On Spinfoams Near a Classical Curvature Singularity
OOn Spinfoams Near a Classical Curvature Singularity
Muxin Han a,b and Mingyi Zhang c a Department of Physics, Florida Atlantic University,777 Glades Road, Boca Raton, FL 33431-0991, USA b Institut f¨ur Quantengravitation, Universit¨at Erlangen-N¨urnberg,Staudtstr. 7/B2, 91058 Erlangen, Germany c Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut),Am M¨uhlenberg 1, 14476 Golm, Germany
E-mail: hanm(AT)fau.edu , mingyi.zhang(AT)aei.mpg.de Abstract:
We apply the technique of spinfoam to study the space-time which, classically,contains a curvature singularity. We derive from the full covariant Loop Quantum Gravity(LQG) that the region near curvature singularity has to be of strong quantum gravity effect.We show that the spinfoam configuration describing the near-singularity region has to beof small spins j , in order that its contribution to the full spinfoam amplitude is nontrivial.The spinfoams in low and high curvature regions of the space-time may be viewed as in twodifferent phases of covariant LQG. There should be a phase transition as the space-timedescribed by spinfoam becomes more and more curved. A candidate of order parameteris proposed for understanding the phase transition. Moreover, we also analyze the spin-spin correlation function of spinfoam, and show the correlation is of long-range in the lowcurvature phase. This work is a first step toward understanding the physics of black holeand early universe from the full covariant LQG theory. a r X i v : . [ g r- q c ] J a n ontents i ε -Regularization, and Small Deficit Angle 64 High Curvature Leads to Small Spins 115 On Large Spin and Small Spin Phases, and Order Parameter 146 Correlation of Spins in the Large Spin Phase 16 The recent studies of spinfoam asymptotics have made a significant progress on understand-ing the semiclassical limit of Loop Quantum Gravity (LQG) (see e.g. [1–6]) . It has beenunderstood that at least at the discrete level, classical 4d geometry emerges from spinfoamamplitude in the regime that the spins j f are uniformly large. The large- j asymptotics ofspinfoam amplitude reproduces the discrete Einstein-Hilbert action at the leading order.In this work, we apply the semiclassical technique and result of spinfoam to the space-time which, classically, contains a curvature singularity. Typical examples are black holespace-times and Friedmann-Robertson-Walker (FRW) space-time of cosmology. The space-time under consideration here has both the low curvature and high curvature regions.The high curvature region encloses the classical singularity where the curvature blows up.The purpose of this paper is to understand the (semiclassical and quantum) behavior ofspinfoam for both low curvature and high curvature regions, as well as the behavior whenthe spinfoam transits from one region to the other.The main results can be summarized as follows: • The low curvature region far from the singularity is described by the large- j spin-foams. In order that the large- j spinfoam has a non-suppress contribution to the fullspinfoam amplitude, the spinfoam configuration must be semiclassical and correspondto a 4d simplicial geometry satisfying (cid:96) P (cid:28) a (cid:28) L (1.1)where the mean lattice spacing of the simplicial geometry is denoted by a . L is themean curvature radius of the geometry. The LQG area spectrum implies a ∼ γj(cid:96) P , See e.g. [7–10] for reviews on LQG. – 1 –here the Barbero-Immirzi parameter γ is set to be of O (1) throughout the paper.Eq.(1.1) is consistent with large- j and low curvature ( L is relatively large). Any large- j spinfoam configuration violating Eq.(1.1) only gives a suppressed contribution tothe spinfoam amplitude. • When the space-time curvature is high, Eq.(1.1) is violated by the small curvatureradius. It turns out that the large- j semiclassical approximation breaks down inthe high curvature region near singularity. The main contribution to the spinfoamamplitude comes from the small- j configurations. The small- j regime of spinfoamamplitude is considered as the quantum regime of the theory, where the quantumgravity effect is strong. Therefore the covariant theory of LQG indicates that the highcurvature region near singularity is a quantum region deviated far away from classicalgravity. It also indicates that the quantum region near singularity is made by a verylarge number of 4-simplices. The spinfoam model becomes refined when approachingthe classical singularity. The physics near the singularity may be understood by thefull nonperturbative theory of LQG, which is well-defined. • The large- j and small- j spinfoams in low and high curvature regions may be viewedas two different phases of covariant LQG. The result suggests that there should be aphase transition of LQG, when the space-time described by spinfoam becomes moreand more curved. Although it is not clear where precisely in the space-time the phasetransition occurs, the analysis suggests that the transition between large- j and small- j phases may happen at certain place where the curvature is still much lower thanthe Planckian curvature, i.e. at L (cid:29) (cid:96) P . So the small- j phase may not only coverthe Planckian curvature region, but also cover a much larger domain. This effectis resulting from the large number of spinfoam degrees of freedom on the refinedtriangulation, which accumulates and produces a strong quantum effect. It mightrelate to the recent proposal in [11], where the proposed quantum region of space-time is even slightly outside the black hole event horizon. It is also likely that thereshould be a domain-wall located at the place where the phase transition occurs. Thedomain-wall separates the low and high curvature regions of the space-time as twophases of spinfoam. It might relate to the proposal of firewall for black hole (see e.g.[12]).The analysis of spinfoam amplitude of low curvature region is carried out in Sections 2and 3. The low curvature region of the spacetime corresponds to the semiclassical regime ofspinfoam amplitude, whose contribution comes from the large- j spinfoam critical configu-ration. The studies of large- j spinfoam asymptotics shows that each simplicial geometry in4d corresponds uniquely to a critical configuration of spinfoam amplitude [2, 3, 13]. Thecontribution of a simplicial geometry to the spinfoam amplitude is obtained by performingthe spinfoam state-sum within a neighborhood at the corresponding critical point in thespace of spinfoam configurations. The correspondence is unique when the spacetime is assumed to be globally oriented and globallytime-oriented – 2 –t is particularly interesting to understand the role played by the sum over spins j in the semiclassical spinfoam amplitude. Although there has been earlier semiclassicalanalysis taking into account of the spin-sum (e.g.[4, 6, 14]), it seems to us that a sufficientunderstanding of the spin-sum in spinfoam amplitude still hasn’t been achieved yet. Oneof the mysteries of the spin-sum comes from the dual role played by the spin j in thespinfoam amplitude. On one hand, j is a scale of the theory since the minimal spacing a the triangulation is given by a ∼ γj(cid:96) P . The semiclassical limit of the theory relates tothe large- j behavior of the spinfoam amplitude. On the other hand, the spin j is also adynamical variable of spinfoam, since it is summed in the spinfoam amplitude. The factthat j is a dynamical scale is a consequence of background independence of LQG, (Seee.g.[15]).Because of the dual role played by the spin j , we propose the following prescription ofthe spin-sum: In order to study the physics at a given (energy) scale corresponding to j ,we should essentially perform the spin-sum within a neighborhood at j . The summed spinsshouldn’t go much beyond the given scale j . To implement this idea, we regularize the sumover j by introducing a decaying factor in the summand to suppress the contributions fromthe j ’s far from j . The regularized spin-sum can be performed explicitly in the spinfoamamplitude. The consequence may be viewed as an analog of Feynman i ε -regularization inquantum field theory (QFT). The suppression regulator δ is sent to be small, in order torecover the large fluctuation of spins.The regularized spin-sum results in a distribution D δ inserted in the spinfoam ampli-tude. Semiclassically, the distribution D δ is supported at the critical configurations whosecorresponding simplicial geometries have small deficit angles Θ (cid:28)
1. The smallness of Θ f is controlled by the small regulator δ regularizing the spin-sum. The contribution from anycritical configurations violating Θ (cid:28) D δ in spinfoam amplitude. Thedeficit angle relates to the curvature of the geometry by Θ ∼ a /L . So the distribution D δ resulting from the spin-sum forces the simplicial geometries emerging from spinfoam ampli-tude to satisfy a (cid:28) L , i.e. the simplicial geometries approximates the smooth geometriesof relatively low curvatures.The discussion in Section 4 is toward a description of classical curvature singularityin covariant LQG. We consider a classical space-time containing both the low curvatureand high curvature regions. The high curvature region encloses a curvature singularity.The low curvature region is emerging from the spinfoam amplitude as a large- j criticalconfiguration satisfying Eq.(1.1). We want to understand how the spinfoam configurationcontinues from the low curvature region to the high curvature region, in order to describethe high curvature region and the singularity using spinfoam.It is not hard to see that Eq.(1.1), in particular a (cid:28) L , is going to be violated, whenwe approach the singularity in the high curvature region. The reason is that L becomessmaller and even L ∼ (cid:96) P in the high curvature region. If the high curvature space-timestill admitted a large- j semiclassical description, the violation of a (cid:28) L would lead to alarge deficit angle. Then its contribution to the spinfoam amplitude would be suppressedby the distribution D δ . Therefore in the high curvature region of the space-time, thelarge- j semiclassical approximation breaks down. The main contribution of the spinfoam– 3 –mplitude comes from the small- j configurations. The quantum gravity effect becomesstrong.In LQG, the idea of quantum region near singularity has been proposed in e.g. [16–20]for loop quantum cosmology and e.g. [21–24] for black holes (including the proposals ofsingularity resolution). However a derivation of this idea from the full LQG theory hasbeen missing. Here we fill this gap and provide a derivation to show that the quantumregion near singularity is indeed predicted by the full LQG. This work is a first step towardunderstanding the physics of black hole and early universe from full LQG theory.It is clear that the distribution D δ from spin-sum plays a crucial role in the deriva-tion. Interestingly, the non-regularized version of D δ ( δ →
0) has been pointed out in theliterature [14, 25, 26]. Its support at small deficit angle leads to the so called, flatness ofspinfoam model. The flatness has been suspected to be a bad property since it seemed toimply that the semiclassical geometries from spinfoam amplitude was always flat. Howeverthe analysis here shows that the flatness property is actually a good property of spinfoammodel. Regularizing the spin-sum leads to D δ which gives a good control of the smalldeficit angle. The “regularized flatness” frees the curvature in the low curvature regionand makes the simplicial geometries approximate the smooth geometries. In the high cur-vature region, the flatness property guarantees the strong quantum effect near curvaturesingularity, such that the physics is deviated away from classical gravity.The large- j spinfoam and small- j spinfoam of low and high curvature regions maybe viewed as two phases of spinfoam model. The continuation of spinfoam from low tohigh curvature regions may be understood as the phase transition from large- j phase tosmall- j phase. The spinfoam model behaves differently in two different phases. In large- j phase, the vacua of spinfoam are the semiclassical 4d simplicial geometries, on whichthe spinfoam degrees of freedom are the excitations producing 1 /j -corrections. In small- j phase, the vacuum of spinfoam is the state with vanishing spin everywhere (no-geometrystate or the so called Ashtekar-Lewandowski vacuum). The spinfoam degrees of freedomon this vacuum are the spin and intertwiner excitations. The phases proposed here mighthave the relation with the recent works [27, 28].It is useful to find an order parameter in order to understand the phase transitionbetween large- j and small- j phases. In Section 5, we proposes a candidate of order param-eter, being the imaginary part Im (cid:104) j (cid:105) of the expectation value of the spin j . The discussionin Section 5 suggests that Im (cid:104) j (cid:105) (cid:28) j phase while it should be finite in thesmall- j phase. In Section 6, we analyze the correlation function of two spins located atdifferent triangles. We find that in the large- j phase, the pair of spins has a strong andlong-range correlation. The correlation function doesn’t decay even for a pair of spinslocated far away.In this paper, the understanding of the phase and their transition is qualitative. Givena space-time with curvature singularity, it is not clear at the moment where precisely thephase transition occurs in the space-time. However the analysis suggests that the transitionbetween large- j and small- j phases may happen at certain place where the curvature isstill much lower than the Planckian curvature, i.e. at L (cid:29) (cid:96) P . So the small- j phasemay not only cover the Planckian curvature region, but also cover a much larger domain.– 4 –his effect may be resulting from the large number of spinfoam degrees of freedom on therefined triangulation, which accumulates and produces a strong quantum effect. A morequantitative understanding of the phase transition is a research undergoing currently, whoseresult will be reported elsewhere. Our analysis here is based on the Lorentzian spinfoam amplitude proposed by Engle-Pereira-Rovelli-Livine (EPRL) [29]. The spinfoam amplitude defined on a simplicial com-plex K can be written in an integral representation [13] Z ( K ) = (cid:88) j f (cid:89) f dim( j f ) A j f ( K )= (cid:88) j f (cid:89) f dim( j f ) (cid:90) SL(2 , C ) (cid:89) ( v,e ) d g ve (cid:90) CP (cid:89) v ∈ ∂f d z vf e S [ j f ,g ve ,z vf ] (2.1)The labels v , e and f are 4-simplices, tetrahedra and triangles in the complex K , or vertices,dual edges and dual faces in the dual 2-complex K ∗ , respectively. Spin j f labels SU(2) irrepsassociated to each triangle f . g ve is an SL(2 , C ) element associated to each half-edge ( v, e ). z vf is a 2-component spinor (modulo complex scaling) associated to each vertex v at theboundary of the dual face f . The spinfoam action S [ j f , g ve , z vf ] is written as S [ j f , g ve , z vf ] ≡ (cid:88) ( ef ) j f (cid:18) ln (cid:104) Z vef , Z v (cid:48) ef (cid:105) (cid:104) Z vef , Z vef (cid:105)(cid:104) Z v (cid:48) ef , Z v (cid:48) ef (cid:105) + i γ ln (cid:104) Z vef , Z vef (cid:105)(cid:104) Z v (cid:48) ef , Z v (cid:48) ef (cid:105) (cid:19) (2.2)where Z vef ≡ g † ve z vf , (cid:104) , (cid:105) is an SU(2) invariant Hermitian inner product between spinors,and γ ∈ R is the Barbero-Immirzi parameter.The asymptotic behavior of the partial amplitude A j f ( K ) has been studied in thelarge- j regime [1–3, 13, 30]. The spins j f ≡ J k f scales uniformly large for all triangles f as J (cid:29)
1. Here J is introduced as the mean value of spins on K . The stationary phaseanalysis can be employed to study the asymptotic behavior of A j f ( K ) since S is linear to j f . The leading contribution of A j f ( K ) in large- j comes from the critical configurations,i.e. the solutions of Re S = 0 and δ g S = δ z S = 0. It turns out that generically once acritical configuration is given, a Lorentzian simpicial geometry can be reconstructed on K (we assume the geometry is non-degenerate), described by the edge lengths together withsome signs labelling the orientations. Here the orientations include both the 4d spacetimeorientation and time orientation [3, 13].In the following discussion, we consider the Lorentzian geometries reconstructed fromthe spinfoam critical configurations, which are globally oriented and time-oriented. Theleading contribution to A j f ( K ), coming from a spinfoam critical configuration, gives theRegge action (discrete Einstein-Hilbert action) of 4d gravity, i.e. A j f ( K ) ∼ exp (cid:18) i J (cid:88) f γk f Θ f + · · · (cid:19) = exp (cid:18) i (cid:96) P S Regge + · · · (cid:19) (2.3)– 5 –y the relation between triangle area and spin a f ∼ γj f (cid:96) P . Θ f is the deficit angle of thesimplicial geometry determined by the critical configuration, which encodes the curvatureof the reconstructed spacetime.“ · · · ” in the above asymptotic formula stands for the ln J and 1 /J corrections. ln J correction relates to the determinant of Hessian matrix H ij ( x ) = ∂ i ∂ j S ( x ) ( x i denotes thespinfoam variables g ve , z vf ). For an integral of type (cid:82) d n x u ( x ) e JS ( x ) ( u ( x ) is a smoothfunction, and corresponds to the integration measure in Z ( K )), the correction of order 1 /J s is given by i − s (cid:88) l − m = s (cid:88) l ≥ m − l l ! m ! n (cid:88) i,j =1 H − ij ( x ) ∂ ∂x i ∂x j l (cid:0) g mx u (cid:1) ( x ) (2.4)where the function g x ( x ) is given by g x ( x ) = S ( x ) − S ( x ) − H ij ( x )( x − x ) i ( x − x ) j .When the triangulation is refined, the number of spinfoam variables g ve , z vf increases. Thenthere will be a large number of terms contributing the above sum (cid:80) ni,j =1 . It is likely thatthe above 1 /J s correction becomes large when the triangulation is refined. So J shouldalso increase while the triangulation is refined, in order to suppress the 1 /J s correctionand keep the Regge action as the leading term. i ε -Regularization, and Small Deficit Angle The semiclassical analysis of the full spinfoam amplitude Z ( K ) is more subtle once thesum of j is taken into account. A naive semiclassical analysis leads to the so called the“flatness” of the spinfoam amplitude. Let us consider the sum of spins only in the largespin regime. We may approximate the spin-sum in Z ( K ) as an integral Z ( K ) ∼ N f J N f (cid:90) (cid:89) f k f d k f (cid:90) SL(2 , C ) (cid:89) ( v,e ) d g ve (cid:90) CP (cid:89) v ∈ ∂f d z vf e J (cid:80) f k f F f [ g ve ,z vf ] (3.1)where the spinfoam action S is rewritten as J (cid:80) f k f F f [ g ve , z vf ]. When J (cid:29)
1, if thestationary phase approximation was employed, the amplitude would be controlled by thedata ( j f , g ve , z vf ) which were the solutions of Re S = 0 and δ k S = δ g S = δ z S = 0. Thesolutions turn out to give the simplicial geometries with γ Θ f = 0 , which seem to allcorrespond to the flat geometry. It seems to imply that semiclassically the amplitudewould be dominated by flat geometry in 4d. This property is usually refered to as the flatness of spinfoam amplitude [14, 25, 31, 32].However at the solutions corresponding to flat geometry, the Hessian matrices aredegenerate, which means that the stationary phase approximation based on Gaussian typeintegral becomes obscure for treating the spin-sum in Eq.(3.1). The solutions are degeneratecritical points because a flat geometry admits too many triangulations with flat 4-simplices.A pair of triangulations can be arbitrarily close to each other (according to a certain normon the parameter space), e.g. a vertex in the triangulation can move continuously while One might replace the spin-sum by integral using Poisson resummation formula, which led to γ Θ f ∈ π i Z [31]. – 6 –he simplicial geometries are always flat. Each triangulation of flat geometry is a criticalpoint for Eq.(3.1). When there are two arbitrarily closed critical points, the critical pointsare in general degenerate.In order to overcome the incapability of the stationary phase analysis, we have to ex-plicitly perform the spin-sum in the spinfoam amplitude. Now we focus on a neighborhoodof a large- j critical configuration ( j f , g ve , z vf ) which corresponds to a globally orientedand time-oriented Lorentzian geometry. We not only consider the integration of g ve , z vf ,but also take into account of the sum over the spins in the neighborhood at j f (cid:29) Z ( j f ,g ve ,z vf ) ( K ) = (cid:90) N ( g ve ,z vf ) (cid:89) ( v,e ) d g ve (cid:89) v ∈ ∂f d z vf e (cid:80) f j f F f (cid:89) f (cid:88) s f (cid:0) j f + 1 + 2 s f (cid:1) e s f F f (3.2)where N ( g ve , z vf ) is the neighborhood at ( g ve , z vf ). s f = j f − j f is the fluctuations of spinsat the large spins j f .It is interesting to understand the sum (cid:80) s f of the perturbations. It has to be essentiallya finite sum by the following reason: The magnitude of { j f } introduces an energy scaleto the system. Because of the LQG area spectrum, γj f (cid:96) P is the area of each plaquette inthe simplicial lattice. When we study the physics at a given energy scale, the energy scalerelates to the size of the lattice plaquette, and relates to a certain magnitude of j f . Weonly consider the fluctuation of j f which doesn’t go much beyond the given scale j f . Inparticular, we don’t consider the deviation of j f which goes much below j f and touchesthe small- j regime. The small- j makes the LQG area closes to the Planck scale, thus is adeep quantum regime.The situation of spinfoam model is very different from the usual context of renormaliza-tion group in QFT. In QFT, one often integrates out the high energy modes to understandthe low energy physics. But here the sum of j f is not a sum over high/low energy modes,but rather a sum over energy scales themselves. The appearance of summing over scalesin the theory essentially because the theory sums all the geometries in a background in-dependent manner. Therefore we wouldn’t expect the physical theory defined at a givenenergy scale came from a sum over all other energy scales (because here it is not a sumover modes at scales but a sum of scales themselves). We also wouldn’t expect the phys-ical theory at a certain scale dominating the contribution in nature. So in our opinion, itdoesn’t make sense to ask whether the contribution from large- j or any scale of j shoulddominate the spinfoam amplitude. Here when we analyze the physics at a given energyscale (corresponding to j f ), we focus on a regime of spin-sum within a neighborhood atthis scale, and ignore the contribution in Z ( K ) from other scales.However it is not completely clear how much should be the size of the neighborhoodat j f . It is difficult to make a precise cut-off of the sum over j f , to decide whether thescales are much beyond j f or not. Therefore instead of making a cut-off, we introduce twodecaying regulators e − δ (1 , f s f in the sum to suppress the large fluctuations, and we define– 7 – regulated distribution: D δ ( F f ) ≡ ∞ (cid:88) s f =0 (cid:0) j f + 1 + 2 s f (cid:1) e s f ( F f − δ (1) f ) + − / (cid:88) s f = −∞ (cid:0) j f + 1 + 2 s f (cid:1) e s f ( F f + δ (2) f ) (3.3)where δ (1 , f >
0. Suppose the real part of Re F f ∈ [ − δ f ,
0] in the neighborhood N ( g ve , z vf ),then δ (2) f > δ f such that exp[ s f ( F f + δ (2) f )] is suppressed while s f goes to −∞ .Recall that we focus on the neighborhood N ( g ve , z vf ) in Eq.(3.2) because we are inthe regime of large j f . We can estimate the relation between δ f or δ (2) f and the scale of j f . Let’s consider a compact neighborhood K in ( g ve , z vf )-space, which is away from thesubmanifold defined by Re F f = 0. Recall that the real part of F f is non-positive Re F f ≤ f . Then there exists a δ f > K , Re F f ≤ − δ f at least for one f . Itis clear that K doesn’t contain any critical point. Given an oscillatory integral (cid:82) K e JS d µ with Re S ≤ K , if there is no critical point of S in the integration domain K [34], (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) K e JS ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) J (cid:19) k sup K | S (cid:48) | − Re S ) k (3.4)the integral decays faster than (1 /λ ) k for all k ∈ Z + , provided that sup([ | S (cid:48) | + Re S ] − k ) isfinite (i.e. doesn’t cancel the (1 /λ ) k behavior in front). Here because in K , Re F f ≤ − δ f at least for one f , | S (cid:48) | − Re S ≥ | S (cid:48) | + k f δ f ≥ k f δ f ⇒ J k sup K | S (cid:48) | − Re S ) k ≤ j f δ f ) k . (3.5)where k f = j f /J . So the integration on K suppresses when j f δ f > . (3.6)It means that we can ignore the contribution from K in ( g ve , z vf )-space in Eq.(3.2). There-fore as j f (cid:29)
1, we restrict our attention to N ( g ve , z vf ) with Re F f ∈ [ − δ f , . In the following we set δ (1 , f ∼ δ f ∼ /J , while δ (2) f > δ f > /J .Perform the sum of s f , D δ ( F f ) becomes D δ ( F f ) = 2 j f + 1 − j f e ( F f − δ (1) f ) / (cid:20) − e ( F f − δ (1) f ) / (cid:21) − j f + 1 − j f e ( F f + δ (2) f ) / (cid:20) − e ( F f + δ (2) f ) / (cid:21) (3.7)It is obvious that D δ ( F f ) has two series of 2nd order poles which are purely imaginary F f − δ (1) f = 4 π i Z , F f + δ (2) f = 4 π i Z (3.8) Introducing two different regulators δ (2) f (cid:54) = δ (1) f because F f is complex valued. This technical imperfec-tion will be alleviated in the formulation of spinfoam using Chern-Simons theory [33]. The contribution from j f far away from j f is suppressed by the decaying regulators. So we essentiallyfocus on a neighborhood of large j ’s. – 8 –ince Re F f ∈ [ − δ f ,
0] in the neighborhood N ( g ve , z vf ), and δ (2) f > δ f , we have Re F f − δ (1) f ≤− δ (1) f < F f + δ (2) f ≥ δ (2) f − δ f >
0. The real parts of F f − δ (1) f and F f + δ (2) f are notzero, which means that the poles of D δ ( F f ) are all falling outside of N ( g ve , z vf ). D δ ( F f )is a smooth function in the domain of N ( g ve , z vf ). The implementation of δ (1 , f might beviewed as an analog of the i ε -regularization of Feynman propagator in QFT.The regularized contribution Z ( δ )( j f ,g ve ,z vf ) ( K ) is defined from Eq.(3.2) by regularizingthe sum over s f Z ( δ )( j f ,g ve ,z vf ) ( K ) = (cid:90) N ( g ve ,z vf ) d g ve d z vf e (cid:80) f j f F f [ g ve ,z vf ] (cid:89) f D δ ( F f [ g ve , z vf ]) . (3.9)Since D δ ( F f ) is a smooth function on N ( g ve , z vf ) and j f (cid:29)
1, the above integral can beanalyzed by the standard stationary phase approximation. There is a singe critical point( j f , g ve , z vf ) inside N ( g ve , z vf ). We have the asymptotic formula with the Regge action asthe leading effective action Z ( δ )( j f ,g ve ,z vf ) ( K ) ∼ e i (cid:80) f γj f Θ f (cid:89) f D δ (cid:0) i γ Θ f (cid:1) (cid:2) O (cid:0) J − (cid:1)(cid:3) (3.10)where Θ f is the deficit angle reconstructed from the critical configuration.At the critical point ( j f , g ve , z vf ), F f takes purely imaginary value F f = i γ Θ f where Θ f is the deficit angle at f . From the expression of D δ ( F f ), it is clear that D δ ( iγ Θ f ) becomeslarge when i γ Θ f approach close to one of the poles F f = 4 π i Z ± δ (1 , f , although the poleshave been regularized away from the purely imaginary axis. Here we are not interested inthe poles F f = 4 π i n f ± δ (1 , f with n f (cid:54) = 0, because the critical points ( j f , g ve , z vf ) closeto these poles doesn’t correspond to a proper simplicial geometry, in the sense that γ Θ f close to 4 πn f ( k f (cid:54) = 0) implies a conical singularity located at f , whose physical meaningis unclear. We expect that the appearance of 4 πn f poles ( n f (cid:54) = 0) is an artifact of Z ( K )being a discrete theory from starting point. For example, unphysical poles of momenta inprinciple also appear in lattice-field-theory propagators, which is an analog to 4 πn f poles( n f (cid:54) = 0) here. But the integration of momenta in lattice field theory is only over theBrillouin zone where only the physical pole is relevant.Now we focus on the neighborhoods at the poles F f = ± δ (1 , f . D δ ( iγ Θ f ) in theasymptotic formula Eq.(3.10) implies that the critical points ( j f , g ve , z vf ) with small deficitangle Θ f (cid:28) a and the mean curvature radius L of the geometry by [35]Θ f ∼ a L (cid:20) o (cid:18) a L (cid:19)(cid:21) (3.11)Therefore when L is fixed, the simplicial geometries close to the continuum limit contributeto Z ( K ) much more than other simplicial geometries. D δ coming from spin-sum forces( j f , g ve , z vf ) to satisfy a (cid:28) L , (3.12)– 9 –n order to have nontrivial contribution to the spinfoam amplitude. The explicit behaviorof D δ (cid:16) iγ Θ f (cid:17) as Θ f (cid:28) D δ (cid:0) iγ Θ f (cid:1) = 4 j f (cid:16) δ (1) f + δ (2) f (cid:17)(cid:16) γ Θ f + iδ (1) f (cid:17) (cid:16) γ Θ f − iδ (2) f (cid:17) + 8 iγ Θ f (cid:16) δ (1) f + δ (2) f (cid:17) + 4 (cid:104) ( δ (1) f ) + ( δ (2) f ) (cid:105)(cid:16) γ Θ f + iδ (1) f (cid:17) (cid:16) γ Θ f − iδ (2) f (cid:17) +regular in Θ f . (3.13)where we see D δ is much greater when Θ f (cid:28) f is finite.The set-up δ (2) f > δ f implies that the removal of regulator δ (2) f has to be done togetherwith large- j limit, by Eq.(3.6). As δ (1 , f become small, the nontrivial contribution of D δ comes from small deficit angle | γ Θ f | ≤ δ (1 , f , and D δ behaves as D δ ∼ j f δ (1 , f + 2 iγ + 1 / δ (1 , f ) . (3.14)The relation j f δ f > a (cid:96) P L ∼ | γj f Θ f | > , (3.15)if we identify a ∼ γj f (cid:96) P and a /L ∼ Θ f ∼ γ − δ f . In this regime, when the number of f is large in K , the effective action in Eq.(3.10) (cid:88) f γj f Θ f = 1 (cid:96) P (cid:88) f a f Θ f (cid:29) , (3.16)and gives a rapid oscillating exponential, unless the Regge equation of motion is satisfiedsuch that Regge action (cid:80) f a f Θ f vanishes.Here we see that the effect of D δ in the asymptotics Eq.(3.10) is to suppress the con-tributions from the critical points ( j f , g ve , z vf ) whose deficit angles Θ f are not small. Weknow that a ( j f , g ve , z vf ) with non-small Θ f corresponds to a simplicial geometry whichdoesn’t approximate any smooth geometry because of Eq.(3.11). By summing over j f ,the appearance of D δ in the asymptotics selects only the simplicial geometries which aregood approximation to the smooth geometries and suppresses the rest. As a result, onlythose ( j f , g ve , z vf )’s with Θ f (cid:28) Z ( K ). As δ (1 , f → D δ pushes the critical points ( j f , g ve , z vf ) contributing Z ( K ) to approach the smooth geome-tries (approximate the smooth geometries arbitrarily well), when the simplicial complex isalso refined accordingly at the same time. At each Z ( j f ,g ve ,z vf ) ( K ) Z ( j f ,g ve ,z vf ) ( K ) ∼ e i (cid:80) f γj f Θ f (cid:39) e i(cid:96) P (cid:82) d x √ − g R (3.17)where the Regge action approaches the Einstein-Hilbert action evaluated at the correspond-ing smooth geometry. Because of Eq.(3.6), δ (1 , f → j f → ∞ ,i.e. the continuum limit and large- j limit are taken at the same time.– 10 –ote that even if the requirement Eq.(3.6) is alleviated (e.g. in the Chern-Simonsformalism [33]), one may still need to increase j f at the same time as refining the triangu-lation. The reason is that when the triangulation is refined, the 1 /j f quantum correctionsmay become larger, since more degrees of freedom are summed. Then j f may have toincreased to suppress the quantum corrections, as mentioned at the end of Section 2.It is important to emphasize that when we set the scale of the theory to be in the large- j regime, the (regularized) spin-sum of the spinfoam amplitude forces the spinfoam criticalconfigurations to correspond to simplicial geometries with small deficit angle Θ f (cid:28)
1, i.e.the resulting simplicial geometries have to satisfy (cid:96) P (cid:28) a (cid:28) L , (3.18)in order to have the nontrivial contribution to the spinfoam amplitude. The previous discussion focuses on the large- j regime of spinfoam amplitude. Eq.(3.18)means that the spinfoam configuration ( j f , g ve , z vf ), which contributes nontrivially to Z ( K ),is a semiclassical space-time with a relatively low curvature. In this section, we considerthe behavior of spinfoam for a space-time containing a high curvature region. A typicalexample is the space-time with curvature singularity.In the following we consider the spinfoam configuration ( j f , g ve , z vf ) have the follow-ing properties: (1) It has a subset of data (¯ j f , ¯ g ve , ¯ z vf ) ⊂ ( j f , g ve , z vf ) being large- j andcritical. The subset of data correspond to a low curvature region of a space-time geometrysatisfying Eq.(3.18) and Einstein equation; (2) In addition to the low curvature region, thespace-time geometry relating to (¯ j f , ¯ g ve , ¯ z vf ) (in its low curvature region) also has a highcurvature region; and (3) ( j f , g ve , z vf ) should have nontrivial contribution to the full spin-foam amplitude Z ( K ) (its contribution is not suppressed). We ask the following question:how does the rest of data ( j f , g ve , z vf ) \ (¯ j f , ¯ g ve , ¯ z vf ) behave in the high curvature region?As an example, we may consider a Schwarzschild black hole, where the large- j criticalconfiguration (¯ j f , ¯ g ve , ¯ z vf ) describes the low curvature geometry outside the event horizon.We are interested in how the data (¯ j f , ¯ g ve , ¯ z vf ) continues to the high curvature region insidethe event horizon, especially near the singularity.As a quick answer, the spinfoam data ( j f , g ve , z vf ) describing a high curvature space-time (near classical singularity) must have small j f , i.e. the spinfoam cannot be semiclas-sical in high curvature region. There are two steps of explanation: • Firstly, in the high curvature region, a spinfoam data ( j f , g ve , z vf ) cannot satisfy both (cid:96) P (cid:28) a and a (cid:28) L . Suppose we keep the data ( j f , g ve , z vf ) satisfying a (cid:28) L in high curvature region (small L ), when the curvature becomes almost Planckian L ∼ (cid:96) P , the lattice spacing a has to approach the Planck scale (cid:96) P , violating (cid:96) P (cid:28) a .Considering a ∼ γj f (cid:96) P , the spin j f has to be small (we always assume γ ∼ o (1)).Another way to conclude, if ( j f , g ve , z vf ) was a large- j critical configuration in highcurvature region, the deficit angle would violate Θ f (cid:28) Secondly, now let’s assume ( j f , g ve , z vf ) to be a large- j critical configuration and Θ f is not small. We apply the analysis in the last section to perform the sum over spinsat the given scale. The regularized spin-sum results in the distribution D δ ( iγ Θ f ) inEq.(3.10). Therefore the large- j critical configuration ( j f , g ve , z vf ) only gives a tinycontribution when Θ f is not small. So it contradicts to our requirement that thecontribution from ( j f , g ve , z vf ) to Z ( K ) is nontrivial. Therefore the only possibilityis that a spinfoam data ( j f , g ve , z vf ) describing the high curvature space-time regionnear a classical singularity is of small- j and is not semiclassical.Therefore we conclude that approaching the high curvature region near a classicalsingularity, the spinfoam amplitude forces the spins j f to become small, in order that thecontribution to the spinfoam amplitude is not suppressed. Thus the spinfoam amplitudeare dominated by small- j contributions in this region. Because of small j f , the regionnear a classical singularity is highly quantum, and is referred to as the quantum gravityregion. Here “small- j ” means that 1 /J correction in Eq.(3.10) is not negligible, so thatthe semiclassical approximation breaks down. Small- j may not necessarily mean that j f ∼ O (1). When the triangulation is sufficiently refined, a not-so-large j f may not suppressthe 1 /J correction due to an increased number of spinfoam degrees of freedom, thus is stillunderstood as small- j .It is clear that the large- j approximation breaks down in the Planckian curvatureregion, where L ∼ (cid:96) P , by the above argument. So the Planckian curvature region isnecessary inside the small- j regime of spinfoam. However, it is not clear where preciselyin the spacetime, the large- j turns to be small. If the small- j region was precisely thePlanckian curvature region, then j f would have to be of O (1) in the small- j region, byEq.(3.18) which should hold outside the small- j region. But it is likely that the regionwhere j f are small is much larger than the Planckian curvature region, due to the largenumber of spinfoam degrees of freedom. Indeed if we travel toward a classical curvaturesingularity, the above argument implies that j f decreases from the low curvature region tothe high curvature region. But when we consider a refined triangulation, it corresponds toa large number of spinfoam degrees of freedom. A slight decreasing of j f may not anymorecapable to suppress the 1 /J correction. It is likely that we arrive the small- j region farbefore we approach the Planckian curvature region. This phenomena may relate to theresults in [11], where the author expect the quantum effect may even appear slightly outsidethe event horizon.In the small- j regime of spinfoam, the semiclassical relation between spinfoam config-uration and discrete geometry is broken down. From spinfoam point of view, the notionof space-time geometry becomes invalid and corrected by large quantum fluctuations (1 /J corrections). To understand the dynamics of this regime, one should study the full non-perturbative behavior of the spinfoam amplitude Z ( K ) in Eq.(2.1). The nonperturbativespinfoam amplitude is well-defined and of nice properties [5, 9, 36–39].We again consider a Schwarzschild black hole space-time. The space-time correspondsto a spinfoam critical configuration (¯ j f , ¯ g ve , ¯ z vf ) in the low curvature region, satisfyingEq.(3.18). – 12 –s shown in fig.(1), the Schwarzschild space-time is naturally divided into three regions:black hole singularity (Planckian curvature region), region inside and near the (event)horizon, region outside and far from the horizon. If we pick up a point and its neighborhoodin the region far away from the horizon, the curvature in this sub-region (denoted byRegion A) is small, as we learned from the Schwarzschild metric, when radius coordinate r is much larger than the Schwarzschild radius r s = 2 GM/c , the Schwarzschild metricbecomes Minkowski metric with an order ( r s /r ) correction. So in this Region A, the meancurvature radius L A is large, and thus the average deficit angle | Θ o | (cid:28)
1. This regioncorresponds to a large- j spinfoam critical configuration (¯ j f , ¯ g ve , ¯ z vf ) satisfying Eq.(3.18).Because of large- j the triangle areas of the triangulation can be relatively large but stillsatisfy Eq.(3.18). Black Hole Singularity(Quantum gravity region) (Event) Horizon Region A(far from horizon)Region B(inside horizon)
AB B'
Figure 1 . A Schwarzschild space-time. Region A is a sub-space-time which is far from the horizon.Region B is a sub-space-time which is inside the horizon. Region B (cid:48) is a sub-space-time which isnear the quantum region of the black hole singularity.
Region B denotes a sub-region inside the horizon (approaching the near-singularityregion), where the background mean curvature becomes larger. The Kretschmann invariantfor Schwarzschild metric at coordinate radius r is R µνρσ R µνρσ = 12 r s /r , where R µνρσ isRiemann curvature tensor. Since Riemann curvature is scaling as the inverse of the squareof the mean curvature radius L B of Region B, i.e. R µνρσ ∼ L − B , L B behaves as L B ∼ r (cid:114) rr s , r < r s (4.1)The background deficit angle Θ in Region B scales as | Θ | ∼ a r s /r .The quantum effect (organized by 1 /J corrections) cannot be neglected when thecurvature approaches Planckian in Region B (cid:48) , i.e. the mean curvature radius is of orderof Planck length L ∼ (cid:96) p . The spins j f become small in this region. However it is likelythat the small- j region may not only cover Region B (cid:48) , but also cover a part of RegionB or possibly even entire Region B, because of the large number of degrees of freedom– 13 –n a refined triangulation. Traveling from Region A to Region B and B (cid:48) , the quantumarea j f becomes smaller. So the size of each 4-simplex in the triangulation shrinks whileapproaching the Planckian curvature region.We can estimate the radius of the Planckian curvature region where the quantumgravity effect is clearly strong. By Eq.(4.1), the minimal radius is r p ∼ (cid:114) r s (cid:96) p (cid:96) p = (cid:114) (cid:126) G Mc (4.2)When M is the mass of the sun M (cid:12) , then r p is r p ∼ (cid:114) (cid:126) G M (cid:12) c ∼ − m (cid:29) (cid:96) p . (4.3)We may consider an extreme case where the small j f are of O (1) in the Planckian curvatureregion. The average area a of the triangle is of order γ − / (cid:96) p . The ratio between r p and a is r p a ∼ (cid:18) γr s (cid:96) p (cid:19) ∼ γ × (4.4)Taking γ ∼ o (1), we then get r p a ∼ , J ∼ o (1) , (4.5)It means that the minimal size of the quantum gravity region with the mass of the sun maybe of the order 10 bigger than the average size of the simplicies that construct Region B (cid:48) .It means that the quantum regions is made by a large number of small quantum simplicies.So the spinfoam is highly refined in the region where quantum gravity effect is strong.Note that in [22] there has been two estimations for the radius of the quantum region(Planck star) by different argument, our estimation coincides with the rough one there.Here we still keep the Schwarzschild metric as the background space-time for high curvatureregion. However, in the quantum gravity region, the notion of metric is actually ill-defined.Schwarzschild metric needs to be corrected in the Region B (cid:48) . The above discussion aboutSchwarzschild is a rough estimation. The numbers computed above should be correctedwhen it is derived in a more rigorous way, which involves a better computation usingspinfoam model in small- j regime. As it has been shown in the above, the physics of space-time near the curvature singularity isdescribed by the spinfoams whose spins are small. Far away from the singularity, the space-time is semi-classical and of low curvature. The corresponding spinfoam configurations areof large spins. It is clear that the spins j f are summed in spinfoam amplitude Z ( K ). Sothe small or large spin mentioned above means the spin-sum is effectively carried out inthe small or large spin regime. – 14 –t is intuitive to consider the high curvature and low curvature regions as two different phases of small- j and large- j . The different phases relate to the different vacua of spinfoammodel. The low curvature region is the vacuum of spinfoam being the large- j criticalconfiguration (¯ j f , ¯ g ve , ¯ z vf ) satisfying Eq.(3.18). The spinfoam degrees of freedom are theexcitations on (¯ j f , ¯ g ve , ¯ z vf ) producing 1 /J -corrections of spinfoam amplitude. The highcurvature region has the vacuum state with vanishing spin everywhere (no-geometry stateor the so called Ashtekar-Lewandowski vacuum). The spinfoam degrees of freedom on thisvacuum are the spin and intertwiner excitations.Finding an order parameter is usually helpful to understand the phases, as well as thetransition between them. Here we find a candidate of order parameter to be the imaginarypart of j f expectation value, Im (cid:104) j f (cid:105) , which is expected to behave differently in differencephases. Firstly we consider the large- j phase. Instead of perform the sum over spins asshowing in Section 2, we integrate the group elements g ve and spinors z vf in the first place.Because the computation is in the large- j regime, the integration of g ve and z vf can beperformed by using the saddle point approximation in the large- j limit. As shown in [31],the spinfoam amplitude expanding at a low curvature critical configuration ( j f , g ve , z vf )can be written as an effective partition function of a spin system. The amplitude is Z ( j f ,g ve ,z vf ) ( K ) = (cid:88) { j f } (2 j f + 1) exp I K [ j f ] (5.1)where I K [ j f ] = I K [ J k f ] is the effective action. Define new variables κ f ≡ k f − k f andexpand I K [ J k f ] around κ f = 0, then the effective action is obtained as I K [ J k f ] = J (cid:16) I + I f κ f + I ff (cid:48) κ f κ f (cid:48) + O (cid:0) κ (cid:1)(cid:17) (5.2)The first three coefficients are computed in [31] to the leading order in 1 /J : I = i γk f Θ f , I f = i γ Θ f − δ , f , I ff (cid:48) = 2(1 + 2i γ − γ − γ )5 + 2i γ n Tef X − e n ef (cid:48) (5.3)where Θ f is the deficit angle given by the critical configuration; n ef is the unit 3-vectornormal determined by ( j f , g ve , z vf ) which is the normal vector of the triangle f in the frameof tetrahedron e [40]. The matrix X e is X ije ≡ (cid:80) f k f ( − δ ij + n ief n jef ). The expectationvalue of (cid:104) j f (cid:105) = j f + (cid:104) κ f (cid:105) to the leading order in 1 /J can be obtain by the equation ofmotion of I K [31]. (cid:104) κ f (cid:105) ∼ (cid:88) f (cid:48) (cid:0) I − (cid:1) ff (cid:48) (cid:0) i γ Θ f (cid:48) − δ , f (cid:48) (cid:1) + O (cid:16) (i γ Θ f − δ , f ) (cid:17) . (5.4)The above is an expansion in the low curvature regime, where γ Θ f (cid:48) ∼ δ , f (cid:48) ∼ /J . Thereforein the low curvature regime, Im (cid:104) j f (cid:105) = Im (cid:104) κ f (cid:105) ∼ /J (5.5)is suppressed by large- J . In particular if we consider a black hole spacetime with asymp-totically flat region, we can set both Θ f and δ , f to be very small, corresponding J beingvery large. Then Im (cid:104) j f (cid:105) is very small in the region.– 15 –hen we approach the high curvature regime, j f becomes small so that the 1 /J cor-rections are not negligible. Then Im (cid:104) j f (cid:105) cannot be suppressed by 1 /J , and likely becomes afinite number. In the small- j phase, we insert j f of a triangle f into the integration formulaEq.(2.1) of Z ( K ). The sum of j f is carried out in the small- j regime. The expectationvalue of j f is written as (cid:104) j f (cid:105) = 1 Z ( K ) (cid:88) { j f (cid:48) } (cid:89) f (cid:48) dim( j f (cid:48) ) j f (cid:90) SL(2 , C ) (cid:89) ( v,e ) d g ve (cid:90) CP (cid:89) v ∈ ∂f (cid:48) d z vf (cid:48) e S [ j f (cid:48) ,g ve ,z vf (cid:48) ] . (5.6)The integrand is a complex function, and the large- j approximation breaks down in thisphase. The large 1 /J corrections suggests that Im (cid:104) j f (cid:105) should be generically nonzero andfinite, although a mathematically rigorous proof of Im (cid:104) j f (cid:105) being finite (i.e. a lower boundof Im (cid:104) j f (cid:105) ) is still lacking for the small- j phase.Therefore the above argument suggests that the quantity Im (cid:104) j f (cid:105) should have twodifferent behavior in the large- j and small- j phases:Im (cid:104) j f (cid:105) ∼ /J (cid:28) j phase (low curvature)Im (cid:104) j f (cid:105) = finite in small- j phase (high curvature) . (5.7)Im (cid:104) j f (cid:105) (cid:28) j interpretation of j f as semiclassical triangle area,while the finite Im (cid:104) j f (cid:105) in small- j phase means that the semiclassical approximation breaksdown.It should be noted that the above argument toward the order parameter is still at thequalitative level. The more detailed investigation is postponed in the future research. The behavior of correlation functions is usually useful to understand the phases and theirtransition. Here we view the spinfoam amplitude as a “statistical system”, and we studythe correlation function of a pair of spins j f , j f (cid:48) at different locations. We find that in thelarge- j phase (low curvature region), the correlation between spins is of long-range, i.e.no matter how “far” away the two different spins are separated, their correlation hardlydecays.Recall Eq.(5.1), the spinfoam amplitude can be written perturbatively in the large spinregime with effective action I K [ J k f ], where κ f is the perturbation of j f at j f : Z ( j f ,g ve ,z vf ) ( K ) = (2 J ) N f e JI (cid:88) { κ f } (cid:18) k f + κ f + 12 J (cid:19) e J (cid:16) I f κ f + I ff (cid:48) κ f κ f (cid:48) + O ( κ ) (cid:17) (6.1)We keep the effective action to the quadratic order in the perturbation κ f , and approximatethe sum (cid:80) { κ f } by an integral. The amplitude Z ( j f ,g ve ,z vf ) ( K ) looks like a path integralover κ f with an external source J I f . – 16 –he (connected) correlations between two spins j f and j f (cid:48) is computed at the leadingorder (cid:104) ( j f − j f )( j f (cid:48) − j f (cid:48) ) (cid:105) = J (cid:104) κ f κ f (cid:48) (cid:105) = ∂ I f ∂ I f (cid:48) Z ( j f ,g ve ,z vf ) ( K ) Z ( j f ,g ve ,z vf ) ( K ) ∼ J I f ( I − ) f f I f ( I − ) f f (cid:48) − J I − ) ff (cid:48) (6.2)Because in the low curvature regime, γ Θ f (cid:48) ∼ δ , f (cid:48) ∼ /J , i.e. I f ∼ /J , the leadingcontribution to the correlation function comes from the second term: (cid:104) ( j f − j f )( j f (cid:48) − j f (cid:48) ) (cid:105) = − J I − ) ff (cid:48) (6.3)The matrix elements of I ff (cid:48) is non-zero only when triangles f and f (cid:48) belong to the sametetrahedron. The non-zero elements of I ff (cid:48) are mainly next to the diagonal. However thematrix is not a block-diagonal matrix. Then its inverse ( I − ) ff (cid:48) is also not block-diagonal.Moreover, the matrix elements ( I − ) ff (cid:48) are generically nonvanishing for an arbitrary pairof f, f (cid:48) . The correlation between two spins are of long-range and strong. The magnitudeof correlation function scales linearly in J .Indeed, to illustrate the inverse of I ff (cid:48) , we consider a tridiagonal matrix (analog of I ff (cid:48) ) and its inverse (analog of ( I − ) ff (cid:48) ) I = a b c a b c . . . . . .. . . . . . b n − c n − a n and ( I − ) ij = (cid:40) ( − i + j b i · · · b j − θ i − φ j +1 /θ n if i ≤ j ( − i + j c j · · · c i − θ j − φ i +1 /θ n if i > j (6.4)where { θ i } i satisfy the recurrence relation θ i = a i θ i − − b i − c i − θ i − for i = 2 , , . . . , n withinitial conditions θ = 1, θ = a . { φ i } i satisfy φ i = a i φ i +1 − b i c i φ i +2 for i = n − , . . . , φ n +1 = 1 and φ n = a n (see [41] for examples of symmetrictridiagonal matrix). Generically, all the matrix elements of I − are nonvanishing.The correlation function in small- j phase (high curvature region) is more difficult tocompute, due to the lack of approximation scheme. However we do believe the spin-spincorrelation function in small- j phase should be of dramatically different behavior from it isin the large- j phase, because the 1 /J correction becomes non-negligible in small- j phase.The further investigation of correlation function is postponed to the future research. Acknowledgments
The authors acknowledge the helpful discussions with Jonathan Engle, Francesca Vidotto,Carlo Rovelli, and Zhaolong Wang, and acknowledge the comments from Daniele Oritiand Edward Wilson-Ewing. MZ acknowledges the funding received from Alexander vonHumboldt Foundation. MH acknowledges the Institute of Modern Physics at Northwestern– 17 –niversity in Xi’an, Yau Mathematical Sciences Center at Tsinghua University in Beijing,and Fudan University in Shanghai, for their hospitality during his visits. MH also acknowl-edges the support from the US National Science Foundation through grant PHY-1602867,and the Start-up Grant at Florida Atlantic University, USA.
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