Physics Letters B 751 (2015) 343–347

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Physics Letters B

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New Hamiltonian constraint operator for loop quantum gravity ∗ Jinsong Yang a,b, Yongge Ma c, a Department of Physics, Guizhou university, Guiyang 550025, China b Institute of Physics, Academia Sinica, Taiwan c Department of Physics, Beijing Normal University, Beijing 100875, China a r t i c l e i n f o a b s t r a c t

Article history: A new symmetric Hamiltonian constraint operator is proposed for loop quantum gravity, which is well Received 15 September 2015 defined in the Hilbert space of diffeomorphism invariant states up to non-planar vertices with valence Accepted 23 October 2015 higher than three. It inherits the advantage of the original regularization method to create new vertices Available online 29 October 2015 to the spin networks. The quantum algebra of this Hamiltonian is anomaly-free on shell, and there is Editor: M. Cveticˇ less ambiguity in its construction in comparison with the original method. The regularization procedure for this Hamiltonian constraint operator can also be applied to the symmetric model of loop quantum cosmology, which leads to a new quantum dynamics of the cosmological model. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The singularity theorem of general relativity (GR) is a strong bang singularities are resolved by quantum bounces in the models signal that the classical Einstein’s equations cannot be trusted [20–22]. However, there are ambiguities in the graph-dependent when the spacetime curvature grows unboundedly. It is widely triangulation construction of this operator. There is no unique way expected that a quantum theory of gravity would overcome the to average over different choices of the triangulation. Moreover, in singularity problem of classical GR. A very lesson that one can order to obtain the on shell anomaly-free quantum algebra of the learn from GR is that the spacetime geometry itself becomes dy- Hamiltonian constraint operator [23], one has to employ degen- namical. To carry out this crucial idea raised by Einstein 100 years erate triangulation at the co-planar vertices of spin networks in ago, loop quantum gravity (LQG) is notable for its nonpertuibative the regularization procedure of the Hamiltonian.1 This treatment and background-independent construction [1–4]. The kinematical implies that the regularization procedure has essentially neglected Hilbert space of LQG consists of cylindrical functions over finite the Hamiltonian at the co-planar vertices before acting the regu- graphs embedded in the spatial manifold. The quantum geomet- lated operator on them. Otherwise, this kind of Hamiltonian con- ric operators corresponding to area [5,6], volume [5,7,8], length straint operator would generate an anomalous algebra in the full [9–11], ADM energy [12] and quasi-local energy [13], etc. have theory, unless one inputs certain unnatural requirement to the in- discrete spectrums. The LQG quantization framework can also be teraction manner of the edges of the graph and the arcs added by generalized to high-dimensional GR [14] and scalar-tensor theories the Hamiltonian operator [24]. The Hamiltonian constraint opera- of gravity [15,16]. A crucial topic now in LQG is its quantum dy- tors proposed recently in [25–27] do not generate new vertices on namics, which is being attacked from both the canonical LQG and the graph of the cylindrical function and hence are anomaly-free the path integral approach of spin foam models. In the canoni- on shell. However this kind of action cannot match the quantum cal approach a suitable regularization procedure was first proposed dynamics of spin foam models where new vertices are unavoid- by Thiemann to obtain well-defined Hamiltonian constraint oper- able in their construction [28]. A regularization of the Hamiltonian ators [17]. The Hamiltonian constraint operators obtained in this constraint compatible with the spinfoam dynamics was consid- way will attach new arcs (edges) and hence create new trivalent ered in [29]. However, the resulted Hamiltonian operator acts non- co-planar vertices to the graph of the cylindrical function upon trivially on the vertices that it created and thus has still an anoma- which they act [17,18]. The quantum dynamics determined by the lous quantum algebra. It is therefore natural to ask the question Hamiltonian constraint operator is well tested in the symmetric whether one can construct some Hamiltonian constraint operator models of loop quantum cosmology (LQC) [19]. The classical big with the following properties: (i) it is well defined in a suitable Hilbert space, symmetric and anomaly-free; (ii) it generates new

* Corresponding author. E-mail addresses: [email protected] (J. Yang), [email protected] (Y. Ma). 1 Thanks to the remark from Thomas Thiemann. http://dx.doi.org/10.1016/j.physletb.2015.10.062 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 344 J. Yang, Y. Ma / Physics Letters B 751 (2015) 343–347 vertices; (iii) its action on co-planar vertices is not neglected by × i   ˙ a  ˙b j  k −1/2 ijkFab(e (t ))e (t )e (t) Xe (t )Xe (t)V (e(t),) some special regularization procedure, and there is no special re- striction on the interaction manner of the edges of the graph and   −1/2 i  ˙a  ˙b + χ e(t ), e(t) N(e(t ))V  ijkF (e(t ))e (t )e (t) the arcs added by its action. We will show that the answer is af- (e(t ),) ab e firmative. An alternative quantization of the Hamiltonian constraint in LQG possessing the above three properties will be proposed. The ×  kj  +  jk  −1/2 · θ(t,t )Xe (t,t ) θ(t ,t)Xe (t ,t) V (e(t),) fγ , (4) regularization procedure of the Hamiltonian operator can also be applied to LQC models.   where θ(t, t ) = 1for t > t and zero otherwise, The Hamiltonian formalism of GR is formulated on a 4-dimen- k := [ T ] jk  := sional manifold M = R × , with being a 3-dimensional spatial Xe (t) tr (he(0,t)τkhe(t,1)) ∂/∂he(0,1) , Xe (t , t) [   T ] := − i manifold. In connection dynamics, the canonical variables on tr (he(0,t )τ jhe(t ,t)τkhe(t,1)) ∂/∂he(0,1) , here τk 2 σk with σk i ˜ b being the Pauli matrices, and T denotes transpose. Partitioning are the SU(2)-connection Aa and the densitized triad E j , with the of the domain [0, 1] as N segments by inputing N − 1points, only nontrivial Poisson bracket {Ai (x), E˜ b(y)} = κβδ3(x, y), where a j 0 = t , t , ···, t − , t = 1, and setting t ≡ t − t − ≡ δ, the ≡ 0 1 N 1 N n n n 1 κ 8π G and β is the Barbero–Immirzi parameter. The Hamilto- integral in (4) can be replaced by the Riemann’s sum in a limit. nian constraint reads Then (4) reduces to ˜ a ˜ b ⎧ 1 Ei E j j = 3 k − + 2 i 2 2 ⎨ N H(N) d xN ijkFab 2(1 β )K[a K ] −iβ 2κ det (q) b p 2   lim lim δ χ e (tm−1), e(tn−1) 2κ δ→0 →0 ⎩ e=e n,m=1 =: H E (N) − T (N), (1)   −1/2 × N(e (t ))V i ≡ i + i j k i i m−1   where F 2∂[a A ]  jk Aa A is the curvature of A , K is the e (t − ), ab b b a a m 1 extrinsic curvature of , and det (q) is the determinate of 3-metric     j × i ˙ a ˙b ≡ i i E ijkFab(e (tm−1))e (tm−1)e (tn−1) qab eaebδij with ea being the co-triad. H (N) and T (N) are called the Euclidean and Lorentzian terms of the Hamiltonian constraints − E × j  k 1/2 respectively. Both H (N) and T (N) depend on the canonical vari- X  (tm−1)Xe (tn−1)V e e(tn−1), ables in non-polynomial ways. Besides the indication of spin foam models, it is argued in [30] that the momentum variables in H E (N) N −1/2 + χ (e(tm−1), e(tn−1)) N(e(tm−1))V also imply the creation of new vertices by its action. Thus we e(tm−1), e = adopt the so-called semi-quantized regularization approach de- n,m 1 veloped in [31] to derive a new Hamiltonian constraint operator, × i ˙a ˙b ijkFab(e(tm−1))e (tm−1)e (tn−1) which creates new vertices as well. The Hamiltonian is not ne- glected at the co-planar vertices of spin networks by the regular- × kj ization. But the result of its action on the co-planar vertices is zero. θ(tn−1,tm−1)Xe (tn−1,tm−1) Hence it has an anomaly-free algebra on shell. − E + jk 1/2 · Let us first consider H (N). By introducing a characteristic θ(tm−1,tn−1)Xe (tm−1,tn−1) V fγ . (5) 3 3 e(tn−1), function χ(x, y) such that lim χ(x, y)/ = δ (x, y) and using →0 the point-splitting scheme, it can be regularized as −1/2 Since the volume operator and hence V vanish at divalent ver- (y,) 1 − E = 3 1/2 i ˜ a tices, for sufficiently small , the only non-vanishing terms of the H (N) lim d xN(x)V (x,) ijkFab(x)E j (x) = = 2κ →0 summation in (5) correspond to those of m n 1. Moreover, the terms corresponding to m = n = 1in the second summation =  −1/2 of (5), which involves only summation over e( e ), vanish due to × 3 ˜ b  d y χ(x, y)Ek (y)V (y,) , (2) i ˙a ˙b = = ijkFab(e(t0))e (t0)e (t0) 0. For e e , we have 2 i   a b 2  3 δ F (e (0))e˙ (0)e˙ (0) ≈ sgn(e , e)tr h  τ , (6) where V (x,) :=  det(q)(x). Since the volume operator has a ab αe e i N large kernel, the naive inverse volume operator is not well defined. + + However, one can use the idea in [32] to circumvent this problem N =− ( 1)(2 1) where 3 with a half-integer representing a spin by defining a permissible inverse square root of volume operator representation of SU(2), αee is a loop formed by adding an arc be-   ˙ a ˙b as tween e (δ) and e(δ), and sgn(e , e) := sgn[abe (0)e (0)] is the orientation factor which can be promoted into its quantum op- −1/2 := ˆ + 3 −1 ˆ 1/2 V (y, ) lim (V (y,) λ p) V (y, ), (3) 1/2 1/2  λ→0  ˆ ≡ erator. From the property of V , we know that V (v,) V v is ˆ independent of . Thus we can take the trivial limit  → 0 and where V (y,) is the standard volume operator in LQG (see [7]) cor- responding to the volume of the cube with center y and radial . obtain −1/2 2 It is easy to see that qualitatively V has the same properties β 2 ˆ ˆ E · := p −1/2 as V . Thus we can promote the classical volume in (2) into its Hδ (N) fγ Nv V v κN quantum version (3) and replace both densitized triads in (2) by v∈V (γ ) corresponding operators Eˆ b(y) =−iβ 2δ/δ Ak (y) where 2 = h¯ κ. k p b p Acting on a cylindrical function fγ , the result formally reads  j k −1/2 × sgn(e , e)ijktr hα  τi J  J V · fγ ⎧ e e e e v 1 1 e∩e=v − 2 2 ⎨ iβ p      −1/2   =: ˆ E lim dt dt χ e (t ), e(t) N(e (t ))V (e (t ), ) Nv H  (7) 2 →0 ⎩  v,e e κ =  0 0 e e v∈V (γ ) e∩e =v J. Yang, Y. Ma / Physics Letters B 751 (2015) 343–347 345

≡ i ≡− i ˆ where Nv N(v) and Je iXe is the self-adjoint right-invariant vertices as well, Hδ(N) acts nontrivially only on non-planar ver- operator. The assignment of αee is diffeormorphism covariant in tices with valence higher than three. Note that we can also employ  the sense that, for any ϕ ∈ Diff () there exists ϕ ∈ Diff() such a fourfold point-splitting scheme as in [31] to regulate the Hamil-   that ϕ (ϕ(γ )) = ϕ(γ ) and ϕ αϕ(e)ϕ(e) = ϕ(αee) [17]. Applying tonian constraint. In this case 1/ det(q) in (1) can be absorbed ˆ E v H  on the n-valent non-planar vertex v of T , the inter- into Poisson brackets, and the final operator takes the same form v,e e γ , j, i  −1/2 twiner i associated to v will be changed to i , while, as a mul- as (7) and (11), except that V v is replaced by v v tiplication operator, hα  will change the spins associated to the e e  ˆ E − 4 × 8 1 1 1 segments of e and e. Hence the action of H  is given by 1/2 IJK ( )ˆ ( )ˆ ( )ˆ v,e e V := −  tr 2 eI 2 e J 2 eK , (14) alt,v 3!E(v) sI ∩s J ∩sK =v

( 1 ) 2 −1 ˆ 1/2 where 2 eˆ := − h [h , V ]. It can be shown that qualita- , (8) I 2 sI sI v iβ p −1/2 −1/2 tively V alt,v has the same key properties as V v [33]. ∞ ˜ ∈{|  − | ···  + } ˜ ∈{| − | ··· + }  ˜ Let  := Cyl (A/G) be the set of smooth cylindrical function where j j , , j , j j , , j , and H( j , j ,  ˜ and  be the space of diffeomorphism-invariant distributions j, j, ···) are coefficients with “···” denoting spins associated to Diff ˆ · ∈ ∈  other edges incident at v. on . Then the number ψ(Hδ(N) f ) for any f  and ψ Diff The above regularization approach can be similarly used to depends only on the diffeomorphism class of the loop assignments. quantize the Lorentzian term in the Hamiltonian (1). The Lorentzian Hence the limit δ → 0can be taken in the Rovelli–Smolin topology term can be regularized as as [24,34] + 2 ˆ  ˆ  2(1 β ) 3 −1/2 [ j k] a (H(N)) · ψ ( f ) := lim (Hδ(N)) · ψ ( f ) = ˜ γ → γ T (N) lim d xN(x)V (x,) (Ka Kb )(x)E j (x) δ 0 2κ →0 ˆ := lim ψ(Hδ(N) · fγ ). (15) δ→0 × 3 ˜ b −1/2 d y χ(x, y)Ek (y)V (y,) . (9) ˆ   However, (H(N)) is not well defined in Diff, since it does not  keep Diff invariant. Taking account of the fact that the assign- ˆ Following the regularization procedure of the Euclidean term, ment of loops by Hδ(N) is diffeomorphism covariant and it acts Eq. (6) is now replaced by nontrivially only on non-planar vertices with valence higher than three, we consider almost diffeomorphism invariant states which 2 [ j  k]  ˙ a ˙b ≈  j k δ Ka (e (0))Kb (e (0))e (0)e (0) sgn(e , e)Ke Ke , (10) are obtained from the spin network states by averaging over their images under diffeomorphisms but leaving fixed sets of non-planar j − := − 1 1 ¯  where K  τ jhs  h  , K , with se as the start- e κβN e se vertices with valence higher than 3in the spatial manifold invari-  ing segment of e with parameter length δ, and we have used ant, parallel to the proposal in [25]. Given a graph γ , we denote j j = 1 { ¯ } = 1 { ¯ } ¯ := its non-planar vertices with valence higher than 3by V np4(γ ), the the identity Ka κβ Aa, K κβN tr (τ j Aa, K ), with K group of all diffeomorphisms preserving V ( ) by Diff() , 3 i ˜ a = 1 { E } np4 γ V np4 (γ ) d x Ka E 2 H (1), V [17]. Replacing the Poisson brackets i β and the diffeomorphism acting trivially on γ by TDiff()γ . For any by commutators times 1/(ih¯ ), we obtain the quantized Lorentzian  fγ ∈ , we define a map η :  →  by term 1 2 := ˆ · 2(1 + β2) β 2 η( fγ ) Uϕ fγ , (16) ˆ p −1/2 Nγ Tδ(N) · fγ = Nv V v ϕ∈Diff() /TDiff() 2κ Vnp4(γ ) γ v∈V (γ ) ˆ where Uϕ is the unitary representation of ϕ, and Nγ is a nor-  j j −1/2 × ˆ ˆ k k · malization factor. We can equip the space np4 := η() with a sgn(e , e)Ke Ke Je Je V v fγ , (11)  natural inner product as η( f )|η(g) := η( f )(g), ∀ f , g ∈ . The e∩e =v new Hilbert space Hnp4 is defined as the completion of np4 . Note where that np4 is also in the dual space of the space of diffeomorphism invariant states up to all vertices of spin networks which was fully ˆ i 1 −1 ¯ˆ ¯ˆ 1 ˆ E ˆ ˆ  K := − τihs h , K , with K := H (1), V . discussed in [25]. Whether (H(N)) can be well defined in H e 2 N e se ¯ 2 δ np4 i pβ ihβ ˆ is a delicate issue. In particular, if the action of Hδ(N) on a non- (12) planar vertex with higher valence could deduce the valence so that  it becomes less than 4or the vertex becomes co-planar, (Hˆ (N)) Hence the total regulated Hamiltonian constraint operator is given would be ill-defined on the resulted almost diffeomorphism in- by variant states.2 Fortunately, we can use the freedom of choosing the spin representations attached to each new added loop in (6) Hˆ (N) · f := N Hˆ E − Tˆ · f =: N Hˆ · f . δ γ v v v γ v v γ to ensure that the valence of any vertex would not be changed by v∈V (γ ) v∈V (γ ) ˆ ˆ  the action of Hδ(N). Then it is straightforward to see that (H(N)) ˆ  (13) is well defined in Hnp4 . By the dual action, (H(N)) will annihi- late the arcs like the one connecting v and v in (8). Since Hˆ (N) Notice that for a given γ , there are indeed finite terms contribu- 2 3 δ vanishes at co-planar vertices, it is obvious that tion to the summation no matter how fine the partition is. Hence the operator (13) is well defined in Hkin. Since the volume oper- −1/2 2 Thanks to the comment from Jerzy Lewandowski. ator and hence V (y,) vanish at internal gauge invariant trivalent 346 J. Yang, Y. Ma / Physics Letters B 751 (2015) 343–347 [ ˆ  ˆ ]· μ¯ (H(M)) ,(H(N)) φ ( fγ ) where α jk is a loop along a square in the j–k plane spanned by V ¯ 1/3 ˆ ˆ ˆ ˆ a face of , each of whose sides has length μV o with respect = φ Hδ (N)Hδ(M) − Hδ (M)Hδ(N) · fγ −1 −1 o μ¯ μ¯ μ¯ μ¯ to q , and h μ¯ = h h h h . Since the area operator ab α j k j k = 0, ∀ fγ ∈ , φ ∈ np4. (17) jk √ in LQG has a minimum nonzero eigenvalue  ≡ 2 3πβ 2 which ∈  p Hence for any ψ  , which is also a distribution on np4 , we μ¯ Diff introduces a nature cut off to the size of the loop α , we obtain have jk  ˆ  ˆ  ˆ  ˆ  3 − 1 − [(H(M)) ,(H(N)) ] · ψ (φ) = ψ [(H(M)) ,(H(M)) )]·φ Hˆ =− V 1/2 sin2(μ¯ c)pˆ pˆ V 1/2, μ¯ 2|p|=. (24) κβ2 μ¯ 2 = ∀ ∈ 0, φ np4. (18) This Hamiltonian constraint operator is obviously simpler than Therefore the quantum algebra of the Hamiltonian constraint op- those appeared in LQC models (e.g. the APS Hamiltonian opera-  ˆ erators (Hˆ (N)) is anomaly-free on shell. A symmetric Hamiltonian tor [20]). Note that the volume operator V acts on its eigenstates ˆ  as constraint operator corresponding to (H(N)) can be defined in √ H np4 as 8πβ 3/2 |v| 2 2 Vˆ |v = 3|v , with K ≡ √ , (25)  1   6 K p (Hˆ sym(N)) := (Hˆ (N)) + ((Hˆ (N)) )† , (19) 3 3 3 2 √   where v := sgn(p)|p|3/2/ 2πβ 2  , and the states |v consist where ((Hˆ (N)) )† denotes the adjoint of (Hˆ (N)) . The action of p ˆ  † Hgrav ((H(N)) ) on an almost diffeomorphism invariant state will cre- of an orthonormal basis of kin . The symmetric Hamiltonian con- ate co-planar vertices and arcs like in (8). Thus it is easy to see straint operator can be defined by ˆ sym  that the quantum algebra of (H (N)) is also anomaly-free. 1 Now we test the above regularization technique of the anomaly- Hˆ sym := (Hˆ + Hˆ †). (26) free Hamiltonian constraint operators in the symmetry-reduced 2 model of LQC. For simplification, we consider only the spatially Then we can write down the action of Hˆ sym on |v as flat and isotropic model. The construction can be generalized to f+4(v)|v + 4 −2 f0(v)|v + f−4|v − 4 , v = 0, the other cosmological models directly. Introducing an elemen- Hˆ sym|v = V i = tary cell adapted to the fiducial triad, the connections Aa and 0, v 0, − a i = 1/3 o i (27) the densitized triads Ei can be expressed as Aa cVo ea and − √ a = 2/3 o o a o i o a Ei pVo q ei , where ( ea, ei ) is a set of orthonormal co- where o triads and triads compatible with the fiducial metric qab [19]. The κβ basic (nonvanishing) Poisson bracket is given by {c, p} = . To 27 8π 3 f M (v) := K p pass to the quantum theory, one constructs a kinematical Hilbert 32κβ3/2 6 grav space H = L2(R , dμ ), where R is the Bohr com- kin Bohr Bohr Bohr × |v|5/6|v + M|1/6 +|v|1/6|v + M|5/6 . (28) pactification of the real line and dμBohr is the Haar measure i on it [19]. The holonomy of Aa along an edge, parallel to the It is straightforward to check that the Hamiltonian operator (26) o a ¯ 1/3 o triad vector ei , of length μV o with respect to qab is given by has correct classical limit and the classical big bang singularity can ¯ ¯ ¯ μ = μc I + μc I × also be avoided by a quantum bounce of this dynamics. hi cos 2 2 sin 2 τi , where is the identity 2 2matrix. Be- cause of homogeneity, the Hamiltonian constraint can be written To summarize, the Hamiltonian constraint of GR is successfully as [19,20] quantized in LQG by adopting the semi-quantized regularization approach. The resulted operator is symmetric and well defined in F i E˜ a E˜ b 1 1 ijk ab j k the Hilbert space H of diffeomorphism invariant states up to H =− H E (1) =− d3x . (20) np4 β2 2κβ2 det(q) non-planar vertices with valences higher than 3. The action of this V Hamiltonian constraint operator creates new trivalent co-planar − vertices to the spin networks but does not change the valence Since det(q) = V V 1 det(oq) where V := |p|3/2 is the physi- o of the acted vertices, and hence it can be symmetric and match cal volume of V, we can write H as the quantum dynamics of spin foam models. Meanwhile, since 1 V =− 3 o −1/2 i ˜ a ˜ b −1/2 our regularization procedure does not involve any triangulation of H d x V ijkFab E j Ek V . (21) 2κβ2 det(oq) the spatial manifold, the ambiguities of choosing certain triangula- V tion are avoided in the new construction. The Hamiltonian is not neglected at the co-planar vertices of spin networks by the reg- Note that E˜ a can be directly quantized as j ularization. But the result of its action on the co-planar vertices a ˆ −2/3 is zero. Hence the quantum algebra of the new Hamiltonian con- E˜ = V det(oq) oea pˆ j o j straint operator is anomaly-free on shell. It is thus possible to find − κβ d solutions of the quantum Hamiltonian constraint in Hnp4 . The reg- := 2/3 o o a × − V o det( q) e ih¯ . (22) j 3 dc ularization procedure for the Hamiltonian operator in full theory is also applied to the isotropic model of LQC. The resulted Hamil- a b ˆ˜ ˆ˜ i tonian constraint operator is simpler than those appeared in LQC After the action of E j and Ek , we can integrate F over the plane ab models. Meanwhile, it inherits the qualitative properties of the pre- spanned by oea and oeb to yield a holonomy along a loop. Promot- j k vious Hamiltonian operators in LQC, so that it has correct classical ing functions into corresponding operators, the regulated operator limit and the classical big bang singularity can also be avoided by corresponding to (20) reads a quantum bounce. 1 1 The authors would like to thank Jerzy Lewandowski, Chopin ˆ μ¯ −1/2 ˆ −1/2 H =− V  tr h μ¯ τ pˆ pˆ V , (23) 2 2 ijk α i Soo, Thomas Thiemann and Hoi-Lai Yu for useful discussions. J.Y. 2κβ N μ¯ jk J. Yang, Y. Ma / Physics Letters B 751 (2015) 343–347 347 is supported in part by NSFC No. 11347006, and by the Institute of [15] X. Zhang, Y. Ma, Phys. Rev. Lett. 106 (2011) 171301. Physics, Academia Sinica, Taiwan. Y.M. is supported in part by the [16] X. Zhang, Y. Ma, Phys. Rev. D 84 (2011) 104045. NSFC (Grant Nos. 11235003 and 11475023) and the Research Fund [17] T. Thiemann, Class. Quantum Gravity 15 (1998) 839. [18] M. Gaul, C. Rovelli, Class. 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