Constraint algebra of general relativity from a formal continuum limit of canonical tensor model

Canonical tensor model (CTM for short below) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. In the classical case, the constraints form a first-class constraint Poisson algebra with structures similar to that of the ADM formalism of general relativity, qualifying CTM as a possible discrete formalism for quantum gravity. In this paper, we show that, in a formal continuum limit, the constraint Poisson algebra of CTM with no cosmological constant exactly reproduces that of the ADM formalism. To this end, we obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism. On the other hand, the cosmological constant term of CTM seems to induce non-local dynamics, and is inconsistent with an assumption about locality of the continuum limit.


Introduction
Tensor models [1][2][3] were originally introduced as models for quantum gravity in D > 2 dimensions, extending the matrix models which are considered to successfully describe quantum gravity in D = 2. Subsequently, tensor models with group-valued indices [4,5], called group field theories [6][7][8], were introduced, which are especially studied in the context of loop quantum gravity. The central idea of tensor models is that Feynman diagrams in tensor models may correspond to dual diagrams of discretized spacetimes. Though the original models were not successful due to some difficulties [2,9], colored tensor models [10,11] with promising properties appeared, and have extensively been analyzed with interesting concrete results (see for instance [12][13][14][15][16] for some recent developments). The colored tensor models also stimulated the renormalization group procedures of group field theories (see for instance [17][18][19][20] for recent developments). There also appeared a new approach to random volumes in terms of matrix models [21,22], which are in relation with colored tensor models.
The analysis of the colored tensor models has shown that leading orders of 1/N * expansions of the partition functions are dominated by branched polymers composed of melonic diagrams [11,23,24]. Naively, this would be an obstacle for a model of our spacetime, since branched polymers do not seem to represent extending entities like our real space, though there have been some interesting directions of study to change the situation by considering higher orders [13,[25][26][27][28]. On the other hand, it might be possible that the existence of a time-like direction is essentially important in quantum gravity, while the tensor models above basically deal with Euclidean signatures. This possibility arises from the fact that Causal Dynamical Triangulation has succeeded in generating de Sitter-like spacetimes [29], while the Euclidean cousin, Dynamical Triangulation, is not successful in this respect † .
Motivated by these considerations, one of the present authors has introduced a rank-three tensor model in Hamilton formalism [32][33][34] (There exists another Hamiltonian approach [35] in the framework of group field theories.). Here the minimum choice of rank-three over matrices has been taken, based on a belief that rank-three is enough to describe any dimensional space; this is a conclusion from the past works by one of the present authors on Euclidean rank-three tensor models [36,37], though these models themselves are not successful due to serious necessities of fine-tuning. A time-like direction has been introduced by constructing Hamiltonian constraint forming a first-class constraint Poisson algebra with kinematical symmetries, which are the analog of spatial diffeomorphism in general relativity. This way of introducing a time-like direction as gauge symmetry would be necessary for such a model aiming for quantum gravity to reproduce general relativity in a (presently unknown) classical limit of continuous spacetime. The requirement of the first-class nature of the constraint algebra is so strong that the constraints and the algebraic structure of such a tensor model are unique under some physically reasonable assumptions [33]. Thus, our tensor model in the Hamilton formalism (canonical tensor model or CTM for short below) has turned out to be formulated as a totally constrained system with first-class constraints, which have a Poisson algebraic structure very similar to the constraint Poisson algebra [38][39][40] of the Arnowitt-Deser-Misner (ADM) formalism of general relativity [41,42].
The subsequent analyses have revealed some remarkable properties of CTM. The N = 1 case exactly reproduces the mini-superspace approximation of general relativity with a cosmological constant [43]. CTM can consistently be quantized [44], and a number of exact physical wave functions for general N, namely the exact solutions to the CTM analogue of the Wheeler-DeWitt equations, have been found [45]. There is an intimate relation between CTM and statistical systems on random networks: the Hamiltonian constraint of CTM generates the renormalization group flow of randomly connected tensor networks [46][47][48]. This insight was remarkably useful in constructing the exact physical wave functions mentioned above [45].
The main purpose of the present paper is to consider a formal continuum limit of CTM to find a relation with general relativity more general than the mini-superspace approximation mentioned above. We will show that, in the formal continuum limit, the first-class constraint Poisson algebra of CTM exactly agrees with that of the ADM formalism of general relativity by properly taking into account a difference of weights between the Hamiltonian constraints of the both theories. The continuum limit contains the following two main assumptions: the indices of tensors can be replaced by continuous D-dimensional coordinates with an implicit assumption of very large N, and one of the dynamical tensors of CTM must have an almost diagonal form. Here, the off-diagonal components of the tensor are essentially important: the lowest orders of a moment expansion for the off-diagonal components will be identified with the (inverse) metric tensor field in general relativity. It should be noted that we take the continuum limit in a formal manner, and the limit must be justified by the dynamics of CTM in future study. We would also like to mention that a similar derivation of the constraint algebra of the ADM formalism from that of CTM was done in a previous work [32] by one of the present authors. The previous work, however, was obviously insufficient, because a specific Gaussian distribution of off-diagonal components was assumed for computational simplicity, and coordinate dependences of variables were not fully accounted. On the contrary, the treatment of this paper is general and thorough. This paper is organized as follows. In Section 2, the formalism of CTM is briefly recapitulated. In Section 3, the first-class constraint Poisson algebra of CTM is computed in the formal continuum limit. Here, we introduce a moment expansion for the off-diagonal components of one of the dynamical tensors of CTM, and use the moments to express the continuum limit of the algebra. In Section 4, we interpret the continuum limit in terms of the ADM formalism of general relativity. This is successfully done by taking into account differences of weights of the gauge parameters between CTM and the ADM formalism, and by introducing an assumed relation between the lowest orders of the moment expansion and the (inverse) metric tensor field. In Section 5, we consider the Hamiltonian constraint obtained by multiplying that of the ADM formalism by a weight of half-density, and study the constraint Poisson algebra among the newly defined Hamiltonian constraint and the momentum constraint. The algebra certainly reproduces the continuum limit of that of CTM, and hence this proves the equivalence of the two. The final section is devoted to summary and discussions.

Canonical tensor model
The set of the dynamical variables of the canonical tensor model (CTM) [32][33][34] in the minimal setting [44] is given by a canonical conjugate pair of symmetric real rank-three tensors, M abc and P abc (a, b, c = 1, 2, . . . , N), satisfying where { , } denotes the Poisson bracket, and the summation is over all the permutations of d, e, f to incorporate the symmetry of the tensors under all the permutations of the indices. The Hamiltonian is given by where ξ a and η [ab] are Lagrange multipliers, repeated indices are summed over, and Here the square brackets in the indices symbolically represent the anti-symmetry, , and J [ab] and H a are the generators of the SO(N)-kinematical symmetry and of the symmetry analogous to the temporal diffeomorphism in general relativity, respectively. Following the naming in the ADM formalism [41,42] of general relativity, we call H a and J [ab] Hamiltonian constraint and momentum constraint of CTM, respectively. λ is a real undetermined constant, which we call cosmological constant. The last naming comes from the fact that the N = 1 case exactly agrees with the mini-superspace approximation of general relativity with a cosmological constant proportional to λ [43].
As in the case of the ADM formalism, H a and J [ab] form a first-class constraint Poisson algebra given by Here, on the right-hand sides, [ , ] denotes the matrix commutator, and [[ξ 1 , It is important to note that the algebra (5) has a structure depending on P on the right-hand side in the first line, and therefore it is not a genuine Lie algebra. This is a similar situation as the constraint algebra [38][39][40] of general relativity in the ADM formalism, and will be essential for (5) to reproduce the constraint algebra of the ADM formalism in a formal continuum limit. The constraints (3), (4) and hence the first-class algebra (5) are unique under some physically reasonable assumptions [33].

Constraint algebra of CTM in a formal continuum limit
In this section, we take a continuum limit of the constraint algebra (5) of CTM by assuming emergence of a continuum space. Note that this assumption is imposed without any justifications. Ideally, such an assumption should be derived as infrared effective dynamics of CTM, but this is out of our reach at present. Namely, the derivation in this section should be considered as a formal continuum limit ignoring the presently unknown real dynamics of CTM. On the other hand, it will suggest a plausible way for emergence of space and general relativity in the framework of CTM, and give directions of future study to finally justify the assumption.
First of all, the assumption will be translated to that the indices of the tensors can be replaced by continuous coordinates of space as where D denotes the spatial dimension. Then, we also assume that index contractions are replaced by integrations as In general, there could exist a non-trivial integration measure as d D x ρ(x), but this could be canceled by a Jacobian | ∂x ′ ∂x | after an appropriate transformation of the coordinate x ′ (x). Note that there are no contradictions in regarding x to be fixed labels as here, since the transformations below apply to the dynamical variables of CTM but not to the indices represented by the coordinates.
A continuum space has an intrinsic concept of locality, and we will pick up a local part of the constraint algebra (5). We will also introduce an assumption about the form of P which is in accord with the locality. We will show that this reduction to a local part of the algebra can consistently be done, except for the cosmological constant term in (3). We will comment on the cosmological constant term in the final section.

{J , J } part
Let us first discuss the Poisson algebra of J . The local part of the algebra with respect to the continuum space would be picked up by putting the D-dimensional delta-function δ D (x − y) into the argument of J (η) in some manner. Since η must be anti-symmetric, the lowest order should be expressed by using the first derivatives of δ D (x − y), and we are lead to the form, where v is an arbitrary vector field on the space, the Greek index µ denotes the spatial indices, µ = 1, 2, . . . , D, and The third equation of (5) implies that the algebra of J is equivalent to the commutator algebra of η. Products of distributions can be computed rather easily by using test functions. In the present case, for a test function f , we obtain So, we obtain where v 1,2 are respectively related to η 1,2 as (8), and Note that (11) has exactly the same form as (10). Therefore, where η 3 is (8) with v = v 3 . Thus, the commutator algebra of η with the form (8) closes, and the representative vectors v form a commutation algebra given in (12).

{J , H} part
Under the assumption of the continuum limit, the vector ξ of H(ξ) should be replaced by a function ξ(x) on the D-dimensional space. Then, the second line of (5) implies that the Poisson bracket between J and H is equivalent to the matrix-vector product η ξ. Thus, by using (8) and (10), we obtain where v is the representative vector of η.

{H, H} part
This is the most non-trivial part of our discussions, and we have to take into account not only the leading order in the expansion regarding locality but also the next leading orders to obtain a non-vanishing result. The next leading orders are dimensionally higher than the leading one by (length) 2 . This would mean that {H, H} part contains an effective length scale, and the way it appears suggests that it is intimately related with the smallest length scale in a space or the Planck length. In this paper, its definitive interpretation is beyond our scope, and is left for future study.
Let us first consider the part independent of the cosmological constant λ, namely [ξ 1 ,ξ 2 ] on the right-hand side in the first line of (5). Let us first try a strictly local form of P , The operationξ on a test function f can be computed as and hence Therefore, P needs to be smeared for [ξ 1 ,ξ 2 ] to be non-vanishing.
To characterize such smearing of P in the order of (length) 2 ‡ , let us introduce the following moments, where δy = y − x, δz = z − x, and we have taken into account P xyz = P xzy , which is one of the permutation symmetries of P . In fact, the remaining permutation symmetry of P gives more restrictions on the moments in (17). To see this, let us consider three independent test ‡ As shown below, β looks like in the first order in (17), but turns out to be in the second order due to the permutation symmetry of P .
functions f i (i = 1, 2, 3). By using (17) and performing partial integrations, we obtain where we have expanded the test functions f 2 (y), f 3 (z) around x, suppressing the obvious argument x, and have used the shorthand notations, f ,µ = ∂ µ f , etc. Here the dots represent terms with higher moments and will be ignored. The third line contains no derivatives of f 1 , and, from the third line to the final ones, we have performed partial integrations to remove the derivatives of f 2 . Since the permutation symmetry of P requires that (18) must be symmetric under the permutations of the test functions, the third and the final lines should have the same expression after the exchange of f 1 and f 2 . This demands γ µν −γ µν = 1 2 γ µν , and the solution is Thus we obtain that the moment expansion (17) is characterized thoroughly by α and γ.
Here, it is important to recall that we are not performing derivative expansions for dynamical variables, but rather we are performing expansions in moments of "fuzziness" of space: we count the number of derivatives on test functions as orders, but not derivatives on dynamical variables such as α, γ. Therefore, the orders associated to α and γ are zeroth and second, respectively, and β should be considered to be in the same order as γ, irrespective of a derivative on γ in (20).
By using (17) and (20), the operationξ in (5) on a test function f can be computed as Then, by using (21), we obtain the commutator [ξ 1 ,ξ 2 ] as where we have discarded terms with orders (γ) 2 and higher. In (5), [ξ 1 ,ξ 2 ] is the argument of the generator J , and, surprisingly, (22) has exactly the form of (10) with This shows that the local form of η in (8) is consistent with the algebra of the Hamiltonian constraints in the second order of the moment expansion of P , except for the part depending on the cosmological constant λ.
As for the term depending on the cosmological constant λ on the right-hand side in the first equation of (5), we obtain Since we may take any x, y, the expression is generally non-local with respect to the continuum space.

Interpretation in terms of geometrodynamics
In Section 3, we have analyzed the structure of the constraint algebra of CTM on the assumption of the emergence of a continuous D-dimensional space with its intrinsic locality. In this section, we will compare the result with the constraint algebra [38][39][40] of the ADM formalism [41,42] of general relativity, and will construct an exact correspondence except for the part proportional to the cosmological constant λ.
In the Hamilton formalism, general relativity can be formulated as a totally constrained system due to the gauge symmetry of the spacetime diffeomorphism. In the ADM formalism, the Hamiltonian is given by where H GR and J GR µ are respectively the Hamiltonian and momentum constraints, and n and w µ are the lapse function and the shift vector, respectively. The constraints satisfy the first-class constraint Poisson algebra, where and L denotes the Lie derivative, An important feature of (26) is that the algebraic structure depends on the dynamical field, the spatial inverse metric g µν (x), as in (29), and the algebra is therefore not a genuine Lie algebra. This structure is an essence of geometrodynamics, as thoroughly discussed in [39]. Dependence on the dynamical variable P exists similarly in the constraint algebra (5) of CTM, and was important in deriving the constraint algebra in the continuum limit of CTM in Section 3.
Let us first compare {J , J } with {J GR , J GR }. By comparing (12) with (31), we can simply identify v = w.
Thus J (η) with (8) of CTM can be identified with the spatial diffeomorphism in the continuum limit.
Next, let us compare {J , H} with {J GR , H GR }. With the identification (32), it is not possible to identify ξ and n, because there is a difference of weights between (14) and (30). Geometrically, n is a scalar, but ξ is a scalar half-density. The difference can be balanced by assuming where g = Det[g µν ]. In fact, by assuming (30) and using L w g µν = ∇ µ w ν + ∇ ν w µ , one obtains where A is a number. For A = 1 4 , (34) agrees with (14) under (32), and therefore we should choose as (33).
Lastly, we compare {H, H} with {H GR , H GR }. By putting (33) into (23), we obtain v µ = αγ µν g 1 4 By comparing this with (29) under (32), we obtain As for the the part proportional to the cosmological constant on the righthand side in the first equation of (5), it seems difficult to associate a geometrodynamical interpretation due to the non-local character shown in (24).
Finally, we will make a comment on the weights appearing in (33) and (36). As shown in (10), with the identification (32), a vector in CTM is translated to a quantity which transforms as a scalar half-density by J (η) with (8). This applies to ξ(x) as in (14) as well as to each index of P . Therefore, we make assignments, where we have introduced [ ] w to denote the weight of a quantity; the weight is defined so that a quantity q transforms as a scalar density, if [q] w = 1. On the other hand, we have since an integration over the space cancels the transformation of a quantity q(x) with [q] w = 1 as where we have ignored possibilities of boundary contributions. By applying (38) and (39) to the quantities in (17), we obtain and therefore [αγ µν ] w = −1. This explains the power of g in (36).

Modified constraint algebra in the ADM formalism
In Section 4, it has been shown that, to relate CTM to the ADM formalism, the gauge parameters ξ and n associated respectively to the Hamiltonian constraints of CTM and the ADM formalism should have different weights as in (33). In the discussions, a factor g 1 4 has been introduced to modify the wight of the gauge parameter n, and g µν (x) in it was treated as a metric tensor field on the space. In the Hamilton formalism, however, g µν (x) is a dynamical variable, and therefore the wight factor should be considered to be a modification of the Hamiltonian constraint rather than the gauge parameter. Then, the additional weight factor multiplied on the Hamiltonian constraint may potentially ruin the correspondence argued in Section 4 on account of the Poisson brackets between the weight factor and the constraints. Therefore, in this section, we will explicitly write down the constraint algebra after the change of the weight of the Hamiltonian constraint in the ADM formalism, and will show that it actually agrees with the continuum limit of the constraint algebra of CTM, namely (14) and (23) with (36) (Since there is no change in the momentum constraint, (12) does not need to be checked.).
Let us define the modified Hamiltonian constraintH (andH) as where g = Det(g µν ) and B is a number. Here, we have changed the weight of the Hamiltonian constraint, and, from (36), we expect that the constraint algebra of CTM can be obtained for B = − 1 4 . The Hamiltonian constraint of the ADM formalism satisfies where C µν (x) = −2K µν (x), being proportional to the extrinsic curvature expressed in terms of g µν (x) and its conjugate momentum π µν (x) [38][39][40]. Here, what matters in the following discussion is only the fact that the right-hand side of (44) is strictly local: it does not contain derivatives of δ D (x − y). From (44), we obtain where C = g µν C µν . Then, we find that where the two terms in the second line have canceled with each other due to the strict local form of (45). Therefore, by using (26), (27), (28), (29), and (46), we obtain where Indeed, for B = − 1 4 , this agrees with (23) under (36). We will next compute {J GR (v),H(ξ)}. Let us first remind the explicit expression of the momentum constraint [38][39][40] in the ADM formalism, where D µ denotes the covariant derivative, and π satisfies Then, for a vector v, we obtain Therefore, where we have used (51), (26), and (30). Then, by substituting an identity, into (52), we obtain Thus, for B = − 1 4 , we certainly obtain (14) of CTM.

Summary and discussions
The canonical tensor model (CTM) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have the Poisson algebraic structure similar to the constraint Poisson algebra of the ADM formalism of general relativity.
In this paper, we consider a formal continuum limit of CTM and have shown that, in the continuum limit, the constraint algebra of CTM coincides with that of the ADM formalism by properly taking into account the weight difference: the Hamiltonian constraints of CTM and the ADM formalism are different with each other by half-density. We have obtained the expression of the (inverse) metric tensor field of general relativity in terms of the dynamical rank-three tensor P of CTM. Here the continuum limit assumes an almost diagonal form of P , and the lowest and the next to the lowest order coefficients of a moment expansion for the offdiagonal components of P give the expression of the (inverse) metric tensor field. This explicit correspondence between P and the metric tensor field would be the most useful achievement of this paper to guide future study of the dynamics of CTM.
A specific form of P , almost diagonal, is assumed in the continuum limit. This is obviously an insufficient treatment, since P is a dynamical variable, and its form must be determined dynamically rather than formally assumed. One possible way to justify it would be to study the exact physical wave functions which have been obtained so far [44,45], and check whether such configurations can appear as peaks of these wave functions. A non-trivial issue in such a study would be that we have to take a certain large N limit, which would be necessary for indices to be replaced by continuous coordinates.
We have seen that the cosmological constant of CTM generates non-local dynamics which is in contradiction with the locality assumption of the continuum limit. Therefore, for the consistency of our discussions, the cosmological constant must vanish. In fact, we have previously shown that the value of the cosmological constant can be changed by shifting P [45]. It would be interesting to study the exact physical wave functions [44,45] to see whether there is a dynamical mechanism which tunes P to cancel the cosmological constant. Note that the dynamics of the cosmological constant term of CTM is obviously different from that of general relativity due to the non-local property, and this would give a certain chance to circumvent the common difficulties to tune the cosmological constant to the observed value [49]. Another possibility is that the cosmological constant term is prohibited by the consistency of the constraint algebra of CTM. As shown in a previous paper [33], this actually occurs, if we impose the generalized Hermiticity condition on the dynamical variables of CTM, instead of imposing the reality condition as in this paper.
We have chosen a local class of momentum constraints of CTM from all, and have shown that they correspond to the momentum constraints in the ADM formalism. However, we have not discussed consequences of ignoring the non-local momentum constraints of CTM. One way to justify the ignorance would be to gauge-fix the non-local ones, while the local ones are left intact. Then, a consequence in quantum case would be that a constraint, sayĈ, would be modified by a similarity transformationĈ eff = √ VĈ √ V −1 [50], where V is the gauge volume of the non-local gauge transformations. In a more general treatment, V could also contain a Jacobian generated from a process of taking the coefficients of the moment expansions as coarse grained collective coordinates for P [50]. Such a similarity transformation does not change the algebraic structure, but there will be some consequences of physical importance. One is that an almost diagonal form will be required forP eff = √ VP √ V −1 instead ofP in the continuum limit. Another will be that Hamiltonian constraintĤ eff = √ VĤ √ V −1 will be changed from the original formĤ. Since the original Hamiltonian constraint of CTM seems lacking a term corresponding to the spatial curvature term in that of the ADM formalism, it would be highly interesting to see whether the corrections will modify the Hamiltonian constraint in a desired manner or not.