglement said But somehow that suffices defining entanglement tosubsystem requires define structure a spacetime. (to subsystem the structure, Indeed, extent and we to inwith which LQFT it the this is topology defined, with (structure a cutoff) of is subregions) closely associated of the preexisting underlying manifold. In relying only onsomehow, this in structure, gravity, and itfruitfully does), doesn’t explored. then require entanglement the entropy Hilbert might be space precisely to definedis factorize and true that (or in if LQFT, entanglement is implied by the underlying spacetime structure — the is removed. be even further removedwe from we having haven’t a evenmore sharp yet needs definition to identified of be aa a learned natural precise subsystem about subalgebra subalgebra structure, structure, the structure. since and algebras if relevant Thus, a to it useful quantum definition gravity. seems of that If entanglement they can have be defined factorization. ment entropy. and the corresponding lacktheory of with a a cutoff, factorization and structure. one finds an Instead, entanglement one entropy that must diverges truncate when the the cutoff gled. Entanglement is defined with respect to the subsystem structure given by a tensor JHEP12(2015)099 ]. 55 recover subalgebras ap- ]. Let us, however, momen- ]. As we have noted, when ] and references therein). An 54 of quantum gravity with AdS 50 48 , – 49 51 approximately definition – 16 – ] states that the number of degrees of freedom in a causal dia- 47 ] and Bousso [ 46 would it do so? It has also been proposed that two copies of a conformal field theory, entangled in Since the boundary theory is a LQFT, it has the kind of subalgebra structure, asso- 5 Conclusions This paper has exploredchanics, a rather starting than point beginning for with spacetime. quantum gravity This grounded approach in is suggested quantum by me- indications bulk physics, if it did indeed describe the detaileda structure thermofield of double bulk physics. state,This leads proposal to seems anentanglement, also AdS and subject thus black appears hole to equally with puzzling. the Einstein-Rosen preceding bridge comments [ regarding geometry from propriate to describing bulk locality.demonstration An of important how challenge, AdS/CFT which works, couldstructure would be in be the a algebra part to of of identifyticularly operators the this of for approximate the approximate subalgebra boundary subalgebras theory.AdS corresponding Thus, radius this to scale, identification, could regions par- be small a as key part compared of to establishing the the utility of AdS/CFT for describing of algebraic structure expected forconjectures gravity about in possible anti subregion-subregion deconsidering Sitter duality gravitationally-dressed space, [ operators though in therenatural have bulk subalgebra been structure, physics, though we weabove, haven’t expect in yet that a in identified bulk the gravity a correspondence theory we limit should described be able to tarily ignore the difficulties foundbulk in theory, trying and to ask reconstructhow the the fine-grained question, structure if of AdS/CFT the does define a theoryciated of with quantum boundary gravity, subregions,clear that whether we the have LQFT described subalgebra structure for of LQFTs. the boundary However, CFT it is is related not to the kind 4 Algebraic aspects and puzzlesIt of has AdS/CFT been proposedboundary that conditions, AdS/CFT though for provides a a critical viewpoint see [ match onto LQFT, includinglimit. perturbative In gravity, particular, in beyondflux the the lines type-III appropriate from question, excitations “correspondence” inboundary inside the of such weak-gravity the a limit, spacetime. diamond gravitational willis Discussion extend left of outside other for aspects that future and work. diamond differences to of the such approaches mond is bounded by the Bekenstein-Hawkingthat entropy of region, the which maximal in area turn surface wouldalgebra bounding mean in that describing one is any dealing such“Holographic with region. spacetime” a approach finite This to type-I quantum proposal gravity von has (seeimportant Neumann [ served question as for a this starting kind point of for approach the is to understand the details of how it would Susskind [ JHEP12(2015)099 ]. 4 – 2 – 17 – If quantum gravity can be formulated in such an intrinsically quantum-mechanical This discussion appears to have important implications for attempts to find a quantum These observations prompt exploration of a possible fundamental role for the algebra If the correct framework for quantum gravity is intrinsically quantum-mechanical, an gebra of observables,this and include possible correspondence refinementity. with of the Indeed, known that long-distance/low-energyspacetime structure. ultimately behavior emerges one of Important might from grav- “quantum guides emergent anticipate a geometry.” that in more the basic familiar quantum geometric algebraic structure of structure, defining such a of having infinitely extendedthe weak energy field, of and thequantum moreover systems. qubit. its strong-field This region is grows a with qualitative difference withframework, behavior it of more will familiar be very important to further characterize the structure of its al- theory of gravity. Therewith is defining not a tensor clearemerge factorizations, primary from and role entanglement. for Moreover, it if entanglement,thinks is one given of takes the difficult a particles difficulty quantum to information asis see perspective a roughly and how qubit?” corresponding spacetime it to itself appears qubits, that could and the asks answer is the that question the “how qubit big is arbitrarily large in the sense tations of local quantum field theoryOne that does have been find parameterized that by locality commutatorsappears bounds can possible [ be to recover small, the inthe subalgebras the correspondence of long-distance/low-energy limit. local limit, quantum so field it theory approximately in of gauge theory. Thisthe is quantum readily structure seen of ifrelativity gravity in one approximately the assumes long-distance/low-energy matches a limit.operators onto principle Even that of generally in of correspondence, fail this quantized where limit, tocreated gravitationally-dressed general in commute spacelike-separated even regions. when Thus, describing this further excitations confirms which and are na¨ıvely quantifies the limi- of observables in quantumrefinements gravity, and of this indicate algebraic the structure.tational importance field, Given of and that that understanding a gravity further apparently particleidentifying cannot is a be inseparable screened, subalgebra from one structure finds its an associated gravi- obstacle with to even regions, a subtlety going beyond that by a tensor factorfield structure for theory the both Hilbert duepresence space, to of such the long a range type-III structure gaugenet is property fields. of problematic (infinite subalgebras even Instead, entanglement), in one which togethertheory focuses correspond with only on to certain the the subregions restricted algebraic of classes structure, of the and operators spacetime. a define commuting Moreover, subalgebras. in gauge itational theory, and by difficultiesto of quantize the approaches that metric. begin with spacetime and then try important question is what mathematicalWhile structure for is finite needed or beyond locally that finite of quantum Hilbert systems space. important additional structure is supplied that the usual locality of local quantum field theory is not a fundamental property of grav- JHEP12(2015)099 D D 65 55 D 85 , Phys. 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