A canonical rank-three tensor model with a scaling constraint
aa r X i v : . [ h e p - t h ] A p r YITP-13-11
A canonical rank-three tensor modelwith a scaling constraint
Naoki
Sasakura ∗ Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan
Abstract
A rank-three tensor model in canonical formalism has recently been proposed.The model describes consistent local-time evolutions of fuzzy spaces through aset of first-class constraints which form an on-shell closed algebra with structurefunctions. In fact, the algebra provides an algebraically consistent discretization ofthe Dirac-DeWitt constraint algebra in the canonical formalism of general relativ-ity. However, the configuration space of this model contains obvious degeneraciesof representing identical fuzzy spaces. In this paper, to delete the degeneracies,another first-class constraint representing a scaling symmetry is added to proposea new canonical rank-three tensor model. A consequence is that, while classicalsolutions of the previous model have typically runaway or vanishing behaviors, thenew model has a compact configuration space and its classical solutions asymp-totically approach either fixed points or cyclic orbits in time evolution. Amongothers, fixed points contain configurations with group symmetries, and may repre-sent stationary symmetric fuzzy spaces. Another consequence on the uniqueness ofthe local Hamiltonian constraint is also discussed, and a minimal canonical tensormodel, which is unique, is given. ∗ [email protected] Introduction
The tensor models have first been proposed [1, 2, 3] as analytical description of the
D > D = 2 dimensional case to the other dimensions. The idea of the tensor models hasalso been applied to the loop quantum gravity as group field theories by considering group-valued indices [4, 5, 6, 7, 8]. In these approaches, the theoretical interpretation of the tensormodels is essentially based on the correspondence between perturbative Feynman diagrams ofthe tensor models and the dual diagrams of simplicial manifolds. In the original tensor modelswith Hermitian tensors, however, the correspondence has delicate issues [2, 9], and it is notknown how to take the large N limit, which was essential in relating the matrix models to D = 2 quantum gravity. On the other hand, another kind of tensor models with unsymmetrictensors, called colored tensor models [10], have been proposed. The colored tensor models havegood correspondence to simplicial manifolds, and various analytical results including the large N limit have been revealed [11]. The colored tensor models have also stimulated developmentsof renormalization of the tensor group field theories [12, 13, 14, 15]. However, the presentsituation of the tensor models as quantum gravity is still uncertain; in Feynman perturbationseries, the large N limit of the colored tensor models is dominated by the “melonic” diagrams[11, 16], which are topologically spheres but look rather singular [17] unlike our actual space.The dominance of the melonic diagrams in the large N limit has also been shown [18] for othernew models which are called multi-orientable tensor models [19].In view of the present unsatisfactory status of the tensor models as quantum gravity in theabove interpretation, it would also be meaningful to pursue another interpretation of the tensormodels. In fact, the present author has proposed the interpretation that the rank-three tensormodels, which have a rank-three tensor as their only dynamical variable, may be regarded asdynamical models of fuzzy spaces [20, 21]. An advantage of this interpretation is that, sincefuzzy spaces can generally describe any dimensional spaces, any dimensional quantum gravitycan be considered to be incorporated in the rank-three tensor models. This is in contrastwith that ranks of tensors are directly related to dimensions in the above interpretation interms of simplicial manifolds. In fact, by semi-classical analyses, the present author hasshown spontaneous generation of various dimensional fuzzy spaces [22] and Euclidean generalrelativity on them from a certain fine-tuned rank-three tensor model [23, 24].However, the above results of the Euclidean tensor model are not satisfactory. The actionis complicated and unnatural. Moreover it must be fine-tuned so that the above physicallywanted results be obtained, but there is no principle to choose the action out of the otherinfinitely many possibilities. This drawback may be solved by a kind of universality throughquantum mechanical treatment. But first of all it is necessary to introduce a notion of timeinto tensor models before discussing quantum mechanics.Thus, to incorporate time into tensor models, the present author has proposed a rank-three tensor model in a canonical formalism [25, 26]. The model is defined as a pure constraintsystem with a set of first-class constraints which form an on-shell closed algebra with structurefunctions. In fact, the algebra has a resemblance to the Dirac-DeWitt first-class constraintalgebra in the canonical formalism of general relativity [27, 28, 29], and the former agrees1ith the latter in a formal limit of vanishing fuzziness. Moreover, there exist a notion of localtime and local time evolutions controlled by local Hamiltonian constraints in the model, asin general relativity. The on-shell closure condition is so strong that the local Hamiltonianconstraints are (two-fold) unique under some physically reasonable assumptions.However, as will be discussed below, the canonical rank-three tensor model above seemsto have some unsatisfactory features concerning the classical solutions. So the main purposeof the present paper is to propose a new canonical rank-three tensor model by adding aconstraint representing a scaling symmetry to the previous model. The scaling symmetry isnatural from the perspective of fuzzy spaces, and the new model has nice features for futurestudy. The paper is organized as follows. In Section 2, the canonical rank-three tensor modelproposed in the previous paper [25] is summarized. In Section 3, the unsatisfactory featuresof the previous model are discussed, and a new model is proposed by adding a new first-classconstraint representing a scaling symmetry. In Section 4, the configuration space and fixedpoints of the classical equation of motion of the new model are discussed. Among others, suchfixed points contain configurations with group symmetries. In Section 5, the uniqueness of thelocal Hamiltonian constraint for the tensor model with a totally symmetric rank-three tensoris discussed. This provides a minimal canonical tensor model. The finial section is devoted tosummary and future prospects. In this subsection, I will summarize the canonical rank-three tensor model proposed in theprevious paper [25].The dynamical variables of the canonical rank-three tensor model are given by the canonicalvariables, M abc , P abc ( a, b, c = 1 , , · · · , N ). They satisfy the generalized Hermiticity condition, X abc = X bca = X cab = X ∗ bac = X ∗ acb = X ∗ cba , (1)where X = M, P and ∗ denotes complex conjugation. The Poisson brackets between them aregiven by { M abc , P def } = δ ad δ be δ cf + δ ae δ bf δ cd + δ af δ bd δ ce , (2) { M abc , M def } = { P abc , P def } = 0 . (3)Here the first Poisson bracket is taken to be consistent with the generalized Hermiticity con-dition (1).The kinematical symmetry of the canonical tensor model is given by the orthogonal group O ( N ), X abc = G ad G be G cf X def , G ∈ O ( N ) , (4)where repeated indices are summed over. In what follows, this convention is used, unlessotherwise stated. 2ith the canonical variables, the Lie generators of the kinematical symmetry are expressedby J [ ab ] = σ X acd Y bcd − X bcd Y acd ) , (5)where the square bracket [ ] in the index symbolically represents the antisymmetry, J [ ab ] = −J [ ba ] . As for X, Y , the following two cases,(i) X = M, Y = P, (6)(ii) X = P, Y = M, (7)can be considered. The numerical factor σ in (5) takes for convenience the values, σ = (cid:26) − , , (8)respectively. With (8), the fundamental Poisson bracket (2) can be expressed as { X abc , Y def } = − σ ( δ ad δ be δ cf + δ ae δ bf δ cd + δ af δ bd δ ce ) (9)for both cases (i) and (ii).The two consistent local Hamiltonian constraints, which have a slight difference in indexcontraction, are given by ∗ H a = X a ( bc ) X bde Y cde , (10) H a = X a ( bc ) X bde Y ced , (11)where X a ( bc ) = ( X abc + X acb ) / H a and J [ ab ] form a Poisson algebra given by { H ( T ) , H ( T ) } = J ([ ˜ T , ˜ T ]) , (12) { J ( V ) , H ( T ) } = H ( V T ) , (13) { J ( V ) , J ( V ) } = J ([ V , V ]) , (14)where H ( T ) = T a H a , (15) J ( V ) = V [ ab ] J [ ab ] , (16)with a real vector T a and an antisymmetric real matrix V [ ab ] = − V [ ba ] . On the right-hand sidesof the Poisson algebra, ˜ T ( bc ) = T a X a ( bc ) , (17) ∗ Strictly speaking, the previous paper [25] only deals with the case (ii). As for the case (i), H a satisfies theconditions of the previous paper, if the time reversal symmetry is replaced with H a → −H a . T is the usual multiplication of a matrix and a vector, and [ , ] denotes the matrix commu-tator. Since the right-hand side of (12) contains ˜ T dependent on X , the algebra has structurefunctions, but not structure constants. This feature makes the apparently simple Poissonalgebra (12), (13), (14) highly non-trivial, and plays an essential role in deriving from thePoisson algebra the Dirac-DeWitt first-class constraint algebra in the canonical formalism ofgeneral relativity [27, 28, 29] by taking a formal limit of vanishing fuzziness [26]. It is alsoan important fact that the multiple possibilities (6), (7), (10), (11) actually lead to the samePoisson algebra (12), (13), (14).The closure of the Poisson algebra (12), (13), (14) on the on-shell subspace defined by J [ ab ] = H a = 0 implies that a canonical rank-three tensor model can consistently be definedas a constraint system with a set of first-class constraints, J [ ab ] = H a = 0. In analogy withgeneral relativity, J [ ab ] and H a may be called the momentum and Hamiltonian constraints,respectively. As explained in Section 2, the canonical tensor model is a pure constraint system with thefirst-class constraints, J [ ab ] = H a = 0. Following the standard method for singular systems,the total hamiltonian is given by H tot = N a H a + N [ ab ] J [ ab ] , (18)where N a , N [ ab ] are arbitrary variables, the actual values of which may be fixed by some gaugefixing conditions. For the choice of the local Hamiltonian (10), the classical equation of motionfor X is given by dX abc dt = { X abc , H tot } ≈ − σ N d (cid:0) X d ( ae ) X ebc + X d ( be ) X eca + X d ( ce ) X eab (cid:1) + N [ de ] ( · · · ) , (19)where ≈ denotes the so-called weak equality, and · · · are the terms representing the infinites-imal O ( N ) transformation. The choice (11) as H a instead of (10) will change the order of abc on the right-hand side of (19), but this is not important for the following discussions.It is not difficult to numerically study the equation of motion (19) simultaneously takinginto account the constraints H a = J [ ab ] = 0 and some appropriate gauge fixing conditions.This has been carried out, and it has turned out that the time-dependence of the classicalsolutions is rather extreme. This can essentially be captured by considering the followingsimplified version of (19), dxdt = x . (20)The behavior in time evolution is obviously given by x ( t ) → ∞ for x (0) > ,x ( t ) = 0 for x (0) = 0 ,x ( t ) → − x (0) < , (21)4or initial values x (0). Thus the point x = 0 is the only fixed point, and x ( t ) either diverges orasymptotically vanishes for non-vanishing initial values. From the numerical study, it seemsthat the original equation (19) has similar properties. There does not seem to exist any otherfixed points but the trivial one ∀ X abc = 0, and X abc seem to either diverge or asymptoticallyvanish for non-trivial initial configurations. These extreme behaviors cast doubts on thephysical sense of the model.On the other hand, in the numerical study, it has often been observed that the ratios X abc /X def have finite and non-vanishing asymptotic values. This suggests that the modelshould be modified so that only the ratios of X abc become the true dynamical variables. Thiscan easily be realized by introducing a gauge symmetry of common rescaling, X abc → γX abc for all X abc , (22)where γ is real and arbitrary.The gauge symmetry (22) is also natural from the perspective of fuzzy spaces [20, 21]. Inthe interpretation, a configuration X abc of the tensor model is assumed to correspond to afuzzy space defined by an algebra of the functions f a on it, f a · f b = X abc f c . (23)Here X abc plays the role of the structure constants of the function algebra. Since the essentialproperties of the functions do not change under the common rescaling f a → γf a for all f a ,imposing the gauge symmetry (22) is a natural requirement.The above discussions imply the necessity of adding a new constraint D = 0 with D = σ X abc Y abc , (24)which generates a scaling transformation, {D , X abc } = X abc , (25) {D , Y abc } = − Y abc . (26)The newly introduced D forms a closed algebra with J [ ab ] and H a as {D , H ( T ) } = H ( T ) , (27) {D , J ( V ) } = 0 . (28)The algebraic on-shell closure of (12), (13), (14), (27), (28) on the constraint subspace H a = J [ ab ] = D = 0 implies that a new canonical tensor model can consistently be defined as aconstraint system H a = J [ ab ] = D = 0 † . † In fact, it seems possible to consider a shifted constraint
D − d = 0 with a non-zero real parameter d . Thisambiguity may be avoided by embedding the algebra into a larger one, which has D as a result of Poissonbrackets between constraints. This possibility is left for future study. The configuration space and classical fixed points
The total hamiltonian of the new system is given by H newtot = N a H a + N D + N [ ab ] J [ ab ] , (29)where N is a new variable. Then the equation of motion is given by dX abc dt = { X abc , H newtot } ≈ − σ N d (cid:0) X d ( ae ) X ebc + X d ( be ) X eca + X d ( ce ) X eab (cid:1) − σ N X abc + N [ de ] ( · · · ) , (30)where H a is taken to be (10).Since the trivial configuration, ∀ X abc = 0, is a fixed point of the classical equation of motion(30), one cannot get to it with a finite time starting from another configuration. Thereforeone can consistently decouple the trivial point from the rest of the configuration space. Byusing D , which generates (25), an arbitrary configuration in the rest space can be gauge fixedas X abc X ∗ abc = 1 . (31)Thus the configuration space of the new model can be represented by the intersection of thecompact space (31) and some other gauge-fixing conditions. In such a compact space, classicalsolutions will in general asymptotically approach either fixed points or cyclic orbits, but willnot have the extreme behaviors as the previous model explained in Section 3.It is not difficult to give a general example for fixed points of the classical equation ofmotion (30). Suppose that there exists an index value 0, which satisfies X ab = x δ ab (32)with a real parameter x . Suppose also a gauge which takes N a = n δ a with a real parameter n and N [ ab ] = 0. Then the equation of motion (30) becomes dX abc dt = − σn (cid:0) X ae ) X ebc + X be ) X eca + X ce ) X eab (cid:1) − σ N X abc , = − σ (3 n x + N ) X abc . (33)Then a fixed point solution can be obtained by x = −N / n . One can further set ∀ Y abc = 0for the equation of motion of Y abc and the constraints to be satisfied.The above setup for fixed point solutions naturally appears for configurations with groupsymmetries. To see this, consider a configuration ¯ X abc which is invariant under a group L embedded in O ( N ) as ‡ l ad l be l cf ¯ X def = ¯ X abc , ∀ l ∈ L ⊂ O ( N ) . (34) ‡ For concrete examples, ¯ X abc can be taken to be C-G coefficients among various representations of groupssuch as the 3 j -symbol of SO(3). L on ¯ X abc is assumed to be reducible to a number of irreduciblerepresentations by the O ( N ) transformation and contain uniquely a one-dimensional trivialrepresentation. Then ¯ X ab = x R ( a ) δ ab , (35)where 0 denotes the index value in the trivial representation, and x R ( a ) are real parameterswhich can depend on each irreducible representation R ( a ) to which the index value a belongs.On such a symmetric configuration, one can in principle take a gauge which is consistentwith the group symmetry. This requires N a = n δ a , and that N [ ab ] take a gauge in which N [ ab ] J [ ab ] generates the infinitesimal transformation of the group symmetry (34) § . Then, since { X abc , N [ de ] J [ de ] }| X = ¯ X ≈ x R ( a ) = −N / n (36)is a fixed point of the classical equation of motion.It is noteworthy that the above solution satisfies a simple usual property of a space, whenit is interpreted as a fuzzy space defined by (23). From (35) and (36), one obtains, after properrescaling of f a with D ¶ , f · f a = f a for all f a . (37)This implies that there exists a constant function f on the space. This is actually non-trivial,since a fuzzy space defined by (23) does not necessarily have such a constant function forgeneral X abc . Because of a vast number of degrees of freedom of tensor models, it should be useful to thinkof a minimal model. This is the tensor model with a real symmetric rank-three tensor, X ∗ abc = X abc , (38) X abc = X bca = X cab = X bac = X acb = X cba . (39)In the canonical formalism, X = M, P . In this section, I will discuss the uniqueness of thelocal Hamiltonian constraint of the canonical real symmetric rank-three tensor model with thenew constraint D = 0.The (two-fold) uniqueness (10), (11) of the local Hamiltonian constraint shown in theprevious paper [25] is only for the canonical rank-three tensor model with the Hermiticity § If the group symmetry does not have infinitesimal transformations, such as in case of a discrete symmetry, N [ ab ] are taken to vanish. ¶ This is equivalent to take a gauge N = − n . D = 0 is not introduced, the most general form of the local Hamiltonianconstraint under the physically reasonable assumptions of the previous paper can be shownto have a one-parameter ambiguity as H sym,no D a = X abc X bde Y cde + λY abb , (40)where λ is an arbitrary real constant. This can easily be shown by applying the former partof the previous paper [25] to this case, and checking the on-shell algebraic closure k .From (25) and (26), one can see that the two terms in (40) are transformed differently by D . Therefore, λ = 0 is required for the algebraic closure of the constraints, H a , J ab , D . Thus H syma = X abc X bde Y cde (41)is the unique local Hamiltonian constraint for the real symmetric rank-three tensor model withthe constraints, H a = J ab = D = 0. The canonical rank-three tensor model proposed in the previous paper has the bad featurethat the solutions to the classical equation of motion have extreme behaviors. There existno other fixed points other than the trivial one, and the classical solutions either divergeor asymptotically vanish in time evolution. These extreme behaviors would become majorobstacles in future study such as of obtaining stationary spaces and quantizing the model.To improve the previous model, this paper has proposed a new canonical rank-three tensormodel by adding a scaling constraint. This constraint is a natural expectation from theinterpretation that the rank-three tensor model describes dynamics of fuzzy spaces. The newconstraint makes the configuration space compact, and the classical solutions asymptoticallyapproach either fixed points or cyclic orbits in general. It is shown that configurations withgroup symmetries provides a general class of such fixed points. These fixed points wouldrepresent stationary fuzzy spaces in physical interpretation of the model.With the scaling constraint, it is also shown that the local Hamiltonian constraint is uniquein the minimal case, namely, the canonical real symmetric rank-three tensor model. This is incontrast with that, without the scaling constraint, the local Hamiltonian has one parameterambiguity. The new canonical symmetric rank-three tensor model will provide the simplestsetup for future study. k The most difficult issue in the previous paper was how to incorporate the complications originating withthe change of orders of the indices of M abc and P abc , since it generates quite a large number of distinct termswhich potentially compose a local Hamiltonian constraint. This issue was treated in the latter part of theprevious paper, after the former part of the analysis ignoring the orders. On the other hand, in the presentcase, M abc and P abc are symmetric and therefore the former part is enough. The conclusion of the former partis that the diagrams G and G are allowed, which correspond to the two terms in (40), respectively.
8t would obviously be interesting to study the large N dynamics of the canonical tensormodels. Since the local Hamiltonians have rather simple polynomial forms, the correspondingLagrangians and hence the Feynman rules will become involved. This in turn would potentiallymake the large N behaviors of the canonical tensor models significantly different from thoseof the unsymmetric tensor models [11, 16, 17, 18]. Or an alternative way of study would beto carry out perturbative expansions around fixed points discussed in Section 4. In this case,the fixed points would provide backgrounds, and the situation would rather have similarity tothe formalism of the tensor group field theories [12, 13, 14, 15]. Acknowledgement
The author would like to thank L. Freidel for discussions on tensor models and various othertopics of quantum gravity, which have much influenced the contents of the present paper,during his stay in YITP as the visiting professor of Kyoto University. The author would alsolike to thank L. Freidel, V. Bonzom, and J.B. Geloun for invitation, hospitality and stimulatingdiscussions on tensor models during his stay in Perimeter Institute.
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