Clock-dependent spacetime

Einstein's theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.


I. INTRODUCTION
In Newtonian paradigm, physical laws govern the evolution of dynamical degrees of freedom with respect to one universal time. Einstein's theory of relativity demotes time from the absolute status in two ways. First, the notion of simultaneity becomes observer-dependent for events that are spatially separated, and there is no universal sense of past, present and future. Second, time evolution is turned into a gauge transformation, and time as a gauge parameter has no physical meaning by itself [1][2][3][4]. One needs to use dynamical variables as clocks to describe the relative evolution of other degrees of freedom [5][6][7]. The theory only predicts correlations among dynamical degrees of freedom.
In quantum gravity, choosing clocks boils down to dividing the kinematic Hilbert space into a sub-Hilbert space for clocks and the rest for the 'true' physical degrees of freedom. In general relativity, while there is no preferred coordinate system, there is still a preferred way of identifying a sub-Hilbert space for each local clock. This is because the theory is covariant only under the transformations that preserve the integrity of local sites. Under diffeomorphism, points in space are permuted, but the quantum information stored at a point is never spread over multiple points.
The preferred set of local kinematic Hilbert spaces is invariant under diffeomorphism, and each local clock variable is chosen from one of the local Hilbert spaces.
Requiring that physical laws are covariant only under the local Hilbert space-preserving transformations may be too restrictive in quantum gravity that has no predetermined notion of locality.
In priori, one partitioning of Hilbert space is no better than others. Furthermore, the fact that locality is a dynamical concept in quantum gravity obscures the distinction between local and non-local transformations. Consider a unitary transformation that mixes local Hilbert spaces associated with multiple sites. For states in which those sites are within a short-distance cutoff scale, the transformation can be regarded as local in space, and we may gauge it as an internal symmetry.
However, it is no longer local for other states in which the sites affected by the transformation are macroscopically apart. Once we gauge general unitary transformations that do not preserve local kinematic Hilbert spaces, the preferred Hilbert space decomposition is lost.
In this paper, we examine consequences of having no preferred Hilbert space decomposition in a recently proposed model of quantum gravity [8]. In the model, geometric degrees of freedom are collective variables of underlying quantum matter [9][10][11]. In the absence of a predetermined Hilbert space decomposition, there exists a greater freedom in how clock variables are identified within the kinematic Hilbert space. Instead of choosing local clock variables from predetermined local Hilbert spaces, in this theory local Hilbert spaces are identified from a choice of clock variables. In other words, the notion of local site is derived from clocks. A set of local observers who use a particular set of local clocks constructs a geometry based on the pattern of entanglement present across the local Hilbert spaces associated with the clocks. Inasmuch as patterns of entanglement depend on partitioning of the total Hilbert space, one state can exhibit different geometries with different choices of local clocks. The spacetime that emerges with respect to one choice of clocks is in general different from the spacetime that arises with respect to another set of local clocks. The purpose of the paper is to show that the spacetimes that emerge from different choices of local clocks can exhibit different dimensions, signatures, topologies and geometries.
The rest of the paper is organized as follows. In Sec. II, we review the theory introduced in Ref. [8] as it forms the basis of the present work. In the review, the gauge symmetry, the constraint algebra, and the way an emergent geometry is identified from the underlying microscopic degrees of freedom are emphasized as they are the key ingredients needed in this paper. Sec. III is the main part of the present paper. The objectives of this section are two-folded. The first is to identify a set of clock variables to construct spacetime from the correlation between the clock variables and the remaining physical degrees of freedom in the semi-classical limit. The second is to examine how different choices of local clocks leads to different spacetimes. In Sec. III A, a procedure that generates gauge invariant states from a set of first-class constraints is discussed. In Sec. III B, the gauge constraints are solved to identify the constraint surface in the semi-classical limit. Sec. III C discusses a gauge fixing prescription that introduces clock variables and the associated local Hilbert spaces. The section also discusses how the correlation between the remaining physical degrees of freedom and the local clocks determines an emergent spacetime. In Sec. III D, two schemes that use different sets of local clocks are applied to one physical state, and extract the spacetimes that emerge from those choices. Sec. IV is a summary with open questions.

II. REVIEW : A MODEL OF QUANTUM GRAVITY WITH EMERGENT SPACETIME
In this section, we review the model introduced in Ref. [8]. We recap the main results that are needed in this paper, and refer the readers to the original paper for details.

• Kinematic Hilbert space
The fundamental degree of freedom is a real rectangular matrix with M rows and L columns with M L 1 : Φ A i with A = 1, 2, .., M and i = 1, 2, .., L. The row (A) labels flavour, and the column (i) labels sites. The full kinematic Hilbert space H is spanned by The global symmetry is O(M ). It rotates the flavour index acting as a left multiplication where H i is the local Hilbert space spanned by ⊗ A Φ A i at site i. The frame can be rotated with SL(L, R) transformations that act as right multiplications on Φ : Φ → Φg, where g ∈ SL(L, R). New basis states defined by Φ ≡ Φg have the same inner product, In general, a state that is unentangled in one frame has non-trivial inter-site entanglement in another frame.

• Gauge symmetry
In the limit that the size of matrix becomes large, the sites can collectively form a space manifold. We identify the emergent geometric degrees of freedom from the microscopic degree of freedom based on the algebra that gauge constraints obey. Just as the momentum and Hamiltonian constraints generate spacetime diffeomorphism in general relativity, the present theory comes with generalized momentum and Hamiltonian constraints.

Generalized momentum
The SL(L, R) group that rotates frame is taken as the gauge group that generalizes the spatial diffeomorphism in the general relativity. The generalized momentum operator that generates SL(L, R) isĜ whereĜ is an operator valued rank 2 traceless tensor given byĜ i and y is a traceless L × L real matrix called the shift tensor. Under the SL(L, R) transformation,Φ andΠ transform as e −iĜyΦ e iĜy =Φg y and e −iĜyΠ e iĜy = g −1 yΠ , where g y = e −y .

Generalized Hamiltonian
The Hamiltonian constraint is written aŝ whereĤ is an operator valued rank 2 symmetric tensor given byĤ ij = TΦTΦΠΠT ji ,α is a constant parameter of the theory, and v is an L × L real symmetric matrix called the lapse

Constraint Algebra
The generalized momentum and Hamiltonian constraints satisfy the first-class quantum algebra : where , and the term that is proportional toĤ mn in Ĥ ij ,Ĥ kl is sub-leading. Eq. (5) is the exact commutator that remains well-defined at the quantum level. In the large M limit, the algebra reduces to the one that includes the hypersurface embedding algebra of the general relativity. This plays the key role in identifying the emergent geometry in this model.

• Emergent geometry
A coordinate system is a mapping r : i → r i from the set of sites in a frame to a manifold M with a region R i r i assigned to site i (see Fig. 1). In the large L limit in which the image of sites is dense in M, Eq. (5) induces an algebra on M. The generators of the induced algebra include the Weyl generator (D), the momentum density (P µ ), the Hamiltonian density (H) and generators for higher spin gauge transformations. Since v transforms as a rank 2 symmetric tensor under SL(L, R) , one can always choose a frame in which v is diagonal.
In this frame, the Hamiltonian constraint can be written aŝ Here H(r i ) = V −1 iĤ ii is the Hamiltonian density, V i is the coordinate volume of R i , and can be expanded in the relative coordinate aŝ where the ellipsis represents higher derivative terms. This leads tô G y = dr D(r)ζ y (r) + P µ (r)ξ µ y (r) + .. , where the derivative expansion is shown to the first order that includes the momentum den- dr ξ µ 1 (r)P µ (r), dr ξ ν 2 (r)P ν (r) = i dr L ξ 1 ξ µ 2 (r) P µ (r), dr ξ µ (r)P µ (r), dr θ(r)H(r) = i dr L ξ θ(r) H(r), where L ξ represents the Lie derivative with respect to the vector field ξ, andF ν (r m ) and G µν (r m ) are given byF The momentum and Hamiltonian densities obey an algebra that generalizes the hypersurface deformation algebra of the general relativity [12,13], provided that the metric is identified as the symmetric part ofĜ µν , where S is the signature of the spacetime direction translated by the Hamiltonian constraint.
The overall sign of the spacetime metric can be chosen either way. In the rest of the paper, we choose the convention in which S = −1.F ν (r) and the anti-symmetric part ofĜ µν represent additional collective fields that generalize the hyper-surface deformation algebra of general relativity.
The contravariant metric in Eq. (13) is given by a second moment ofĈ iikkn m , which measures a multi-point correlation in the system. If the range of entanglement and correlation is large in the coordinate distance, the second moment becomes large accordingly, which results in a small proper distance between points in space. The metric identified from the constraint algebra naturally captures the intuition that two sites that are strongly entangled are physically close [14][15][16][17][18][19][20][21][22][23]. On the other hand, the metric captures only a specific pattern of entanglement, and there also exist non-geometric entanglements. For example, there exist finely tuned states in which two points that are infinitely far still have O(M ) entanglement through other channels such as the higher-order moments ofĈ iikkn m [8]. In this sense, EPR is strictly 'bigger' than ER in the present theory [24].
States for which there exist coordinate systems with well-defined metric in the large M and L limit form a special set of states, and are referred to have local structures. For a state with a classical local structure, there exists a coordinate system associated with a well- is invertible and smooth on the manifold, and ĝ µν − ĝ µν 2 → 0 in the large M and L limit. The dimension, topology and geometry of the manifold are properties of state.
H v in Eq. (4) is a non-local Hamiltonian as a quantum operator, but it is relatively local [25] in the following sense [31]. Suppose Ψ has a local structure in a frame. To this state,Ĥ v with a lapse tensor diagonal in that frame acts as a local Hamiltonian to the leading order in whereĤ Ψ ef f is a Hamiltonian that is local in the manifold associated with the local structure of Ψ . The discrepancy betweenĤ v andĤ Ψ ef f in Eq. (14) is sub-leading in 1/M . This can be understood by writing the Hamiltonian with the lapse tensor v = I aŝ where all repeated indices are summed over. In the large M limit, the fluctuation ofΠ i AΠ j A is small, and the replacement of the operator with its expectation value is valid to the leading order in the large M limit. The second term in the Hamiltonian can be viewed as a hopping term between sites j and k whose hopping amplitude is proportional to the expectation value A is short-ranged as a function of r i − r j in a manifold, the Hamiltonian effectively behaves as a local Hamiltonian in the manifold. The Hamiltonian acts in a state-dependent manner to the leading order in the large M limit, and the local properties of the effective Hamiltonian are inherited from the state [26].

A. Gauge invariant states
The physical Hilbert space is given by the set of gauge invariant states that satisfŷ for any lapse tensor v and shift tensor y. A gauge invariant state can be constructed by projecting an arbitrary state to the physical Hilbert space. The projection can be implemented with a series of gauge transformations applied to an initial trial state χ as where ε is a non-zero constant, Dv ≡ Basis states of V Γ can be labeled by a set of collective variables. The bigger Γ is, the less collective variables are needed to span V Γ . If Γ is too big, there are too few kinematic collective variables to support non-trivial physical degrees of freedom after the gauge degrees of freedom are removed.
One simple choice of Γ that supports a minimal number of non-trivial physical degrees of freedom . Here S f L is the permutation group acting on the first L flavours. Two O(N/2) groups generate flavour rotations within the remaining two sets of N/2 flavours. Basis states for V Γ * can be written as where s, t 1 , t 2 are collective variables that label the basis states; s is L × L matrix and t 1 , t 2 are L × L symmetric matrices. In Eq. (18) This follows from (sP s −1 )s = sP ∼ s. The unbroken gauge group is related to the even site-permutation group through a similarity transformation. Therefore, s acts as a Stueckelberg field that breaks the generalized spatial diffeomorphism to the discrete permutation group. On the other hand, t ij 1 and t ij 2 are bi-local fields that generate inter-site entanglement. The mutual information between sites i and j for state in Eq. (18) to the leading order in the small t ij c limit [27]. Geometry is determined from the connectivity formed by these bi-local fields. Generic choices of t ij c would break SL(L, R) completely. If t ij c depends only on r i − r j in a coordinate system, the global translational symmetry in the manifold remains unbroken.
General states in V Γ * can be written as where χ(s, t 1 , t 2 ) is a wavefunction of the collective variables. In Eq. (19), the integrations over and their conjugate variables. By taking the small ε limit after the large Z limit is taken first, Eq.
(17) can be written as 2 ). (20) Here S is the action for the collective variables and their conjugate momenta, H and G are the induced Hamiltonian and momentum constraints, respectively, Here conjugate to s. p 1 and p 2 are symmetric L × L matrices conjugate to t 1 and t 2 , respectively. While s, t 1 and t 2 represent the 'sources', the conjugate variables represent the corresponding operators, In total, there are D k = 2L 2 + 2L(L + 1) kinematic phase space variables. Dx ≡ ∞ l=1 Dx (l) and x(τ ) = x (l) with τ = lε for x = s, q, t c , p c , v, y. τ is the parameter that labels the evolution of dynamical variables along gauge orbits.
All gauge invariant states have an infinite norm with respect to the inner product of the underlying Hilbert space. The non-normalizability of gauge invariant states is attributed to the fact that gauge orbits defined in the infinite-dimensional kinematic Hilbert space are non-compact [8].
This is fine because the dynamical variables include both clocks and physical degrees of freedom, and a gauge invariant state encodes the information about an entire spacetime history. In the large N limit with L >> 1, the path integration in Eq. (20) is well approximated by the saddle-point approximation. In this paper, we study the classical dynamics of the theory in the semi-classical limit. In particular, we identify a set of local clocks from the dynamical variables, and construct a spacetime from the correlation between the clocks and the remaining dynamical variables. We will see that different choices of local clocks lead to different spacetimes.

B. Constraint surface
From now on, we denote the saddle-point configuration as {q, s, t 1 , t 2 , p 1 , p 2 }, using the same collective variables that appear in the path integration. As an initial state in Eq. (17), we consider a semi-classical state in which both the collective variables and their conjugate momenta are well defined. An example is the gaussian wavepacket considered in Ref. [8]. Let us denote a semi-classical state whose collective variables are peaked at {q, s, p 1 , t 1 , p 2 , t 2 } as Ψ q,s,p 1 ,t 1 ,p 2 ,t 2 .
Because of the permutation symmetry S f L in Γ * , Ψ q,s,p 1 ,t 1 ,p 2 ,t 2 = Ψ P q,sP T ,p 1 ,t 1 ,p 2 ,t 2 for any P ∈ S f L . To the leading order in 1/N , the application of the gauge transformation results in where x = x + ε x, tr {Gy} for arbitrary traceless matrix y (shift tensor) and symmetric matrix v (lapse tensor) [33]. Herẽ Plugging Eq. (27) into Eq. (28), we obtain a quadratic matrix equation fort 1 , where The solution to Eq. (29) is written as where O is an orthogonal matrix that should be chosen so thatt 1 is symmetric. For every orthog- where g(τ 1 ) =P τ e τ 1 0 dτ y(τ ) ∈ SL(L, R) .P τ orders the matrix multiplication so that e dτ y(τ ) with smaller τ are placed to the left of the terms with larger τ . For q with det q ≥ 0, the choice of with q d (0) = [det q(0)] 1/L and g f ∈ SL(L, R) leads to For a given g f , Eq. (35) completely fixes the gauge freedom associated with SL(L, R) : g(τ 1 ) in Eq. (34) is the only element in SL(L, R) that satisfies Eq. (35). This gauge fixing amounts to locking site indices (columns) with reference to the flavour indices (rows). We refer to the frame in which q = q d g f as g f -frame. The path that connects an initial configuration to the one that satisfies Eq. (35) is denoted as path I in Fig. 3.

Fixing the lapse tensor
In priori, there is no preferred frame, and any g f can be used in Eq. takes diagonal forms in both frames. This is illustrated in Fig. 4.
The dynamical information of the theory is encoded in the correlation between the clocks and the remaining physical degrees of freedom. The state of physical degrees of freedom given as a function of state of the clocks is a prediction of the theory. To extract this correlation, we impose the following gauge fixing conditions on q and p 1 , where g f ∈ SL(L, R) and q d , p d are real variables. p 1,ii serves as the local clock at site i in the To make sure that the gauge fixing condition for q is maintained along the evolution, the shift is chosen to be where A ≡ 1 L tr {A}. This guarantees that q is proportional to g f at all τ irrespective of the lapse tensor. To transform p 1 (τ 1 ) to the desired form of p 1 (τ 2 ) = p d I, we write the equation of motion for p 1 as where for τ 1 ≤ τ ≤ τ 2 . Eq. (39) is a set of L(L+1) 2 linear equations for v(τ ) at each τ , and admits a unique solution in general. It is straightforward to show that with this choice of the lapse tensor p 1 (τ 2 ) = p d I at τ 2 = τ 1 + ln 2. This is denoted as path II in Fig. 3.
For a given initial state, the physical variables obtained at τ 2 depend on p d and g f . Therefore, the physical variables at τ 2 can be written as Within the constraint surface, the spatial metric in Eq. (13) is given by [8] g µν = −2α l,n U nl U lm (r µ nm r ν lm + r ν nm r µ lm ) with r µ nm = r µ n − r µ m to the leading order in 1/N . Consequently, Eq. (41) gives the spatial metric g µν (r, p d ; g f ) that depends on space (r) and time (p d ) in the g f -frame. The correlation between the spatial metric and the physical clocks describes a spacetime that emerges for the set of observers who use local clocks chosen in the g f -frame.
For some p d , there may be no lapse and shift tensors that brings the initial state to the one that satisfy the gauge fixing condition in Eq. (36). It is also possible that a constant p 1 surface intersects with a gauge orbits multiple times. In this case, p 1 can not be used as a time variable globally [2,3]. Here we don't attempt to find a global time variable. We will be content with the fact that p 1 serves as a set of clocks locally in the phase space.

D. Multi-fingered internal time
The fact that one can choose any g f ∈ SL(L, R) in Eq. (36) encodes the freedom in choosing a frame in which local clocks are defined. Under a rotation of frame, a state in a local Hilbert space can be transformed to a linear superposition of states that belong to multiple local Hilbert spaces.
As a result, one state can exhibit different local structures in different frames. To illustrate this through a concrete example, let us consider a semi-classical state with condition as is shown in Fig. 5(a). This state breaks SL(L, R) down to the discrete translation (i x , i y ) → (i x + 1, i y ) and the permutation group that interchanges i y . In this section, we consider the spacetime that emerges for a set of local observers who use the diagonal elements of p 1 as local clocks in the I-frame. This frame is defined by the gauge fixing conditions, We choose the lapse tensor, with the shift tensor given in Eq. (37). With this choice, the equation of motion for p 1 becomes In the I-frame, it is convenient to introduce the one-dimensional coordinate system, r i = i x for each decoupled ring. In this coordinate system, p 2,ij andt ij 2 in Eq. (42) are short-ranged in r i − r j . Sincet 1 and s are determined from p 2 ,t 2 , q from Eqs. (27) and (30),t ij 1 and s(q −1 ) T ij also decays exponentially in r i − r j in the one-dimensional manifold. Accordingly, the lapse tensor v ij in Eq. (44) also decays exponentially in r i − r j . This guarantees thatĤ v is relatively local in the one-dimensional coordinate system [8]. Consequently, the decoupled chains remain decoupled under the Hamiltonian evolution to the leading order in 1/M . Furthermore, the Hamiltonian acts as a local one-dimensional Hamiltonian within each chain [26]. As a result, the emergent space- Here ζ is a constant, and A ζ = The new choice of clocks leads to a different decomposition of the kinematic Hilbert space into local Hilbert spaces. In order to extract the spacetime that emerges in this new frame, we transformation that brings q into the form in Eq. (47), the collective variables become A site with coordinate (i x , i y ) in the g o -frame is composed of a linear superposition of sites with (i x , i y −1), (i x , i y ), (i x , i y +1) in the I-frame. Because one site in the g o -frame is delocalized across three neighbouring chains of the I-frame, the chains are no longer decoupled in the g o -frame. Due to the interchain entanglement, Eq. (48) has a two-dimensional local structure, as is shown in Fig.  5(b). Now, we apply the second set of gauge transformations to enforce the gauge fixing condition for p 1 . As explained in Sec. III C 2, this is achieved with the lapse tensor that satisfies Eq. (39) and the shift tensor given in Eq. (37). To understand the nature of this second gauge transformation, we use the two-dimensional coordinate system, r i = (i x , i y ). This coordinate system makes the twodimensional local structure manifest. In other words, p c,ij ,t ij c and s(q −1 ) T ij connect a site with its neighbours in the two-dimensional manifold, and decay exponentially in r i − r j . As a result, the lapse tensor v ij that satisfies Eq. (39) also decays exponentially in r i − r j . This implies that the Hamiltonian acts as a two-dimensional local Hamiltonian along the gauge orbit that connects Eq. (48) with the one that satisfies the gauge fixing condition in Eq. (47). Therefore, the state obtained at the end of the second gauge transformation at τ 2 also supports a two-dimensional local structure in the g o -frame. The physical variables (q d , p 2 ,t 2 ) at τ 2 viewed as functions of p d describe a three-dimensional spacetime. can be generalized to higher dimensions, and we expect that anisotropic spacetimes close Lifshitz transitions generically exhibit multiple time directions [29].
This example shows that one state can exhibit spacetime manifolds with different dimensions, signatures, topologies and geometries in different frames. This is possible because the enlarged gauge symmetry generated by SL(L, R) can not only permute sites but also change the very notion of local sites by constructing new sites out of linear superpositions of old sites.
If one chooses local clocks in an arbitrary frame, the state generally does not retain any local structure. Even for a state that has a local structure in one frame, a well-defined spacetime manifold does not emerge if a collection of clocks are chosen in another frame that is related to the first frame through a non-local transformation [39]. The emergence of a well-defined spacetime hinges both on local structure of the state and on the choice of local clocks that are compatible with the local structure of the state.

IV. SUMMARY AND DISCUSSION
In this paper, we consider a theory of quantum gravity that does not have a preferred decomposition of the kinematic Hilbert space into local Hilbert spaces. The theory is covariant under a gauge symmetry larger than diffeomorphism, where the extra gauge symmetry includes transformations that mix local kinematic Hilbert spaces. This gives rise to a greater freedom in choosing a collection of local clocks with respect to which the evolution of other physical degrees of freedom is tracked. It is shown that dimension, signature, topology and geometry of spacetime depend on the choice of local clocks. Just as a gem reveals different facets in different cuts, one state can exhibit different spacetimes with different choices of clocks. We expect that this is a generic feature of theories that do not have a preferred Hilbert space decomposition.
Another consequence of the enlarged gauge symmetry is the presence of extra propagating modes besides the spin 2 gravitational mode. They are represented by the higher-spin fields associated with the bi-local collective fields. Higher-spin gauge fields are Higgsed in states that break SL(L, R) to the global translation symmetry, as is the case for the states considered in Sec.
III D [8]. It will be of interest to understand the physical spectrum of the theory.
over v and y.
[34] The square root of a symmetric matrix can be defined as follows. A real symmetric matrix X can be written as X = O X D X O T X , where D X is a diagonal matrix and O X is an orthogonal matrix. Its square root is given by [35] Because p 1,ij = 1 N L+N/2 b=L+1 Φ b i Φ b j , off-diagonal elements of p 1 generate inter-site entanglement.
[36] There always exist frames in which p 1 is diagonal. Suppose that p 1 = X in the g f -frame, where X is a general L × L symmetric matrix. Under a frame rotation, q = q d g f g and p 1 = g T Xg, where g ∈ SL(L, R). One can always choose g such that p 1 = p d I, where p d is a real number. Now the clock takes the diagonal form in the g f -frame, where g f = g f g.
[37] We assume that L is the square of a whole number.
[38] According to Eq. (13), the contravariant metric is given by the second moment ofĈ iikkn m .
In the presence of the translational invariance, the uniform metric can be written as g µν = 4α ∂U k ∂kµ ∂U k ∂kν + U k ∂ 2 U k ∂kµ∂kν k=0 [8]. With the reflection symmetry, ∂U k ∂kµ k=0 = 0, and the metric is given by the second derivative of U k .
[39] Equivalently, a state that is short-range entangled in one basis can exhibit long-range entanglement if one chooses non-local basis.