Emergent gravity from relatively local Hamiltonians and a possible resolution of the black hole information puzzle

In this paper, we study a possibility where gravity and time emerge from quantum matter. Within the Hilbert space of matter fields defined on a spatial manifold, we consider a sub-Hilbert space spanned by states which are parameterized by spatial metric. In those states, metric is introduced as a collective variable that controls local structures of entanglement. The underlying matter fields endow the states labeled by metric with an unambiguous inner product. Then we construct a Hamiltonian for the matter fields that is an endomorphism of the sub-Hilbert space, thereby inducing a quantum Hamiltonian of the metric. It is shown that there exists a matter Hamiltonian that induces the general relativity in the semi-classical field theory limit. Although the Hamiltonian is not local in the absolute sense, it has a weaker notion of locality, called relative locality : the range of interactions is set by the entanglement present in target states on which the Hamiltonian acts. In general, normalizable states are not invariant under the transformations generated by the Hamiltonian. As a result, a physical state spontaneously breaks the Hamiltonian constraint, and picks a moment of time. The subsequent flow of time can be understood as a Goldstone mode associated with the broken symmetry. The construction allows one to study dynamics of gravity from the perspective of matter fields. The Hawking radiation corresponds to a unitary evolution where entanglement across horizon is gradually transferred from color degrees of freedom to singlet degrees of freedom. The underlying quantum states remain pure as evaporating black holes keep entanglement with early Hawking radiations in the singlet sector which is not captured by the Bekenstein-Hawking entropy.


JHEP10(2018)043
There have been many efforts to understand gravity as an emergent phenomenon from various angles [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The anti-de Sitter space/conformal field theory (AdS/CFT) correspondence [21][22][23] is a concrete example of emergent gravity, where a gravitational theory in the bulk emerges from a quantum field theory defined at the boundary of the anti-de Sitter space. However, the AdS/CFT correspondence does not directly apply to our universe which does not have a spatial boundary. For our universe, it seems more natural that the bulk spacetime emerges from a theory defined at a temporal boundary in the past or future. The program of the de Sitter space/conformal field theory (dS/CFT) correspondence aims to make this scenario concrete with the guidance from the AdS/CFT correspondence [24][25][26].
In this paper, we study the possibility in which time and gravity emerge from quantum matter, employing a more microscopic perspective built from the quantum renormalization group (RG) [27,28]. Quantum RG provides a prescription to construct holographic duals for general quantum field theories based on the intuition that the emergent space direction in the bulk corresponds to a length scale [29][30][31][32][33][34][35]. The basic object in quantum RG is wavefunctions defined in the space of couplings. Instead of specifying a quantum field theory in terms of classical values of all couplings allowed by symmetry, a theory is represented as a wavefunction defined in a much smaller subspace of couplings. The subspace is chosen so that all symmetry-allowed operators can be constructed as composites of those operators sourced by the couplings in the subspace. Then, general theories can be represented as coherent linear superpositions of theories defined in the subspace. As a result, the couplings in the subspace are promoted to fluctuating variables. Metric, which sources the energy-momentum tensor, also becomes a dynamical variable whose fluctuations account for composite operators made of the energy-momentum tensor. While the Wilsonian RG flow is a classical flow defined in the full space of couplings, the same exact RG flow can be represented as a quantum evolution of the wavefunction defined in the subspace. The classical flow of the Wilsonian RG is replaced by a sum of all possible RG paths defined in the subspace of couplings. The weight for each RG path is determined by an action which includes a dynamical gravity [36].
In order to realize an emergent time in a manner that a space-like direction emerges in quantum RG, we consider wavefunctions of ordinary quantum matters defined on a space manifold instead of wavefunctions of couplings defined on a spacetime manifold. Metric in Lorentzian quantum field theories determines connectivity of spacetime by setting the strength of derivative terms in local actions. In quantum states of matter fields we consider here, metric with the Euclidean signature is introduced as a collective variable that controls entanglement of matter fields in the space manifold. Namely, local actions for Lorentzian quantum field theories are replaced with short-range entangled states of quantum matters. The metric in quantum states of matter plays the role of a variational parameter that sets the notion of locality ('short-rangeness') in how matter fields are entangled in space. More specifically, we consider a set of wavefunctions of matter fields parameterized by Riemannian metric. The space spanned by those states forms a sub-Hilbert space in the full Hilbert space of the matter field.

JHEP10(2018)043
With wavefunctions of couplings replaced by wavefunctions of matter fields, we consider a unitary evolution of the quantum states. Although an unitary evolution is not same as RG flow, one may still view the former as a coarse graining process in which information accessible to local observers decreases in time through scrambling. In particular, we consider an evolution generated by a Hamiltonian which maps the sub-Hilbert space into the sub-Hilbert space. Since the sub-Hilbert space is parameterized by spatial metric, the Hamiltonian of the matter fields induces a quantum Hamiltonian of the metric. In this way, one can induce quantum theories of metric from matter fields. The main goal of this paper is to address the following questions: 1. Can a matter Hamiltonian induce a quantum theory that becomes Einstein's general relativity at long distances in the classical limit?
2. Is the matter Hamiltonian that gives rise to the general relativity local?
3. What is the nature of time in the emergent gravity?
4. How does a quantum state of matter maintain its purity under an evolution that is dual to a black hole evaporation?
The short answers to these questions are 1. Yes, one can engineer a matter Hamiltonian whose induced dynamics agrees with the general relativity in the semi-classical field theory limit.
2. No, the Hamiltonian is not local in the usual sense. However, it possesses a relative locality in that the range of interactions depends on states on which the Hamiltonian acts.
3. Time arises as a Goldstone mode associated with a spontaneous breaking of the symmetry generated by the Hamiltonian constraint. 4. During black hole evaporations, quantum states stay pure by transferring entanglement from color degrees of freedom to singlet sectors.
The rest of the paper gives long answers to the questions. Here is an outline that may serve as a summary of the paper.
In section 2, we sketch the main idea that is used in the explicit examples constructed in the following sections. This section constitutes a conceptual guide for the rest of the paper. Section 3 is a warm-up which discusses a toy model from which a minisuperspace quantum cosmology emerges. Although there is no extended space in the toy model, it still contains the essential idea on how time emerges. In section 3.1, we introduce a set of quantum states for N variables. The states in the set are parameterized by two collective variables. Those states labeled by the collective variables span a sub-Hilbert space in the full Hilbert space of the N fundamental variables. Throughout the section, we will focus on the sub-Hilbert space. It becomes the Hilbert space for the induced cosmology in which the two collective variables become the scale factor of a universe and a scalar field, JHEP10(2018)043 respectively. The inner product between states in the sub-Hilbert space, which is inherited from the one defined in the full Hilbert space, provides a notion of distance between states with different collective variables. With increasing N , two states with different collective variables become increasingly orthogonal.
In section 3.2, we construct a Hamiltonian for the N variables that is an endomorphism of the sub-Hilbert space, that is, an operator that maps the sub-Hilbert space into the sub-Hilbert space. Through the evolution generated by the Hamiltonian, a state with a definite collective variable evolves into a linear superposition of states with different collective variables in the sub-Hilbert space. The evolution is naturally described as a unitary quantum evolution of wavefunctions defined in the space of the collective variables. Therefore, one can identify a Hamiltonian for the collective variables induced from the matter Hamiltonian. The key result of this subsection is that there exists a Hamiltonian for the N variables which gives rise to a minisuperspace Wheeler-DeWitt Hamiltonian for the collective variables.
In section 3.3, we address the issue of time. In general relativity which includes minisuperspace cosmology, Hamiltonian is a constraint which generates time reparameterization transformations. It is usually assumed that 'physical states' are the ones that are invariant under diffeomorphism and are annihilated by the Hamiltonian constraint. This gives rise to the problem of time because 'physical states' are stationary, and no change is generated under Hamiltonian evolution. In the present theory of induced quantum cosmology, the problem of time is avoided because there is no normalizable state that satisfies the Hamiltonian constraint in the sub-Hilbert space. This is shown by diagonalizing the matter Hamiltonian numerically. Nonetheless, there exist semi-classical states which are normalizable. They satisfy the Hamiltonian constraint to the leading order in the large N limit, yet break the constraint beyond the leading order. While the semi-classical states do not satisfy the Hamiltonian constraint exactly, they are legitimate states as quantum states of matter. A semi-classical state 'picks' a moment of time spontaneously because it is forced to have a finite norm. Non-trivial time evolution of the semi-classical states can be understood as Goldstone modes associated with the weak spontaneous symmetry breaking. In the large N limit, the time evolution of semi-classical states coincides with the classical minisuperspace cosmology.
In the following section, we generalize the discussion on the emergent minisuperspace cosmology to a fully fledged gravity in (3 + 1)-dimensions. The starting point is an N × N Hermitian matrix field defined on a three-dimensional spatial manifold. In section 4.1.1, we define a Hilbert space for the induced gravity from the matter field. The full Hilbert space of the matter field is spanned by eigenstates of the matrix field. Within the full Hilbert space, we focus on a sub-Hilbert space spanned by gaussian wavefunctions. Those gaussian wavefunctions, which are singlet under a SU(N ) internal symmetry, are parameterized by a Riemannian metric and a scalar field. Namely, we consider a set of SU(N ) invariant wavefunctions in which metric and a scalar field enters as collective variables (equivalently, variational parameters) that control how the matter field is entangled in space. General states within the sub-Hilbert space are given by linear superpositions of states with different collective variables.

JHEP10(2018)043
Section 4.1.2 is devoted to the inner product. The inner product in the full Hilbert space is defined in terms of normal modes of an elliptic differential operator associated with a fiducial Riemannian metric. Although a fiducial metric is introduced to define the inner product in the full Hilbert space, the fiducial metric decouples in the inner product between normalized states in the sub-Hilbert space. Based on this, we show the following properties of the inner product. First, the induced inner product in the sub-Hilbert space is invariant under diffeomorphisms of the collective variables. Second, we show that two states in the sub-Hilbert space are orthogonal unless the two have metrics that give same local proper volume. Third, even for states with same local proper volume, the overlap decays exponentially as the difference in the collective variables increases in the large N limit. This is explicitly shown for states whose metrics are close to the flat Euclidean metric.
In section 4.2, we show that the metric, as a variational parameter, controls the number of degrees of freedom that generate entanglement in space. As quantum states of matter field defined in continuum, the number of degrees of freedom per unit coordinate volume is infinite. Nonetheless, the wavefunctions in the sub-Hilbert space have a short-distance cut-off scale which regularizes the amount of entanglement. It is shown that the von Neumann entanglement entropy of states in the sub-Hilbert space has two contributions. One is the color entanglement entropy which is generated by the matter field in a classical configuration of the collective variables. The color entanglement entropy of a region in space obeys the area law, where the area is measured with the metric associated with the state in the unit of the short-distance cut-off. The amount of color entanglement a region in the manifold can support is not fixed. Rather it is a dynamical quantity that is determined by the metric. With increasing proper volume, the entanglement entropy increases accordingly. In the presence of fluctuations of the collective variables, correlations in the fluctuations give rise to an additional contribution to the von Neumann entanglement entropy, called singlet entanglement entropy. These two contributions can be approximately separated in semi-classical states where fluctuations of the collective variables are small.
In section 4.3, we consider endomorphisms of the sub-Hilbert space. We show that there exist Hermitian operators for the matter field which induce the momentum density operator and a regularized Wheeler-DeWitt Hamiltonian density operator for the collective variables. Those operators for the matter fields generate an evolution of the collective variables once the lapse and the shift are fixed. At long distances and in the large N limit, the evolution coincides with the time evolution of the classical Einstein's gravity in a fixed gauge. The matter Hamiltonian that gives rise to the general relativity has no absolute locality because the Hamiltonian, as a quantum operator, contains operators with arbitrarily long ranges. Nonetheless, it is relatively local in that the range of interactions that survive when applied to a state is limited by the amount of entanglement present in the state. The notion of locality in the Hamiltonian is determined relative to states to which the Hamiltonian is applied.
The fact that the general relativity can emerge from matter fields provides an opportunity to examine the black hole information puzzle from the perspective of the underlying matter field. In section 5, we consider a formation and evaporation of a black hole in the induced theory of gravity. By construction, the time evolution is unitary. The discussion JHEP10(2018)043 is centered on how purity of a quantum state can be in principle maintained during an evolution. As a black hole evaporates, the color entanglement entropy across the horizon, which is identified as the Bekenstein-Hawking entropy [37,38], decreases in time. On the other hand, the singlet entanglement entropy increases because Hawking radiation is emitted in the singlet sector. As a result, entanglement is gradually transferred from the color sector to the singlet sector. This leads to a 'neutralization' of entanglement entropy. The full quantum states remain pure as black holes keep the entanglement with early Hawking radiation in the singlet sector which is not captured by the Bekenstein-Hawking entropy. The failure of the Bekenstein-Hawking entropy to account for all available states in a black hole is attributed to a lack of equilibrium. It is argued that localization can arise dynamically because both states and Hamiltonian effectively flow under the time evolution generated by the relatively local Hamiltonian. In section 6, we conclude with a summary and discussions on open questions.

The main idea
In this section, we sketch the main idea of the paper that is summarized in figure 1. Our starting point is a Hilbert space of matter fields defined on a spatial manifold with a fixed topology. Within the full Hilbert space (H) of the matter fields, we consider a sub-Hilbert space (V) that is spanned by a set of short-range entangled states. H is equipped with an inner product, which gives a unique inner product in V. Each basis state of V is associated with a set of collective variables. Among the collective variables is spatial metric. In this discussion of the conceptual idea, we focus on the metric, ignoring other collective variables. A spatial metric is assigned to each basis state such that the von Neumann entanglement entropy of a basis state for any region in space is proportional to the proper area of the boundary of the region measured with the metric for the basis state (see figure 2). Two states which support different amounts of entanglement in a region are assigned to have different metrics so that they give different proper sizes of the region in proportion to the entanglement. In this sense, the metric can be viewed as a collective variable which controls the amount of entanglement in quantum states of the matter fields. A general state in V can be expressed as a linear superposition of the basis states, χ = Dg µν g µν χ(g µν ).
Here g µν is the basis state associated with g µν (x), χ(g µν ) is a wavefunction defined in the space of spatial metric, and Dg µν is a measure that is defined based on the inner product in V. In the limit that the number of matter fields is large, two states with different metrics become orthogonal. This way, the sub-Hilbert space of the matter field is identified as a Hilbert space for spatial metric.
Next we study dynamics of the matter field within the sub-Hilbert space by considering a HamiltonianĤ that maps V into V. If an initial state χ is prepared to be in V, e −itĤ χ can be written as a linear superposition of g µν (x) . Because the unitary time evolution is a linear map acting on the sub-Hilbert space, one can identify a differential operator H g µν , ∂ ∂gµν that acts on the wavefunction of metric such that e −iĤt χ = Dg µν g µν e −iH gµν , ∂ ∂gµν t χ(g µν ). We identify H g µν , ∂ ∂gµν as an induced Hamiltonian JHEP10(2018)043 Figure 1. In the full Hilbert space H of matter fields, a sub-Hilbert space V is spanned by a set of basis vectors, where each basis vector is associated with a spatial metric. A general state in V can be written as a linear superposition of the basis vectors, χ = Dg µν g µν χ(g µν ), where χ(g µν ) is a wavefunction defined in the space of spatial metric. An endomorphic Hamiltonian H generates a map from V into V. Therefore, χ = e −iĤt χ can be also written as χ = Dg µν g µν χ (g µν ). The linear map between χ(g µν ) and χ (g µν ) can be written as is identified as an induced Hamiltonian for the metric. of the metric. By requiring that the induced Hamiltonian becomes the Wheeler-DeWitt Hamiltonian in the classical limit, we construct a matter Hamiltonian that induces the general relativity.
The matter Hamiltonian that induces the general relativity turns out to be a nonlocal Hamiltonian. Yet, it has a weaker notion of locality called relative locality. While the Hamiltonian is non-local as a quantum operator, the range of interaction that survives when applied to a state is determined by the entanglement present in the target state. There is an intuitive way to understand this. The Hamiltonian that induces the general relativity can not be local because one can not have a local gradient term in the Hamiltonian without JHEP10(2018)043 introducing a fixed background. 1 Therefore, any background independent theory can not have an absolute notion of locality. On the other hand, the general relativity is reduced to a local effective field theory when fluctuations of the metric are weak. Small fluctuations of metric propagate on top of a saddle point configuration that is dynamically determined, and the notion of locality in the effective field theory is determined by the saddle point metric. In the present construction, the metric is determined by the entanglement present in quantum matter. Therefore, the notion of locality should be set by the amount of entanglement present in states of matter fields.
In the following two sections, we work out examples which elucidate the idea outlined in this section. In section 3, we provide a toy example of quantum mechanical system in zero space dimension. In this model, the spatial metric is reduced to one scale factor, and a minisuperspace quantum cosmology emerges. In section 4, we generalize the construction to a fully fledged gravity in three space dimension. In section 5, we discuss possible implications of the induced gravity for the black hole information puzzle.

Emergent minisuperspace cosmology
Based on the general idea outlined in the previous section, in this section we consider a quantum mechanical system of N variables from which a minisuperspace quantum cosmology emerges. We start by defining a sub-Hilbert space of the N variables which becomes the Hilbert space for two collective variables: a scale factor and a scalar. The sub-Hilbert space is spanned by a set of basis vectors labeled by the two collective variables. After examining the kinematic structure of the sub-Hilbert space, we explicitly construct a Hamiltonian of the N variables that induces the Wheeler-DeWitt Hamiltonian of the minisuperspace cosmology for the scale factor and the scalar. Finally, we address the problem of time in quantum cosmology. By numerically diagonalizing the matter Hamiltonian, we show that there is no normalizable state that satisfies the Hamiltonian constraint within the sub-Hilbert space. From this, we conclude that the requirement that a physical state should have a finite norm forces quantum states spontaneously break the Hamiltonian constraint, and the subsequent time evolution arises as a Goldstone mode associated with the broken symmetry.

Hilbert space
We consider a system of N compact variables whose Hilbert space is spanned by φ | 0 ≤ φ a < 2π, a = 1, 2, .., N with the inner product φ φ = N a=1 δ(φ a − φ a ). Within the full Hilbert space, we consider states which are invariant under permutations of the N flavors. Furthermore, we focus on wavefunctions that depend only on the first harmonics of φ a through O c = N a=1 cos φ a and O s = N a=1 sin φ a . Wavefunctions that depend on O c and O s can be spanned by two-parameter family of 'plane waves', e i(kcOc+ksOs) . We denote the sources for O c and O s as k c = e 3α cosh 3 √ 2 σ and k s = e 3α sinh 3 √ 2 σ to label the basis states 1 One alternative possibility for a ultra-local Hamiltonian has been considered in ref. [28].  in terms of two non-compact variables (α, σ) as where the wavefunction is written as Here the normalization is chosen such that α, σ α, σ = 1. V denotes the sub-Hilbert space spanned by The wavefunction Ψ(φ; α, σ) can be viewed as a tensor which depends on φ, α, σ as is shown in figure 3. If we considered more general wavefunctions that include higher harmonics, we would have to introduce more collective variables to span the extended Hilbert space. However, we focus on the two-parameter family of basis states in our discussion to keep the form of wavefunction simple. We are mainly interested in constructing a simple example of emergent cosmology to demonstrate the proof of principle discussed in section 2.
The overlap between states in V is given by where I 0 (x) is the modified Bessel function, and is a measure of distance between two states. For N 1, the Bessel function decays Roughly speaking, two states with R α ,σ ;α,σ greater than N −1/2 are orthogonal.
In the large N limit, the overlap is proportional to the delta function upto a multiplicative factor,

Hamiltonian for induced quantum cosmology
We emphasize that φ a 's, which we call 'matter fields', are the only fundamental degrees of freedom. {α, σ} parameterizes collective modes of the matter fields. If a Hamiltonian for the matter fields generates a dynamical flow within V, the dynamical flow can be understood as an evolution generated by an induced Hamiltonian for the collective variables. In the following, we construct a Hamiltonian for the matter fields which induces a minisuperspace quantum cosmology for the collective variables, where α and σ become the scale factor of a flat universe and a scalar field, respectively. We first look for a HamiltonianĤ(α, σ) whose action on α, σ induceŝ where h α,σ is a Wheeler-DeWitt differential operator for the minisuperspace cosmology of a flat three-dimensional universe, with a potential V (σ). It is not difficult to construct a Hamiltonian that does the job for a given state α, σ . We try the standard quadratic kinetic term with a potential term, whereπ a is the conjugate momentum forφ a with the commutation relation [π a , ApplyingĤ(α, σ) to α, σ , one indeed obtains eq. (3.10) with V (σ) = 1 18 cosh 3 √ 2σ − 1 and κ 2 = 1 9 √ N . Eq. (3.10) implies that the action ofĤ(α, σ) on α, σ is equivalent to a differential operator acting on the collective variables. This proves thatĤ(α, σ) α, σ is in N to the leading order in the large N limit. 2 H(α, σ), as a quantum operator of the matter fields, depends on α, σ. This means thatĤ(α, σ) can not generate the desired dynamics for general states in V, that is, H(α, σ) α , σ = h α ,σ α , σ if (α , σ ) = (α, σ). In order for the Hamiltonian flow to stay within V for arbitrary initial states in V, one effectively has to choose different Hamiltonians for states with different collective variables. Such a 'state-dependent' operator can be realized through a linear map in the large N limit because states with different collective variables are orthogonal in the large N limit as is shown in eq. (3.6). Based on this intuition, we consider the following Hamiltonian, It is noted thatP α,σ ≡ α, σ α, σ becomes orthogonal projection operators in the large N limit.Ĥ is made of the projection operator andĤ(α, σ). In the large N limit, the projection operator first picks a state with a definite {α, σ} beforeĤ(α, σ) is applied. 3 This way, the operator tailored for each set of collective variables is applied to the state with the corresponding collective variables. BecauseĤ(α, σ) depends on α, σ, one may regardĤ as a state dependent operator whose action on the Hilbert space depends on states it acts on [39]. However, it is still a linear operator. 4 Eq. (3.10) allows us to writeĤ aŝ It is noted that h † α,σ differs from h α,σ only by terms that are at most O(1). For general states constructed from linear superpositions of α, σ ,

17)
H acts asĤ  . The filled box represents an operator that acts on a quantum state of the matter field, χ = dαdσ α, σ χ(α, σ). If the operator is an endomorphism of V, it can be represented as an operator (represented by the empty box) that acts on the wavefunction χ(α, σ) for the collective variables.
Therefore, the Hamiltonian for the matter fields translates to a linear operator acting on the wavefunction of the collective variables. This is illustrated in figure 4. The induced Hamiltonian for the collective variables consists of two parts. The first is the convolution of the The convolution smears out sharp features which vary at scales shorter than ∆R α,σ;α ,σ ∼ N −1/2 in the space of α, σ. For slowly vary- , and the convolution merely changes the normalization of χ(α, σ). The second is the minisuperspace Wheeler-DeWitt Hamiltonian,H α,σ . Combined, H α,σ can be understood as a regularized Wheeler-DeWitt Hamiltonian.

Emergent time as a Goldstone mode
Eq. (3.18) implies thatĤ induces the Wheeler-DeWitt Hamiltonian of a minisuperspace cosmology. In gravity, time evolution is a part of diffeomorphism, and states that are invariant under diffeomorphism are annihilated by the Hamiltonian. A state that satisfies the Hamiltonian constraint represents a whole history rather than a moment of time.
Recovering time from a stationary state is the problem of time in gravity [40,41]. States that satisfy the Hamiltonian constraint correspond to quantum states with zero energy. For a finite N , however, there is no guarantee that there exists a state with zero energy in V. This is because the configuration space is compact, and the energy level is discrete. In order to check this explicitly, we diagonalizeĤ in eq. (3.14). The matrix elements of the Hamiltonian is written as (3.20) Explicit integrations over α, σ result in With increasing L, the bandwidth of the eigenvalues keeps increasing, which reflects the fact that the spectrum is unbounded both from the above and below in the continuum limit. On the other hand, the eigenvalue that is smallest in magnitude saturates to a nonzero value. This suggests that there is no state in V with zero energy in the continuum limit.
We note that the matter fields are subject to strong all-to-all interactions.
We numerically diagonalizeĤ for N = 3. Indeed, all eigenstates which have nonzero projection in V have non-zero eigenvalues, as is shown in figure 5. This may seem contradictory because H α,σ χ 0 (α, σ) = 0 is a hyperbolic equation, which can be solved once a boundary condition is provided. Solutions to the Wheeler-DeWitt equation formally give zero energy states. The reason why such states do not appear in the spectrum is because they are not normalizable [40,41].
The fact that there is no normalizable state which satisfies the Hamiltonian constraint means that a physical state in V inevitably breaks the time translational symmetry by the virtue of having a finite norm. This is analogous to the fact that states that are invariant under spatial translations in the Euclidean space are not normalizable, and physical states (such as wave packets) necessarily break the translational symmetry. However, there exist normalizable semi-classical states which are annihilated byĤ to the leading order in 1/ √ N and break the symmetry only weakly, whereᾱ,σ,π,π σ are classical coordinates and momenta which satisfy ∆ determines the uncertainty of α and σ. With κ 2 ∆ 1, eq. (3.22) represents a semi-classical state in which both coordinates and momenta are well defined with δα δπ ∼ δσ δπ σ ∼ κ 2 ∼ 1 √ N , and the Hamiltonian constraint is satisfied to the leading order in 1/ √ N . Because semi-classical states are not exactly annihilated byĤ, they spontaneously break the symmetry generated by the Hamiltonian. The spontaneous symmetry breaking amounts to picking a moment of time in a history. The following evolution of the state generated byĤ creates one-parameter family of states. The evolution can be viewed as a Goldstone mode associated with the spontaneously broken symmetry. This is illustrated in figure 6. We call the parameter along the orbit t. However, t itself is not a physical observable because there is no independent way of measuring t in a closed quantum system. It is merely a parameter that labels a sequence of states generated by the Hamiltonian evolution. What is physical is relation between physical observables, e.g., the value of σ when α takes a certain value. Now we examine how semi-classical states evolve underĤ. Since eq. (3.22) has a fast oscillating phase factor, the convolution integration in eq. (3.19) gives rise to a suppression JHEP10(2018)043 in the norm of the wavefunction, and it is used that χ(α, σ) is sharply peaked at (ᾱ,σ). As a result, H α,σ becomes the Wheeler-DeWitt Hamiltonian upto a multiplicative factor that depends on the wavefunction, By choosing a lapse that absorbs A χ , the state after an infinitesimal step of the parameter time can be written as Here n (1) determines the speed of the flow along the orbit. O(κ 2 ) represents sub-leading terms that are generated from the measure and the smearing. The measure for the conjugate momenta has been defined as DπDπ σ ≡ µ(α, σ) −1 dπdπ σ . In the large N limit, eq. (3.26) remains a semi-classical state centered at a different classical configuration. In the next step, we choose the lapse n (2) A −1 χ(dt) . Repeating these steps, one obtains a state at parameter time t, where n(τ ) is a time-dependent speed of time evolution which can be chosen at one's will, and This is a minisuperspace quantum cosmology for the three-dimensional flat universe with one scalar field. In the large N limit, the classical path dominates the path integration.
There is a sense in which the emergent time in the present theory resembles an internal time generated by relative motions of a subsystem in stationary states [42]. To make the connection, one views Ψ(φ; α, σ) as a wavefunction of an enlarged system that includes not only the matter fields but also the collective variables as independent dynamical degrees of freedom. In this case, eq. (3.10) is understood as the Wheeler-DeWitt equation for the whole system (with a wrong sign in the kinetic term for the matter field). Although the full state is stationary, one defines a time flow in terms of the evolution of the matter fields relative to the collective variables. What is different in the present construction are JHEP10(2018)043 two-fold. First, the collective variables are not independent dynamical degrees of freedom. Instead, they describe collective excitations of the matter fields. Accordingly, the quantization of the collective variables follow from that of the matter fields. Second, the inability to find normalizable states in the Hilbert space of the matter fields provides a dynamical mechanism to pick a moment of time in the induced theory of cosmology.

Emergent gravity
In this section, we extend the discussion on the emergent minisuperspace cosmology to gravity in (3 + 1) dimensions. The biggest difference from the previous section is that we are now dealing with an infinite dimensional Hilbert space. To be concrete, we consider an N × N matrix field defined on a three dimensional manifold. Within the full Hilbert space, we define a sub-Hilbert space of the matter field that becomes a Hilbert space for two collective fields: a spatial metric and a scalar field. 5 The sub-Hilbert space is spanned by a set of basis vectors each of which is labeled by the metric and the scalar field. As variational parameters of wavefunctions of the matter field, the spatial metric sets the notion of locality in how matter fields are entangled in space, while the scalar field determines the range of mutual information in each basis state. After we discuss the covariant regularization of the wavefunctions and the inner product within the sub-Hilbert space, we explain the connection between the collective variables and entanglement in details. Building on the intuitions we learned from the previous two sections, we then construct a matter Hamiltonian that induces the general relativity at long distances in the large N limit.

Construction of a Hilbert space for metric from matter fields
In this subsection, we define a sub-Hilbert space of a matrix field and an inner product that is invariant under spatial diffeomorphisms.

Hilbert space
We consider an N ×N Hermitian matrix field Φ(x) defined on a compact three dimensional manifold. The full Hilbert space of the matrix field is spanned by the eigenstates of the field operator,Φ ab (x) Φ = Φ ab (x) Φ . In order to define an inner product in the infinite dimensional Hilbert space, we need to introduce a discrete basis that spans the space of Φ ab (x). For this, we choose an elliptic differential operator whose eigenvectors form a complete basis,

JHEP10(2018)043
Here ∇ E µ is the covariant derivative defined with respect to a Riemannian metric, g E,µν (x), which is parameterized by a triad, In eq. (4.3), the local Euclidean index i is raised or lowered with δ ij = δ ij , and repeated indices are summed over i = 1, 2, 3. σ(x) is a scalar that determines the 'mass' in the unit of a fixed length scale, l c . λ (E,σ) n is the n-th eigenvalue and f For a choice of (E µi , σ), the set of eigenvectors f ab,n represents the amplitude of the n-th normal mode in the basis of f In order to define an inner product, we choose a fiducial triad and scalar, (Ê µi ,σ). In terms of the normal mode associated with K (Ê,σ) , the inner product is defined to be Two states with different amplitudes in any of the normal modes are orthogonal. The inner product defines a natural measure for a functional integration of the matter field in terms of the normal modes as This guarantees that for general functional f (Φ). Obviously, the inner product and the measure depends on the choice of the fiducial triad and scalar, (Ê µi ,σ). A measure defined in terms of a different triad and scalar field (E, σ) is related to eq. (4.5) through a Jacobian, where J In general, the Jacobian is not unity. However, in special cases with |E(x)| = |Ê(x)|,

JHEP10(2018)043
we consider a sub-Hilbert space, V spanned by a set of basis states that are labeled by Here Ψ(Φ; E, σ) is a short-range entangled wavefunction of the matter field in which the metric (g E,µν ) and the scalar field (σ) set local structures of entanglement. Wavefunctions for such short-range entangled states can be written as an exponential of a local functional, in which the triad and the scalar enter as variational parameters. For simplicity, we choose dt t e −λt is the incomplete Gamma function and l c is the cut-off length scale. Γ l 2 c K (E,σ) is a regularized derivative operator that creates local entanglement at distance scales larger than l c . It has the following asymptotic behaviors, where γ E is the Euler-Mascheroni constant. For modes with eigenvalues λ behaves as a two-derivative term at long distances while it becomes a constant a short distances. Only those modes with wavelengths larger than l c have non- we obtain (4.14) Here Tr (..) denotes the trace of differential operators. In eq. (4.13), eq. (4.7) is used. It is noted that the particular choice in eq. (4.11) is not crucial. In order to include more general wavefunctions, one needs to introduce more collective variables which source JHEP10(2018)043 For e σ 1, the kernel disperses quadratically in k at small momenta before the dispersion is lost at large momenta with k l −1 c . For e σ 1, the momentum dependence is suppressed at all momenta, and the wavefuction becomes a direct product state in space. L can be understood as local tensors that generate short-range entangled states, Here E µi and σ play the role of variational parameters (see figure 8) [28]. The metric sets the notion of distance in how matter fields are entangled in eq. (4.11). Because the proper cut-off length scale below which the matter field is unentangled is measured with the metric, the metric controls the number of degrees of freedom that participate in entanglement. On the other hand, σ determines the range of mutual information. In the large σ limit, Ψ(Φ; E, σ) becomes a direct product state in real space. The precise connection between entanglement and the collective variables will be established in section 4.2.
We note that Ψ(Φ; E, σ) depends on triad only through g E,µν . Because metric is gives a physically distinct state of the matter field. diffeomorphism generated by an infinitesimal vector field, ξ µ , 7  Therefore Ẽ ,σ represents a state in which the matter field is shifted in space, and is in general distinct from E, σ as a quantum state of the matrix field (see figure 9).

JHEP10(2018)043
in eq. (4.19) in terms of the measure associated with E µi or E µi . In terms of the measure associated with (E, σ), eq. (4.19) can be written as (4.20) The fiducial metric drops out in eq. (4.20). This has an important consequence: the inner product between states in V is invariant under spatial diffeomorphisms, Two states with |E(x)| = |E (x)| are not orthogonal in general. Nonetheless, E , σ E, σ decays exponentially in h µν and δσ in the large N limit. This is because each of the N 2 components of the matrix field contributes an overlap which is less than 1 when the collective variables do not match. Since l c is the only scale, M(x) ∼ N 2 l 3 c (1 + O(l c ∇)) in eq. (4.22). This form of M(x) is confirmed through an explicit computation of the overlap between states with metrics close to the Euclidean metric in appendix C. Two states whose collective variables differ by or more over a proper volume larger than l 3 c are nearly orthogonal even when |E(x)| = |E (x)| (see appendix C). With increasing N , the overlap approaches the delta function upto a normalization factor. In the large N limit, the overlap can be formally written as  whereμ −1 (E, σ) is a measure determined from the determinant of eq. (C.15). The full expression forμ(E, σ) can be in principle computed from eq. (4.20). Here we don't need an explicit form of the measure.
The overlap provides the natural measure for the functional integration over the collective variables. We define the measure from the condition that for any E µi and σ . Formally, the measure is written as , where the last factor divides out the SO(3) gauge volume. The measure defined by this condition is invariant under diffeomorphism. This can be checked from a series of identities, where χ(E, σ) is invariant under local SO (3) transformations. Its tensor representation is shown in figure 10.

Metric as a collective variable for entanglement
In this subsection, we discuss the physical meaning of the metric and the scalar field as collective variables for the matter field. In particular, we show that the metric controls the JHEP10(2018)043 Figure 11. A tensor representation of the density matrix of region A in space.
number of degrees of freedom that are entangled in space, and the scalar field determines the rate at which the mutual information decays in space. Being a wavefunction defined in continuum, the size of the Hilbert space per unit coordinate volume is infinite. However, the number of degrees of freedom that contribute to entanglement is controlled by the proper volume measured in the unit of the short-distance cut-off, l c . The metric sets the notion of distance in the short range entangled states of the matter. Let us consider a region A in space. For general states in eq. (4.27), the density matrix of the region is given by whereĀ is the complement of A (see figure 11). The replica method allows one to express the von Neumann entanglement entropy as where Z n = Tr (ρ n A ). The entanglement entropy for general states depends both on Ψ(Φ; E, σ) and χ(E, σ) in a complicated way, where the former represents the wavefunction of the matter field for a fixed collective variable (E, σ) and the latter encodes fluctuations of the collective variables. Here we focus on χ(E, σ) that is peaked at a classical configuration (Ē,σ) with small fluctuations around it. For such semi-classical wavefunctions for the collective variables, the entanglement entropy can be approximately decomposed into two contributions, (4.31) Here S Φ (A) is the entanglement entropy of the matter degrees of freedom defined in the classical background collective variables (Ē,σ),

(4.33)
On the other hand, S E,σ (A) is the entanglement generated by correlations between fluctuations of the collective variables, where . Here E 0 = E n and σ 0 = σ n . The derivation of eqs. (4.31)-(4.35) is given in appendix D. Here we provide an intuitive explanation of the result. When χ(E, σ) ∝ δ(g E,µν − gĒ ,µν )δ(σ −σ), there is no fluctuations in the collective variables. In this case, the entanglement entropy is given by that of Ψ(Φ;Ē,σ). Because Ψ(Φ;Ē,σ) is written as an exponential of a local functional, the entanglement entropy is related to the 'free energy' difference caused by a Dirichlet boundary condition as is shown in eq. (4.33) [43]. Now, suppose the wavefunction for the collective variables has a small but nonzero width around the semi-classical configuration. As a simple example, let us assume that there are only two configurations of the collective variables, χ(E, σ) = Aδ(g E,µν − g E 1 ,µν )δ(σ − σ 1 ) + Bδ(g E,µν − g E 2 ,µν )δ(σ − σ 2 ), where (E 1 , σ 1 ) and (E 2 , σ 2 ) are distinct from each other but are close to their average, (Ē,σ). On the one hand, there is an entanglement generated by the matter field whose wavefunction is well approximated by Ψ(Φ;Ē,σ). This entanglement is given by S Φ . However, Ψ(Φ;Ē,σ) does not capture the entire correlation present in the system. There is an additional correlation generated by fluctuations of the collective variables. Since Ψ(Φ; E 1 , σ 1 ) and Ψ(Φ; E 2 , σ 2 ) are almost orthogonal when N is large, these fluctuations of the collective variables give rise to an additional entanglement which is captured by S E,σ .
In the limits that e σ 1 and the linear proper size of A is much larger than l c , S Φ is proportional to the area of ∂A and the number of matter fields. When the metric is flat and σ is constant, one can compute S Φ explicitly. Consider a region, A = {(x 1 , x 2 , x 3 ) 0 < x 1 < l, 0 ≤ x 2 < l c , 0 ≤ x 3 < l c } in T 3 with the flat metric gĒ ,µν = a 2 δ µν as is shown in figure 12. In the small l c limit with fixed al c , the entanglement entropy of region A is given by (see appendix E for derivation) where A ∂A is the area of ∂A measured with the metric gĒ ,µν , and The entanglement entropy is given by the proper area of the boundary measured in the unit of κ 2 . κ is much smaller than the cut-off scale l c in the large N limit. Although eq. (4.36) has been derived in the flat metric, the same formula is expected to hold for general metrics to the leading order in the limit that the curvature is much smaller than l −1 c . This is because the leading order contribution, which is divergent in the l c → 0 limit, comes from short-wavelength modes for which geometry can be regarded locally flat and the WKB approximation is valid.
The entanglement entropy of a fixed region A increases as the proper area of ∂A increases. This can be understood in terms of mode softening with increasing proper volume. The eigenvalue for the mode with momentum k µ = 2π lc (n 1 , n 2 , n 3 ) with integer n µ is given by λ k = 2π alc 2 (n 2 1 + n 2 2 + n 3 3 ) + e 2σ l 2 c . Because of the cut-off scale, only the modes with wave-numbers, n i < a contribute significantly to the entanglement entropy as is shown in appendix E. With increasing a, more modes become soft and contribute to entanglement. Therefore, the number of degrees of freedom that generate entanglement in space is a dynamical quantity rather than a fixed number. This has an important consequence. There is no fundamental limit in the amount of information a 'finite' region in space can hold because the proper size of the region is a dynamical variable which can be as large as it can be. This may sound unphysical until we think about our universe, which was once of the Planck size yet contained the vast amount of information on the current universe.

JHEP10(2018)043
In terms of N counting, S E,σ (A) is O(1) while S Φ (A) is O(N 2 ) for semi-classical states. However, the singlet entanglement entropy can be larger than the color entanglement entropy in some states. This is because the singlet entanglement entropy can scale with the volume of a region if there exist long-range correlations in fluctuations of the collective variables while color entanglement entropy scales with the area of its boundary. This point will become important in the discussion on black hole evaporation in section 4.
In the von Neumann entanglement entropy, σ doesn't enter to the leading order in the small l c limit. However, the scalar field plays a more distinct role in determining mutual information. The mutual information between two regions A and B is defined to be I  σ (A, B)) is the contribution from color degrees of freedom (singlet collective variables). In order to examine the relation between the color mutual information and the collective variables, it is useful to write the expression for the color entanglement entropy as . λ j is an N × N Hermitian Lagrangian multiplier which enforces the Dirichlet boundary condition at the boundary. This is only schematic because the measure for x∈∂A hasn't been specified. However, it is still useful in understanding the connection between the mutual information and the collective variables. Suppose A and B represent infinitesimally small balls centered at x and y, respectively. In the limit that the proper distance between x and y is large, the color mutual information is dominated by the connected correlation function between the fundamental fields inserted at x and y which exhibits the slowest decay in eq. (4.38). A straightforward calculation shows that the color mutual information scales as where G[x, y;Ē,σ] is the correlation function of the fundamental field. In the small e σ limit, G[x, y;Ē,σ] ∼ 1 dx,y , where d x,y is the proper distance between x and y measured with the metric gĒ ,µν . For fixed x and y in the manifold, the proper distance between the points is controlled by the metric, and so does the mutual information. For example, states that support small (large) color mutual information between two points give large (small) proper distance between the points. When e σ is not negligible, the Green's function decays exponentially at large distances, G[x, y;Ē,σ] ∼ e − y x e σ lc ds , where ds is the infinitesimal proper distance along the geodesic that connects x and y. This shows that σ determines the range of entanglement, while the metric sets the notion of locality in how matter fields are entangled in space. In this construction, the connection between entanglement and geometry [44][45][46][47][48][49] has been encoded as a kinematic building block of the theory.

Relatively local Hamiltonian
Having understood the kinematic structure of the sub-Hilbert space, now we construct a Hamiltonian of matter field which induces the Wheeler-DeWitt Hamiltonian of the general relativity in the sub-Hilbert space.

JHEP10(2018)043
Hermitian operators that map V to V generate unitary evolutions of the collective variables. One example of such endomorphisms is the momentum density operator for the matter fields,Ĥ whereπ ba (x) is the conjugate momentum ofΦ ab (x) with the commutator [π ab (x),Φ cd (y)] = −iδ ad δ bc δ(x − y). Due to eq. (4.18), the action ofĤ µ (x) on E, σ is equivalent to a differential operator that induces a diffeomorphism of the collective variables, where .  Here it is used that DEDσ is invariant under diffeomorphism. 8 Similarly, a Hamiltonian for the matter field whose trajectories stay within V induces a quantum Hamiltonian for E µi and σ. Our goal is to construct a Hamiltonian for the matter field which induces the Einstein's general relativity at long distances in the large N limit. Our strategy is to start with a regularized Wheeler-DeWitt Hamiltonian for the collective variables, and reverse engineer to find the corresponding Hamiltonian for the matter field. We look for a Hamiltonian density whose action on E, σ leads tô whereh E,σ (x) is a regularized Wheeler-DeWitt differential operator [50][51][52] for the collective variables, : Here G ijkl = 1 4 δ ik δ jl − 1 2 δ ij δ kl is the supermetric for the kinetic term of the triad. F (σ) represents a nonlinear term in the kinetic energy of the scalar. R is the curvature scalar for the three-dimensional metric g E,µν . V (σ) is a potential for the scalar. U 3 (g E , σ) represents terms that involve more than two derivatives for g E,µν and σ, where the higher-derivative terms are suppressed by (l c ∇) compared to the two-derivative terms. F (σ), V (σ) and U 3 (g E , σ) are included for generality, but we do not need to specify their forms for our purpose. It is important to note that the second order functional derivatives in eq. (4.45) needs to be regularized as the derivatives acting on one point in space is ill-defined. Here the derivatives are regularized through a point-splitting scheme based on the heat-Kernel regularization, , . (4.46) K µiνk (y, z; x, t) and K(y, z; x, t) spread the two differential operators over the cut-off length scale l c centered at x. In the heat kernel regularization scheme [53][54][55], the kernels satisfy the diffusion equation, with the boundary condition, K µiνk (y, z; . In eq. (4.47) and eq. (4.48), ∇ y (∇ z ) represents the covariant derivative acting on coordinate y (z). In the Euclidean space, the regulators become

JHEP10(2018)043
where d x,y is the proper distance between x and y. In eq. (4.45),κ is the Planck scale for the induced gravity, which is a free parameter for now. Below, we show thatκ should be order of κ in the large N limit if the underlying matter Hamiltonian has a well-defined large N limit. The Hamiltonian density for the matter field that satisfies eq. (4.44) is given bŷ to the leading order in the large N limit, wherê .T µi (x) andÔ σ (x) in eq. (4.50) scale as O N 0 in the large N limit. In eq. (4.49), the double-trace operators [33,34,36,56,57] are responsible for generating the kinetic terms inh E,σ . In order for the leading kinetic term and the potential term in eq. (4.49) to scale uniformly in the large N limit, one needsκ κ ∼ O(N 0 ). This implies that a matter Hamiltonian which scales as O(N 2 ) in the large N limit induces a gravity with the Planck scale κ ∼ lc N , which also controls the color entanglement entropy through eq. (4.37). From now on, we focus on such Hamiltonians, and setκ = κ.
Eq. (4.44) implies that the evolution of E, σ generated byĤ(x; E, σ) is reproduced by the differential operatorh E,σ (x) acting on the collective variables. However,Ĥ(x; E, σ) is defined with reference to (E, σ), and it does not act on states with different collective variables in the same way. We encountered the same issue in section 3. In order to induce a background independent Hamiltonian for the collective variables, we use the strategy introduced in the minisuperspace cosmology. Namely, we make a Hamiltonian to effectively depend on the collective variables so thatĤ(x; E, σ) associated with a specific collective variable is only applied to the corresponding state, E, σ . This can be implemented by the following Hamiltonian density, Here h.c. represents the Hermitian conjugate of the fist term.Ĥ(x, E, σ) is given by eq. (4.49) withκ = κ.P E,σ is an operator that satisfieŝ P E,σ DE Dσ E , σ χ(E , σ ) = E, σ χ(E, σ).  applied (see figure 14). For any E, σ ,Ĥ(x) satisfieŝ where , l c κ 2 δ δσ(x) . (4.54) The construction ofP E,σ and the derivation of eq. (4.54) are in appendix G. In the limit that l c κ 2 δ δE µi (x) , l c κ 2 δ δσ(x) 1, the higher derivative terms in eq. (4.54) can be ignored, and H(x) induces the Wheeler-DeWitt Hamiltonian for the collective variables at long distance scales, (4.55) Unlike the case with the minisuperspace cosmology discussed in the previous section, it is hard to perform the functional integrations over the collective variables explicitly in eq. (4.51). In the following, we discuss general features of the Hamiltonian, focusing on its locality. By choosing a space dependent speed of local time evolution, we construct a Hamiltonian,Ĥ n (t) ≡ dx n(x, t)Ĥ(x), (4.56) where n(x, t) in general depends both on space and time. As a Hamiltonian for matter fields, one can ask how local the Hamiltonian is in space. In order to answer this question, we first focus onĤ(x; E, σ) which is a part ofĤ(x). In eq. (4.49), the point-splitting of the kinetic terms and the higher-derivative terms are controlled by l c . Therefore,Ĥ(x; E, σ) is local at length scales larger than l c , if distances are measured with the metric g E,µν . However, there is no absolute sense of locality inĤ n (t) because the metric that determines proper distances is not fixed. Instead,Ĥ(x) is given by the sum ofĤ(x; E, σ)P E,σ over different metrics. To understand this point, let us consider two points, say x A and x B in the manifold. Because the metric inĤ(x; E, σ) is determined by the state to which JHEP10(2018)043 Figure 15.Ĥ(x A ; E, σ) has a spread over ∆x ∼ lc √ g E,xx along x direction in the manifold. In the presence of a large (small) mutual information between x A and x B , g E,µν is small (large) and the spread of the operator inserted at x A can (can not) reach x B . Therefore, the strength of coupling between x A and x B depends on entanglement of target states which determines the metric. a target state is first projected byP E,σ , the strength of the coupling between x A and x B in the Hamiltonian is determined by the entanglement present in the target state. A state which supports small mutual information between the two points is projected to a state in which the proper distance between the points is large. Accordingly, the operators in eq. (4.49) are spread over a small coordiniate distance, and the coupling between the points is weak. Conversely, for a state which supports large mutual information between the two points, the metric in E, σ gives a small proper distance and a large coupling in H n (t). This is illustrated in figure 15. There is no absolute locality because the metric with which locality is defined varies with states. Since the coupling between any two points can be large for long-range entangled states,Ĥ n (t) is not a local Hamiltonian as an operator. This is expected because there is no fixed notion of distance in any theory of background independent gravity [16]. Since locality of the Hamiltonian is determined relative to target states, we callĤ n (t) relatively local. We emphasize that this conclusion on relative locality holds generally for Hamiltonians which induce background independent gravity irrespective of specific choice of Ψ(Φ; E, σ).
Ideally, one would hope to fix the regularization scheme and the higher-derivative terms in eq. (4.45) such thatĤ(x) andĤ µ (x) satisfy the hypersurface embedding algebra at the quantum level [58]. The commutators that involve the momentum constraint satisfy the algebra easily. However, it is not clear whether there exists a regularized matter Hamiltonian which obeys the closed algebra at the quantum level. What is guaranteed in this construction is thatĤ(x) andĤ µ (x) satisfy the closed algebra to the leading order in the large N limit within states with slowly varying collective variables in space. Now we viewĤ(x) andĤ µ (x) as generators of symmetry. States that are invariant under the symmetry, if exist, satisfy  [40,41]. Therefore we consider normalizable states that break the symmetry spontaneously. In particular, we consider normalizable semi-classical states which satisfy the constraints to the leading order in N but break the symmetry only to the sub-leading order, Hereḡ µν (x),σ(x),π µν (x) andπ σ (x) are classical collective variables and their conjugate momenta, and χ n is a normalization constant. e − l 2 and (4.58) become the classical constraint equations [50], to the leading order in 1/N, ∆/l c , (l c ∇), (l c π). If the classical collective variables and the conjugate momenta satisfy eq. (4.60), semi-classical states obey the momentum and Hamiltonian constraints approximately. For such states with weakly broken symmetry, the constraints generate non-trivial evolution by creating Goldstone modes associated with the spontaneously broken symmetry, where dt is an infinitesimal parameter. n (1) (x) and n (1)µ (x) are arbitrary functions of x which control the local speed of the unitary transformation and the shift respectively. From eq. (4.43) and eq. (4.55), we obtain H[E, σ, π, π σ ] = : After repeating this step infinitely many times in the dt → 0 limit, one obtains The theory describes gravity coupled with a scalar field in a fixed gauge. In the large N limit, the saddle-point configuration which satisfies the classical field equations dominates the path integration. The theory in eq. (4.65) has three length scales. One is the cut-off length scale, l c which controls the relative locality of the theory. At length scales larger than l c , where lengths are measured with a saddle-point configuration of the dynamical metric, the local field theory is valid for the description of fluctuations of the collective variables above the saddle-point configuration. At shorter length scales, non-local effects kick in through the higher-derivative terms in the Hamiltonian. The other length scale is the Planck scale, κ ∼ lc N below which quantum fluctuations of the collective variables become important. Another scale is the curvature of the saddle-point geometry set by the vacuum energy. Here we assume that V (σ) is chosen such that the cosmological constant is much smaller than l −4 c , which in general requires a fine tuning of the potential. In the large N limit, the semi-classical field theory approximation is valid for modes whose wavelengths are larger than l c .

Entanglement neutralization
The discussion in the previous section shows that a relatively local Hamiltonian for the matter field induces a quantum theory for the collective variables, which reduces to Einstein's gravity coupled with a scalar field at long distances in the large N limit. Given an initial state χ , the state at parameter time t is given by where P T time-orders the evolution operator. The state at time t can be written in the basis of E, σ as χ(t) = DEDσ E, σ χ(t; E, σ). If the initial state is chosen to be JHEP10(2018)043 a semi-classical state in eq. (4.59), χ(t; E, σ) is sharply peaked at a saddle-point path in the large N limit. The saddle-point path, {Ē µi (x, t),σ(x, t),π µi (x, t),π σ (x, t)} solves the classical field equation with the initial condition, gĒ ,µν (x, 0) =ḡ µν (x),σ(x, 0) =σ(x), E ν i (x, 0)π µi (x, 0) =π µν (x) +π νµ (x),π σ (x, 0) =π σ (x). Suppose the initial condition of the semi-classical state is chosen such that the classical solution describes a gravitational collapse of a spherically symmetric mass shell, which forms a macroscopic black hole with mass M in the asymptotically flat Minkowski space. We assume that the size of the horizon is much larger than the cut-off length scales, r H = 2M κ 2 l c right after the black hole is formed. Across the horizon, the underlying quantum state supports color entanglement, We identify eq. (5.2) as the Bekenstein-Hawking entropy [37,38]. On the other hand, the singlet entanglement entropy is negligible when the black hole is just formed.
In describing the consequent evolution of the black hole, we choose n µ (x, t) = 0 and n(x, t) > 0 at all x, t in eq. (5.1). Far away from the black hole, the lapse is chosen to approach a non-zero constant. Inside the black hole, the lapse is chosen such that time slices do not hit the 'singularity', and the theory at each time slice stays within the realm of the semi-classical field theory. As time progresses, the space that connects the interior of the black hole and the asymptotic region is stretched as is illustrated in figure 16.
For a large but finite N , one should include quantum fluctuations of the collective variables in eq. (4.64). Due to quantum fluctuations, the black hole emits Hawking radiation, and its mass decreases in time. Letḡ µν (x, t) describe an evaporating black hole geometry that satisfies the classical field equation in the presence of a time dependent black hole mass and the energy-momentum tensor of the Hawking radiation. The rate at which the black hole mass decreases should be self-consistently determined by the Hawking radiation created by small fluctuations around the classical geometry, g µν (x, t) =ḡ µν (x, t) + δg µν (x, t), σ(x, t) =σ (x, t) + δσ(x, t). For large black holes, the Hawking radiation should be well approximated by the adiabatic approximation because the rate at which the mass decreases is much smaller than r −1 H . As the black hole evaporates, the horizon shrinks and the color entanglement entropy decreases. On the other hand, the singlet entanglement entropy increases because Hawking radiation is emitted in the form of fluctuations of the collective variables. This is easy to understand in the weakly coupled effective theory for the collective variables. From the perspective of the fundamental matrix field, it is not obvious why Hawking radiation is emitted only in the singlet sector. However, the underlying theory for the matrix field in eq. (4.51) is likely to be a strongly coupled field theory, and it is conceivable that there is only O(1) Hawking radiation [59]. If there was Hawking radiation of O(N 2 ) color degrees of freedom, the induced theory for the collective variables could not be the semi-classical general relativity which we know is the correct description of eq. (4.51). Therefore the increasing entanglement between the Hawking radiation and the degrees of freedom inside the horizon should be in the singlet sector. In this regard, black hole evaporation can be JHEP10(2018)043 Figure 16.
A collapsing mass shell (red dashed line) and the horizon (blue solid line) in the in-going Eddington-Finkelstein coordinate system, where the transverse directions (θ, φ) are suppressed. The metric outside the collapsing mass shell is given by the Schwarzschild metric, The curves represent time slices which march forward with a constant lapse far from the black hole while avoiding the singularity inside the horizon. ∂ ∂r ∂ ∂t represents the vector field that is tangential (perpendicular) to each time slice. It is noted that ∂ ∂r , which is distinct from ∂ ∂r , is chosen to be space-like everywhere.
viewed as an entanglement neutralization process in which entanglement across horizon is transferred from color degrees of freedom to singlet degrees of freedom.

Late time evolution
The fate of χ(t; E, σ) in the large t limit largely depends on which of the following two possibilities is realized. The first possibility is that eq. (5.1) evolves to a state which ceases to support a well-defined horizon as early as the Page time. The second is that χ(t; E, σ) remains sharply peaked around the time dependent classical metricḡ µν (x, t) with a well defined horizon. Resolving this issue is a complicated dynamical question. It may well be that the answer depends on details of initial states. In this section, we consider consequences of the second possibility, assuming that there exists some initial states which continue to support a well-defined horizon throughout the evolution before the size of black hole reaches the cut-off length scale. The reason we focus on the second possibility is because the black hole information puzzle arises in that case [38]. Our goal here is to understand how the puzzle can be in principle resolved in the current framework.
In the context of the present induced gravity, the 'information puzzle' [38,[60][61][62][63][64][65] can be phrased as the statement that the small color entanglement entropy, which is identified as the Bekenstein-Hawking entropy, can not account for the large entanglement created by the Hawking radiation. However, this is not necessarily paradoxical because the color entanglement entropy captures only a part of the full entanglement. The other part is the singlet entanglement entropy which is supported by correlations between fluctuations of the collective variables. In this theory, the time evolution is unitary by construction. In order to support the entanglement with the Hawking radiation outside the horizon, at least e S H states need to be excited inside the horizon. In the large N limit, modes with wavelengths larger than l c are JHEP10(2018)043 described by the weakly coupled field theory, and they have Gaussian fluctuations which are order of δh µ ν , δσ ∼ 1 N . 9 The volume of the throat inside the horizon is V ∼ M 7/2 κ 5 l 3/2 c at t = T . If all field theory modes with wavelengths larger than l c are excited, the total number of states that are available inside the horizon is e S col with For a macroscopic initial black hole with r H l c , S col S H . The number of field theory modes available inside the horizon is much larger than what the color entanglement entropy (Bekenstein-Hawking entropy) accounts for near the end of evaporation process. 10 The states counted in eq. (5.4) include highly excited states. However, one does not need all of them. In order to account for S H ∼ N 2 r H lc 2 , it is enough to excite modes with wavelengths larger than λ ∼ l c r H lc 1/2 . In the r H l c limit, only those excitations whose wavelengths are much larger than l c are needed to account for the entanglement with the early Hawking radiation.
The large number of singlet states that are not captured by the Bekenstein-Hawking entropy suggests that the quantum state inside the horizon is far from equilibrium. The lack of equilibrium can arise dynamically because of the relative nature of the Hamiltonian. Since the strength of coupling between two given points in the manifold is determined by states, points that were once connected by a strong coupling can dynamically decouple at a later time if the state at later time supports little color entanglement. In this sense, not only states but also the Hamiltonian effectively flow in time under the evolution generated by the relatively local Hamiltonian. The growth of a long geometry inside the horizon with a small contact with the exterior is a form of dynamical localization where the coupling across the horizon becomes weaker in time.

Summary and discussion
In this paper, it is shown that a quantum theory of gravity can be induced from quantum matter. Metric is introduced as a collective variable which controls entanglement of matter fields. There exists a Hamiltonian for matter fields whose induced dynamics for metric coincides with the general relativity at long distances in the large N limit. The Hamiltonian that gives rise to the background independent gravity is non-local. However, it has a relative locality in that the range of interactions is controlled by entanglement present in target states. Within the induced theory of gravity, a black hole evaporation can be understood as a unitary process where entanglement of matter is gradually transferred from color degrees of freedom to singlet collective degrees of freedom. We close with some remarks and speculations on open questions. 9 For example, we can choose F (σ) ∼ e 3σ , V (σ) ∼ l −2 c e 4σ such that δh µ ν ∼ 1 N , δσ ∼ e −3/2σ N for e σ 1. 10 It is noted that a renormalization of the Newton's constant by 1/N corrections is not enough to incorporate the singlet entanglement entropy within the color entanglement entropy. JHEP10(2018)043 Figure 18.
One possible outcome of evolving the state in figure 17(b) further in time is a topological phase transition where the region inside the horizon is disconnected from the outer universe.

Baby universe as a dynamical localization
What happens if we continue to evolve the state in eq. (5.1) beyond time T ? It is hard to answer the question without considering the full theory beyond the local semi-classical approximation. Here we consider possibilities that do not modify the usual rules of quantum mechanics. If the long throat inside the horizon remains attached to the outer space, it gives rise to a long-lived (or stable) remnant (for a review on remnant, see ref. [66] and references there-in). If the region inside the horizon becomes geometrically disconnected from the exterior, a baby universe can form as in figure 18 [67,68]. From the perspective of the underlying matter fields, a baby universe corresponds to a dynamical localization where the region inside the horizon dynamically decouples from the exterior. Here localization is driven not by disorder but by the relative nature of the Hamiltonian, where the strength of couplings between the interior and the exterior of the horizon dynamically flows to zero at late time.
Since the proper volume inside the horizon can be arbitrarily large, there can be infinitely many different remnants or baby universes. Although this seems unphysical, this is allowed within the present theory because the number of degrees of freedom within a given region of the manifold is not fixed. Any background independent quantum theory of gravity should include such states in the Hilbert space. The presence of infinitely many internal states does not necessarily lead to an infinite production rate if the matrix element between a state with a smooth geometry and a state with a remnant or a baby universe is exponentially suppressed as the volume of the 'hidden' space increases. For example, let E µi (x) be the flat Euclidean geometry, g E,µν = δ µ,ν , and E µi (x) represent a geometry which coincides with the Euclidean metric for |x| > R but has a long funnel with proper volume V R 3 for |x| < R. LetÔ be an operator that has a support within |x| < R. The matrix element is given by . (6.1) If the matrix element is small enough, the net production rate can be suppressed. Figure 19.

JHEP10(2018)043
A pictorial representation of the overlap between 0 and E, σ . Since 0 is annihilated by the Hamiltonian and momentum constraints, the overlap is invariant under the insertions of the evolution operator, which gives rise to a path integration of the metric and the scalar field in the bulk.

dS/CFT
By construction,Ĥ µ (x) andĤ(x) satisfy the closed algebra [58] to the leading order in the 1/N and the derivative expansions. However, the commutator between two Hamiltonian constraints may have an anomaly that involves higher derivative terms and 1/N corrections. Whether one can choose a regularization scheme for the matter Hamiltonian such that the algebra is closed exactly is an open question [69][70][71][72].
Suppose there exists a state 0 which is annihilated byĤ(x) andĤ µ (x). The existence of such states doesn't necessarily require that the algebra is closed at the operator level. Although 0 is not normalizable in general, the overlap with a normalizable state, E, σ is well defined. The overlap, which can be viewed as a wavefunction of universe, is invariant under the insertion of the evolution operator generated by the Hamiltonian and momentum constraints [28], 0 E, σ = 0 e −idt dx [n(x,τ)Ĥ(x)+n µ (x,τ )Ĥµ(x)] E, σ . where the bulk action is given by eq. (4.65). Therefore, the overlap is given by the (D + 1)dimensional path integration with a Dirichlet boundary condition for the collective variables as is represented in figure 19. The bulk path integral can be viewed as the gravitational dual for the generating functional of the non-unitary boundary field theory in the dS/CFT correspondence [24,25]. If the lapse and the shift are integrated over, the bulk path integration becomes a projection operator which imposes the Hamiltonian and momentum constraints on the boundary state, E, σ .

Non-gaussian states
The standard lore in the AdS/CFT correspondence is that a bulk theory includes only a small number of fields if the dual boundary theory is in a strong coupling regime and the majority of operators have large scaling dimensions [73]. The number of dynamical fields one has to keep in the bulk is determined by the number of independent operators

JHEP10(2018)043
from which all other operators can be constructed as composite operators [27]. Although there are in general infinitely many such operators, if most of them acquire large scaling dimensions they correspond to heavy fields in the bulk, which can be integrated out without sacrificing locality in the bulk. From this perspective, it is rather surprising that a simple bulk theory is obtained from the Gaussian wavefunction defined at a temporal boundary. The reason why we have only dynamical metric and a scalar field in the bulk is because the initial states and the Hamiltonian for the matter field are fine-tuned so that the time evolution generates deformations contained within the sector of the energy-momentum tensor and one scalar operator only. From the point of view of RG flow, we are in the basin of attraction toward a multi-critical point via a fine tuning. In general, there exist 'relevant' perturbations which take the flow away from the multi-critical point once perturbations are turned on. In the present work, we simply didn't consider such perturbations in the initial state. In order to suppress other operators without fine tuning, one probably needs non-Gaussian wavefunctions which describe strongly coupled boundary theories.
We prove the title of this appendix. The vanishing overlap between states with different local proper volumes is the result of mismatch in the gaussian wavefunctions for large momentum modes. For modes with momenta much larger than curvature scales, we can use the semi-classical approximation to represent eigenmodes in terms of wavepackets. To keep track of position of wavepackets, we divide the spatial manifold into blocks. Suppose we choose a coordinate system with 0 ≤ x 1 , x 2 , x 3 < l c for a spatial manifold with T 3 topology. The j-th block is denoted as S j = {x|x µ j ≤ x µ < x µ j + l, with µ = 1, 2, 3}, where x µ j = l j µ with j µ being integers. l ≡ lc P with an integer P is chosen to be small enough that g E,µν (x), σ(x), g E ,µν (x), σ (x) can be regarded to be constants in each block. A wavepacket is labeled by a block index j and quantized momentum k allowed within the block, Here momentum is quantized as k µ = πnµ l with positive integer n µ . The normalization is chosen such that dx|E|f Field configurations which vanish at the boundaries of the blocks can be decomposed in terms of the wavepack- . The wave packets contribute to the overlap in eq. (4.19) as → 0 in the large k limit, the contribution from large momentum modes becomes Since there are infinitely many momentum modes with g µν E (x j )k µ k ν , g µν E (x j )k µ k ν l −2 c in each block, E , σ E, σ vanishes if there is any block in which |E(x j )| = |E (x j )|. Here we ignored the modes that describe fluctuations of Φ(x) at the boundaries between blocks. Those modes do not change the conclusion because they form fluctuations of measure zero.

C Overlap
In this appendix, we compute the overlap in eq. (4.20) between states associated with metrics close to the Euclidean metric. Since the overlap is zero for |E(x)| = |E (x)|, we consider the case with |E(x)| = |E (x)|. In this case, the exponent in the integrand of eq. The Kernel with perturbations in the collective variables is written as K = K + δK and K (E ,σ ) = K + 2δK, where K ≡ K (E,σ) . The exponential of the perturbed Kernel can be expressed as the contribution from the collective variables. It is the last three terms that generate non-trivial entanglement. Now we insert the following expression for the identity inside the integration of eq. (D.3), where x, y run over the entire space, and The functional integration of Φ in the last line is unconstrained, and eq. (4.24) has been used. The integrations over E j 2 , σ j 2 result in χ E j (xĀ), σ j (xĀ); E j−1 (x A ), σ j−1 (x A ) σ 2,..,n (x)=σ 1 (x),g E 2,..,n ,µν (x)=g E 1 ,µν (x),x∈∂A × n j=1 D (Ê,σ) Φ j e −2 j tr x,y∈Ā dxdy Φ j (x)t E j ,σ j (x,y)Φ j (y) × e −2 j tr x,y∈A dxdy Φ j (x)t E j ,σ j (x,y)Φ j (y) × e − j tr x∈Ā,y∈A dxdy Φ j (x) t E j ,σ j (x,y)+t E j ,σ j (y,x) Φ j (y) It is noted that the delta functions in eq. (D.4) twists the boundary condition for the collective variables in eq. (D.6), and force σ j (x) and g E j ,µν to be independent of j in ∂A.
If χ(E, σ) is sharply peaked at a classical configuration,Ē(x),σ(x) (and its SO(3) gauge orbits), eq. (D.6) can be approximately factorized into the contribution from the matter fields and the contribution from the collective variables as Z n ≈ Z Φ n (Ē,σ)Z E,σ n , (D.7)

E Computation of the entanglement entropy
Here we derive eq. (4.36) for the state with the Euclidean metric in eq. (C.8) and a constant σ. In the three torus, we compute the entanglement entropy for region A which is parameterized by the coordinate, A = {(x 1 , x 2 , x 3 ) 0 ≤ x 1 < l, 0 ≤ x 2 < l c , 0 ≤ x 2 < l c }.