Holographic networks for (1+1)-dimensional de Sitter spacetime
HHolographic networks for (1 + 1) -dimensional de Sitter spacetime
Laura Niermann and Tobias J. Osborne
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, 30167 Hannover, Germany (Dated: February 19, 2021)Holographic tensor networks associated to tilings of (1 + 1)-dimensional de Sitter spacetime areintroduced. Basic features of these networks are discussed, compared, and contrasted with conjec-tured properties of quantum gravity in de Sitter spacetime. Notably, we highlight a correspondencebetween the quantum information capacity of the network and the cosmological constant.
I. INTRODUCTION
The duality between quantum gravity theories in anti-de Sitter spacetime (AdS) in the semiclassical limit andstrongly interacting conformal field theories (CFT) – theAdS/CFT correspondence – has provided many deep in-sights into the structure of quantum gravity. Since theoriginal incarnation [1, 2] of this duality there has been aCambrian explosion of studies exploring holographic cor-respondences in a variety of settings. These lines of en-quiry have ramified throughout high energy physics andquantum many body theory, branching out from the ini-tial works of Gubser, Klebanov, and Polyakov [3], andWitten [4], and providing an intricate yet coherent un-derstanding of quantum gravity in AdS.Observational evidence [5], however, strongly favoursthe hypothesis that we live in a universe which is asymp-totically de Sitter (dS). It is therefore of critical im-portance to transfer our holographic knowledge and ex-pertise concerning quantum gravities formulated in AdSto the dS setting. This process has been ongoing butprogress has been slow in comparison to the AdS case.Key works on quantum gravity in dS include early papersof Banks [6], Bousso [7–9], Witten [10], and Balasubra-manian et al [11]. The application of holographic ideaswas then initiated in the crucial paper of Strominger [12].Since then there have been a steady stream (see, e.g.,[13, 14] and references therein) of works investigating thisduality. Progress has been impeded by a multitude of ob-structions, not least of which is that the observables ofdS, living at timelike infinity, are radically different tothose of AdS which live at spatial infinity.While a complete quantum gravity theory in dS in(3 + 1) dimensions is still largely out of reach, there hasbeen considerable recent progress in understanding dSin low dimensions. In particular for the case of Jackiw-Teitelboim theory [15–17] in (1 + 1) dimensions therehas been recent progress in understanding the structureof the Hilbert space and unitarity of the theory [18–20].Although there are no bulk gravitons for such theoriesthey do share the property that all the observables liveat the temporal boundaries.From the perspective of the present work, however, themost remarkable feature of quantum gravity in dS is theproposed finite-dimensionality of its kinematical Hilbertspace. This surprising and counterintuitive conclusionwas argued persuasively by Bousso [7–9] (building on ear- lier work of Banks [6]) and points to a deep connection– the Λ − N correspondence – between the cosmologi-cal constant and the dimensionality of Hilbert space. Iftrue, this correspondence suggests that tools arising inquantum information theory, particularly in the theoryof quantum entanglement, may be useful in exploringquantum gravity in dS.There has already been very active interest in exploringconnections between quantum gravity and quantum in-formation theory. This direction originates, in part, fromworks of Ryu and Takayanagi [21, 22], who expressed thequantum entanglement entropy of a region in the bound-ary CFT in terms of the area of a specific bulk minimalsurface. These ideas have been considerably expandedand developed since then commencing with the proposalsof Van Raamsdonk [23] and Swingle [24] and culminat-ing in the ER=EPR proposal of Susskind and Maldacena[25]. The tools of quantum entanglement and Hamilto-nian complexity theory are now also being used to prof-itably explore the quantum dynamics of black holes [26–28].Another central tool in quantum information theory,quantum error correction (QEC), has also played an im-portant role in understanding bulk/boundary correspon-dences. This was initiated by Almheiri, Harlow, andDong [29] who argued that bulk local operators shouldmanifest themselves as logical operators on subspacesof the boundary CFT of AdS spacetime. This paperwas an important inspiration for the construction of aremarkable toy model of holographic duality known asthe holographic code [30], a discrete model of the kine-matical content of the AdS/CFT correspondence builtfrom special tensors coming from QEC known as per-fect tensors . Holographic codes have proved to be a veryhelpful rosetta stone for quantum information theoristsand high energy physicists and there has been consider-able progress investigating and generalising holographiccodes [30–35] and perfect tensors [36–41]. Exploitingpowerful results [42, 43] of Jones on unitary represen-tations of a discrete analogue of the conformal groupknown as Thompson’s group T [44, 45], the kinemati-cal holographic state has been upgraded to yield a fulldynamical toy model of the AdS/CFT correspondence.Our paper represents an initial exploratory step to-wards transferring recent techniques in the study of holo-graphic codes to the de Sitter setting. The primary moti-vation for the current investigation lies in the observation a r X i v : . [ h e p - t h ] F e b that tensor networks such as tree-tensor networks andMERA have a causal structure which naturally matchesthat of spacetimes with Λ > partial isometry , from the past boundary to the fu-ture boundary on a finite-dimensional physical subspaceof an infinite-dimensional boundary Hilbert space. Wethen identify a fundamental correspondence between thequantum information capacity of the network and thecosmological constant. Finally, discrete analogues of dif-feomorphisms are introduced via Thompson’s group T ,and their action on the physical subspace characterised.The tensor networks we propose here have a treelikestructure reminiscent of the proposals involving p -adicnumbers [52, 53] and resemble the eternal symmetreeproposal [54]. Our proposal differs from constructionsinvolving p -adic numbers, however, because the action ofrelevant symmetry groups (i.e., in our case Thompson’sgroup T ) is not equivalent. II. DE SITTER SPACETIME
In this section the basic properties of de Sitter space-time are introduced and reviewed. Readers familiar withde Sitter spacetime solutions may safely skip this section.In 1917 de Sitter reported two solutions for Einstein’sfield equations [55, 56], namely maximally symmetricspacetimes with constant positive (respectively, negative)curvature. In the case of constant curvature the Riemanntensor R ρσµν , is already determined by the Ricci scalar R .Since [57] the Ricci scalar R needs to be constant for aspacetime with constant curvature, de Sitter and anti-deSitter spacetimes are solutions of the field equations forempty spacetime with cosmological constant Λ = R .Spacetimes having constant positive curvature R > de Sitter (respectively, for negative curvature
R < anti-de Sitter ).Based on observations in 1998 concerning type Ia su-pernova [5] it has been hypothesised that the cosmologi-cal constant Λ of our universe is small and positive. Thus,de Sitter spacetime could be a natural limiting space-time for our universe (see, e.g., [58]). These observationsconsiderably motivate us to adapt or transfer the under-standing of quantum gravity for anti-de Sitter spacetimeto the de Sitter case most relevant for our universe.The most straightforward way to define d -dimensionalde Sitter spacetime dS d with one temporal and d − d +1)-dimensional Minkowski spacetime: itis then a single-sheeted hyperboloid in Minkowski space- Figure 1: dS embedded in Minkowski spacetime (greylines are null geodesics, black circles are constant timeslices)time. The points of d -dimensional de Sitter spacetimefulfil the hyperboloid condition: − x + x + · · · + x d = (cid:96) . (1)The parameter (cid:96) on the RHS is the de Sitter radius , whichdescribes the size of space at the time where its size isminimal. We parametrise the temporal coordinate x sothat the radius is smallest at x = 0. If not otherwisespecified, we assume the de Sitter radius is given by (cid:96) = 1.The metric of d -dimensional de Sitter spacetime is givenby d s = − d x + d x + · · · + d x d . (2)For the rest of this paper we focus on the (1 + 1)-dimensional case, depicted in Fig. 1 as a single-sheetedhyperboloid in Minkowski spacetime. The geodesics usedin the figure are reviewed in Appendix A 2. It is worthnoting that d -dimensional de Sitter spacetime has two temporal boundaries and no spatial boundary. (AdS has,by contrast, a single spatial boundary.) The temporalboundaries lie in the temporal future (respectively, past)infinity. This structure has far-reaching consequences forthe formulation of quantum gravity. A. Null geodesics in two-dimensional de Sitterspacetime
Geodesics are an important tool to describe and anal-yse the structure of spacetime. The geodesics we aremost interested in here are null geodesics, which describethe propagation of light rays. Two-dimensional de Sit-ter hypersurface is ruled by null geodesics (there are twodistinct straight lines running through every point in thesurface).The null geodesics in de Sitter spacetime may beconstructed from null geodesics in ( d + 1)-dimensionalMinkowski spacetime by imposing the hyperboloid con-straint Eq. (1): x ( s ) = x x x = su ± vsv ∓ us , s ∈ R , u + v = 1 . (3)The condition u + v = 1 can be observed by writingthe hyperboloid condition in matrix form as x T ηx = 1 , with η = diag( − , , · · · , . (4)We now see that the constraint u + v = 1 on the pa-rameters of the null geodesic x ( s ) arises as follows: x T ( s ) ηx ( s ) = − s + ( u ± vs ) + ( v ∓ us ) = − s + ( u + v ) + s ( u + v ) = 1 . Each null geodesic is specified by its intersection pointwith the time slice x = 0 and a sign. The sign distin-guishes between the two different classes of null geodesics,which we call anticlockwise - and clockwise -pointing nullgeodesics, respectively. This is depicted in Fig. 1, whereone observes the direction of propagation around the spa-tial coordinate of the null geodesic on the hyperboloid iseither clockwise or anticlockwise.The parameters for the clockwise and anticlockwisepointing null geodesics only differ by a sign. This is whyin the sequel we usually only consider the anticlockwisecase as a representative of null geodesics in general. Thecalculations are very similar for clockwise-pointing nullgeodesics. B. Global and conformal coordinates
It is convenient to introduce a coordinatisation to eas-ily work with de Sitter spacetime. The most straight-forward coordinate system for d -dimensional de Sitterspacetime is furnished by global coordinates , so named be-cause they describe the entire single-sheeted hyperboloidembedded in Minkowski spacetime (for further details oncoordinate systems for de Sitter spacetime see, e.g., [59]for further details). Global coordinates are defined via x = sinh( τ ) x j = ω j cosh( τ ) , j = 1 , , . . . , d, (5)where τ ∈ R is the temporal variable and the angularvariables ω i are defined according to ω = cos( θ ) ,ω = sin( θ ) cos( θ ) , ... ω d − = sin( θ ) · · · sin( θ d − ) cos( θ d − ) ,ω d = sin( θ ) · · · sin( θ d − ) sin( θ d − ) , θ π − π π T I − I + Figure 2: dS in conformal coordinates where θ = 0 and θ = 2 π are identified (grey lines are null geodesics).where 0 ≤ θ j < π for 1 ≤ j < d − ≤ θ d − < π .For the two-dimensional case, this simplifies somewhatand only one angle variable θ with 0 ≤ θ < π is neces-sary to define global coordinates (here θ = 0 and θ = 2 π are identified): x = sinh( τ ) ,x = cos( θ ) cosh( τ ) ,x = sin( θ ) cosh( τ ) . (6)The de Sitter metric Eq. (2), parametrised in global co-ordinates, takes the formd s = − d τ + cosh ( τ ) d θ . (7)(This is reviewed in Appendix A 1.) Null geodesics maybe parametrised according to (cid:18) θ ( t ) τ ( t ) (cid:19) = (cid:18) ± arctan t + θ arcsinh t (cid:19) , t ∈ R . (8)It is useful to introduce a new temporal variable to de-scribe the causal structure of de Sitter spacetime. Thisalso makes the temporal boundaries of de Sitter space-time somewhat more amenable to direct study. The cor-responding new coordinates are called conformal coordi-nates . This coordinate system exploits the same spatialvariables as global coordinates, however, a new temporalvariable T is introduced viacos T = 1cosh τ , where − π < T < π . (9)This coordinate transformation maps the temporal infini-ties in global coordinates ( τ = ±∞ ) to T = ± π , respec-tively. Accordingly, the temporal variable in conformalcoordinates is always finite, even at temporal infinity.In conformal coordinates a point in two-dimensional deSitter spacetime may be parametrised with the spatialcoordinate θ and the temporal coordinate T : p = ( θ, T ) . (10)In this way (1 + 1)-dimensional de Sitter spacetime maybe represented by a rectangle (see Fig. 2). Geodesics inconformal coordinates are straight lines tilted at 45 ◦ , justas they are in Minkowski spacetime (this is reviewed inAppendix A 3). A null geodesic in conformal coordinatesmay be parametrised via: (cid:18) θ ( s ) T ( s ) (cid:19) = (cid:18) θ ± ss (cid:19) , − π < s < π . (11)A key advantage of the conformal coordinate system isthat it allows us to transparantly understand the causalstructure of de Sitter spacetime. Further, many calcula-tions are easier to perform on a rectangle, as opposed toa hyperbolic surface. C. Causal structure and causal diamonds
In this section causality relations and the accessibilityof information in de Sitter spacetime are considered. Tothis end we introduce causal diamonds , which are specialsubsets of de Sitter spacetime.In order to explain the physical relevance of causal dia-monds consider an experiment moving along a worldlinecommencing at a spacetime location p and concludingat location q . A causal diamond is then that subset ofde Sitter spacetime where the information influencing —and influenced by — the experiment between point p andpoint q is accessible to an external observer. For a reviewsee [57].There are two restrictions that have to be fulfilled fora point to lie within the causal diamond defined by thepoints p and q , namely, that the information needs to liewithin the causal future J + ( p ) of point p and within thecausal past J − ( q ) of point q : C ( p, q ) = J + ( P ) ∩ J − ( q ) . (12)In this way a causal diamond C ( p, q ) is defined only bythe points p and q , and not by the details of the exper-iment’s worldline. See Fig. 3 for an illustration. Notethat any continuous non-spacelike worldline from point p to point q lies within the causal diamond.The largest influence a local experiment can have onobservers within de Sitter spacetime is then found bysending p to the temporal past infinity and q to the tem-poral future infinity: p → x − ∈ I − q → x + ∈ I + . Everything an observer can do and observe is thus deter-mined by the causal diamond C ( x − , x + ) defined by justthese two points x ± lying within the temporal bound-aries. Such a causal diamond is of maximal size if thedefining points have the same θ coordinate. It is impos-sible to cover the entirety of de Sitter spacetime using justone causal diamond; no single observer can acquire infor-mation about an arbitrary point in de Sitter spacetime, 0 θ π − π π T p q J + ( p ) J − ( q ) C ( p, q ) Figure 3: A causal diamond is the intersection of thecausal future of point p and the causal past of point q .even in principle! Contrast this with Minkowski space-time: here any local experiment may, in principle, influ-ence, and be influenced by, any point within the space-time. Causal diamonds are a central building block insection III A, where a tessellation for a two-dimensionalde Sitter spacetime is constructed.A striking consequence of the causal structure of de Sit-ter spacetime, as emphasised by Witten [10], is that thereis no global positive conserved energy quantity. This isa consequence of the absence of a global timelike Killingvector field (although it is possible to define local time-like Killing vector fields), so there is no global genera-tor of time-translation symmetry. Therefore it is at bestquestionable – and likely impossible – whether one candefine a unitary time-translation operator with a corre-sponding Hamiltonian generator on the quantum gravityHilbert space. This does however leave open the possi-bility that time translation might be implemented quan-tum mechanically via a dissipative process described viaa completely positive map.If we take all of these observations at face value we areforced to accept that the observables of quantum gravityin dS live on I ± . This is in stark contrast to the AdScase where we have a wealth of observables on the spa-tial boundary which we may identify with observablesof a conformal field theory on the holographic bound-ary. These features (or bugs, depending on your pointof view) make reasoning about quantum gravity in dSrather different to its AdS counterpart.The – here assumed – impossibility of defining a Hamil-tonian for de Sitter spacetime is one core reason why inthis paper we consider the quantum mechanics of de Sit-ter spacetime only at its boundaries. The model we con-struct is defined on a Hilbert space attached to temporalinfinity. As a result, our model gives us only indirect in-sight, which must be holographically reconstructred, onevents within de Sitter spacetime at finite times.0 θ π − π π T − π D D D D D D D D D D D D D D D D C ± −
11 01 11
Figure 4: Recursive construction of a tessellation of dS with causal diamonds using Farey numbers. III. A HOLOGRAPHIC NETWORK FOR DESITTER SPACETIME
There are profound reasons to believe that a quan-tum theory of gravity can be formulated holographically[60], a programme which has been pursued most prof-itably in the AdS case. Our goal in this section, moti-vated by the proposal of [30] for AdS, is to commencethe exploration of toy holographic formulations of de Sit-ter spacetime using tensor networks associated to tessel-lations in as direct a way as possible. In this way weconstruct a holographic toy model for two dimensionalde Sitter spacetime. It should be noted, however, thatbecause de Sitter spacetime has two temporal boundariesthe model we construct differs drastically from the onepresented in [30], and it does not directly fit the defini-tion of either a holographic code or state. This is whywe tentatively denote the holographic model we deriveas a holographic network . We ultimately interpret thenetwork as the propagator from the past boundary of deSitter spacetime to its future boundary.
A. A tessellation for de Sitter spacetime
As a first step toward defining a holographic tensornetwork for (1 + 1)-dimensional de Sitter spacetime weneed to introduce a tessellation . The kind of tessellationwe describe here was proposed by Aicardi [61], and iscomprised of causal diamonds C ( p, q ). The endpoints ofthe two fundamental causal diamonds defining the tes-sellation lie within the temporal infinities I + and I − .The tessellation, shown in Fig 4, is described recur-sively, starting with two distinguished tiles defining the fundamental regions . Note that, in contrast to regulartessellations of, e.g., Euclidean space, the resulting tes-sellation is not invariant under a discrete subgroup ofthe spacetime isometry group. Further, the fundamentalregions are distinguished in that they are the only tilesthat extend from positive to negative infinity.The causal diamonds comprising the tessellation have either one or both of their endpoints lying within thetemporal boundaries. The set of all these endpoints, the boundary of the tessellation, may be identified with therational numbers Q according to the following prescrip-tion. Firstly, the two fundamental tiles, denoted D and D (as depicted in Fig. 4), are defined to be the followingcausal diamonds D = C (cid:16)(cid:16) π , − π (cid:17) , (cid:16) π , π (cid:17)(cid:17) and D = C (cid:18)(cid:18) π , − π (cid:19) , (cid:18) π , π (cid:19)(cid:19) . (13)The endpoints π/ π/ S ⊂ C ) of these diamonds in I + are identified (for rea-sons that will become clear) with the rational numbers1 / / Cayley transformation w ( z ) ≡ i z − iz + i . (14)Note that in this way we have identified the positive andnegative infinity of Q so that 1 / − / generation of tiles we introduce thefollowing Farey mediant operation pq ⊕ rs = p + rq + s ; (15)we take the preexisting boundary points 0 / ± / − ⊕
01 = −
11 and 01 ⊕
10 = 11 . (16)These rational numbers then induce via the Cayley trans-formation two new boundary points for the new causaldiamonds of the first generation: C = C (cid:16)(cid:16) arg( w (1)) , − π (cid:17) , (cid:16) arg( w (1)) , π (cid:17)(cid:17) = C (cid:18)(cid:18) π , − π (cid:19) , (cid:18) π , π (cid:19)(cid:19) (17) C = C (cid:16)(cid:16) arg( w ( − , − π (cid:17) , (cid:16) arg( w ( − , π (cid:17)(cid:17) = C (cid:16)(cid:16) π , − π (cid:17) , (cid:16) π , π (cid:17)(cid:17) (18)The tiles for the first generation are defined as the setdifference from the causal diamonds defined above andthe tiles of previous generations: D = (cid:0) C ∪ C (cid:1) / (cid:0) D ∪ D (cid:1) . (19)This procedure continues iteratively to define tiles ofhigher generations:-
10 01 10 -
10 01 10 -
10 01 10 -
10 01 10 -
11 11 -
11 11 -
11 11 - -
12 12 21 - -
12 12 21 - - - -
13 13 23 32 31
The resulting sequences of completely reduced rationalnumbers pq and rs satisfying | ps − qr | = 1 are called Fareynumbers . Each Farey number is then mapped to thecircle via the Cayley transformation to yield the futureboundary points of the subsequent generations. The pastboundary points have the same θ value which is foundby mirroring through the T = 0 time slice. The result-ing Farey tessellation is illustrated in Fig. 6. The Fareymediant operation, applied recursively, exhausts the ra-tional numbers Q so that the future and past boundariesof the tessellation are Q .The main reason for choosing the Farey tessellationis that it admits a natural action of the infinite groupPSL(2 , Z ): it preserves the relation | ad − bc | = 1, andhence mediancy. We later identify PSL(2 , Z ) as the groupof isometries compatible with our tessellation (in a senseto be described).It turns out to be convenient to identify the Farey tes-sellation of de Sitter spacetime with a tessellation basedon dyadic rational numbers a/ n , a ∈ Z , n ∈ Z + . This isachieved by applying the Minkowski question mark func-tion ?( x ) — a homeomorphism of S — to the boundarypoints of the tessellation. The question-mark functionmay be defined recursively on the unit interval. The basecases are given by?(0) = 0 = 01 and ?(1) = 1 = 11 , (20)which are then extended recursively to all of the Fareynumbers pq and rs satisfying | ps − qr | = 1 via the rule(see, e.g., [62] for further details):? (cid:18) pq ⊕ rs (cid:19) = 12 ? (cid:18) pq (cid:19) + 12 ? (cid:16) rs (cid:17) , (21)In this fashion one generates a correspondence betweenthe dyadic rationals and the rational numbers in the unitinterval:0 10 10 10 1?( ) = ?( ) = ?( ) = ?( ) = ?( ) = ?( ) = ?( ) = The Minkowski question mark function may be extendedto Q via ?( x + 1) ≡ ?( x ) + 1. As a result, the function(?( x ) − x ) is a periodic function with zeroes at the inte-gers.To get a tessellation that is defined on the dyadic ra-tional numbers, the Minkowski question mark functionis applied to the Farey numbers that define the bound-aries of the tiles in the Farey tessellation. To define atessellation based on dyadic rationals we need to identifythe dyadic rationals on the real axis R with the dyadicrationals the unit interval, which is naturally identifiedwith the unit circle. ψ ( x ) x − − − − − .
751 0 . . ψ ( x ) maps points from the real axisto the unit interval.This is achieved with a particular function ψ which isdefined as follows. The images of subsequent integersare given by: ψ : Z → [0 , , n (cid:55)→ | n | +1 , n < n (cid:55)→ , n = 0 n (cid:55)→ n +1 − n +1 , n > , (22)which may be visualised as − − − − / / / / / R The piecewise-linear function ψ is then fully specified byrequiring its graph interpolates linearly between the in-tegers. The resulting function ψ ( x ) is plotted in Fig. 5.Thus the dyadic rationals induce a tessellation of dSwith the following iterative description. The boundarypoints of the fundamental tiles on the unit interval are 0,1 / D = C (cid:16)(cid:16) π , − π (cid:17) , (cid:16) π , π (cid:17)(cid:17) and D = C (cid:18)(cid:18) π , − π (cid:19) , (cid:18) π , π (cid:19)(cid:19) . (23)The first generation of tiles are defined by the region D ≡ (cid:0) C ∪ C (cid:1) \ (cid:0) D ∪ D (cid:1) = D ∪ D ∪ D ∪ D , (24)0 θ π − π π T − π Figure 6: Tessellation of dS (color indicates whichgeneration the tile belongs to)where C = C (cid:16)(cid:16) , − π (cid:17) , (cid:16) , π (cid:17)(cid:17) and C = C (cid:16)(cid:16) π, − π (cid:17) , (cid:16) π, π (cid:17)(cid:17) . (25)Accordingly, the first generation of the tessellation (i.e.,the set = D ) is comprised of four tiles. Each tile D i maybe interpreted as a new causal diamond. This proceduremay be repeated recursively, giving rise to the tiles D nj of the n th generation D n ≡ n (cid:91) j =1 C nj \ (cid:32) n − (cid:91) k =1 D k (cid:33) , (26) D n = n +1 (cid:91) j =1 D nj , (27)where C nj ≡ C (cid:18)(cid:18) π ( j − n − , − π (cid:19) , (cid:18) π ( j − n − , π (cid:19)(cid:19) . (28)The points p and q defining the causal-diamond tiles D nj ≡ C ( p, q ) of the dyadic tessellation lie on constant-time slices determined by the generation (note that thisis not the case for the Farey tessellation): T n = π n − n , n ≥ − π n − n , n < . (29)The time slices T n admit an interesting interpretation:in the limit T → ∞ the size of de Sitter spacetime doublesfor each consecutive time slice. This may be observed asfollows. The size of de Sitter spacetime at the time τ inglobal coordinates is given by d dS ( τ ) = 2 π cosh τ. (30)The ratio of the size d dS ( τ n ) [63] of de Sitter spacetimeat the consecutive time slices τ n and τ n +1 is given by d dS ( τ n +1 ) d dS ( τ n ) = cos (cid:0) π (1 − − n ) (cid:1) cos (cid:0) π (1 − − ( n +1) ) (cid:1) n →∞ −→ . (a) View of the dyadic tessellation from the side.(b) View of the dyadic tessellation from the top.(c) View of the Farey tessellation from the top. Figure 7: Tessellations of dS on the hyperbolic sheetembedded in Minkowski spacetime.For each consecutive generation in the limit T → ∞ boththe number of tiles and the size of the spacetime doubles.In this way the length of the tiles in the tessellation closeto the boundary approaches a constant. This is mostclearly observed when one embeds de Sitter in Minkowskispacetime, see Fig. 7. B. Construction of a holographic network
In this section a holographic tensor network corre-sponding to (1 + 1)-dimensional de Sitter spacetime withde Sitter radius (cid:96) = 1 is constructed. For a review of thebasics of tensor networks see Appendix B.In order to build our tensor network we consider atomscomprised of isometries and unitaries (of which perfecttensors are a subset [30]). While the motivation for usingperfect tensors comes from the AdS setting where (pla-nar) perfect tensors ensure that the microscopic quantumtensor network inherits a residual rotation invariance, itturns out that for most of our observations we won’t ac-tually need that our tensors obey all of the conditionsrequired of perfect tensor. Indeed, only the conditionthat they are isometries in the usual sense is necessaryfor a majority of our conclusions.Two particular tensors are relevant for us: a 3-leg ten-sor V βγα and a 4-leg perfect tensor U γδαβ : V βγα ≡ V αβ γ = αβ U γδαβ ≡ U = α βγ δ βγ We assume, for simplicity, that the indices of both U and V run over a set of size D . Thus U and V are thefollowing two maps: U : C D ⊗ C D → C D ⊗ C D , and V : C D → C D ⊗ C D . (31)This assumption entails no particular loss of generality(with a little work one may allow for the dimension D ofthe legs to vary from one spacetime location to another).The 4-leg tensor U is a unitary transformation: U † U = U † U = = I . (32)The isometric tensor V is required to fulfil V † V = V † V = = I . (33)In the following the tensor-network labels U and V , aswell as the index labels for the legs, are suppressed. Figure 8: Holographic tensor network M correspondingto the tessellation Fig. 6 of dS (with (cid:96) = 1). Thisnetwork should be understood as an infinite networkextending between the temporal boundaries.To construct our holographic tensor network M for (cid:96) = 1 de Sitter spacetime we associate tensors with thecorners of the tiles as depicted in Fig. 8. The tensornetwork structure is obtained by contracting the legs oftensors corresponding to consecutive generations alongthe edges of the tiles. The resulting network is depictedin Fig. 8. Note that the type of tensor placed at the cor-ner of a tile corresponds to the number of neighbouringtensors. In this way we have associated the 4-leg tensorwith only the initial generation. (Later we consider theconstruction of holographic tensor networks correspond-ing to dS spacetimes with radius (cid:96) >
1, which feature agreater number of 4-leg tensors.) The only uncontractedlegs of the resulting infinite tensor network are associatedwith the temporal boundaries of the tensor network.
C. Kinematic Hilbert space
One of the most important features of de Sitter space-time is that all the observables live on the temporalboundaries I ± . As a consequence, configurations —kinematical states — which determine the expectationvalues of the observables are associated with the tem-poral boundaries. This simple observation has profoundconsequences for the structure of the kinematical Hilbertspace. Indeed, we see little alternative except to asso-ciate the Hilbert space of states of dS with the boundaries I ± and hence introduce two infinite-dimensional Hilbertspaces H in , the input space, and H out , the output space,associated with these boundaries, respectively.Our focus in this paper is on the infinite tensor network M in Fig. 8 to be understood as an operator from the(as yet to be defined) infinite-dimensional Hilbert space H in to H out . There are considerable technicalities thatmust be overcome to discuss these Hilbert spaces at thelevel of mathematical rigour. We do not undertake thisinvestigation here, and the arguments in this paper areat a physical level of rigour. (The interested reader isdirected to [64] and [65] where the construction of theseHilbert spaces as inductive limits are described in detail.)Instead of introducing the formalism of inductive lim-its, we equivalently reason about our infinite tensor-product Hilbert spaces, and the network M , by appealingto cutoff versions of the network: we frequently draw onlya couple of generations of the network and use the isomet-ric and unitary property of U and V to deduce algebraicfacts about the infinite network M , a strategy successfulbecause these conditions can lead to an infinite numberof cancellations, particularly in equations involving M and its adjoint M † .We describe the cutoff Hilbert spaces correspondingto H in and H out as follows. Choose a nonnegative in-teger n ≥
0: we begin by discretising the spatial coor-dinate θ by breaking the circle into 2 n dyadic intervals2 π [ j − n , ( j + 1)2 − n ), where j ∈ { , , . . . , n − } . Thesediscretisations are associated, respectively, with the timeslices T ± n . Attached to each such time slice T n is thefinite Hilbert space H ± n ≡ n − (cid:79) j =0 C D . (34)The tensor network defined by our tessellation induces,for all m ≤ n , a family of linear maps M m,n : H m → H n , (35)given by drawing the tensor network for the appropriatenumber of generations and taking the dangling legs toact on the respective input and output spaces H m and H n . Note that, for 1 < m < n the operator M m,n is anisometry obeying M † m,n M m,n = I H m . (36)Conversely, for m < n < −
1, the operator M m,n is anisometry satisfying M m,n M † m,n = I H n . (37)The tensor network M then acts “in the limit” m →−∞ and n → ∞ . To make proper mathematical senseof this statement one needs the technology of inductivelimits. However, this is equivalent to the expedient ofalways working in the largest input and output Hilbertspaces (corresponding to the finest required discretisa-tions) needed to express all the physical statements re-quired. With a little care the Hilbert spaces H in “ ≡ ” H −∞ and H out “ ≡ ” H ∞ may be simply regarded as the infinitetensor product of C D [66].The actual definition of H in and H out is intimately tiedto the tensor network M as we use the family M m,n tobuild an equivalence relation on the cutoff Hilbert spaces H m . Suppose that 1 < m < n : we say that | φ m (cid:105) ∈ H m and | ψ n (cid:105) ∈ H n are equivalent , written | φ m (cid:105) ∼ | ψ n (cid:105) , if | ψ n (cid:105) = M m,n | φ m (cid:105) . (38)Physically we think of | ψ n (cid:105) as a fine graining of | φ m (cid:105) .(The analogous definition holds for the case m < n < − H m as a subspace of the fine-grained Hilbert space H n . Thedefinition of H out now results by requiring it to be thesmallest Hilbert space which contains all of H m , m > M m, ∞ . D. Basic properties of the tensor network
De Sitter spacetime is characterised by an initial con-traction for
T <
0, followed by expansion for
T > loss , destruction , or deletion of quantum information. Asthe spacetime contracts there is simply less volume andhence less information may be stored in the quantum de-grees of freedom comprising it. Conversely, for T >
0, dSis expanding and new quantum degrees of freedom arebrought into existence. There are many ways to modelloss/deletion and creation processes in quantum mechan-ics. The simplest strategy, and the one employed here,is to model such processes with isometries . (Anotherpossibility, which doesn’t violate unitary so directly, isto model a loss process with a completely positive map .This, however, does introduce a new mystery: where doesthe information go?)To make this discussion more concrete we consider theHilbert space H in for the degrees of freedom on the tem-poral boundary I − , H for the degrees of freedom at T = 0, and H out for I + . According to the above argu-ment the contraction process should be represented at amicroscopic level by an isometry A † : H in → H and theexpansion process by an isometry B : H → H out . Hencethe entire spacetime history should be represented by theoperator W = BA † : H in → H out (39)which can be thought of as a scattering process. Oper-ators W which are such compositions of isometries areknown as partial isometries : recall that a bounded lineartransformation C : H → H is called a partial isometry,if P = C † C is a projection. Choosing C = W we deducethat W † W = AB † BA † = AA † is a projection (similarly,so is W † ). In this way we intepret the operator W asa kind of restricted propagator from the Hilbert spaceof the temporal past infinity to the Hilbert space of thetemporal future infinity. In general the operator W failsto be a propagator in the usual sense because W † W isonly a projection rather than the identity required for W to be unitary.The projection P = W † W arising from a partial isome-try W : H in → H out singles out a distinguished subspaceof H in via the projection P , i.e., H phys ≡ P H in , whichwe term the subspace of physical states . A physical state | φ phys (cid:105) ∈ H phys propagates without loss of norm throughthe spacetime network.0Consider the tensor network M depicted in Fig. 8:to check whether M † M is a projection we exploit theidentities (32) and (33): M † M = = =Hence the tensor network M is indeed a partial isometryas M † M is a projection:( M † M ) = = = M † M A key property of the tensor network M represent-ing the evolution from H in to H out is that the subspace H phys ≡ ( M † M ) H in of physical states determined by M is finite dimensional . If the dimension of the legs of U and V is D we have that M † M is a projection onto a D -dimensional subspace of H in . We later connect thisobservation with the Λ - N correspondence.The tensor network M exhibits an information bottle-neck at T = 0: the infinite-dimensional input Hilbertspace H in has to pass through the finite-dimensional H phys subspace. It is notable that this phenom-ena whereby a finite-dimensional subspace canonicallyemerges from an infinite-dimensional ambient Hilbertspace is most transparently observed in the tensor-network representation.The discussion in this section mirrors that of Wit-ten [10] concerning the nonperturbative definition of theHilbert space for dS. Indeed, we are motivated to con-jecture that the subspace H phys introduced here is a mi-croscopic realisation of Witten’s nonperturbative Hilbertspace H . Further, the partial isometry W is a micro-scopic realisation of the matrix M constructed by Wit-ten.So far our discussions have centred around a tensornetwork representation for (cid:96) = 1. The question of howto represent dS spacetimes with larger radii and, further,the limit (cid:96) → ∞ now emerges. There is a natural con-struction, as we detail in the next subsection.Before we turn to this construction we pause to high-light the general nature of the arguments in this sec-tion: the essential ingredient is that a tensor networkrepresenting dS may be written as a composition of twoisometries, namely, an isometry, representing contrac-tion, mapping from the (large) input Hilbert space rep-resenting I − to a smaller intermediate Hilbert space H composed with an isometry acting from H to the fi-nal Hilbert space for I + . This abstract structure is not Figure 9: Tessellation and holographic network for dS with (cid:96) = 2. The blue lines (edges of D ) divide theinitial causal diamonds.restricted to (1 + 1) dimensions. Indeed, with little mod-ification one may observe the same structure of tensornetworks representing dS in higher dimensions. E. Holographic networks for de Sitter spacetimewith (cid:96) (cid:54) = 1 : a Λ -qubit correspondence We have, so far, only considered tessellations and ten-sor networks corresponding to a (1 + 1)-dimensional deSitter spacetime with radius (cid:96) = 1. In this section we ex-plain how to generalise the construction to model (1+1)-dimensional de Sitter spacetimes with different de Sitterradii (cid:96) (cid:54) = 1. According to this definition one obtains a cor-respondence between the network quantum informationcapacity of the network and the cosmological constant.This is reminiscent of Bousso’s Λ- N correspondence [9].We illustrate our definition for de Sitter radii (cid:96) = 2and (cid:96) = 4; the construction can be readily generalised todescribe de Sitter spacetimes with radii (cid:96) = 2 n . The def-inition is intended to remain consistent with the causalstructure of the tensor network for (cid:96) = 1. To achievethis we modify the tessellation by dividing the two funda-mental regions into four smaller causal tiles, respectively.In this way, the tensor network is only locally modifiedaround the time slice T : this is physically plausible asthe finiteness of the de Sitter radius only obtrudes itselfmost profoundly around T , and its effects are progres-sively less significant towards the temporal infinities. Theconstruction of the modified tensor network then pro-ceeds analogously to Sec. III B and is depicted in Fig. 9.This tensor network is a partial isometry, just as for (cid:96) = 1(see Appendix C).The construction of a tensor network for radius (cid:96) = 4and larger proceeds iteratively as above: the tiles of thefirst two generations, i.e., the fundamental and first, inthe tessellation are each subdivided into four diamonds.Around T = 0 this new tessellation now has four timesas many tiles as the initial tessellation. Positing thateach tile represents a quantum spacetime degree of free-dom leads us to conclude that the new tessellation repre-sents a spacetime four times as large as the original. Thecorresponding holographic tensor network is depicted in1Figure 10: Tessellation and holographic network for dS with (cid:96) = 4. The blue (edges of D ) and light blue(edges of D ) lines divide the initial causal diamonds.Fig. 10. The resulting network is also a partial isometry.Given our definition for (cid:96) > M (cid:96) is a partial isometry,one obtains a corresponding completely positive (CP) –but not trace-preserving – map via E (cid:96) ( ρ ) ≡ M (cid:96) ρM † (cid:96) . (40)This superoperator takes density operators ρ in for theinitial Hilbert space H in to density operators for H out .A CP map can be understood as representing a com-munication process from a sender to a receiver. In thiscontext it makes sense to ask what the corresponding quantum capacity is of this process, i.e., how many qubits Q (1) (cid:96) of quantum information can be sent (perhaps bet-ter: stored ) in a single use, without error through the CPmap. This is easily obtained in our context as Q (1) (cid:96) = (cid:98) log (dim( H (cid:96) phys )) (cid:99) . (41)This follows because M (cid:96) is a partial isometry and it ispossible to directly encode 2 Q (1) (cid:96) qubits into the subspace H phys which is then transmitted noiselessly. Note that[67] Q (1) (cid:96) (cid:46) ( (cid:96) + 1) log ( D ) . (42)This quantity admits a transparent physical interpreta-tion: it simply counts the number of qubits that can besent without disturbance through the bottleneck of thetensor network at T .The correspondence (42) implies a fundamental rela-tionship between the cosmological constant Λ and the in-formation carrying capacity of the network representingthe spacetime. In our case this relationship reads Q (1) ∝ √ Λ . (43)This is a direct consequence of the equation [68] (cid:96) √ (cid:114)
1Λ (44) for the dS radius [69].If we take the correspondence (43) at face value andextrapolate it to the logical extreme one concludes thata positive cosmological constant determines the informa-tion carrying capacity of a spacetime and, conversely, aspacetime with a finite quantum communication capacitycorresponds to a spacetime with a positive cosmologicalconstant. A most dramatic consequence of the correspon-dence (43) is that our apparently continuous and infiniteuniverse is effectively described by a finite-dimensional
Hilbert space carrying the quantum information frompast to future. (However, one should be appropriatelycautious: the example of AdS does strain this conclusionsomewhat, i.e., what is implied by a negative cosmologi-cal constant? We have no answer to this, except to notethat the structure of the observables of AdS is radicallydifferent to Λ > N correspondence, introduced by Bousso [9]. Here a corre-lation between the cosmological constant Λ and the num-ber N of quantum degrees of freedom describing a space-time is posited. The argument put forward by Bousso tojustify the Λ − N correspondence builds on the key obser-vation that the total entropy perceived by an observer O carrying out an experiment commencing at p and endingat q is bounded by that of the causal diamond C ( p, q ).Bousso’s argument may be understood in our contextas follows: since the laboratory for a local observer O can only execute unitary operations local to O then, ifwe assume that the observer O initialises their laboratoryapparatus in a pure state | Ω (cid:105) at p , the total amount ofentropy that O can create by losing halves of entangledpairs before the experiment concludes at q is bounded bythe number of qubits crossing the boundary of J − ( q ).The amount of entropy that O can receive before theexperiment is complete is bounded by the number ofqubits crossing the boundary J + ( p ). The sum of theseentropies is thus determined by the flux of qubits throughthe boundary of the causal diamond. Exploiting the co-variant entropy bound [7] one can then argue that theentropy of a causal diamond is bounded by its area. F. Causal structure of the tensor network
It may be observed that as the de Sitter radius (cid:96) isincreased toward infinity, the tensor network resultingfrom the above procedure has the structure of an in-creasingly large regular grid of unitary operators aroundthe timeslice T preceded, and followed, by tree-like ten-sor networks. In the limit (cid:96) → ∞ we should, and do,recover Minkowski spacetime in the sense that the ten-sor network is essentially comprised of a regular grid ofunitaries, a quantum cellular automaton (QCA) [70–75].Such networks also arise when applying the Lie-Trotterdecomposition to the dynamics of quantum spin chains.Due to the natural causal structure exhibited by QCA,namely that information propagation is bounded by a2speed of light, they provide a natural candidate for atensor-network realisation of flat Minkowski spacetime.This proposal has been explored recently in a variety ofsettings [47, 48, 76–80].It turns out that the tensor-network ansatz we pro-pose here, namely a QCA preceded and followed by tree-like tensor networks, is most closely related to that of a Lorentzian MERA introduced in [51, 77]. In these pa-pers the Lorentzian MERA has been argued to capturethe dynamics of quantum systems on dS . The emer-gence, in the limit (cid:96) → ∞ , of a QCA structure in ourtensor-network ansatz, and the connection to LorentzianMERA, provides further supporting evidence that ourtensor network provides a microscopic model for de Sit-ter spacetime.As our tensor network is comprised of unitaries andisometries it exhibits, similar to MERA, a natural causalstructure. One can confirm this structure using a vari-ety of notions of quantum causal influence , see [47, 78]and references therein for a cross-section of approaches.For example, exploiting the notion of pure causality in-troduced by B´eny in [47], one directly recovers the causalstructure of de Sitter spacetime. IV. SYMMETRIES OF DE SITTER SPACETIME
In this section we discuss the symmetries of de Sitterspacetime. Ultimately our goal here is to understand theaction of these symmetries on the kinematical Hilbertspaces H in and H out associated with our tensor network.To this end we first review the action of isometries on thetemporal boundaries.We can infer the isometries of dS by first noting thatthe metric of three dimensional Minkowski spacetime R , (in which an embedding of de Sitter spacetime isfound) is preserved by a transformation O ∈ O(1 ,
2) [81]: η = O T ηO. (45)This transformation induces a symmetry of (1 + 1)-dimensional de Sitter spacetime because the equationdefining the embedding of dS is preserved: x T ηx = x T O T ηOx = ( Ox ) T ηOx = y T ηy = 1 . (46)The action of isometries O ∈ O(1 ,
2) on dS can beequivalently specified in terms of their action on nullgeodesics, which are specified by the parameters u and v (see Sec. II A). This, in turn, allows us to describe the ac-tion of isometries on the temporal boundaries. With thisaction in hand we are then able to describe the actionof isometries on the kinematic Hilbert spaces H in and H out . To simplify our discussion, we restrict our atten-tion in the following to the subgroup SO(1 ,
2) of properisometries preserving orientation.
A. Symmetry action on the future boundary of deSitter spacetime
The temporal boundaries of (1+1)-dimensional de Sit-ter spacetime are circles. Accordingly, it can be helpfulto describe the symmetry action of an isometry O on theboundary as a linear fractional transformation of the cir-cle or M¨obius transformation . That this is possible is aconsequence of a sporadic isogeny of SL(2 , R ) to the sym-metry group SO(1 , g = (cid:18) a bc d (cid:19) , g ∈ SL(2 , R ) , (47)is identified with a corresponding symmetry transforma-tion in h ( g ) ∈ SO(1 , h ( g ) aresomewhat complicated, and we refer to Appendix D forfurther details. Note that this correspondence is 2 to 1:the element g (cid:48) = − g yields the same transformation as g . In this way the projective special linear group PSL(2 , R ) = SL(2 , R ) / {± I } (48)emerges as the natural subgroup to identify with theisometries of dS.There is an identification between elements ofPSL(2 , R ) and elements of the M¨obius group of linearfractional transformations of the upper half plane H = { z ∈ C : Re z ≥ } (49)given by f : H → H , z (cid:55)→ αz + βγz + δ , (50)where α , β , γ and δ are real coefficients satisfying the con-dition αδ − βγ = 1. Happily, as we describe below, thisidentification is compatible with the action of SO(1 , I ± .A temporal boundary of de Sitter spacetime may berescaled to the unit circle S , which may be understoodas the boundary of the unit disk D = { z ∈ C : | z | ≤ } (51)in the complex plane. We now identify the unit circlewith the upper half plane via the Cayley transform w and its inverse: w : H → D , z (cid:55)→ i z − iz + i , (52) w − : D → H , z (cid:55)→ z + i iz . (53)The Cayley transform identifies the real axis R (theboundary of the upper half plane) with the unit circle S (the boundary of the unit disk); a boundary point of3de Sitter spacetime can now be identified with the com-plex number z = u + iv where u, v ∈ R , (54)with u + v = 1.We now obtain the induced action of a M¨obius trans-formation f on the temporal boundary I + according to p = w ◦ f ◦ w − : D → H → H → D . (55)The induced image of a point z in S under f is thus z (cid:48) = p ( z ) ⇒ u (cid:48) = Re p ( z ) v (cid:48) = Im p ( z ) . (56)An explicit computation yields: u (cid:48) = 4 αδu + 2 βδ (1 − v ) − u + 2 αγ ( v + 1)2 u ( αβ + γδ ) + ( α + γ )( v + 1) − ( β + δ )( v − v (cid:48) = 4 αβu + 2 α ( v + 1) − β ( v − u ( αβ + γδ ) + ( α + γ )( v + 1) − ( β + δ ) ( v − − ,
2) on the temporal boundary foundby a direct computation involving null geodesics.The action of an isometry on the future boundary I + of de Sitter spacetime may be immediately extended toobtain an action on the past boundary I − . In order todo so, we identify a point x in I − with a point x (cid:48) ∈ I + by transporting it along a null geodesic. The symmetryaction is applied to x (cid:48) and the result transported back I − via a null geodesic. This action is well defined becausenull geodesics travelling in either direction between I + and I − transport a past boundary point to the samefuture boundary point. Explicitly, the maps θ ± π : I ∓ →I ± transporting points between the temporal boundariesare given by θ ± π ( x ) = x ± π . Hence we obtain for anisometry p on the future boundary the induced action onthe past boundary I − , depicted in Fig. 11, via q = θ − π ◦ p ◦ θ π . (58) B. Symmetry action of PSL (2 , Z ) on a tessellation Just as general isometries of R n are incompatible witha regular grid, the action, via M¨obius transformations,of the spacetime isometries of dS cannot be compatiblewith our tessellation. This is because boundary points ofthe tessellation are, in general, sent to points lying out-side the tessellation. We can, however, identify a discretesubgroup of SL(2 , R ) which is compatible with the tessel-lation in the sense that the set of boundary points of thetessellation is left invariant. As we’ll see, the tessellationitself is not invariant; this has important consequencesfor the interpretation of our tensor network. 0 θ π − π π T I − I + p qθ π θ − π x x Figure 11: Symmetry action q on the past boundary I − of dS We have seen, in the previous section, how the isome-try group SL(2 , R ) acts on the temporal boundaries of dS.However, the boundary points of our tessellation (in theoriginal non-dyadic formulation) are precisely the ratio-nal numbers lying in the unit circle Q . Since the preim-age of a rational point under the Cayley transform w isrational, we see that the subgroup of isometries compat-ible with the tessellation is then PSL(2 , Q ). It turns outthat the action of this group is actually equivalent to thatof PSL(2 , Z ): consider a M¨obius transformation f associ-ated with an element of PSL(2 , Q ) with arbitrary rationalentries. Clearing denominators allows us to write f ( z ) = a a z + b b c c z + d d = a b c d z + a b c d a b c d z + a b c d . (59)Here the parameters a , a , b , b , c , c , d and d aswell as their product are integers. Thus any transforma-tion in PSL(2 , Q ) is equivalent to a corresponding trans-formation in PSL(2 , Z ). This justifies using PSL(2 , Z ) asthe group of isometries of the Farey tessellation.Since it has been convenient to formulate our tessel-lation in terms of the dyadic rationals, we also describehow to obtain an induced action of f ∈ PSL(2 , Z ) on thesubset of all dyadic rationals in R . This is achieved bycomposing with the Minkowski question mark function ?and its inverse ? − , i.e., the action is described via(? ◦ f ◦ ? − )( x ) . (60)The steps of this symmetry transformation – plotted inFig. 13 for a pair of illustrative examples – are illustratedin Fig. 12. The resulting symmetry transform dilates andcompresses spacetime along light rays.The function (? ◦ f ◦ ? − )( x ) is a piecewise-linear home-omorphism of R . In order to obtain an action on theboundaries of de Sitter spacetime we further identify thedomain and range of this function with the circle. This isachieved with the function ψ that was introduced earlier(see Fig. 5) and identifies the dyadic rationals on thereal axis R with the dyadic rationals in the unit interval.4 RRRR ? − ? f ? dyadic rationals(before transformation)Farey numbers(before transformation)Farey numbers(after transformation)dyadic rationals(after transformation)Figure 12: Symmetry transformation on the dyadicrational numbers induced by a M¨obius function f . y x − − − − − − − − (a) transformation with f ( x ) = − x y x − − − − − − − − (b) transformation with f ( x ) = x − x Figure 13: Different symmetry transformations(? ◦ f ◦ ? − )( x ) with M¨obius functions f which act onthe dyadic rational numbers of the real axis. y x − − − − ψ −→ ψ ( y ) ψ ( x )0 . . . . (a) transformation f ( x ) = − x y x − − − − ψ −→ ψ ( y ) ψ ( x )0 . . . . (b) transformation f ( x ) = x − x Figure 14: Symmetry transformations (? ◦ f ◦ ? − )( x ) fordifferent M¨obius functions f mapped to the unitinterval using ψ ( x ).In this way we finally obtain an action on the definingboundary points of our tessellation. Using the piecewise-linear function ψ , (? ◦ f ◦ ? − ) can now be mapped to theunit interval. The result is plotted in Fig. 14.The projective special linear group PSL(2 , Z ) may bepresented as (cid:104) a, b | a = b = 1 (cid:105) , (61)where the generators a and b are 2 × a = (cid:18) −
11 0 (cid:19) and b = (cid:18) − − (cid:19) . (62)The transformations considered in Fig. 13 and Fig. 14correspond precisely to a and b according to f ( x ) = − x and f ( x ) = x − x . By composing these two functions (andtheir inverses) we hence obtain a representation of the ac-tion of PSL(2 , Z ) on the circle via piecewise-linear func-tions compatible with the dyadic rationals. The resultingaction on the tessellation is described in the next section. V. ASYMPTOTICALLY DE SITTERSPACETIMES
De Sitter spacetime in (1 + 1) dimensions arises as asolution to 2D Einsten gravity, however this theory istrivial as it reduces to the Euler characteristic. Further,it is impossible to couple to matter fields without con-fining to the ground state or putting up with deleteriousbackreation. A natural generalisation of two-dimensional5gravity which avoids these difficulties is furnished by the
Jackiw-Teitelboim (JT) model [15–17]. This theory al-lows for more possibilities and has nearly-dS solutions.Although JT gravity does not have bulk gravitons, itdoes have boundary gravitons arising from fluctuationsof the asymptotic boundaries. Indeed, when one care-fully formulates JT theory in a de Sitter background oneneeds to include contributions from fluctuating space-time boundaries; the resulting Schwarzian theory has at-tracted considerable interest recently in the context ofthe AdS/CFT correspondence with a microscopic reali-sation via the Sachdev-Ye-Kitaev model. The asymptoticsymmetries of the Schwarzian theory are realised by thegroup (diff + ( S ) × diff + ( S ) / PSL(2 , R ). [19]In this section we propose a tessellation realisation ofasymptotically de Sitter spacetime, and highlight the roleof Thompson’s group T as a candidate for an analogue ofthe corresponding asympototic symmetries. We extendthe action of PSL(2 , Z ) on our tessellation to the action ofThompson’s group T . As a consequence, we then define a nearly-dS tessellation to be the resulting holographicnetwork. A. Thompson’s group T : tessellations for nearlydS Thompson’s group T is a remarkable group of homeo-morphisms of the circle. The role of Thompson’s group T as a discretised analogue of diff( S ) compatible with tes-sellations of S was emphasised by Vaughan Jones [42],with later work supporting this idea in the context of theAdS/CFT correspondence [64, 65]. Following these pro-posals, we exploit Thompson’s group T as an analogue ofdiff( S ) in the context of tessellations of dS. As we argue,this allows for a microscropic realisation of a dS / TN correspondence.Here we recall the definition of Thompson’s group T ;we follow [44] and [45]. Definition V.1.
Thompson’s group T is given by a set ofpiecewise linear homeomorphisms of the unit circle, withthe group operation given by function composition. Theelements of Thompson’s group T are required to satisfythe following conditions: • they are differentiable except at finitely manypoints; • points where they are not differentiable lie at dyadicrational numbers; • when they are differentiable, the derivatives arepowers of two.It may be shown that Thompson’s group T is a finitelypresented infinite group generated by the following threefunctions A ( x ), B ( x ), and C ( x ) (and their inverses):
01 0 1 A ( x ) x
01 0 1 B ( x ) x
01 0 1 C ( x ) x
1. The subgroup
PSL(2 , Z ) and its action on tessellations A remarkable fact [82], supporting the proposal thatThompson’s group T supplies an analogue of diff( S ) ap-propriate for tessellations of dS, is the observation thatthe elements S ( x ) = ( C − ◦ A − )( x ) and C ( x ) generatea subgroup isomorphic to PSL(2 , Z ). These two elementsare defined via S ( x ) = (cid:40) x + , x < x − , ≤ x (63)and C ( x ) = x + , x ≤ x − , ≤ x ≤ x − , ≤ x (64)with function graphs01 0 1 S ( x ) x
01 0 1 C ( x ) x As one may readily observe, these two elements are pre-cisely given by S ( x ) = ψ ◦ ? ◦ − x ◦ ? − ◦ ψ − and C ( x ) = ψ ◦ ? ◦ x − x ◦ ? − ◦ ψ − . (65)It may be graphically verified that the functions S ( x )and C ( x ) also fulfil the relations presented in Eq. (61).To describe this we exploit a representation of the func-tions S ( x ) and C ( x ) via rectangle diagrams [44]. Here thetop of the rectancle is identified with the domain of thefunction and the bottom with the range. Lines are drawnfrom top to bottom, indicating the image of given pointsin the domain. The action is then inferred by linearlyinterpolating the action between the lines. By stackingmultiple rectangle diagrams composition of functions inThompson’s group T may be computed. The elements S and C are represented by the following rectangle dia-grams:6 S ( x ) = C ( x ) =Exploiting this graphical representation the identities( S ◦ S )( x ) = x and ( C ◦ C ◦ C )( x ) = x (66)may be verified:( C ◦ C ◦ C )( x ) =( S ◦ S )( x ) =( C ◦ C ◦ C )( x ) ==( C ◦ C ◦ C )( x ) =( C ◦ C ◦ C )( x ) ==Now we have identified the role played by S and C wecan define the action of PSL(2 , Z ) on our tessellations.We do this by transforming the points x + ∈ I + and x − ∈ I − defining a causal diamond according to (notethat θ − π = θ π = S ) x + → S ( x + ) , x − → ( S ◦ S ◦ S )( x − ) = S ( x − ) , (67a) x + → C ( x + ) , x − → ( S ◦ C ◦ S )( x − ) . (67b)Applying this action to all the causal diamonds definingour tessellation yields, in the case of S and C , the trans-formed tessellations shown in Fig. 15. The full action ofPSL(2 , Z ) is obtained analogously. Although the genera-tor S ( x ) seemingly has no effect on the initial tessellation,this is not the case: it acts as an involution by mappingeach point to the opposite side of the spacetime.The action of the generator C ( x ) on our tessellation isdepicted in Fig. 15b. (Even though only a finite numberof tiles are depicted, one must imagine that the tessella-tion actually extends to infinity.) As one may observe,this transformation yields a tessellation with tiles dis-torted along null geodesics. The distortion arises becausethe induced transformations on the temporal boundaries I + and I − are not identical. In this way causal diamondsare both translated and distorted. Note, however, thatonly a finite number tiles are distorted by the transfor-mation. The remaining tiles are left invariant, and thetessellation is hence almost invariant. This residual ac-tion has consequences for the construction of the physi-cal Hilbert space, and supplies additional constraints on (a) Tessellation after action of the generator S ( x )(b) Tessellation after action of the generator C ( x ) Figure 15: Tessellations after the action oftransformations from the subgroup PSL(2 , Z ).the tensors comprising the holographic network to ensurethat the resulting quantum mechanic system is invariantunder the full isometry group PSL(2 , Z ).A useful mnemonic to understand the action ofPSL(2 , Z ) is to think of the tessellation itself as “repre-senting” the process which propagates information, car-ried by null geodesics, around dS from the past to thefuture boundaries. Thinking of this process as a function f : I − → I + we should, according to an isometry, e.g., C ∈ PSL(2 , Z ), transform the function f according to f (cid:55)→ C ◦ f ◦ ( S ◦ C − ◦ S ) . (68)In the case f = S the action is trivial on f . However,the action is nontrivial in general on tessellations and, aswe’ll see, on the physical Hilbert space H phys .
2. The action of Thompson’s group T : tessellations ofnearly dS Here we describe how to extend the action of PSL(2 , Z )described above to an action of Thompson’s group T . Wepropose that the resulting tessellations should be thoughtof as discretised versions of nearly dS .The definition of an asymptotic symmetry of a space-time, such as dS , is nontrivial. Here we broadly fol-low the book of Wald [83] and the thesis of J¨ager [84],we use the definition provided in [84]. However, sincewe are only using these notions as a motivation for ourdefinition, we won’t dwell on the subtle details. It suf-fices, as motivation, to say that asymptotic symmetriesof nearly dS are determined by diffeomorphisms of the7boundaries I ± , which are conformal with respect to theinduced boundary metric. In our case, the conformalitycondition is trivial (our boundary manifolds are one di-mensional), and we understand asymptotic symmetriesas diffeomorphisms of the past and future boundaries.We should mod out by all isometries to find those sym-metries which are truly nontrivial, hence one finds thatthe group of asymptotic symmetries of dS is given by(diff( S ) × diff( S )) / PSL(2 , R ).Employing the idea that T is the correct substitutefor diff( S ) in the context of a tessellation leads us topostulate that the correct group to describe asymptoticsymmetries of the tessellation is hence G ≡ ( T × T ) / PSL(2 , Z ) . (69)This group acts as follows. Given a representative ( f, g ) (cid:63) ( h, S ◦ h ◦ S ), where f, g ∈ T and h ∈ PSL(2 , Z ), and thegroup operation (cid:63) is elementwise composition( f, S ◦ g ◦ S ) (cid:63) ( h, S ◦ h ◦ S ) ≡ ( f ◦ h, S ◦ g ◦ h ◦ S ) , (70)we obtain a new tessellation by applying f ◦ h to thefuture boundary and S ◦ g ◦ h ◦ S to the past boundary.This operation shifts the endpoints of causal diamonds,leaving us with a new tessellation. Commencing withthe original tessellation, one can generate via this actiona family of tessellations, and thereby, tensor networks,corresponding to elements of the asymptotic symmetrygroup. B. Transforming holographic networks
Given a symmetry transformed tessellation one can di-rectly construct the corresponding holographic networkaccording to Sec. III B: tensors are placed at the cornersand legs from neighbouring generations are contracted tobuild a tensor network. We write M f for the tensor net-work arising in this way from the action of f ∈ PSL(2 , Z ).To explain the construction of M f we focus on theelement C ∈ PSL(2 , Z ), with the generalisation to ar-bitrary f ∈ PSL(2 , Z ) left to the reader. Recall fromSec. III C that H in should be thought of as the “limit” n → ∞ of the finite-dimensional cutoff Hilbert spaces H − n ≡ (cid:78) n − j =0 C D . In this way the operator M is reallythought of as a family of linear operators M m,n mappingfrom the cutoff initial Hilbert space H m associated to atime slice T m to the cutoff final Hilbert space H n asso-ciated with the time slice T n . An explicit example is thetensor network for (cid:96) = 2: Here m = − n = 4, i.e., M − , : ( C D ) ⊗ → ( C D ) ⊗ . Acting with C ∈ PSL(2 , Z ) and building theresulting tensor network yields the following operator:This operator still has 32 input and output legs. How-ever , due to the distortion induced by C on the past andfuture boundaries, these legs are no longer associated toa regular cutoff. That is, the transformation produces anonregular grid. To proceed we exploit the fact that thisoperator is really just a member of an infinite family ofphysically equivalent operators acting on arbitrary cut-offs. We obtain an equivalent operator compatible with aregular (but finer) cutoff via a “partial UV completion”where we add in the tensors (shown in blue) necessary todefine an operator which does act on a regular grid:In this case we have described this network as an operator M Cm (cid:48) ,n (cid:48) with m (cid:48) = − n (cid:48) = 6. (It is possible to givea more compact description with m (cid:48)(cid:48) = − n (cid:48)(cid:48) = 4by removing a layer of isometries; this would then be themost compact description.) Note that the action of theisometry entails an effective change of scale; it is often notpossible to define a transformed tensor network M f whileretaining the original cutoff. The final step to defining M C is then to take the limit m (cid:48) → −∞ and n (cid:48) → ∞ ,i.e., M C “=” lim m (cid:48) →−∞ n (cid:48) →∞ M Cm (cid:48) ,n (cid:48) . (71)It turns out that one can express the transformed op-erator M f as M f ≡ U ( f ) M U † ( θ − π ◦ f ◦ θ π ) , (72)where U ( f ) is a unitary operator whose action we nowdescribe. (The following discussion is an abbreviated ver-sion of the discussion in Sec. 4.4 of [64], and the readeris invited to skim over that paper for a more compre-hensive discussion of unitary representations of Thomp-son’s group T .) We focus again on the case f = C for8simplicity and describe how to construct the unitary op-erator U ( C ) : H in → H in . Intuitively U ( C ) is just theoperator which moves the qudits comprising H in aroundaccording to the function C . E.g., a qudit at location θ = × π gets moved to θ = × π , etc. To actu-ally describe this action we choose an element | φ m (cid:105) of H m ⊂ H in . (Recall that, as described in Sec. III C, twoelements | ψ m (cid:105) and | ψ (cid:48) m (cid:48) (cid:105) with m (cid:48) < m are equivalentin H in if M m (cid:48) ,m | ψ (cid:48) m (cid:48) (cid:105) = | ψ m (cid:105) .) We, for concreteness,set m = − | φ (cid:105) ∈ ( C D ) ⊗ . This initial state isthought of as that of a quantum spin system with the 4spins regularly distributed around the circle:After applying C the 4 spins are moved around and di-lated/contracted: e.g., the second spin is now associatedto the last interval with half its original length, and thethird spin is now the first spin of an interval of twice itsoriginal length. Thus we must now assign this permutedstate | φ (cid:48) (cid:105) of the spins to a nonregular grid :This nonregular grid is no longer comparable with theoriginal regular grid: to find a state of a regular grid weneed to fine-grain the state | φ (cid:48) (cid:105) of the nonregular gridusing the V tensor: | φ (cid:48)(cid:48) (cid:105) ≡ [( V ⊗ V ) V ] ⊗ V ⊗ I | φ (cid:48) (cid:105) (73)This new state is now associated to a regular grid with 8intervals:Thus U ( C ) acts on elements of H − via | φ (cid:105) U ( C ) (cid:55)→ [( V ⊗ V ) V ] ⊗ V ⊗ I | φ (cid:48) (cid:105) , (74)where | φ (cid:48) (cid:105) is the cyclically permuted version of | φ (cid:105) .This action is, in general, nontrivial. To see this wecompare the original state to the transformed version | φ (cid:48)(cid:48) (cid:105) ≡ U ( C ) | φ (cid:105) , i.e., we would like to compute the innerproduct (cid:104) φ | U ( C ) | φ (cid:105) (75)This is not possible within the Hilbert space of the orig-inal grid of 4 spins. Instead we must first fine-grain theoriginal state | φ (cid:105) to an equivalent state of a grid of 8 spinsand then compute the overlap: (cid:104) φ | U ( C ) | φ (cid:105)≡(cid:104) φ | ( V † ⊗ V † ⊗ V † ⊗ V † )[( V ⊗ V ) V ] ⊗ V ⊗ I | φ (cid:48) (cid:105) = (cid:104) φ | V ⊗ I ⊗ V † | φ (cid:48) (cid:105) . (76)Depending on | φ (cid:105) and V this overlap may be nontrivialand the action is nontrivial. The action of U ( f ) for f ∈ T on any state in H − m may be computed in a similar fash-ion and furnishes a unitary representation of Thompson’sgroup T on H in . (An analogous construction works on H out .)It now may be argued that, with this unitary repre-sentation in hand, the tensor network corresponding to atessellation transformed by f ∈ T is given by M f ≡ U ( f ) M U † ( S ) U † ( f ) U ( S ) † . (77)(Here we have used the fact that θ π = S and the factthat U ( f ) U ( g ) = U ( f ◦ g ).) This result allows us torapidly deduce that the transformed tensor network M f is a partial isometry from H in to H out .Note that, when the building-block tensors U and V inEq. (31) are arbitrary, there is no reason to expect that M = M f . This means that, in general, the family ofprojections P f ≡ ( M f ) † M f (78)acting on H in project onto differing subspaces H f phys ≡ P f H in . Either one regards this a gauge degree of free-dom, i.e., one posits that the subspaces H f phys are gaugeequivalent, or one demands that H f phys = H phys (79)for all f ∈ PSL(2 , Z ). This latter point of view imposesadditional constraints on U and V .The conditions on U and V so that Eq. (79) exactlyholds are not presently clear, and we haven’t yet assem-bled enough information to formulate a precise conjec-ture. Instead we find a class of examples where one canargue that H (cid:48) phys ≡ (cid:92) f ∈ PSL(2 , Z ) H f phys (cid:54) = ∅ . (80)These examples are furnished in any braided fusion cat-egory by choosing U to be the braiding operation=and the isometry V to satisfy the pivotality condition=For such tensors one can construct an invariant state | Ω (cid:105) ∈ H in as the infinite regular binary tree9Invariance of | Ω (cid:105) under M follows directly from the isom-etry conditions for U and V and pivotality: M =
Invariance under the action of C relies on the braidingand pivotality conditions: C(M) = which simplifies to | Ω (cid:105) thanks to the following sequenceof moves = = Thus we have shown that H (cid:48) phys is nonempty. VI. CONCLUSIONS AND OUTLOOK
In this paper we have proposed a family of holographictensor networks associated to tessellations of de Sitterspacetime in (1 + 1) dimensions. These tensor networkstake the form of partial isometries mapping from an infinite-dimensional kinematical Hilbert space H in asso-ciated to the negative temporal boundary I − to a kine-matical Hilbert space H out associated to I + . The asso-ciated projection on H in has finite rank and singles outa finite-dimensional subspace of physical states of H in .The de Sitter radius is modelled by subdividing the ini-tial tessellation and implies a direct connection betweenthe quantum information carrying capacity of the ten-sor network and the cosmological constant. The actionof isometries on the network and resulting Hilbert spacewas described, along with conditions on the constituenttensors ensuring that (a subspace of) the physical Hilbertspace is invariant. Finally, asymptotic symmetries wereidentified with Thompson’s group T .This paper only just barely scratches the surface ofpossibilities. Many fascinating and challenging problemsremain. A partial list of intriguing questions includes • Quantum error correction and dS: the holographictensor networks we’ve described here are related to concatenated quantum error correcting codes . Thisresemblence is only superficial at the present stage,and to make more direct contact one requires tessel-lations leading to higher-order tensors (e.g., involv-ing 6-leg tensors). Such a generalization is naturaland should enable results concerning bulk recon-struction. • Bulk reconstruction: can one reconstruct bulk per-turbations at the temporal boundaries? This wouldprobably require Witten’s superobservers , but per-haps employing ideas from quantum error correc-tion could help here. • The tensor networks proposed here enjoy cer-tain degenerate entropy/area relationships. Canone microscopically elucidate a general Ryu-Takayanagi formula for dS? • Higher dimensions: We have only considered (1+1)dimensions here. It is possible to define analogoustensor networks in higher dimensions for noncom-pact spatial manifolds. Regular tessellations forcompactified spatial manifolds are impossible, andnew ideas are needed here. • Black holes: can one associate tensor networks toSchwarzschild de Sitter solutions? • Wheeler de Witt: there is a candidate tensor net-work for the Wheeler de Witt solution (apply thetensor network for dS to a standard holographiccode). Does this microscopic model supply us withnew insights? • JT gravity: is there a natural microscopic realisa-tion of JT gravity using the nearly dS tessellationswe’ve defined here? • Quantum information capacity and the cosmologi-cal constant: is there a deeper connection between0the cosmological constant and the quantum infor-mation capacity of spacetime?We are optimistic that the microscopic perspective intro-duced in this paper will enable progress on some of thesechallenging problems.
ACKNOWLEDGMENTS
Helpful discussions with Jordan Cotler and DenizStiegemann are gratefully acknowledged. This work wassupported, in part, by the Quantum Valley Lower Sax-ony (QVLS), the DFG through SFB 1227 (DQ-mat),the RTG 1991, and funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) un-der Germany’s Excellence Strategy EXC-2123 Quantum-Frontiers 390837967.
Appendix A: Properties of geodesics1. Metric in global coordinates
We write global coordinates and obtain the inducedmetric of dS with respect to this coordinate system asfollowsGlobal coordinates: x x x = sinh τ cos θ cosh τ sin θ cosh τ (A1)Metric de Sitter: d s = − d x + d x + d x (A2)Considering the infinitesimalsd x = d x d τ d τ + d x d θ d θ = cosh τ d τ d x = d x d τ d τ + d x d θ d θ = cos θ sinh τ d τ − sin θ cosh τ d θ d x = d x d τ d τ + d x d θ d θ = sin θ sinh τ d τ + cos θ cosh τ d θ we obtaind s = − cosh τ d τ + (cos θ sinh τ d τ − sin θ cosh τ d θ ) + (sin θ sinh τ d τ + cos θ cosh τ d θ ) = − cosh τ d τ + cos θ sinh τ d τ + sin θ cosh τ d θ + sin θ sinh τ d τ + cos θ cosh τ d θ = (cid:0) − cosh τ + cos θ sinh τ + sin θ sinh τ (cid:1) d τ + (cid:0) sin θ cosh τ + cos θ cosh τ (cid:1) d θ = − d τ + cosh τ d θ = g µν d x µ d x ν (A3)In matrix form: g µν = (cid:18) − τ (cid:19) and g µν = (cid:18) − τ (cid:19) . (A4)
2. Geodesics in de Sitter spacetime embedded inMinkowski spacetime
The geodesics of de Sitter spacetime embedded inMinkowski spacetime can be parametrised using globalcoordinates: x x x = sinh( τ )sin( θ ) ± cos( θ ) sinh( τ )cos( θ ) ∓ sin( θ ) sinh( τ ) (A5)We consider the case of the anticlockwise geodesic.Geodesics embedded in Minkowski spacetime are definedin (3). The anticlockwise geodesicis is x ac ( s ) = su + vsv − us ⇒ x ac , = sx ac , = u + vsx ac , = v − us (A6)The parameters u and v ( u + v = 1) have the followingproperties such that the geodesic fulfils the hyperboloidcondition from Eq. (4): u = x ac , − x ac , x ac , x , v = x ac , + x ac , x ac , x , (A7)We want to show that the proposed anticlockwisegeodesic from Eq. (A5) actually is a geodesic x ac = sinh( τ )sin( θ ) + cos( θ ) sinh( τ )cos( θ ) − sin( θ ) sinh( τ ) = x x x (A8)This is solved for cos( θ ) and sin( θ ). They are identifiedwith the parameters u and v . We know that x = sinh τ . x = sin θ + cos θ sinh τ ⇔ sin θ = x − cos θ sinh τx = cos θ − sin θ sinh τ ⇔ cos θ = x + sin θ sinh τ Thus x = sin θ + ( x + sin θ sinh τ ) sinh τ = sin θ + x x + sin θx ⇒ sin θ = x − x x x x = cos θ − ( x − cos θ sinh τ ) sinh τ = cos θ − x x + cos θx ⇒ cos θ = x + x x x With u = sin θ and v = cos θ this fulfils the condition u + v = 1.1
3. Geodesics in global and conformal coordinates
In this appendix we explicitly verify that the nullgeodesics described in this paper satisfy the geodesicequation. These calculations are elementary, however werepeat them here for completeness.First, the geodesic equation is derived in global coor-dinates. Therefore, the Christoffel symbols need to becalculated:Γ σµν = 12 g σκ (cid:18) ∂g νκ ∂x µ + ∂g µκ ∂x ν − ∂g µν ∂x κ (cid:19) (A9)When calculating the Christoffel Symbols, it can be usedimmediately, that the off diagonal elements of the metricare zero. The only derivative that does not vanish is ∂g θθ ∂τ = sinh( τ ). We get the following Christoffel Symbols:Γ τ ττ = 12 g ττ (cid:18) ∂g ττ ∂τ + ∂g ττ ∂τ − ∂g ττ ∂τ (cid:19) = 0Γ τ θτ = Γ τ τθ = 12 g ττ (cid:18) ∂g ττ ∂θ + ∂g θτ ∂τ − ∂g θτ ∂τ (cid:19) = 0Γ τ θθ = 12 g ττ (cid:18) ∂g θτ ∂θ + ∂g θτ ∂θ − ∂g θθ ∂τ (cid:19) = 12 ( − (cid:18) − ∂ cosh τ∂τ (cid:19) = cosh τ sinh τ Γ θθθ = 12 g θθ (cid:18) ∂g θθ ∂θ + ∂g θθ ∂θ − ∂g θθ ∂θ (cid:19) = 0Γ θθτ = Γ θτθ = 12 g θθ (cid:18) ∂g τθ ∂θ + ∂g θθ ∂τ − ∂g θτ ∂θ (cid:19) = 12 1cosh τ (cid:18) ∂ cosh τ∂τ (cid:19) = sinh τ cosh τ = tanh τ Γ θττ = 12 g θθ (cid:18) ∂g τθ ∂τ + ∂g τθ ∂τ − ∂g ττ ∂θ (cid:19) = 0The general form of the geodesic equation isd x λ d t + Γ λµν d x µ d t d x ν d t = 0 (A10)The geodesic equations for τ and θ can be writtendown: d τ ( t )d t + Γ τ θθ d θ ( t )d t d θ ( t )d t = 0 ⇒ d τ ( t )d t + cosh τ ( t ) sinh τ ( t ) (cid:18) d θ ( t )d t (cid:19) = 0 (A11)d θ ( t )d t + 2 Γ θθτ d θ ( t )d t d τ ( t )d t = 0 ⇒ d θ ( t )d t + 2 tanh τ ( t ) d θ ( t )d t d τ ( t )d t = 0 (A12) This geodesic equation is solved by the following nullgeodesic in global coordinates: (cid:18) θ ( t ) τ ( t ) (cid:19) = (cid:18) ± arctan t + θ arcsinh t (cid:19) t ∈ R (A13)This null geodesic in global coordinates can be trans-formed to conformal coordinates:cos( T ) = 1cosh( τ ) ⇒ T = ± arccos (cid:18) τ (cid:19) The resulting null geodesic in conformal coordinates is (cid:18) θ ( t ) T ( t ) (cid:19) = (cid:32) ± arctan t + θ ± arccos (cid:16) t ) (cid:17)(cid:33) = (cid:32) ± arctan t + θ ± arccos (cid:16) √ t +1 (cid:17)(cid:33) (A14)The parameter t can be substituted with t = tan s with − π < s < π . (cid:18) θ ( s ) T ( s ) (cid:19) = (cid:32) ± arctan(tan s ) + θ arccos (cid:16) √ tan s +1 (cid:17) (cid:33) (A15)For − π < s < π this can be further simplified: (cid:18) θ ( s ) T ( s ) (cid:19) = (cid:18) ± s + θ ± arccos( ± cos s ) (cid:19) , − π < s < π T ( s ) = ± arccos( ± cos s ) = (cid:40) ±| s | , + ± ( π − | s | ) , − we can choose one valid solution for the geodesic. Herebywe know, that the variable T and the parameter s liein the boundaries − π ≤ T ≤ π . The geodesic can bedescribed as (cid:18) θ ( s ) T ( s ) (cid:19) = (cid:18) θ ± ss (cid:19) , − π < s < π ◦ . Appendix B: Basics on tensor networks
Our review of the basic notions of tensor networksfollows the paper by Bridgeman and Chubb [85].The basic building block of any tensor network is atensor. A key defining property of a tensor is its rank:A d -dimensional vector is a rank-1 tensor which is anelement of C d . Similarly, a ( n × m )-matrix is a rank-2tensor which is an element of C n × m . This motivates the2definition of a rank- r tensor with dimensions d × · · · × d r which is an element of C d ×···× d r . The number of indicesof the tensor in index notation and the number of legsin the tensor network notation correspond to the rank ofthe tensor. To illustrate the tensor network notation, welook at an example of a rank-4 tensor:= X ρσµν X ρσ µ ν We interpret lower tensor legs as incoming legs andupper tensor legs as outgoing legs. The direction of thelegs of the tensor is hereby associated with covariant andcontravariant indices in the Einstein notation. The in-coming and outgoing tensor legs are associated with dif-ferent Hilbert spaces. Each tensor is proportional to amap from the Hilbert space associated with the incom-ing tensor legs to the Hilbert space associated with theoutgoing tensor legs. A tensor X † which is adjoint tothe tensor X can be expressed as follows: In the tensornetwork notation upper and lower legs are flipped, suchthat the tensor is mirrored along the constant time sliceit sits on. =( X ρσµν ) † = X σµνρ X † ρσ µ ν Definition B.1. An isometry is a linear map T : H A →H B between the Hilbert spaces H A and H B which pre-serves the inner product.A linear map T : H A → H B can be expressed as fol-lows in index notation, where {| a (cid:105)} is the complete or-thonormal basis of the Hilbert space H A and {| b (cid:105)} is thecomplete orthonormal basis of the Hilbert space H B : T : | a (cid:105) (cid:55)→ (cid:88) b | b (cid:105) T ba (B1)The linear map T is an isometry if and only if (cid:88) b T † a (cid:48) b T ba = δ a (cid:48) a (B2)Graphically, this is represented as follows: TT † = baa (cid:48) aa (cid:48) Definition B.2. A perfect tensor is a tensor T a ··· a n with n indices that is proportional to an isometric tensor from A to A C for any bipartition of its indices into a set A andits complementary set A C with | A | ≤ | A C | .In order to build tensor networks we introduce somebasic tensor operations. Definition B.3.
The tensor product is a generalisationof the outer product of vectors. It is defined as theelement-wise product of the values of each tensor com-ponent:[ A ⊗ B ] i , ··· ,i r ,j , ··· ,j s := A i , ··· ,i r · B j , ··· ,j s In tensor network notation, the tensor product is repre-sented by placing two tensors next to each other.
Definition B.4.
The (partial) trace is a joint summa-tion over two indices of a given tensor that have the samedimension. The following example shows the trace oper-ation for a tensor T where the dimensions d x and d y areequal:[tr x,y ( A )] i , ··· ,i x − ,i x +1 , ··· i r ,j , ··· ,j y − ,j y +1 , ··· j s = d x (cid:88) α A i , ··· ,i x − ,α,i x +1 , ··· i r ,j , ··· ,j y − ,α,j y +1 , ··· j s In tensor network notation, this summation is repre-sented by joining the corresponding tensor legs together: ii jk A tr i = (cid:80) i ii jk A = jk A ii jj A tr = (cid:80) i,j ii jj A = A Definition B.5. A contraction is a tensor product (twotensors are placed next to each other) followed by a tracebetween corresponding indices of the two tensors. In ten-sor network notation this can be represented as follows: (cid:80) i,j i ji j = Appendix C: Tensor networks with (cid:96) (cid:54) = 1 are partialisometries
It can easily be shown, that the tensor networksrepreselting de Sitter spacetime with (cid:96) (cid:54) = 1 are also3partial isometries. The following calculation shows thisfor the tensor network for (cid:96) = 2: M † M = = == =An analogous calculation can be carried out for largervalues of (cid:96) . Appendix D: Matrix representation for the action ofisometries on the future boundary of de Sitterspacetime
In this appendix we explicitly describe an isogeny ofthe group SL(2 , R ) to isometry group SO(1 ,
2) of de Sitterspacetime. We further calculate the induced action of anelement of SL(2 , R ) on null geodesics, and thereby, on thetemporal boundaries.We commence by first finding an explicit expressionfor the parameters s , u and v defining a null geodesic interms of the embedding coordinates of the null geodesic(we consider both anticlockwise pointing null geodesics x ac ( s ) and clockwise pointing null geodesics x c ( s )): x ac ( s ) = su + vsv − us ⇒ x ac , = sx ac , = u + vsx ac , = v − us (D1)Hence we find u = x ac , − x ac , x ac , x , , and v = x ac , + x ac , x ac , x , . (D2)Similarly, for clockwise pointing null geodesics x c ( s ) = su − vsv + us ⇒ x c , = sx c , = u − vsx c , = v + us (D3)we have u = x c , + x c , x c , x , , and v = x c , − x c , x c , x , . (D4)We now analyse the action of a spacetime isometry onthe future boundary of de Sitter spacetime by exploitinga sporadic 2-to-1 homomorphism [86] h : SL(2 , R ) → SO(1 , . This homomorphism is constructed via an auxiliary vec-tor space V defined by the space of real-valued 2 × (cid:104) x, y (cid:105) = tr( xy ) . (D5)We choose a basis of V in terms of Pauli-type matrices: e = (cid:18) − (cid:19) , e = (cid:18) (cid:19) , e = (cid:18) − (cid:19) . (D6)With respect to this basis the bilinear form has ma-trix elements (cid:104) e j , e k (cid:105) , and is manifestly equivalent to theMinkowski metric with matrix representation − , (D7)allowing us to identify Minkowski spacetime R , with V .The action of SL(2 , R ) on the space V is defined by M (cid:55)→ g · M = gM g − , (D8)where g ∈ SL(2 , R ). This action preserves the bilinearform defined in Eq. (D5), because the trace is cyclic and g multiplied with its inverse yields the identity: (cid:104) M, N (cid:105) = 12 tr(
M N ) (cid:55)→
12 tr( gM g − gN g − )= (cid:104) g · M, g · N (cid:105) = 12 tr( M N ) = (cid:104)
M, N (cid:105) . (D9)Thus, according to this homomorphism, SL(2 , R ) isidentified with a copy of the special orthogonal groupSO(1 , g for which g · M = M ⇔ gM g − = M. (D10)Thus the kernel is given by the set of all g ∈ SL(2 , R )commuting with all elements M ∈ V . This is only truefor the elements {± I } , so we have a double covering.We now explicitly calculate the matrix elements of thetransformation h ∈ SO(1 ,
2) corresponding to a givenelement g ∈ SL(2 , R ) with matrix representation g = (cid:18) a bc d (cid:19) . Any M ∈ V can be expressed as a linear combination ofthe basis elements e j . In order to express the function h with regard to this basis, we exploit the homomorphism: e j (cid:55)→ gσ i g − ≡ h j e + h j e + h j e (D11)This leads to the following matrix representation:4 h ( g ) = (cid:0) a + b + c + d (cid:1) (cid:0) a − b + c − d (cid:1) − ab − cd (cid:0) a + b − c − d (cid:1) (cid:0) a − b − c + d (cid:1) cd − ab − ac − bd bd − ac bc + ad . (D12)In order to analyse the action of the isometry group SO(1 , h to a null geodesic x ( s ): x ( s ) = su + vsv − us → x (cid:48) ( s (cid:48) ) = h ( g ) x ( s ) = s (cid:48) u (cid:48) + v (cid:48) s (cid:48) v (cid:48) − u (cid:48) s (cid:48) = x (cid:48) x (cid:48) x (cid:48) (D13)Thus x (cid:48) = (cid:0) ( sv + u ) (cid:0) a − b + c − d (cid:1) + s (cid:0) a + b + c + d (cid:1)(cid:1) + ( su − v )( ab + cd ) (cid:0) ( sv + u ) (cid:0) a − b − c + d (cid:1) + s (cid:0) a + b − c − d (cid:1)(cid:1) + ( v − su )( cd − ab )( v − su )( ad + bc ) + ( sv + u )( bd − ac ) − s ( ac + bd ) . We are interested in the symmetry action the temporal boundaries, from which we infer the transformation rules ofthe tessellation and therefore our holographic network. By considering the limit s (cid:48) → ∞ we obtain the action of h ( g )on the temporal future boundary I + : x (cid:48) ( s ) s (cid:48) = u (cid:48) s (cid:48) + v (cid:48) v (cid:48) s (cid:48) − u (cid:48) −→ s (cid:48) →∞ v (cid:48) − u (cid:48) . (D14)With this expression for the symmetry action on the future boundary of de Sitter spacetime, the parameters u (cid:48) and v (cid:48) can easily be calculated using the relation derived in Eq. (D2). The new parameters u (cid:48) and v (cid:48) satisfy the condition u (cid:48) + v (cid:48) = 1. u (cid:48) = x (cid:48) − x (cid:48) x (cid:48) x (cid:48) ) = − ( − acu + adv + bcv + bdu ) (cid:0) u (cid:0) a − b + c − d (cid:1) − v ( ab + cd ) (cid:1) (cid:16) ( u ( a − b + c − d ) − v ( ab + cd )) + 1 (cid:17) + u (cid:0) a − b − c + d (cid:1) + v (2 cd − ab )2 (cid:16) ( u ( a − b + c − d ) − v ( ab + cd )) + 1 (cid:17) (D15a) v (cid:48) = x (cid:48) + x (cid:48) x (cid:48) x (cid:48) ) = (cid:0) u (cid:0) a − b − c + d (cid:1) − abv + 2 cdv (cid:1) (cid:0) u (cid:0) a − b + c − d (cid:1) − v ( ab + cd ) (cid:1) ( u ( a − b + c − d ) − v ( ab + cd )) + 1+ u ( bd − ac ) + v (2 ad − ( u ( a − b + c − d ) − v ( ab + cd )) + 1 . (D15b) [1] J. Maldacena, Int. J. Theor. Phys. , 1113 (1999).[2] J. Maldacena, Adv. Theor. Math. Phys. , 231 (1998).[3] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys.Lett. B , 105 (1998).[4] E. Witten, Adv. Theor. Math. Phys. , 253 (1998), hep-th/9802150.[5] B. P. Schmidt, N. B. Suntzeff, M. M. Phillips, R. A. Schommer, A. Clocchiatti, R. P. Kirshner, P. Garnavich,P. Challis, B. Leibundgut, J. Spyromilio, A. G. Riess,A. V. Filippenko, M. Hamuy, R. C. Smith, C. Hogan,C. Stubbs, A. Diercks, D. Reiss, R. Gilliland, J. Tonry,J. Maza, A. Dressler, J. Walsh, and R. Ciardullo, ApJ , 46 (1998).[6] T. Banks, Int. J. Mod. Phys. A , 910 (2001). [7] R. Bousso, J. High Energy Phys. (07), 004.[8] R. Bousso, J. High Energy Phys. (06), 028.[9] R. Bousso, J. High Energy Phys. (11), 038.[10] E. Witten, arXiv (2001), arXiv:hep-th/0106109.[11] V. Balasubramanian, P. Horava, and D. Minic, J. HighEnergy Phys. (05), 043.[12] A. Strominger, J. High Energy Phys. (10), 034,hep-th/0106113.[13] V. Balasubramanian, J. de Boer, and D. Minic, Phys.Rev. D , 123508 (2002).[14] D. Anninos, T. Hartman, and A. Strominger, Class.Quantum Gravity , 015009 (2016).[15] R. Jackiw, Nucl. Phys. B , 343 (1985).[16] R. Jackiw, Theor. Math. Phys. , 979 (1992).[17] C. Teitelboim, Phys. Lett. B , 41 (1983).[18] J. Maldacena, G. J. Turiaci, and Z. Yang, J. High EnergyPhys. (1), 139.[19] J. Cotler, K. Jensen, and A. Maloney, J. High EnergyPhys. (6).[20] J. Cotler and K. Jensen, arXiv (2019), arXiv:1911.12358.[21] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 045(2006).[22] M. Rangamani and T. Takayanagi, Holographic entan-glement entropy (Springer Berlin Heidelberg, New York,NY, 2017).[23] M. Van Raamsdonk, Gen. Relativ. Gravit. ,10.1007/s10714-010-1034-0 (2010).[24] B. Swingle, Phys. Rev. D , 10.1103/Phys-RevD.86.065007 (2012).[25] J. Maldacena and L. Susskind, Fortschr. Phys. , 781(2013).[26] D. Harlow, Rev. Mod. Phys. , 015002 (2016).[27] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle,and Y. Zhao, Phys. Rev. D , 086006 (2016).[28] P. Hayden and J. Preskill, J. High Energy Phys. (09), 120.[29] A. Almheiri, X. Dong, and D. Harlow, J. High Energ.Phys. (4), 163.[30] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, J.High Energ. Phys. (6), 149.[31] P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter,and Z. Yang, J. High Energ. Phys. (11), 9.[32] N. Bao, C. Cao, S. M. Carroll, A. Chatwin-Davies,N. Hunter-Jones, J. Pollack, and G. N. Remmen, Phys.Rev. D , 125036 (2015).[33] Z. Yang, P. Hayden, and X.-L. Qi, J. High Energ. Phys. (1), 175.[34] A. Bhattacharyya, Z.-S. Gao, L.-Y. Hung, and S.-N. Liu,J. High Energ. Phys. (8), 86.[35] A. May, J. High Energ. Phys. (6), 118.[36] D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, andK. ˙Zyczkowski, Phys. Rev. A , 032316 (2015).[37] M. Enr´ıquez, I. Wintrowicz, and K. ˙Zyczkowski, J. Phys.:Conf. Ser. , 012003 (2016).[38] Z. Raissi, C. Gogolin, A. Riera, and A. Ac´ın, J. Phys. A:Math. Theor. , 075301 (2018).[39] Y. Li, M. Han, M. Grassl, and B. Zeng, New J. Phys. ,063029 (2017).[40] A. Peach and S. F. Ross, Class. Quantum Gravity ,10.1088/1361-6382/aa6b0f (2017).[41] W. Donnelly, D. Marolf, B. Michel, and J. Wien, J. HighEnerg. Phys. (4), 93.[42] V. F. R. Jones, arXiv (2014), arXiv:1412.7740. [43] V. F. R. Jones, Commun. Math. Phys. ,10.1007/s00220-017-2945-3 (2018).[44] J. W. Cannon, W. J. Floyd, and W. R. Parry, Enseign.Math. , pp. 215 (1996).[45] J. Belk, Thompson’s Group F , Ph.D. thesis, Cornell Uni-versity (2004), arXiv:0708.3609.[46] B. Czech, L. Lamprou, S. McCandlish, and J. Sully, J.High Energ. Phys. (10), 175.[47] C. B´eny, New J. Phys. , 023020 (2013).[48] B. Czech, L. Lamprou, S. McCandlish, and J. Sully, J.High Energ. Phys. (7), 100.[49] X.-L. Qi, arXiv (2013), arXiv:1309.6282.[50] N. Bao, C. Cao, S. M. Carroll, and A. Chatwin-Davies,Phys. Rev. D , 123536 (2017).[51] A. Milsted and G. Vidal, arXiv (2018),arXiv:1812.00529.[52] S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, andP. Witaszczyk, Commun. Math. Phys. , 1019 (2017).[53] S. S. Gubser, arXiv (2017), arXiv:1705.00373.[54] D. Harlow, S. H. Shenker, D. Stanford, and L. Susskind,Phys. Rev. D , 063516 (2012).[55] W. de Sitter, Proc. Kon. Ned. Acad. Wet. , 1217(1917).[56] W. de Sitter, Proc. Kon. Ned. Acad. Wet. , 229 (1917).[57] S. W. Hawking and G. F. R. Ellis, The Large Scale Struc-ture of Space-Time (Cambridge University Press, 1975).[58] V. Suneeta, J. High Energy Phys. (09), 040.[59] M. Spradlin, A. Strominger, and A. Volovich, in
Unityfrom Duality: Gravity, Gauge Theory and Strings: LesHouches Session LXXVI, 30 July–31 August 2001 , LesHouches - Ecole d’Ete de Physique Theorique, editedby C. Bachas, A. Bilal, M. Douglas, N. Nekrasov, andF. David (Springer Berlin Heidelberg, 2002) pp. 423–453,arxiv:hep-th/0110007.[60] M. Van Raamsdonk, in
New Frontiers in Fields andStrings: TASI 2015 Proceedings of the 2015 TheoreticalAdvanced Study Institute in Elementary Particle Physics (World Scientific, 2017) pp. 297–351, arXiv:1609.00026.[61] F. Aicardi, arXiv (2007), arXiv:0708.1571.[62] P. Viader, J. Parad´ıs, and L. Bibiloni, J. Number Theory , 10.1006/jnth.1998.2294 (1998).[63] Here τ n denotes the time coordinate in global coordinatescorresponding to T n .[64] T. J. Osborne and D. E. Stiegemann, J. High EnergyPhys. , 1.[65] T. J. Osborne and D. E. Stiegemann, arXiv (2019),arXiv:1903.00318.[66] Care must be exercised here: the infinite tensor productHilbert space should be regarded as the kinematical spacebuilt from a given reference state | Ω (cid:105) and all states builtfrom it by applying local operations.[67] Recall that D is the dimension of the legs of the tensors.[68] We are unwilling, at this premature stage, to posit theconstant of proportionality.[69] R. Bousso, arXiv (2002), arXiv:hep-th/0205177.[70] B. Schumacher and R. F. Werner, arXiv (2004),arXiv:quant-ph/0405174.[71] P. Arrighi, C. B´eny, and T. Farrelly, Quantum Inf. Pro-cess. , 88 (2020).[72] P. Arrighi and C. Patricot, J. Phys. A , L287 (2003).[73] A. Bibeau-Delisle, A. Bisio, G. M. D’Ariano, P. Perinotti,and A. Tosini, Europhys. Lett , 50003 (2015).[74] A. Bisio, G. M. D’Ariano, and P. Perinotti, Found. Phys. , 1065 (2017). [75] F. Debbasch, Ann. Phys. , 340 (2019).[76] J. de Boer, M. P. Heller, R. C. Myers, and Y. Neiman,Phys. Rev. Lett. , 061602 (2016).[77] A. Milsted and G. Vidal, arXiv (2018),arXiv:1807.02501.[78] J. Cotler, X. Han, X.-L. Qi, and Z. Yang, J. High EnergyPhys. (7), 42.[79] R. S. Kunkolienkar and K. Banerjee, Int. J. Mod. Phys.D , 1750143 (2017).[80] A. Bhattacharyya, L. Cheng, L.-Y. Hung, S. Ning, andZ. Yang, Phys. Rev. D , 086007 (2019).[81] K. Nomizu, Hokkaido Math. J. , 253 (1982). [82] A. Fossas, arXiv (2010), arXiv:1006.0508.[83] R. Wald, General Relativity (Univ. of Chicago Press,1984).[84] S. J¨ager,
Conserved quantities in asymptotically de Sitterspacetimes , Diplomarbeit, Georg-August-Universit¨at zuG¨ottingen (2008).[85] J. C. Bridgeman and C. T. Chubb, J. Phys. A: Math.Theor. 10.1088/1751-8121/aa6dc3 (2017), 1603.03039.[86] P. Garrett (2015), available online at