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.

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.
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 Sec. II, 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.
Sec. III 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 Sec. III A, 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, 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 Sec. III B, 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 Sec. III C, 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 Sec. IV A 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.
Sec. IV A 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

II. THE MAIN IDEA
In this section, we sketch the main idea of the paper that is summarized in Fig. 1 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 HamiltonianĤ 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 ∂gµν is identified as an induced Hamiltonian for the metric.
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 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 non-local Hamil- tonian. 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 introducing a fixed background [74]. 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 Sec. III, 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 Sec. IV, we generalize the construction to a fully fledged gravity in three space dimension. In Sec. V, we discuss possible implications of the induced gravity for the black hole information puzzle.

III. 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.

A. 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 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 Fig. 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 Sec. II. 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 exponentially 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, where The overlap defines a natural measure in the space of α, σ, which guarantees that lim N →∞ DαDσ α , σ α, σ = 1 (9) for any α and σ .

B. 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 ,φ b ] = −iδ a,b .
Requiring thatĤ(α, σ) α, σ is in V fixes U (φ, α, σ) to be Eq. (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 V. The induced H(α, σ), as a quantum operator of the matter fields, depends on α, σ. This means that 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. (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.
H 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 [76]. 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 Eq. (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 , FIG. 4: 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.

C. Emergent time as a Goldstone mode
Eq. (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. (14). The matrix elements of the Hamiltonian is written as Explicit integrations over α, σ result in where 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 Fig. 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, 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.
whereᾱ,σ,π,π σ are classical coordinates and momenta which satisfy ∆ determines the uncertainty of α and σ. With 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 oneparameter family of states. The evolution can be viewed as a Goldstone mode associated with the spontaneously broken symmetry. This is illustrated in Fig. 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. (22) has a fast oscillating phase factor, the convolution integration in Eq. (19) gives rise to a suppression in the norm of the wavefunction, 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. (26) remains a semiclassical state centered at a different classical configuration. In the next step, we choose the lapse 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.
(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 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.

IV. 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 [78]. 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.

A. 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, with 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. (31), 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 ab,n represents the amplitude of the n-th normal mode in the basis of f (E,σ) n (x) . 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. (33) through a Jacobian, where J In general, the Jacobian is not unity. However, in special cases with Within the full Hilbert space, we focus on singlet states that are invariant under global SU (N ) In particular, we consider a sub-Hilbert space, V spanned by a set of basis states that are labeled by {E µi (x), σ(x)}, 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 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-negligible entanglement in space. A plot of e −Γ[l 2 c K (E,σ)] is shown in Fig. 7. S 0 [E, σ] is chosen to enforce the normalization condition, we obtain Here Tr (..) denotes the trace of differential operators. In Eq.
generally gives a physically distinct state of the matter field.
We note that Ψ(Φ; E, σ) depends on triad only through g E,µν . Because metric is invariant under there is a gauge redundancy in labeling states in V in terms of triad. Each gauge orbit generated by SO(3) transformations corresponds to one state in V. Unlike the SO(3) gauge transformation, a diffeomorphism of the collective variables generates a different state of the matter field in general. In order to see this, we note that Ψ(Φ; E, σ) is invariant upto a multiplicative factor under diffeomorphisms of the collective variables and the matter field. Under a diffeomorphism generated by an infinitesimal vector field, the wavefunction is transformed as 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 Fig. 9).

Inner product
The inner product between states in V is written as While both D (Ê,σ) Φ and Ψ * (Φ; E , σ )Ψ(Φ; E, σ) depend on the fiducial metric, E , σ E, σ does not because the dependence on the fiducial metric in the measure is canceled by the normalization factor in Eq. (42). This can be seen by rewriting the functional integration in Eq. (47) in terms of the measure associated with E µi or E µi . In terms of the measure associated with (E, σ), Eq. (47) can be written as The fiducial metric drops out in Eq. (48). This has an important consequence : the inner product between states in V is invariant under spatial diffeomorphisms, Once the Gaussian integration is performed in Eq. (48), the overlap can be written as to the quadratic order in the difference of the collective variables, v a (x) = (h E,µν (x), δσ(x)) with index a running over different collective variables, where h µν = g E ,µν − g E,µν and δσ = σ − σ. 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. (C15). The full expression forμ(E, σ) can be in principle computed from Eq. (48). 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 {Ẽ µi ,σ} is related to {E µi , σ} through a diffeomorphism. For the first equality, we use the fact that Eq. (53) holds for any E µi and σ. The second equality is a simple change of variables.
For the third equality, we use the fact that the inner product is invariant under diffeomorphism. Eq. General states in V can be expressed as linear superpositions of E, σ , where χ(E, σ) is invariant under local SO(3) transformations. Its tensor representation is shown in Fig. 10. It is normalized such that DE Dσ DEDσ χ * (E , σ ) E , σ E, σ χ(E, σ) = 1. In the large N limit, E , σ E, σ is sharply peaked at g E ,µν = g E,µν , σ = σ, and the normalization condition reduces to DEDσ |χ(E, σ)| 2 = 1. Similar to Eq. (16), we define the Hermitian conjugate of a differential operator acting on the collective variables from

B. 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 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. (55), the density matrix of the region is given by whereĀ is the complement of A (See Fig. 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, Here S Φ (A) is the entanglement entropy of the matter degrees of freedom defined in the classical background collective variables (Ē,σ), where On the other hand, S E,σ (A) is the entanglement generated by correlations between fluctuations of the collective variables, where withχ(E, σ) = e − 1 2 S 0 [E,σ] χ(E, σ). Here E 0 = E n and σ 0 = σ n .
The derivation of Eqs. (59)-(63) 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. (61) [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, 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 Fig. 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. (64) 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.  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. (66). 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, 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.

C. 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.
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, 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. (46), the action ofĤ µ (x) on E, σ is equivalent to a differential operator that induces a diffeomorphism of the collective variables, where Eq. (69)  (68) results in a shift in wavefunctions of the collective variables as is illustrated in Fig. 13, Here it is used that DEDσ is invariant under diffeomorphism[81].
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 vari- : 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. (73) 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, , 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, ∂ ∂t K(y, z; x, t) = ∇ 2 y + ∇ 2 z K(y, z; x, t) with the boundary condition, K µiνk (y, z; . In Eq. (75) and Eq. (76), ∇ y (∇ z ) represents the covariant derivative acting on coordinate y (z). In the Euclidean space, the regulators become where d x,y is the proper distance between x and y. In Eq. (73),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. (72) is given bŷ to the leading order in the large N limit, wherê (78) scale as O (N 0 ) in the large N limit. In Eq. (77), 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. (77) 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. (65). From now on, we focus on such Hamiltonians, and setκ = κ. Eq. (72) 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 Sec. III. 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. (77) with κ = κ.P E,σ is an operator that satisfieŝ In Eq. (79), a general state is first projected to the state with each collective variable (E, σ), and then the Hamiltonian associated with the collective variables,Ĥ(x; E, σ) is applied (see Fig. 14).
For any E, σ ,Ĥ(x) satisfiesĤ where The construction ofP E,σ and the derivation of Eq. (82) are in Appendix G. In the limit that 1, the higher derivative terms in Eq. (82) can be ignored, and H(x) induces the Wheeler-DeWitt Hamiltonian for the collective variables at long distance scales, 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. (79). 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, 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. (77) 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 inĤ n (t). This is illustrated in Fig. 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. (73) 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 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 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. (88), 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. Eq. (83), we obtain where 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. (93) has three length scales. One is the cut-off length scale, l c which con- 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 .

A. 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 a semi-classical state in Eq. (87), 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),Ē 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 ∂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.
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. (95) 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. (94). 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 Fig. 16.
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. (79) 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. (79). 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 viewed as an entanglement neutralization process in which entanglement across horizon is transferred from color degrees of freedom to singlet degrees of freedom.

B. 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. (94) 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. Under the time evolution, the size of the horizon continues to shrink, and so does the color entanglement entropy. At t = T , the color entanglement entropy across the horizon becomes On the other hand, the Hawking radiation generates a large entanglement across the horizon, 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 described by the weakly coupled field theory, and they have Gaussian fluctuations which are order of δh µ ν , δσ ∼ 1 N [82]. 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

VI. 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 What happens if we continue to evolve the state in Eq. (94) 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 Fig. 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 If the matrix element is small enough, the net production rate can be suppressed.

B. 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], where the bulk action is given by Eq. (93). 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 Fig. 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 .

C. 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 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.
[78] The collective variables are fields in this case.
[83] 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.
given by I µναβ (p) = I 4,1 (p) g µν E g αβ E + I 4,2 (p) g µα E g νβ E + g µβ E g να Therefore, ln E , σ E, σ ∼ − N 2 Now we insert the following expression for the identity inside the integration of Eq. (D3), 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. (D4) twists the boundary condition for the collective variables in Eq. (D6), and force σ j (x) and g E j ,µν to be independent of j in ∂A.
(D8) can be decomposed into modes with a Dirichlet boundary condition and modes localized at the boundary as the coordinate, A = {(x 1 , x 2 , x 3 ) 0 ≤ x 1 < l, 0 ≤ x 2 < l c , 0 ≤ x 2 < l c }. In the presence of the Dirichlet boundary condition, Φ(0, x 2 , x 3 ) = Φ(l, x 2 , x 3 ) = 0, the eigenmodes in region A are modified such that where (k 1 , k 2 , k 3 ) = n 1 π l , 2n 2 π lc , 2n 3 π lc with eigenvalue λ k = k 2 + l −2 c e 2σ . Here n 1 represents positive integers while n 2 , n 3 are general integers. The free energy from region A is given by in the limit that a 1. Integration over t gives Eq. (64) in the e σ → 0 limit.