A model of quantum gravity with emergent spacetime

We construct a model of quantum gravity in which dimension, topology and geometry of spacetime are dynamical. The microscopic degree of freedom is a real rectangular matrix whose rows label internal flavours, and columns label spatial sites. In the limit that the size of the matrix is large, the sites can collectively form a spatial manifold. The manifold is determined from the pattern of entanglement present across local Hilbert spaces associated with column vectors of the matrix. With no structure of manifold fixed in the background, the spacetime gauge symmetry is generalized to a group that includes diffeomorphism in arbitrary dimensions. The momentum and Hamiltonian that generate the generalized diffeomorphism obey a first-class constraint algebra at the quantum level. In the classical limit, the constraint algebra of the general relativity is reproduced as a special case. The first-class nature of the algebra allows one to express the projection of a quantum state of the matrix to a gauge-invariant state as a path integration of dynamical variables that describe collective fluctuations of the matrix. The collective variables describe dynamics of emergent spacetime, where multi-fingered times arise as Lagrangian multipliers that enforce the gauge constraints. If the quantum state has a local structure of entanglement, a smooth spacetime with well-defined dimension, topology, signature and geometry emerges at the saddle-point, and the spin two mode that determines the geometry can be identified. We find a saddle-point solution that describes a series of (3+1)-dimensional de Sitter-like spacetimes with the Lorentzian signature bridged by Euclidean spaces in between. Fluctuations of the collective variables are described by bi-local fields that propagate in the spacetime set up by the saddle-point solution.


I. INTRODUCTION
According to Einstein's theory of general relativity, gravity originates from dynamical geometry [1]. While the theory has been extremely successful in explaining a myriad of phenomena in the classical regime, understanding the quantum nature of gravity remains an outstanding problem [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. One notoriously difficult problem in quantum gravity is to tame quantum fluctuations at short distance scales while preserving the essential aspects of the general relativity at long distances [18,19]. The crucial feature that a successful quantum theory of gravity should reproduce in the continuum limit is the diffeomorphism invariance, which largely fixes the theory at long distance scales.
Quantizing fluctuations of geometry in a fixed dimension, either in the form of metric or a new degree of freedom, has provided important insights into quantum gravity. However, this may not give the complete picture. If metric is dynamical, it is natural to posit that dimension and topology of spacetime are also dynamical. In the presence of strong quantum fluctuations of geometry, there is in priori no reason why topology and dimension of spacetime remain well-defined. Ideally, a full theory of quantum gravity should be able to describe phenomena in which not only geometry but also dimension and topology evolve dynamically. A background independent theory in the strongest sense should include all of dimension, topology and geometry as dynamical degrees of freedom : the entirety of spacetime should emerge [20].
Background independent theories can not be local theories [21] because a fixed notion of locality can not be defined without specifying dimension, topology and geometry in the background [22]. Nonetheless, the success of local quantum field theories as a low-energy description of our universe implies that local effective theories should arise as approximate descriptions within states that describe classical spacetimes [23,24]. The degree of locality in the effective theory should be determined with respect to the metric of the classical geometry. Because the classical geometry is state dependent, so is the locality of the effective field theory. Background independent theories from which local effective theories emerge, while being non-local in the strict sense, should have a weaker notion of locality called relative locality [25]. This may be an essential ingredient that needs to be taken into account in understanding quantum gravity.
Understanding what aspect of state determines the notion of distance is pertinent to the question of identifying the microscopic nature of gravity. In this regard, the AdS/CFT correspondence [8][9][10] provides a clue : geometry is nothing but a coarse grained variable that controls entanglement (among other things) in quantum field theories [26][27][28][29][30][31][32]. This suggests an approach to quantum gravity that we adopt in this paper. Here, the fundamental degrees of freedom are ordinary quantum matter defined on a set. The set is what is to become a spatial manifold, but it does not have a fixed structure of manifold. Dimension, topology and geometry of the set are all collective degrees of freedom of the matter defined on the set. In particular, distances between points in the set are determined from the amount of entanglement between points. Two points that are strongly (weakly) entangled are deemed to be close (far) [25,33]. The pattern of entanglement determines the connectivity among points in the set. The connectivity, in turn, determines a manifold, if its pattern exhibits a local structure.
Within this framework, a Hamiltonian that governs the dynamics of the underlying quantum matter naturally induces dynamics of collective variables that describe dimension, topology and geometry of the manifold. As much as the underlying quantum matter is dynamical, the emergent manifold is fully dynamical. One main goal in this program is to construct a Hamiltonian for the matter such that the induced Hamiltonian for the collective variables reduces to that of the general relativity in a classical limit. Such a Hamiltonian must be relatively local if it is to induce background independent dynamics for the collective variables and admit a local effective theory description for perturbative fluctuations around semi-classical states [25]. Since geometric distance is nothing but a collective property of the underlying matter, the effective strength of interactions between points should be state dependent. In a state that describes a geometry that gives rise to a small (large) proper distance between two points, the interaction between them should be strong (weak) in the relatively local Hamiltonian.
Recently, a simple relatively local model has been constructed. In the background independent model, the collective variables that describe dynamics of dimension, topology and geometry are classical in the large N limit, where N is the number of flavours of underlying quantum matter [34].
In different states, the same Hamiltonian acts as local Hamiltonians defined on manifolds with different dimensions, topologies and geometries. However, the model is not a theory of gravity yet because it lacks the diffeomorphism invariance. The goal of this paper is to construct a relatively local model of quantum gravity. For other related approaches to quantum gravity, see Refs. [33,[35][36][37].

A. Conceptual overview
In this section, we provide a conceptual overview without technical details. the Hilbert space are generated by the special linear group, SL(L, R) that multiplies the matrix from the right. Once a frame is chosen, the total Hilbert space can be written as a direct product of the local Hilbert spaces defined in that frame [80]. This is illustrated in Fig. 1.
In a frame, the collection of sites can form a spatial manifold for L 1. However, no structure to the leading order in d(r i , r j ) for any i and j, where d(r i , r j ) is the proper distance between r i and r j in the Riemannian manifold and ξ is a constant.
of manifold, not even its existence, is fixed in the theory. Instead, the existence of spatial manifold and its structure, if exists, depend on quantum state of the matrix in the following way. Suppose we compute the mutual informations between all pairs of sites in a frame. The mutual information between sites i and j is given by I ij = S i + S j − S i∪j , where S A is the von Neumann entanglement entropy of subset A. Then we ask if there exists a Riemannian manifold into which the sites can be embedded such that − ln I ij is proportional to the proper distance between the images of i and j in the Riemannian manifold to the leading order in the proper distance. If there exists such a Riemannian manifold, we say that the state has a local structure (see Fig. 2). Roughly speaking, a state with a local structure is a short-range entangled state when viewed as a state defined on a Riemannian manifold in which sites are embedded. A classical notion of manifold exists only for states with local structures. Any spatial manifold can emerge in this manner. Some examples are given in Fig. 4. Here, dimension, topology and proper volume of space are order parameters that encode different patterns of entanglement.
In order to identify physical Hilbert space, we need to construct constraints that generate gauge symmetry. We start with the gauge group that generalizes the spatial diffeomorphism. The immediate issue that we face is that the theory does not assume a manifold with fixed dimension and topology. Because a spatial manifold with any dimension can dynamically arise, the gauge group must be general enough to include spatial diffeomorphism in arbitrary dimensions. With this in and is a space symmetry. As SL(L, R) can rotate a frame into any other frame, it can generate diffeomorphism in any manifold. SL(L, R) generators that give rise to smooth diffeomorphisms in the continuum limit can be identified in any dimension as is discussed in Sec. III B. Therefore, we take SL(L, R) as the generalized spatial diffeomorphism group.
An unavoidable consequence of the enlarged gauge symmetry is that SL(L, R) also includes transformations that are non-local in a given coordinate system chosen by the local structure of a state. Under an SL(L, R) transformation that mixes two columns through linear superpositions, the set of local sites is not preserved. A local Hilbert space in one frame is made of linear superpositions of states defined in multiple local Hilbert spaces in another frame. Since the total Hilbert space can be written as direct products of local Hilbert spaces in any frame, there is no preferred frame. Consequently, whether a state has a local structure or not is not gauge invariant. Even if a state has a local structure of entanglement in one frame, it does not have a local structure in a rotated frame in general. In order to define a gauge invariant notion of local structure, we first need to choose a frame in a gauge invariant manner. Although there is no preferred frame fixed in the background, one can define a set of local Hilbert spaces in terms of physical degrees of freedom within the theory. This can be done in states in which the O(M ) flavour symmetry is spontaneously broken to a smaller group by a non-zero expectation value in an L × L block of the M × L rectangular matrix. If the square matrix formed by the L × L block is non-singular, a frame can be defined in terms of the row vectors in the block. Since it provides a physical reference with respect to which the notion of local Hilbert spaces are defined, the local structure defined in this frame is gauge invariant.
To fully specify the physical Hilbert space, one also needs to define a Hamiltonian constraint.
In the general relativity, the Hamiltonian density forms the representation of a scalar density under the spatial diffeomorphism group. In the present theory, the Hamiltonian should form a representation under the generalized spatial diffeomorphism group, SL(L, R) . The minimal representation that an O(M ) invariant Hamiltonian forms under SL(L, R) is a rank 2 symmetric tensorial representation. Accordingly, the lapse function is generalized to a rank 2 symmetric tensor, which is called lapse tensor.
In the general relativity, the lapse function controls the lapse of proper time at each location in space, and there are one scalar function worth of ways to generate multi-fingered time evolutions.
In the present theory, there are more ways of generating time evolutions because of the off-diagonal elements of the lapse tensor. A general lapse tensor can be rotated into a diagonal form using an SL(L, R) transformation. The off-diagonal elements of the lapse tensor encode the information on the frame in which lapse tensor is put into a diagonal form. Therefore, the lapse tensor determines not only the speed of local time evolutions but also the frame in which the time evolution is generated. In priori, one can describe time evolution in any frame, and there is no preferred lapse tensor. A gauge invariant notion of time evolution can only be defined in terms of physical clocks made of dynamical degrees of freedom within the theory. If a set of unentangled clocks are prepared out of local degrees of freedom in a frame, the Hamiltonian with a lapse tensor diagonal in that frame should evolve those clocks independently. When the whole system that includes the clocks and other degrees of freedom is evolved with the Hamiltonian, the evolution of the other degrees of freedom relative to the local clocks describes the physical time evolution. The correlation between the evolution of the local clocks and the remaining degrees of freedom is the gauge invariant content of the theory. Different choices of local clocks give rise to different relative motions.
In order to make sure that the Hamiltonian is background independent, and evolves unentangled set of clocks independently, the Hamiltonian must be relatively local in the frames in which the lapse tensor is diagonal. For the background independence, any two sites can couple with each other as there is no fixed notion of locality. However, the effective strength of coupling between two sites should be determined from the entanglement present between the sites because the proper distance between the points is determined from the entanglement. If two points are weakly (strongly) entangled, the coupling between them are weak (strong In order to describe the process of gauge projection, it is useful to consider a sub-Hilbert space that a given 'initial state' explores through the evolution generated by gauge transformations. The projection of a state with a finite norm in the kinematic Hilbert space to a gauge invariant state can be expressed as a path integration over the collective variables that describe fluctua-tions of spacetime. In the path integration for the collective variables, spacetimes with different dimensions, topologies and geometries are summed over non-perturbatively. In the limit that M L 1, the path integration can be replaced with a saddle-point. If the state with a finite norm has a D-dimensional local structure, a (D + 1)-dimensional spacetime manifold emerges at the saddle-point. The collective variables, which are bi-local in space, can be viewed as an infinite tower of local fields that includes a spin 2 mode. In the classical limit, the constraint algebra for the collective variables reduces to the hypersurface deformation algebra of the general relativity to the leading order in the gradient expansion if all other modes except for the spin 2 mode is turned off. From the constraint algebra, one can identify the emergent metric degree of freedom as a composite of the dynamical collective variables. The metric identified from the algebra confirms the idea that the proper distance between sites is determined from the entanglement such that strongly entangled sites are physically close to each other. As a bi-product, we also point out that the emergent geometry captures only a partial information of the full entanglement pattern. There exists non-geometric entanglement that is encoded in higher-spin modes.
The saddle-point configuration provides a classical spacetime on which fluctuations of collective variables propagate. The propagating modes are bi-local fields that are described by an effective theory whose locality is determined with respect to the classical geometry set by the saddle-point configuration.

B. Outline
Here is an outline of the rest of the paper. In Sec. II, the kinematics of the theory is discussed.
We define the full Hilbert space from which the kinematic Hilbert space and the physical Hilbert space are to be defined. We also define the inner product, and introduce the notion of frame. In Sec. III, we first review the Hamiltonian formulation of the general relativity as we will use the Hamiltonian formalism in this paper. We then construct the generalized momentum and Hamiltonian constraints. From an explicit computation of the commutators between the constraints, it is shown that the constraints obey a first-class operator algebra. In Sec. IV, we define the kinematic   it is shown that the constraint algebra of the present theory reduces to that of the general relativity in a special case. Based on the algebra that the momentum and Hamiltonian constraints obey in the continuum limit, the emergent metric degree of freedom is identified in terms of the collective variables. In the limit that the size of the matrix is large, the dynamical collective variables become classical. In Sec. VI, the saddle-point equation of motion for the collective variables is derived.
The equation of motion is solved both numerically and analytically in Sec. VII for an initial state that exhibits a three-dimensional local structure. We find a solution which describes a series of  We consider an M × L real rectangular matrix, Φ A i with A = 1, 2, .., M and i = 1, 2, .., L in the M L 1 limit. The full Hilbert space is spanned by The inner product between basis states is given by The conjugate momentum denoted asΠ i A satisfies the standard commutation relation, represents the M × L (L × M ) operator valued matrix. The row index A is referred to as flavour index. In this paper, we consider a model that has the O(M ) flavour symmetry generated bŷ where the flavour indices are raised or lowered with the Euclidean metric :Φ Ai =Φ A i . The flavour symmetry acts on Φ (Π) from the left (right) as whereõ is an anti-symmetric matrix and can be constructed as composites of the following bi-linears, whereΦ T (Π T ) denotes the transpose ofΦ (Π). Products of operator valued matrices are defined in the usual way, e.g., Henceforth, all repeated indices are understood to be summed over unless mentioned otherwise.

B. Frame
The column index i is referred to as site index as it labels points of space in the model of gravity to be constructed. Once we identify i as a site index, it is natural to write the total Hilbert space as a direct product of local Hilbert spaces as where H i is the local Hilbert space spanned by ⊗ A Φ A i . Such a decomposition of the total Hilbert space is called a frame.
The total Hilbert space can be decomposed in different frames. For example, one can use a different set of basis states that are related to the original basis states through whereΦ A i = g I i Φ A I and g ∈ SL(L, R). The inner product is preserved because where det g = 1 is used. This allows one to represent Φ = ⊗ A,I Φ A I , where ⊗ A Φ A I spans the local Hilbert space H I at site I in the rotated frame. The total Hilbert space can be written as a direct product of the local Hilbert spaces in the rotated frame as H = ⊗ I H I .
A frame, denoted as X, is defined by a set of L linearly independent row vectors in R L : ., e L ) = 1, i = 1, 2, .., L}, where V ( e 1 , e 2 , .., e L ) is the Euclidean volume of the parallelepiped formed by the L vectors. This is illustrated in Fig. 3. If X is a frame, X g = { e i g i = 1, 2, .., L} is also a frame for any g ∈ SL(L, R). A frame defines a set of local Hilbert spaces of which the total Hilbert is decomposed as a direct product. Associated with a frame, one can define a set of local observers : a local observer at site i in frame X has access to H X i , where H X i is the local Hilbert space defined at site i in frame X. The Hilbert space accessible to a local observer in one frame is comprised of linear superpositions of states accessible to multiple local observers in another frame. There is a priori no preferred frame and thus no preferred set of local observers.

C. Local structure
In a given frame, one can define entanglement formed across local Hilbert spaces. In the presence of a local structure of entanglement, a spatial manifold can be defined from the pattern of entanglement. A state is defined to have a local structure in a frame if there exists a mapping from the sites to a Riemannian manifold such that the mutual information between any two sites decays exponentially in the proper distance between the images of the sites in the Riemannian manifold to the leading order in the proper distance (see Fig. 2) [81]. The dimension, topology and metric of the manifold are collective properties of a state. A state with a local structure can be regarded as a short-range entangled state with respect to the corresponding spatial manifold. For general states, local structure does not exist. The existence of local structure is a dynamical property that only a sub-set of states possess. Dimension, topology and geometry are order parameters that differentiate different local structures.
In order to illustrate the idea, we consider a set of O(M ) invariant states labeled by a collective variable, where t ij is a complex L × L collective variable. If the off-diagonal elements of t are smaller in magnitude than the diagonal elements with Imt ii > 0, the mutual information between sites i and j is given by [38] I ij = 2M − ln |t ij | 2 4Imt ii Imt jj + 1 |t ij | 2 4Imt ii Imt jj + ... (9) to the leading order in t ij Imt ii .
Here ... represents higher order terms that include n+m>0 k 1 ,..,kn l 1 ,..,lm . Sites that are not directed connected by a non-zero collective variable are entangled via multiple legs of the bi-local collective variables. Obviously, inter-site entanglement can not exist if t is diagonal. The off-diagonal elements of the collective variable describe 'bonds' that create inter-site entanglement, where the strength of the bond between sites i and j is proportional to the magnitude of t ij . If the shortranged entanglement bonds form a regular lattice (similar to the way a lattice is formed by chemical bonds in solids), the corresponding state has a local structure that exhibits a manifold with a well-defined dimension and topology. We will later see how the emergent geometry is determined from the collective variables as well. Intuitively, the geometry is determined such that the proper distance between two points gets smaller if the two points are connected by stronger entanglement bonds (larger t ij ).
As a second example, let us consider the collective variable given by where with x being the largest integer equal to or smaller than x. In this state, the entanglement bonds form a square lattice. It exhibits a two-dimensional local structure with the topology of a disk. Eq.
(13) is a natural coordinate system in which the local structure is manifest. These are illustrated in The existence of local structure depends on the choice of frame. Under a change of frame, the collective variable is transformed as t IJ = g I i g J j t ij , where g ∈ SL(L, R). Even if a state exhibits a local entanglement structure in one frame, it does not have a local structure in another frame if the latter is related to the former through a transformation that is non-local with respect to the locality defined in one frame. Generic states with bonds that form a global network do not exhibit a local structure.

III. GAUGE SYMMETRY
In this section, we construct the generators of the gauge symmetry that generalizes the spacetime diffeomorphism of the general relativity. Since we are going to use the Hamiltonian formalism, we first review the Hamiltonian formulation of the general relativity.

A. Review of the Hamiltonian formalism of the general relativity
In (3 + 1) dimensions, the action of the general relativity can be written as [39] Here a four-dimensional spacetime is decomposed into a stack of three-dimensional spatial manifolds that are labeled by coordinate time τ . A point within each time slice is labeled by r. g µν and π µν with µ, ν = 1, 2, 3 are the spatial metric and its conjugate variable, respectively. The sym- is the momentum density that generates spatial diffeomorphism within a spatial manifold. ξ µ (r) is the shift that specifies an infinitesimal spatial diffeomorphism.
H(r) is the Hamiltonian density that generates local time translation. θ(r) is the lapse that determines the position dependent time translation. The shift and the lapse can be chosen arbitrarily.
Consequently, the momentum and Hamiltonian, become constraints. The key property of the general relativity is that the entire dynamics is generated by the constraints that satisfy the hypersurface deformation algebra [40], Here L ξ represents the Lie derivative with respect to the vector field ξ and The signature of spacetime is chosen to be (S, +, +, +).
The discrepancy between the two is generated by the shift given in Eq. (19) to the order of 2 due to the Jacobi identity. This relation has some dynamical information because Eq. (19) depends on the spatial metric and the signature of spacetime. In other words, the discrepancy is compensated by different shifts in different states. This will be important in identifying the emergent metric degree of freedom in our theory later in this paper.
The Hamiltonian and momentum form the first-class constraint algebra classically, which is crucial to guarantee that the constraints are preserved under the evolution generated by the constraints themselves. The constraint algebra largely fixes the form of the Einstein-Hilbert action.
Up to two derivative order, the Einstein-Hilbert action is the only theory that satisfies Eqs. (16), (17) and (18) [40,41]. In quantum gravity, the constraints are to be promoted to operators that satisfy a first-class operator algebra. The challenge is to regularize the constraints in a way that they satisfy a first-class algebra at the quantum level which is reduced to Eq. (16)-(18) in the classical limit.
In the following two subsections, we construct momentum and Hamiltonian constraints that generate generalized spacetime diffeomorphism in the absence of manifold with fixed dimension and topology. We impose the O(M ) flavour symmetry, and the constraints are built out of the bi-linears in Eq. (4).

B. Momentum constraint
Because dimension and topology of spatial manifold are not fixed, spatial diffeomorphism needs to be generalized to a group that includes diffeomorphism in any dimension in the limit that L is large. For any pair of sites i and j, there should exist a generator that maps i to j because there are states with local structures in which the two sites are close to each other. The desired gauge group is the special linear group (SL(L, R) ) introduced in Sec. II B that generates rotations of frame.
The first operator in Eq. (4) generates the general linear transformation. The Hermitian generator of GL(L, R) is given byĜ where i, j = 1, 2, ..., L. The GL(L, R) generators can be decomposed into (L 2 − 1) generators for and one for the global dilatation,Ĝ Because the inner product is preserved under SL(L, R) as is shown in Eq. (7), we pick SL(L, R) as the gauge group that generalizes the spatial diffeomorphism. SL(L, R) generators can be written asĜ where y is a traceless L × L real matrix. The SL(L, R) transformations act on Φ (Π) from the right (left) as where g y = e −y ∈ SL(L, R) . Under SL(L, R) , Φ (Π) transforms covariantly (contravariantly).
Now we examine how SL(L, R) acts on the matrix in states which have local structures. In the presence of a local structure, one can define a manifold into which sites are embedded. Let r i represent the point in the spatial manifold associated with site i. The matrix Φ A i is then viewed as a field Φ A (r i ) defined at position r i . For an infinitesimal SL(L, R) transformation with g y = e − y in Eq. (24), the field transforms as Let us consider field configurations that change slowly on the manifold in the continuum limit with L 1. In this case, a gradient expansion can be used to write . [82], and Eq. (25) becomes Here represents a scalar and a vector fields, respectively, associated with y. In Eq. (26), ... represent higher derivative terms. The scalar field determines the position dependent rescaling of the field (Weyl transformation) [42]. The vector field describes spatial diffeomorphism. From this, we see that SL(L, R) includes spatial diffeomorphism for slowly varying fields in states that possesses local structures. We callĜ and y the momentum constraint and the shift tensor, respectively.
A few remarks are in order. First, the diffeomorphism induced by SL(L, R) is an active transformation. Eq. (26) shows how the field is actively 'dragged' under SL(L, R) in a fixed coordinate system. Second, SL(L, R) includes smooth diffeomorphism of any dimension in the large L limit.
Once a D-dimensional coordinate system is chosen by the local structure of a state, there exists a set of shift tensors that generate general diffeomorphism in D-dimensional. For example, the state in Eq. (8) with Eq. (10) has the one-dimensional local structure. The shift tensor given by gives rise to a vector field ξ(r i ) = ξ i in the coordinate system given by Eq. (11). This generates an one-dimensional diffeomorphism in the continuum limit. The state with Eq. (12) has a twodimensional local structure which is manifest in the coordinate system given by Eq. (13). The shift tensor, gives rise to a two-dimensional diffeomorphism generated by the vector field ξ µ (r i ) = (ξ 1 i , ξ 2 i ) on the two-dimensional manifold in the continuum limit. These examples show that SL(L, R) include general diffeomorphism in arbitrary dimensions. Third, D-dimensional diffeomorphisms act locally only in states with a D-dimensional local structure. In general, a SL(L, R) transformation that generates a D-dimensional diffeomorphism acts as a non-local transformation in states with local structures with different dimensions. For example, the shift tensor in Eq. (30) generates a non-local transformation in the one-dimensional manifold given by Eq. (11), while it generates a local diffeomorphism in the two-dimensional manifold in Eq. (13). A transformation that maps site 1 to site L is quasi-local in Fig. 4(b) but non-local in Fig. 4(a) in the continuum limit.

C. Hamiltonian constraint
Having identified the generator for the spatial diffeomorphism, we now construct the Hamiltonian density. In the general relativity, the Hamiltonian density forms a representation of scalar density under spatial diffeomorphism. Each element in the representation generates one of many- Even if the kinetic term is local in one frame, it is generally not in other frames. In order to construct a kinetic term that forms a representation of SL(L, R) , we need to include the full L × L matrix,ΠΠ T which forms a rank 2 contravariant symmetric representation of SL(L, R) . The most general kinetic operator is labeled by a rank 2 symmetric tensor v aŝ Under SL(L, R) , v transforms as v → g −1T y vg −1 y . The set ofĥ 1,v with symmetric v forms a representation under SL(L, R) . We refer to v as the lapse tensor as it plays the role of the lapse function in the general relativity. Before we discuss the meaning of the lapse tensor further, let us complete the construction of the Hamiltonian. We need to add a hopping operator that becomes gradient term (|∇Φ| 2 ) in the continuum limit. As a part of the Hamiltonian that is added to Eq.
(31), it should also form a rank 2 symmetric contravariant tensorial representation of SL(L, R) .
The bilinear, Φ A i Φ A j is a candidate of the hopping term, but it is a covariant tensor not a contravariant tensor. In order to convert it into a contravariant tensor, the site indices should be raised witĥ ΠΠ T . The minimal hopping term that includes Φ A i Φ A j and transforms in the desired representation isĥ The factor of M −2 is introduced in Eq. (32) to make sure that bothĥ 1,v andĥ 2,v scale as O(M ) in the large M limit. We combineĥ 1,v andĥ 2,v to write the Hamiltonian asĤ v =α 1ĥ1,v +α 2ĥ2,v , whereα 1 andα 2 are dimensionless parameters. We chooseα 2 > 0 such that the Hamiltonian is bounded from below for large Π. Furthermore,α 1 < 0 is chosen so that the space of configurations that satisfy the constraintĤ v = 0 is non-trivial in the classical limit [83]. Without loss of generality, one can setα 1 = −1. The full Hamiltonian with lapse tensor v is written aŝ In order to understand the meaning of the Hamiltonian, it is convenient to go to the frame in which the lapse tensor is diagonal. A non-singular lapse tensor can be written as where n v is a positive number, g v ∈ SL(L, R) and S v is a diagonal matrix whose elements are either 1 or −1. See Appendix A for the proof. Then, Eq. (33) can be written aŝ whereΦ =Φg −1 v andΠ = g vΠ . Henceforth, let us omit the prime signs. Here S i determines the direction of local time evolution at each site i. The first term in the bracket of Eq. (35) is an ultra-local kinetic term. The second term describes a hopping process, where a particle jumps from sites j to k (and vice versa) with a hopping amplitude proportional to (Π j The hopping amplitude between two sites is given by the amplitudes of bi-local operators that connect the two sites through a third site i as is shown in Fig. 5. In the large M limit, the bi-linear operator (Π j AΠ i A ) has a well-defined expectation value for O(M ) invariant states. For the state in Eq. (8), the expectation value of the bi-local operator is given by the collective variable The collective variable t ij in turn controls the mutual information between sites i and j through Eq. (9). Therefore, Eq. (35) describes a relatively local kinetic term whose hopping amplitude is determined from the entanglement present between sites [34]. Because entanglement is state dependent, so are the hopping amplitudes and the graph that is formed by the network of hopping amplitudes. Since the underlying matrix is dynamical, the emergent manifold that is formed by the entanglement bonds is fully dynamical. In a globally entangled state, the Hamiltonian acts as a non-local Hamiltonian with global hoppings. In a state with a local structure, the Hamiltonian acts as a local Hamiltonian in the corresponding dimension set by the local structure [25]. For this reason, the dimension, topology and geometry are all dynamical in this theory [84].
For each choice of the lapse tensor, the Hamiltonian is relatively local in the frame in which the lapse tensor is diagonal. The gauge freedom in the choice of lapse tensor includes not only the freedom to choose site dependent speed of time evolution in a given frame [85] but also the freedom to rotate the frame in which the lapse tensor is diagonal. The space of lapse tensors in the present theory is much larger than that of lapse functions in the general relativity. In the general relativity, The commutator between Hamiltonians is more complicated. But, the form of the Hamiltonian in Eq. (33) suggests that the commutator is proportional to the generators of GL(L, R) because the only non-trivial commutator arises from i . An explicit calculation shows that the commutator actually depends only on the SL(L, R) generator and the Hamiltonian itself (see in the large M limit. The first two terms in Eq. (40) are O(M ). The last term that depends onĤ is O(1), and is sub-leading in the large M limit [86]. The last term is generated as operators are ordered such thatĜ appears at the far right in the first two terms. This ordering makes it manifest that states annihilated byĜ andĤ are automatically annihilated by their commutators.
Therefore, no additional constraints are needed to define the space of gauge invariant states. In short, the momentum and Hamiltonian constraints form a first class algebra [43]. On the other hand, quantum states to which probabilities can be assigned should be normalizable. Wavefunctions of normalizable states are localized within compact regions in the gauge orbit, breaking gauge symmetry [25]. The incompatibility between gauge invariance and normalizability gives rise to a non-trivial evolution as normalizable states are projected toward a gauge invariant state [44,45]. Let us denote a normalizable state as χ . The projection of χ to a gauge invariant state, 0 is given by This can be viewed as the wavefunction of 0 written in the basis of χ [46]. In the following, we show that Eq. (42) q represents an L × L matrix that acquire non-zero expectation values. A factor of √ N is introduced as we will consider the large N limit with q ∼ O(1). Under generalized spatial diffeomorphisms, q is transformed as q → q g, where g ∈ SL(L, R), and plays the role of a Stueckelberg field. If q contains L independent row vectors, one can define a frame in terms of those vectors.
The local structure defined in this frame is gauge invariant. φ and ϕ represent N/2 × L matrices that have zero expectation value. Henceforth, we use q, φ, ϕ in place of Φ .
In the frame chosen by the Stueckelberg field, sites are labeled by distinguishable physical flavour. However, it is not necessary to have distinguishable sites to define a frame. It is sufficient to have L unordered independent vectors. In order to define an unordered set of row vectors, we consider states in which S f L symmetry is unbroken, where S f L is the permutation that acts on the first L flavours. We refer to S f L as the flavour permutation group. We denote the sub-Hilbert space with unbroken S f L × O(N/2) × O(N/2) as V. General states in V can be parameterized in terms of collective variables. In order to construct basis states for V, it is convenient to introduce , where the flavour a is summed from 1 to N/2 in the last two operators. Therefore, states in V can be spanned by the following basis states labeled by three collective variables, is invariant under flavour permutations, so is s, t 1 , t 2 , for any P f ∈ S f L . For a later use, we also introduce permutations of sites which act on the site index of the collective variables as The site permutation group with the even parity, denoted as S g L , is a subgroup of the generalized spatial diffeomorphism group, SL(L, R) .
Here Ds ≡ i,α ds i α , Dt ≡ i≥j dt ij 1 dt ij 2 , and the integrations of ds i α and dt ij c are defined along the real axis. χ(s, t 1 , t 2 ) is a wavefunction of the collective variables. The states in Eq.
(48) form a complete basis of V. This sub-Hilbert space forms the kinematic Hilbert space of the present theory.

C. Path integration of collective variables
Now we consider the projection of a normalizable state in V to a gauge invariant state in Eq. (42) in the limit that N L 1. Thanks to the gauge invariance of 0 , the overlap in Eq. (42) is invariant under gauge rotation [38], where v (1) is a lapse tensor and y (1) is a shift tensor. IfĜ andĤ are applied to the right in Eq.
It is straightforward to check that Eq. (53) is reduced to Eq. (50) upon integrating out the collective variables. Upon integrating over s (1) and t (1) c , which play the role of dynamical sources, one obtains the delta functions that enforce the constraints for the conjugate variables, q (1) = q, The following integration over q (1) and p    . The resulting state can be again expressed as a linear superposition of s, t 1 , t 2 , which is expressed as an integration over a yet another set of dynamical collective variables. Repeated insertions of the Hamiltonian and momentum give rise to a path integration of the collective variables [47], where Here Ds ≡ ∞ l=1 ds (l) and s(τ ) = s (l) with τ = l . Dq, Dt, Dp, Dv, Dy, and q(τ ), t c (τ ), p c (τ ), v(τ ), y(τ ) are similarly defined. Here τ is a parameter time that labels different stages of evolution generated by the momentum and Hamiltonian constraints. Here q and p c play the role of generalized coordinates and s and t c are their conjugate momenta [87]. The sum over the shift and lapse tensors in Eq. (54) represents different paths in which the state in the kinematic Hilbert space is projected to the gauge invariant state. A particular path of the shift and lapse tensor represents one of the multi-fingered time evolutions. This is illustrated in Fig. 7 The collective variables t c (τ ), p c (τ ), s(τ ), q(τ ) are generalized gravitational degrees of freedom that describe fluctuations of dimension, topology and geometry of spacetime. In the following section, we derive the precise relation between the collective variables and the metric of the emergent geometry.
where δ kl ij = 1 2 δ k i δ l j + δ l i δ k j . To the leading order in 1/N , the Poisson brackets of Eq. (51) and Eq. (52) are given by (see Appendix D) where with In the last equation of Eq. (60), the indices within brackets are symmetrized, e.g., where F = {s, q, t, p} denote the collective variables, and η j i (τ ) and ρ ij (τ ) are infinitesimal gauge parameters. The action is invariant off-shell as the equation of motion is not needed for the invariance of the action. Besides the spacetime diffeomorphism, the theory is also invariant under the reversal of the parameter time, Let us count the number of physical degrees of freedom. q and s are L × L matrices, and

V. SPACETIME DIFFEOMORPHISM AND EMERGENT GEOMETRY
The path integral in Eq. (54) consists of two parts. The first is the integration over v(τ ) and y(τ ). Each path of the lapse and shift tensors selects one of the multi-fingered time evolutions.
Because of the gauge invariance in Eq. (62), the shift and lapse tensors need to be fixed through a gauge fixing condition. This will be discussed in the next section. The remaining path integration is over the collective variables. Each path describes a history of the collective variables that represents a spacetime which emerges dynamically. In the large N limit, the fluctuations of the collective variables become small, and the saddle-point approximation can be made. If the initial state χ has a local structure in a frame, a spacetime manifold with well-defined dimension, topology and geometry emerges at the saddle point. In this section, we discuss how the geometry of the emergent manifold is determined from the collective variables. For this, we first extract the constraint algebra of the general relativity in Eqs. (16)- (17) from Eqs. (57)-(59).

A. Momentum density
We first identify the generators of smooth spacetime diffeomorphism for states that have local structures. Let r µ i be the mapping from sites to a manifold that is determined from the local structure in a frame. The tensor G i j can be viewed as a bi-local field that depends on two positions on the manifold. If the collective variables change slowly in the manifold, G i j varies slowly as a function of r i and r j , and can be expanded in coordinates. By expanding G i j around j = i, we write the SL(L, R) generator with shift tensor y as where ζ y and ξ µ y represent the scalar and vector fields defined in Eqs. (27)-(28) associated with shift tensor y. In the continuum limit, Eq. (65) is written as G y = dr D(r)ζ y (r) + P µ (r)ξ µ y (r) + .. , where V i is the coordinate volume assigned to site i in the manifold (see Fig. 4 57) implies {G x , G y } P B = G yx−xy , which can be written as dr D(r)ζ x (r) + P µ (r)ξ µ x (r) + .. , dr D(r )ζ y (r ) + P ν (r )ξ ν y (r ) + ..
O(∂ 2 ) denotes terms that involve two or more derivatives. Eqs. (70) and (71) imply that P µ defined in Eq. (68) generates spatial diffeomorphism under which D and P µ transform as a scalar density and a vector density, respectively. Eq. (16) is indeed reproduced to the leading order in the derivative expansion.

B. Hamiltonian density
Now we write the Hamiltonian in the continuum limit. In a frame chosen by the local structure of a state, we divide the lapse tensor into the diagonal components and the off-diagonal components as In the continuum limit, Eq. (72) can be written as Here, θ v (r i ) = v ii (75) are identified as the Hamiltonian density and the lapse function of the general relativity. The off-diagonal Hamiltonian and the off-diagonal lapse function are given by In the second term of Eq. drdr H (2) (r, r )λ v (r, r ) where the lapse function and the off-diagonal lapse function associated with vx + x T v are given by where L ξx and L ξx represent the Lie derivative acting on r and r , respectively. This is shown in  17) is reproduced from Eq. (78).

From Eq. (59), it is expected that the Poisson bracket between the Hamiltonian densities is
proportional to D and P µ to the leading order in the derivative expansion. Combining Eqs. (59), (73) and (66), we obtain dr θ u (r)H(r) + .. , dr θ v (r )H(r ) + .. where The derivation can be found in Appendix E 4. The difference between two evolutions generated by Hamiltonians with different lapse tensors is given by an Weyl transformation and a spatial diffeomorphism. In order to identify the metric tensor, we decompose G µν into the symmetric and anti-symmetric parts as The symmetric tensor is identified as −Sg µν in Eq. (19). Here g µν is the spatial metric whose overall sign is chosen such that the signature of the first spatial component is positive. S is the signature of time relative that of the first spatial component. b µν is the anti-symmetric component In the generalized constraint algebra summarized in Eqs. (69), (77) and (79) 77) and (79) reduces to the hypersurface deformation algebra of the general relativity in Eqs. (16), (17) and (18) up to the two derivative order in the gradient expansion.

A. Symmetry of semi-classical states
We view the path integration in Eq. (54) as the evolution of the initial state χ in Eq. (48) under the change of parameter time τ . As an initial state, we consider In the 1 to q, s represent the same physical state, χ P f q,sP T f ,pc,tc (s, t 1 , t 2 ) ∼ χ q,s,pc,tc (s, t 1 , t 2 ), where P T f is the transpose of P f . If q contains L linearly independent row vectors, the state can not be invariant under any infinitesimal SL(L, R) transformations. However, a discrete subgroup of SL(L, R) can be still preserved. Under the even site permutation group defined in Eq. (47), the wavefunction is transformed as χ q,s,pc,tc (s, t 1 , t 2 ) → χ q,s,pc,tc (P g s, P g t 1 P T g , P g t 2 P T g ) = χ qPg,P T g s,P T g pcPg,P T g tcPg (s, t 1 , t 2 ).
If there exists a P f that satisfies the site permutation in Eq. (86) can be undone by a flavour permutation through Eq. (85), χ q,s,pc,tc (P g s, P g t 1 P T g , P g t 2 P T g ) ∼ χ q,s,pc,tc (s, t 1 , t 2 ).
The unbroken subgroup of SL(L, R) is given by the set of P g that satisfies Eq. (87) for some P f .
The subgroup is denoted as I q,s,pc,tc .
First, let us consider the case with p c = t c = 0. With a generic choice of q and s, I q,s,pc,tc = ∅. In this case, SL(L, R) is completely broken spontaneously. Site permutation symmetry can be preserved maximally if both q and s T are proportional to the identity matrix. In this case, I q,s,pc,tc = S g L because P f = P g satisfies Eq. (87) for any P g ∈ S g L . In the continuum limit, S g L becomes a group that includes the spatial diffeomorphism group. However, there are various choices of q and s in which smaller site permutation groups are preserved. If (sq) i j depends only on r i − r j in a manifold associated with a local structure, the state has an unbroken discrete translational symmetry. To see this, it is convenient to go to the frame in which q is proportional to the identity matrix. In this frame, the first L flavour indices can be identified with the site indices, and it is easy to see that a site translation can be compensated by a translation in the space of flavour. With p c and t c turned on, the site permutation group is generally broken to a smaller subgroup. If t ij c and p c,ij support the same local structure, and depend only on r i − r j in the manifold, the discrete translational symmetry remains preserved. This discrete space symmetry can be enhanced to the continuous symmetry in the long distance limit. In Sec. VII, we will discuss an example that preserves a discrete translational and rotational symmetry that is enhanced to a continuous group at long distances.

B. Saddle-point equation of motion
In the large N limit, the path integration is dominated by the semi-classical path that satisfies the saddle-point equation of motion, for any traceless tensor y and symmetric tensor v at all τ . In the phase space of the collective variables, the classical constraint hypersurface is defined bȳ where β is an arbitrary constant. Thanks to the first-class nature of the constraints, Eqs.
This vanishes on the constraint hypersurface for any value of β.
A few remarks are in order regarding the constraints that classical variables satisfy. First, the fact that the initial classical data should satisfy the constraints is a common feature of constraint systems including the general relativity. In the present theory, this is imposed through the condition that the state in the kinematic Hilbert space has a nonzero projection with a gauge invariant state in the large N limit. Second, this is closely related to the well-known source-operator relation in the AdS/CFT correspondence [9,10,48]. In the AdS/CFT correspondence, sources fixed at a UV boundary (say through the Dirichlet boundary condition) fix their conjugate variables to avoid singularity in the bulk. In the present case, the collective variables (sources) and their conjugate momenta (operators) should satisfy the constraints to make sure that the overlap between the late time state given by and the gauge invariant state is non-zero in the large N limit, where T time-orders the operators.
If the collective variables do not satisfy the constraints classically, the state in Eq. (95) will have a catastrophic collapse upon 'measured' by 0 at a late time in Eq. (54). This will manifest itself as a singularity in the long time limit. Third, the fact that the classical variables satisfy the constraints guarantees that χ is invariant under the gauge transformation to the leading order in the large N limit. The gauge symmetry is broken at the sub-leading order. This sub-leading gauge symmetry breaking is what makes sure that the state is normalizable and evolves non-trivially under time evolution [25].

C. Gauge fixing
The shift and lapse tensors are arbitrary, and we can choose them through a gauge fixing. It is convenient to use some physical degrees of freedom as a set of local clocks that fixes a frame in which a gauge invariant local structure is defined and time evolution is generated. To be concrete, let us consider an initial state with detq(0) > 0. Without loss of generality, we can choose our initial frame such thatq(0) is proportional to the identity matrix. Furthermore, we can make sure thatq(τ ) remains proportional to the identity matrix at all τ by choosing the shift as and <W >= tr{W} L . In this gauge,q(τ ) remains proportional to the identity matrix at all τ , and q(τ ) can be written asq where q d is a single variable that represents the diagonal elements. This amounts to choosing a frame at each moment of time in terms of the L independent vectors ofq. This frame defines a gauge invariant set of local Hilbert spaces and a local structure associated with them. The gauge freedom associated with the lapse tensor can be fixed so that the lapse tensor is diagonal in the frame in which q is diagonal. This describes the time evolution of the system relative to a set of local clocks defined in the frame chosen by q. Here, we choose a uniform lapse tensor as v = I.

VII. TRANSLATIONALLY INVARIANT SOLUTION
In this section, we solve Eqs. (105)-(108) for an initial state that exhibits a D-dimensional local structure with a translational invariance. We choose a state with t 1 ,t 2 =t, wheret andp describe a D-dimensional manifold with topology, T D . For those states, it is natural to label sites in terms of D integers that form the D-dimensional hyper-cubic lattice, r = (n 1 , n 2 , .., n D ) for 1 ≤ n i ≤ L 1/D with the identification n i ∼ n i + L 1/D . If the state has the D-dimensional translational invariance, collective variables can be represented in the momentum space, where k = 2π L 1/D (l 1 , l 2 , .., l D ) with − L 1/D 2 ≤ l i < L 1/D 2 . For simplicity, we consider the case with the reflection symmetry and the discrete rotational symmetry. This guaranteest k =t k ,p k =p k , s k =s k , andt k =t Rk ,p k =p Rk ,s k =s Rk . Here k = (k 1 , .., k l−1 , −k l , k l+1 , .., k D ) for some l.
R is a π 2 -rotation on any of the principal planes in the D-dimensional hypercubic lattice. The hyper-cubic lattice breaks the continuous rotational symmetry to the discrete group. This is a spontaneously broken symmetry that is determined from the pattern of entanglement in the state. Nonetheless, we expect that the continuous rotational symmetry emerges at long distance scales. To see this, one can expand the collective variables in powers of k, where t µ 1 µ 2 ..µn represents a field with spin n. Due to the reflection symmetry, all odd spin fields vanish. Furthermore, the discrete rotational symmetry guarantees that t µν ∝ δ µν , and any spin 2 field should respect the full rotational symmetry. This implies that the spatial metric should be invariant under the full rotational symmetry as well. Higher spin fields do break the continuous rotational symmetry, but they become less important in the small k limit.
The collective variables at each k satisfy whereŪ k =s ks−k + 4t 2 kp k + 1 4p <s >= 1 L ks k , and we use (p −1 ) k = (p k ) −1 . The constraints in Eqs. (92) and (93) implȳ for all k. We can use the constraints to solve s k andt k in terms ofp k andq d as where with γ ≡ 1 α (4 −α − 4αβ 2 ). Here we considerα > 0 and β > 0 with 1 α (4 −α − 4αβ 2 ) > 0. Here T + [p k ,q d ] and T − [p k ,q d ] represent two possible branches oft k that satisfy the constraints for givenp k andq d . The time evolutions ofp k anq d can be determined by Eq. (101) and Eq. (103), The metric in Eq. (82), which is independent of r, becomes (see Appendix F) By using Eq. (93), Eq. (125) can be written as Here we use Eq.
where a(τ ) is the scale factor. With the choice of the positive signature for the spatial metric (a > 0), the signature of time is given by and the inverse of the scale factor becomes Due to the rotational symmetry, one can use any spatial direction to define S and a in Eq. (128) and Eq. (129).
According to Eq. (129), the proper size of the emergent space becomes smaller forp k that varies more sharply in the momentum space. This can be understood intuitively.p k that changes sharply in the momentum space leads top r,r that decays slowly in r − r . The slowly decaying collective variables in the real space creates entanglement bonds that connect sites that are far from each other in coordinate. The long-ranged entanglement bonds bring those sites close in physical distance because they become strongly coupled under the relatively local Hamiltonian.
This results in the decrease of the scale factor of space. k. If the band becomes completely flat, the collective variables become ultra-local, and the state becomes unentangled to the leading order in 1/N . This is a fragmented space [38,49]. According to Eq. (129), the proper volume of space diverges as far as its second derivative vanishes at k = 0 with or without a global flattening. At local Lifshitz critical points, the scale factor of the universe diverges although there still exists non-zero inter-site entanglement mediated by higher order k dependence ofp k . In this case, the non-trivial spatial entanglement is not encoded in the metric but in higher spin fields. This shows that the metric carries only a partial information on the pattern of entanglement. The full structure of entanglement is encoded in the complete set of collective variables. It is also these higher spin fields that carry the information that the continuous rotational symmetry is spontaneously broken to a discrete symmetry by the hypercubic lattice. Colloquially speaking, the pattern of entanglement fixes the emergent geometry, but not the other way around.

A. Numerical solution
Collective variables with different k remain coupled with each other through <s > in Eqs.
(113)-(116). This makes it hard to solve the equations of motion in a closed form. In order to gain some insight, we first solve the equations of motion numerically for a finite system with L = 10 6 and N = ∞. We choose the Hamiltonian withα = 1. For an initial state, we consider a state which has a three-dimensional local structure with the discrete translation, the π 2 -rotation and the reflection symmetry of the cubic lattice. We choosep r ,r (0) that is non-zero only for nearest neighbour r and r. In the momentum space, this gives In Fig. 9, we show the evolution ofp k (τ ) as a function of τ along one direction of k. At τ = 0,p k is convex near k = 0. As τ increases, the k dependence becomes weaker, andp k becomes flatter as a function of k. At a critical time τ * 1 ≈ 1.065,p k becomes independent of k and completely flat at all k. As time increases further,p k becomes concave in k. This is a global  Fig. 10(a). As τ increases further,p k undergoes a second Lifshitz transition at τ * 2 ≈ 3.6. This time, the Lifshitz transition is local : ∂ 2 ∂kµ∂kνp k at k = 0 changes sign from negative to positive while the band does not become globally flat. The profile ofp k near the local Lifshitz transition is shown in Fig. 10(b). As time keeps increasing, the second set of global and local Lifshitz transitions occur at τ * 3 ≈ 6.365 and τ * 4 ≈ 11.035, respectively as is shown in Fig. 10(c) and Fig.  10(d). This evolution ofp k near the second global Lifshitz transition is almost identical to the evolution near the first transition. This 'universality' can be understood from an analytic solution that is valid near the global Lifshitz transitions. This will be discussed in the following section. Although the profile ofq d (τ ) is not exactly periodic, it follows an oscillatory pattern. In Fig.   11(a), we plotq d (τ ) as a function of τ . During one oscillation ofq d , the collective variable undergoes four Lifshitz transitions, alternating between global and local Lifshitz transitions as is shown in Fig. 9. The global Lifshitz transitions coincide with the points at whichq d vanishes. This concurrence will be explained through analytic solutions in the next section. At the Lifshitz critical points (either global or local), the scale factor diverges, and the signature of time changes as is shown in Fig. 11(b) and Fig. 11(c). During the periods in which S < 0, we have de Sitter-like spacetimes with the Lorentzian signature [50]. In the smallq d limit, Eqs. (122) and (120) becomes To the leading order inq d , Eq. (123) and Eq. (124) become k . Ifq d (τ ) = 0 at τ = τ * , the solution to Eq. (132) is given bȳ To keep track of momentum dependece ofp k , it is convenient to consider the equation of motion for δp k =p k −p 0 . To the leading order in (τ − τ * ) and δp k , the equation of motion for δp k is given by To the leading order in (τ − τ * ), the solutions to Eq. (135) is obtained to be where f k is a function of k that is determined by matchingp k (τ ) away from τ = τ * . In Eq. (136) we use the fact that < p(τ * ) − 1 2 >=p 0 (τ * ) − 1 2 , which holds because δp k (τ * ) = 0. Eq. (136) implies that near τ = τ * the momentum dependence in δp k linearly vanishes in τ − τ * so thatp k becomes independent of k at τ * . Forα > 0 andp 0 (τ * ) > 0, the signature and the scale factor of the universe are given by This explains why the signature changes and the scale factor diverges at the global Lifshitz critical points.
Near the local Lifshitz transitions, the collective variables varies in time such that only the second derivative in k vanishes linearly in time as where B is a constant and τ * is the critical point. This also causes the change in the signature of time, and the divergence of the scale factor as a ∝ 1 |τ −τ * | at the local Lifshitz critical points. The emergent geometry is insensitive to the terms that are quartic and higher order in k, and contains only a partial information on the pattern of entanglement of the microscopic degree of freedom.

C. Effective theory
In this section, we derive the effective theory that describes fluctuations of the collective vari- where U = s s +ss T + √ 2 2t +pt + 4tp +t + 2tpt + − 1 4 t ± , p ± , s , q are all L × L matrices which become bi-local fields in the continuum. Due to the local structure,t r 1 ,r 2 ,p r 1 ,r 2 ,s r 1 r 2 decay exponentially in r 1 − r 2 , andt k ,p k ,s k are analytic functions of momentum. This guarantees that the effective theory that describes propagation of the fluctuating modes is local. We can write the effective theory in the gradient expansion. To organize the gradient expansion, it is useful to note thats k andt k are both determined fromp k andq d through Eqs. (120) and (121).
Since the spatial metric is directly related toŪ k through Eq. (125), it is convenient to takeŪ k and q d as independent variables, and writes k ,t k ,p k as functions ofŪ k andq d as U k is an analytic function of k. In real space,Ū r 1 ,r 2 decays exponentially over the coordinate scale that corresponds to a unit proper distance [88]. Consequently,s,t,p all decay exponentially in real space in the same manner. ExpandingŪ k ,s k ,t k ,p k , y k in k, we writē .ȳ k is the shift tensor in Eq. (96) written in the momentum space.
up to two derivative order, where ∇ 2 . Herē Q = 1 αŪ is used. In Eq. (147), all bi-local fields depend on τ . The propagating degrees of freedom are kinematically non-local, namely, bi-local in this case [51,52]. Nonetheless, the theory is dynamically local [53] in that the theory does not allow bi-local objects to suddenly jump from one location to another location non-locally. The locality of the effective theory is guaranteed by the fact that the saddle-point configurations of collective variables decay exponentially in the relative coordinate due to the local structure. As a result, the gradient expansion is well defined.
At length scale larger than the scale set by the local structure, the higher order derivative terms are negligible.
The effective action includes terms that are cubic and higher order in the bi-local fields. For example, the cubic action takes the form of where f i (r A , r B ) represents one of the bi-local fields and their derivatives. Again the higher derivative terms are suppressed by the scale associated with the local structure, and the interaction terms are also local. In the large N limit, the interactions are suppressed, and the bi-local fields are weakly interacting. To the leading order in 1/N , the bi-local objects freely propagate in the spacetime that is determined from the saddle-point configuration of the collective variables. This is illustrated in Fig. 12.
While the effective theory in Eq. (147) is local, it is not a local field theory of point-like particles. If one starts with a state with one bi-local excitation, its size can grow in time without a bound because there is no potential that confines the end points in the large N limit. This is because two end points of the bi-local fields propagate freely to the leading order in the 1/N expansion.
The sub-leading interactions such as the one in Eq. (148) can create bound states which behave as point particles. Only at length scales larger than the length scale of the bound state, one may use a local field theory description of point particles. However, the legnth scale of the bound state scales with N in the large N limit. For this reason, the present theory is far from the theory of pure gravity.
As is discussed in IV C, only L(L + 1) + 2 phase space variables represent physical degrees of freedom. One possible gauge choice is to determine t + and the traceless part of s to solve the Hamiltonian constraint and the momentum constraint, respectively. Furthermore, one can fix p + and the traceless part of q to fix the gauge associated with the Hamiltonian and the momentum constraints, respectively. This leaves t − , p − and the trace parts of q and s as physical degrees of freedom.

VIII. SUMMARY AND DISCUSSION
In this paper, we present a model for a background independent quantum gravity in which dimension, topology and geometry are all dynamical. The fundamental degree of freedom is a rectangular matrix. Matrix elements within each column define a local Hilbert space. The pattern of entanglement across local Hilbert spaces determines a spatial manifold if it has a local structure. A state has a local structure if it is short-ranged entangled when viewed as a state defined on a Riemannian manifold in which the column indices of the matrix is embedded. The theory does not have a manifold with a fixed dimension, and the spatial diffeomorphism of the general relativity is generalized to a larger group that includes diffeomorphism in arbitrary dimensions in the large L limit. The Hamiltonian that generates a background independent dynamics is relatively local in that the effective interaction between sites is determined by the pre-existing entanglement between the sites. The generalized momentum and Hamiltonian constraints obey a first-class constraint algebra that is reduced to that of the hypersurface deformation algebra of the general relativity in a special case. Using the constraints, we express the projection of a state with a finite norm to a gauge invariant state as a path integration of collective variables that describe fluctuating spacetime. In the limit that the size of the matrix is large, the path integration can be replaced with a saddle-point that satisfies the classical equation of motion. The equation of motion is solved both numerically and analytically for a state that has a three-dimensional local structure with a translational symmetry. We obtain a solution that describes a series of (3 + Suppose that t c and p c has a local structure. Namely, there exists a mapping from sites to a Riemannian manifold such that t c,ij and p ij c decays exponentially in the proper distance between r i and r j , where r i is the mapping from site i to the manifold. In this case, Eq. (150) can be viewed as the generating function of a local (non-unitary) field theory defined on the Riemannian manifold. The equivalence between Eq. (150) and Eq. (54) shows that the generating function of the non-unitary field theory is given by the path integration of the quantum gravity in which the de Sitter-like spacetime emerges at the saddle-point.

B. AdS/CFT
In order to make a connection with the AdS/CFT correspondence [8][9][10], one should consider a unitary field theory defined on a manifold with the Lorentzian signature instead of the Riemannian manifold. If t c and p c in Eq. (150) exhibit a local structure, the local structure determines the signature of the manifold on which the field theory is defined. If there is a translational invariance, the signature of each direction in the manifold is determined by the sign of the second derivative of the bi-local fields in the momentum space. If the second derivative of the collective variable has the opposite sign in one direction compared with other directions, this gives rise to a Lorentzian metric. In this case, Eq. (150) corresponds to the generating function of a Lorentzian field theory in the continuum limit. It is useful to continue to think of the generating function as an overlap of two wavefunctions, one given by Ψ 1 = 1 and the other given by Ψ 2 = e iS , where S is the action of the Lorentzian quantum field theory [38]. The wavefunctions are defined on spacetime (not just in space). The Hamiltonian is then replaced with a generator of coarse graining. The fact that Ψ 1 is invariant under the transformation generated by the coarse graining implies that Ψ 1 is a fixed point action that is scale invariant. In particular, Ψ 1 = 1 represents the trivial insulating fixed point.
Just as the overlap between a gauge invariant state and a state with a finite norm is invariant under the gauge transformation generated by the Hamiltonian and momentum constraints in Eq. (49), the generating function is invariant under an insertion of e −iĤτ , whereĤ is the generator of coarse graining [38]. An evolution generated by successive applications of coarse graining gives rise to the exact Wilsonian renormalization group (RG) flow [60]. The exponent of Ψ 2 (τ ) = e −iĤτ Ψ 2 represents the effective action defined at a length scale τ . In the effective action, all couplings that are allowed by symmetry are generally turned on. While the flow of Ψ 2 (τ ) can be tracked in terms of the classical couplings that parameterize the renormalized action, it is more natural to view the RG flow as an evolution of the wavefunction in the vector space. In particular, one can choose a set of basis wavefunctions that spans the full space of wavefunctions. Because the set of single-trace operators generate all multi-trace operators, the basis states can be chosen so that their wavefunctions only include the single-trace operators. As a result, Ψ 2 (τ ) can be represented as a linear superposition of states whose actions include only single-trace operators. In this way, the classical Wilsonian RG flow defined in the space of all couplings is replaced with a quantum evolution of wavefunction defined in the space of the single-trace couplings [47]. The full Wilsonian renormalization group flow is projected to the space of single-trace operators at the expense of promoting the sources for the single-trace operators to dynamical variables. Since the source for the single-trace energy-momentum tensor is also promoted to dynamical variable, the path integration of the dynamical source represents a dynamical gravity in the bulk. This may eventually provide a non-perturbative formulation of quantum gravity in the anti-de Sitter space.
One missing piece in the puzzle is to construct a UV finite coarse graining operator such as the one in Eq. (33) that gives rise to a background independent gravity in the bulk.

C. Open questions
Here we list some open questions.
• Physical spectrum In general, saddle-point configurations do not have temporal Killing vector. It would be interesting to find a saddle-point configuration that support a temporal Killing vector and extract the physical energy spectrum of the propagating modes in this theory [61,62] • Local field theory of point particles Background independent theories of quantum gravity include kinematically non-local objects. In the present theory, it is the bi-local field. The ultimate goal is to construct a background independent theory that reduces to a field theory of a small number of pointlike particles at low energies. However, the present theory does not achieve this goal because point-like particles emerge only through the interactions between bi-local fields that are suppressed by 1/N . This is because the present theory is similar to vector models [38,51,52,[63][64][65][66][67][68][69][70][71][72][73][74][75] although the microscopic degree of freedom is a matrix. One index of the matrix is used to generate an emergent space, and the internal symmetry acts only on the remaining one index. In order to construct a quantum theory of gravity that reduces to the Einstein's general relativity with a small number of additional fields, one needs a mechanism that keeps dynamical objects finite to the leading order in 1/N . For this, tensor models may be a natural direction [76][77][78][79]. In tensor models with rank greater than 2, where one index labels sites and the remaining indices label internal flavour (or color), one expects to have multi-local fields as emergent degree of freedom. In this case, a non-zero tension may naturally arise to keep kinematically non-local objects finite. In relation to a non-perturbative formulation of the AdS/CFT correspondence for gauge theories, we expect that the background independent coarse grainer takes a form of a relatively local tensor model with rank

3.
Appendix A: Derivation of Eq. (34) Using the singular value decomposition, a symmetric matrix can be written as where O v is an orthogonal matrix and is a diagonal matrix. If v is non-singular, d i = 0. One can then write the diagonal matrix as Therefore, v can be written as

Appendix B: Non-normalizability of gauge invariant state
In this appendix, we prove that all gauge invariant states have infinite norm with respect to the norm defined in Eq. (1). Consider a gauge invariant wavefunction Ψ(Φ A i ) defined in the space of {Φ A i }. Suppose that the wavefunction has a support away from the origin in the M L-dimensional space. Without loss of generality, let us choose the frame such that a point in the support is given by Φ A i = Φ 0 δ A 1 δ 1 i , where Φ 0 is a positive constant. Under the SL(L, R) transformation generated byĜ 1 2 +Ĝ 2 1 , the point can be mapped into any point on the hyperbola defined by (Φ 1 1 ) 2 − (Φ 1 2 ) 2 = Φ 2 0 with Φ 1 1 > 0. In order for the wavefunction to be invariant under the transformation, the wavefunction must have amplitude Φ 0 along the entire hyperbola. Because the hyperbola is unbounded, the norm of the wavefunction is infinite. The only state which does not have support away from the origin is Ψ(Φ A i ) ∝ Π A,i δ(Φ A i ). However, this has an infinite norm as well because the wavefunction is constant in the conjugate space.
To prove Eq. (58), we first note Here U = ss T + c [4t c p c t c − it c ] and Q = q T q + c p c . v andṽ are covariant and contravariant symmetric tensors that are independent of s, q, t c , p c , respectively. Using the distribution rule, we write the Poisson bracket between the momentum and the Hamiltonian constraints as Here G x , U A P B ≡ G i j , U lm P B x j i A ml and G x , Q B P B ≡ G i j , Q lm P B x j i B ml . A and B are covariant and contravariant tensors that are made of v, x, s, q, t, p. From Eq. (D2), we obtain In order to compute the Poisson bracket between Hamiltonians, we use From Eq. (D5), we obtain By choosing (v 1 ) i j = δ ij i j , (v 2 ) l k = δ lk l k , we obtain Eq. (59).