EPRL/FK Asymptotics and the Flatness Problem
EEPRL/FK Asymptotics and the Flatness Problem
José Ricardo Oliveira ∗ School of Mathematical Sciences, University of Nottingham † University ParkNottingham NG7 2RD, UK
April 18, 2017
Abstract
Spin foam models are an approach to quantum gravity based on the concept of sum overstates, which aims to describe quantum spacetime dynamics in a way that its parent framework,loop quantum gravity, has not as of yet succeeded. Since these models’ relation to classical Einsteingravity is not explicit, an important test of their viabilitiy is the study of asymptotics - the classicaltheory should be obtained in a limit where quantum effects are negligible, taken to be the limitof large triangle areas in a triangulated manifold with boundary. In this paper we will brieflyintroduce the EPRL/FK spin foam model and known results about its asymptotics, proceedingthen to describe a practical computation of spin foam and semiclassical geometric data for a simpletriangulation with only one interior triangle. The results are used to comment on the "flatnessproblem" - a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK’s classical limit onlydescribes flat geometries in vacuum.
Contents ∆ j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Constructing EPRL spin foam variables from geometrical data . . . . . . . . . . . . . . 20 Spin foam models are an approach to quantum gravity heavily inspired in Loop Quantum Gravity(LQG)[1], which aimed to address the parent theory’s issues with describing dynamics while providinga clear picture of the quantum geometry of a general relativistic spacetime. While LQG’s proposed ∗ [email protected] † Paper written as a result of PhD work under the supervision of Prof. John W. Barrett. The author acknowledgesfunding by the Fundação para a Ciência e Tecnologia (FCT), scholarship SFRH/BD/77045/2011. a r X i v : . [ g r- q c ] A p r anonical quantization of the first order version of Einstein’s general relativity gives us a background-independent model with mostly well understood kinematics, its dynamics is encoded in a time evolutionequation named the Hamiltonian constraint - time being defined via a 3+1 ADM decomposition ofspacetime[2] used to derive a Hamiltonian form for the Holst-Palatini action. Until today, solvingthe Hamiltonian constraint remains an open problem, due to two main issues. One is the problemof time - defining the dynamics of a system which is manifestly diffeomorphism-invariant by its timeevolution is possible in the classical theory, by considering the time variable in an ADM decompositionof a solution of Einstein’s equations, but in a quantum version of the theory these solutions only de-termine probabilities of different spacetimes occurring, and therefore defining time in this way wouldbe ambiguous. While the former is a more conceptual problem with known workarounds[15], thereis also a more serious practical issue - quantizing the Hamiltonian constraint and writing down therespective operator. There are several ambiguities in doing so, and while there are proposals for it,such as Thiemann’s[16], it remains as an open problem, especially because it has proven difficult toverify the viability of a given proposal.The spin foam approach originated from an attempt to enunciate a path integral formulation of LQG.It uses the basis of spin network states, taking them as quantum states of a triangulated manifoldwhich are summed over to form a partition function. Dynamics is determined by the probability am-plitudes attributed to each state. Therefore, the problem to solve in the spin foam program is to definea set of amplitudes which is consistent with GR. Ponzano and Regge formulated a suitable model forthree-dimensional gravity[3], but the four-dimensional problem is much more difficult in nature - 3dgeneral relativity in vaccuum is purely topological, with no dynamical degrees of freedom, while the4d theory is not[4].The first concrete attempt at devising a spin foam model for 4d gravity was the Barrett-Cranemodel[24], which gave a set of bivector variables, obtainable from the spin foam parameters, andequivalent to a set of variables describing the Euclidean geometry of a triangulation. The modelwas later abandoned as it was found that the bivectors were over-constrained by the requirement ofsimplicity. The idea of enforcing that specific constraint only in a weak “expectation value” senseinstead of the strong sense led to two independent proposals (Engle/Pereira/Rovelli/Livine and Frei-del/Krasnov), which turned out to be equivalent for Immirzi parameter < γ < , and gave rise tothe EPRL/FK model. Additionally, the Ooguri[5] and Crane-Yetter[6] models are often mentioned astriangulation-independent models that do not describe gravity.Study of asymptotics becomes necessary for two main reasons. The first, and most evident, is todetermine if a given model reduces to classical General Relativity in the (cid:126) → limit. Secondly, itseems apparent that diffeomorphism invariance in GR should be realized as triangulation indepen-dence in the spin foam model, but this condition is manifestly not satisfied in either the Barrett-Craneor EPRL/FK models. While this is a major issue in itself, it can be argued that the triangulationinvariance requirement can be “relaxed” somewhat, by only enforcing it in the semiclassical limit. Theimplication that there would be “preferred coordinate choices”, realized as preferred triangulations, inthe full quantum theory is certainly an uncomfortable one, but not necessarily invalid, since the lengthscale which one would have to probe to “see” the triangulation structure of spacetime, if it exists, iswell below anything feasible with the current means.In Section 2, we briefly introduce the basic concepts of general spin foam models in four dimensions,and fully define the EPRL/FK model in the Euclidean setting with an Immirzi parameter < γ < .Section 3 is dedicated to the asymptotics of the EPRL/FK model in an arbitrary simplicial complexwith boundary, consisting of a short review of past work and results, as well as more detailed consid-erations about minute details in the formalism and the key tool used to derive a semiclassical limit,the stationary phase method, leading into some new insight on the “flatness problem”.Finally, section 4 includes a thorough calculation of the classical geometry of a simplicial complexdubbed ∆ , consisting of three 4-simplices, describing the methods used which apply to any Regge-likeboundary data and presenting the results obtained from two examples with specified boundary. Section5 is reserved for discussion of the results and their implications about the validity of the model.2 Spin Foam Models and EPRL/FK
Spin foams are constructed from arbitrary spin network states ψ Γ ( { g l } ) over graphs Γ embedded ina manifold M (which corresponds to the spatial slice of spacetime), where g l are elements of a gaugegroup G which in gravity is the relativistic symmetry group of the theory (in general it could be any Liegroup). The edges l of Γ have spins j l associated to them, corresponding to irreducible representationsof G , while the graph’s vertices v are labelled by intertwiners i v . Now if we picture the extra timedimension and imagine the graph evolving into it, it will form a so-called where the edgesare foliated into faces f and the vertices into new edges e . The graph can change topologically withtime, and there will be new vertices v , signalling points in spacetime where one edge breaks intoseveral, or vice-versa with two or more edges joining into one. The “time-evolved” graph is called thespin foam, and can be generally defined by • an arbitrary 2-complex; • representation spins j f for each face f of the 2-complex; • intertwiners i e for each edge e .In four dimensions, the geometrical picture associated to spin foam gravity can be described intuitivelywith the existent duality between 2-complexes and triangulations of a 4-dimensional manifold. Indeed,a spin foam model in four dimensions can be defined as a state sum whose quantum states are config-urations of a -dimensional simplicial complex ∆ with its -simplices σ v , tetrahedra τ e and triangles δ f coloured by a set of geometrical variables c [7]. ∆ can be associated with its dual 2-complex asfollows: simplicial complex dual 2-complex σ v vertex vτ e edge eδ f face f The state sum is defined for a given simplicial complex, and is a weighted sum over all possiblecolourings, with amplitudes attributed to each face, edge and vertex. Z = (cid:88) colourings c (cid:89) f W f ( c ) (cid:89) e W e ( c ) (cid:89) v W v ( c ) (1) W f , W e , W v are the face, edge and vertex amplitudes of each configuration, respectively. Defininga particular spin foam model corresponds to setting these amplitudes. We now state them for theEPRL/FK model[8, 9] in Euclidean signature. Vertex amplitude W v We follow the construction of W v given in[13]. The colourings for the Euclidean EPRL/FK model areSU(2) quantum numbers j f for each face and SU(2) intertwiners ˆ ι e for each edge, given by ˆ ι e ( k ef , n ef ) = ˆ SU (2) dh e (cid:79) f ∈ e h e | k ef , n ef (cid:105) (2)where | k, n (cid:105) ≡ | k, (cid:126)n, θ n (cid:105) are the Livine-Speziale coherent states[10] in the spin- k representation ofSU(2) . They minimize the uncertainty ∆( J ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:68) (cid:126)J (cid:69) − (cid:68) (cid:126)J (cid:69) (cid:12)(cid:12)(cid:12)(cid:12) in the direction of angular momentum (cid:126)n , and their definition is | k, n (cid:105) ≡ G ( (cid:126)n ) | k, k (cid:105) (cid:126)z (3)where | k, k (cid:105) (cid:126)z is the maximum angular momentum eigenstate of ˆ J z and G ( (cid:126)n ) ∈ SU (2) rotates (cid:126)z into (cid:126)n . There is a phase ambiguity in this definition that cannot be resolved in a canonical way, since the Note that a priori k f (cid:54) = j f . | n (cid:105) ∈ S ⊂ C to S to obtain therotation axis (cid:126)n . It will become apparent in a later section that this ambiguity is not reflected in anycalculations, as all related phase factors cancel out.For the intertwiner definition to make sense there must be an ordering of the faces in a tetrahedron[11].Setting an ordering for the points in a 4-simplex, σ v = ( p , p , p , p , p ) ≡ (1 , , , , , is equivalentto doing the same for the tetrahedra in it, since the tetrahedron t e i can be defined as the one thatdoes not contain the point i . The operation ∂ i ( v , ..., v n ) ≡ ( − i ( v , ..., ˆ v i , ..., v n ) ∂ n +1 ( v , ..., v n ) ≡ ∂ n ( v , ..., v n ) (4)induces an ordering in a ( n − -simplex from that of a n -simplex. Using it, we can establish a co-herent ordering of tetrahedra and triangles starting from what was defined for the 4-simplex. We canalso define the orientation of a simplex - ( v , ..., v n ) is positively oriented if it is an even permutationof (1 , ..., n ) , and negatively oriented otherwise. Since ∂ satisfies ∂ i ∂ j = − ∂ j ∂ i , a consequence of thedefinition is that if f = t e ∩ t e , then the orientations of f induced by t e and t e are opposite. Thishas an intuitive explanation if one considers the normal vectors to each tetrahedron.The construction of the 4-vertex amplitude is based on the spin network basis states of Loop Quan-tum Gravity[13], and it relies on defining a Spin(4) (that is, the Euclidean isometry group SO(4))intertwiner ι e from ˆ ι e , using the decomposition SU(2) × SU(2) = Spin(4). First note that ˆ ι e ∈ Hom
SU(2) C , (cid:79) f ∈ e V k ef , (5)since it is a SU(2)-invariant vector of (cid:78) f ∈ e V k ef , where V k ef is the vector space associated with the k ef -spin (irreducible unitary) representation of SU(2). One can construct an injection φ : Hom
SU(2) C , (cid:79) f ∈ e V k ef → Hom
Spin(4) C , (cid:79) f ∈ e V j − f ,j + f (6)such that φ (ˆ ι e ) = ι e is the Spin(4) intertwiner. This is done by using the Clebsch-Gordan maps C j − f ,j + f k ef : V k ef → V j − f ⊗ V j + f ≈ V j − f ,j + f and constraining the values of j ± f via the Immirzi parameter: j ± f = | ± γ | j f relates them to the original SU(2) quantum number (which is itself constrained bythis relation, since j ± f ∈ N ). ι e ( j f , n ef ) ≡ (cid:88) k ef ˆ Spin(4) dg ( π j − f ⊗ π j + f )( g ) ◦ (cid:79) f ∈ e C j − f ,j + f k ef ◦ ˆ ι e ( k ef , n ef ) , (7)where g = ( g + , g − ) , g ± ∈ SU(2) and π j ± f : Spin(4) → V j ± f , such that ( π j − f ⊗ π j + f )( g ) : V j − f ⊗ V j + f → V j − f ,j + f . The integration over Spin(4) is there, once again, to ensure group invariance of the intertwiner. The vertex amplitude W v is then a closed spin network (more details on graphical calculus in[18] for theLorentzian case) constructed by taking (cid:78) e =1 ι e and “joining the extremities”, for each face, of the twoedges that share it, as illustrated in the figure below (each face corresponds to 2 × (cid:15) -inner product (cid:15) k : V k ⊗ V k → C . (8) The sum over k ef is there because the edge amplitude has the practical effect of selecting these numbers. For ageneral W e , they are summed over (as happens in the FK model for γ > ) (cid:15) / , given in our convention by the matrix (cid:15) ab = (cid:20) i − i (cid:21) . The spin network diagram can now be evaluated using the Kaufmann bracket[17]with parameter A = − . In practice this means that each pair of crossing lines with spins k , k addsa sign ( − k k . These signs result in an overall sign ( − χ in the amplitude.Finally, W v takes the form (now introducing the dependence in v ) W v = ( − χ (cid:88) { k ef } ˆ Spin(4) (cid:89) e ∈ v dg + ve dg − ve ˆ ( S ) (cid:89) ef dn ef (cid:79) f K vf ◦ (cid:32)(cid:79) e ˆ ι e (cid:33) (9)where K vf = (cid:16) (cid:15) j − f ⊗ (cid:15) j + f (cid:17) ◦ (cid:20)(cid:18)(cid:16) π j − f ( g − ve f ) ⊗ π j + f ( g + ve f ) (cid:17) ◦ C j − f ,j + f k ef (cid:19) ⊗ (cid:18)(cid:16) π j − f ( g − ve (cid:48) f ) ⊗ π j + f ( g + ve (cid:48) f ) (cid:17) ◦ C j − f ,j + f k e (cid:48) f (cid:19)(cid:21) . (10) In this expression, e, e (cid:48) are the edges that share the face f . Edge amplitude W e The edge amplitude is taken in modern models to be a selection rule for the values of k ef , and is theonly difference between the EPRL and FK models. Its choice depends on the value of the Immirziparameter. • for γ < , both EPRL and FK select the choice k ef = j f = j + f + j − f : W γ< e = d ˆ ι e (cid:89) f ∈ e δ k ef , j + f + j − f (11) • for γ > , EPRL select k ef = j f = j + f − j − f , W EPRL , γ< e = d ˆ ι e (cid:89) f ∈ e δ k ef , j + f − j − f (12)while FK’s amplitude is a weighed sum over all possible values of k ef , peaking at k ef = j f = j + f − j − f (the expression in brackets is a squared 3j-symbol): W EPRL , γ< e = d ˆ ι e (cid:89) f ∈ e (cid:88) k ef d k ef (cid:20)(cid:18) j + f j − f k ef j + f − j − f j − f − j + f (cid:19)(cid:21) . (13) Face amplitude W f Fixing the face amplitude has been an open problem since the inception of spin foam models, since thestructure of Loop Quantum Gravity does not seem to impose any particular choice for it. It is oftenassociated with the quantized area of a triangle (see for example [1]). While several choices have beenproposed in the literature, the most common being simply the dimension of the SU(2) representation5ssociated to the face, W f = 2 j f + 1 (indeed, in [19] it is argued it is the correct choice), in the fol-lowing we shall keep it as general as possible depending only on the face quantum numbers, W f ≡ µ ( j f ) .For the rest of this study we will use the EPRL prescription, so that the partition function is (consid-ering a manifold with boundary and fixed boundary data satisfying Regge-like conditions[14]) Z ( j f B , g ve B , n ef B ) = ( − χ (cid:88) j f (cid:89) f µ ( j f ) ˆ (cid:89) ve dg + ve dg − ve ˆ (cid:89) ef dn ef ˆ (cid:89) e dh e (cid:79) f K f ◦ (cid:32)(cid:79) e ˆ ι e ( j + f ± j − f , n ef ) (cid:33) . (14) It can now be established that the de facto variables of the model are the face SU(2) quantum numbers,Spin(4) elements for each half-edge ( ve ) and the coherent state vectors (cid:12)(cid:12)(cid:12) j + f ± j − f , n ef (cid:69) for each edgeconnected to the vertex containing f , for each f . In order to study the asymptotics of the model, we use the partition function written in a path integralform, Z = (cid:88) c e S [ c ] . (15)We will review the derivation of this form for the EPRL/FK model[22], but it is worth noting thatBonzom[20] has extended the process for any SFM under some general assumptions.Introducing in (14) the expressions for ˆ ι e and K f , (cid:15) -inner products of coherent states appear. Theycan be written in terms of the standard Hilbert inner product by introducing the antilinear structuremap J : V k → V k defined by (cid:15) k ( v k , v (cid:48) k ) = (cid:104)J v k | v (cid:48) k (cid:105) . (16) J has several properties: it commutes with SU(2) group elements, satisfies J = ( − k and, since J ( (cid:126)n · (cid:126)J ) = − ( (cid:126)n · (cid:126)J ) J , it takes a coherent state for the vector (cid:126)n to one for − (cid:126)n . We should also noticethat the orientation requirements described above (4) are the basis for a supplementary requirementon the n ef , which we will call here the weak gluing condition, | n ef (cid:105) v = J | n ef (cid:105) v (cid:48) (17)for a tetrahedron that is shared by two vertices. Using this notation the partition function becomes Z = ( − χ (cid:48) (cid:88) j f (cid:89) f µ ( j f ) ˆ (cid:89) ve dg + ve dg − ve ˆ (cid:89) ef dn ef (cid:89) e dh e (cid:89) vf P vf (18)where P vf = (cid:104) k ef , J n ef | π k ef ( h − e ) C k ef j − f j + f π j − f ( g − ev g − ve (cid:48) ) π j + f ( g + ev g + ve (cid:48) ) C j − f j + f k e (cid:48) f π k e (cid:48) f ( h e (cid:48) ) | k e (cid:48) f , n e (cid:48) f (cid:105) (19)can be interpreted as a propagator between two coherent states in the two edges sharing the face f . Now the Clebsch-Gordan maps are SU(2)-invariant, which means that the h e can be commutedwith the C ’s into the Spin(4) terms, which take the form π j ± f ( h − e g ± ev g ± ve (cid:48) h e (cid:48) ) . The h e can then beeliminated by a change of variables ˜ g ± ve = g ± ve h e , and the corresponding integrations over them add upto a prefactor Vol(SU(2)) .The action of the Clebsch-Gordan maps is simple in the EPRL prescription. In particular for γ < (thecase γ > is slightly more complicated in analysis but similar in result), we have k ef = k e (cid:48) f = j − f + j + f :the C-G maps project to the highest spin subspace of V j − f ⊗ V j + f . Remembering the property of coherentstates that | k, n (cid:105) ∼ ⊗ k (cid:12)(cid:12)(cid:12)(cid:12) , n (cid:29) ≡ ⊗ k | n (cid:105) , (20)6hich is a fully symmetric state and that the highest spin subspace is precisely the one obtained byfull symmetrization, we conclude that C j − f j + f k ef | k ef , n ef (cid:105) = | k ef , n ef (cid:105) = ⊗ k | n ef (cid:105) . (21)Therefore the propagator simplifies to P vf = (cid:104)J n ef | g − ev g − ve (cid:48) | n e (cid:48) f (cid:105) j − f (cid:104)J n ef | g + ev g + ve (cid:48) | n e (cid:48) f (cid:105) j + f , (22)and with some simple algebra we can now write Z = ( − χ (cid:48) (cid:88) j f µ ( j f ) ˆ (cid:89) ve dg + ve dg − ve ˆ (cid:89) ef dn ef e S , (23)where the “action” is S = (cid:88) f (cid:88) v ∈ f j ± f log (cid:104)J n ef | g ± ev g ± ve (cid:48) | n e (cid:48) f (cid:105)≡ (cid:88) f S f (24)Since, by the discussion above, the boundary data are considered to be fixed for the “path-integral”approach, while only the interior data are dynamical, it is important to separate the action into itsboundary and interior parts, S = S I + S B = (cid:80) f I S f + (cid:80) f B S f . In section 3 we will see how the actionhere written can be related to that of Regge calculus in the large-j regime, the base point of theasymptotics discussion. The semiclassical limit in quantum gravity is commonly taken in the literature as the limit of largeareas, since the discrete area spectrum of LQG is asymptotically indistiguishable from the continuousclassical spectrum when the corresponding quantum number j f is large (i.e. ∆ jj → j →∞ ) . Mathemati-cally this is imposed by making the transformation j f → λj f , ∀ f in the regime λ → ∞ . For the EPRLmodel this means that its action is proportional to λ , so that the partition function is (roughly) of theform I λ = ˆ d n z g ( z ) e λF ( z ) , λ → ∞ . (25)This suggests the use of the stationary phase method to derive an approximation of I λ in the large λ limit. The main principle of the stationary phase method is that due to the large argument of the exponentialin the integrand, the contributions to the integral near certain critical points are much larger thaneverywhere else, and the integral can be estimated by considering the function only near those points.Critical points are given by the following conditions: • (cid:60) ( F ( z )) is at its absolute maximum, so that (cid:12)(cid:12) e λF ( z ) (cid:12)(cid:12) is maximized; • the oscillation is minimized, i.e. the variation of arg (cid:0) e λF ( z ) (cid:1) in a neighbourhood of the pointin question is the slowest. At a first order level this is obtained by extremizing the action, i.e. ∂ i f ( z ) = 0 , ∀ i. , so that the variation of (cid:61) ( F ( z )) near a critical point z is at least second orderin z − z , rather than first.While not a rigorous proof (see [25, 26] for more detailed mathematical treatment), the essentials ofthe method can be understood with the following argument. That we need to maximize the real partof F ( z ) should be obvious in the large λ regime, so assume in the following that F ( z ) = if ( z ) , f ∈ R ,7nd for simplicity g ( z ) ≡ (the only condition on g is that it allows for convergence of the integral,which won’t be a problem in the cases we are interested in considering). Take a Taylor expansion of f around an arbitrary point z : f ( z ) ≈ f ( z ) + ∂f∂z i (cid:12)(cid:12)(cid:12)(cid:12) z ( z − z ) i + 12 ∂ f∂z i ∂z j (cid:12)(cid:12)(cid:12)(cid:12) z ( z − z ) i ( z − z ) j + 13! ∂ f∂z i ∂z j ∂z k (cid:12)(cid:12)(cid:12)(cid:12) z ( z − z ) i ( z − z ) j ( z − z ) k + O ( z ) ≡ f ( z ) + D i ( z )( z − z ) i + H ij ( z )( z − z ) i ( z − z ) j + T ijk ( z )( z − z ) i ( z − z ) j ( z − z ) k + O ( z ) (26)The stationary phase method assumes that when z are critical points, the integral (25) is estimatedby the formula I λ ≈ ˆ dz ˆ U ( z ) d n z e iλf ( z ) (27)where U ( z ) is a neighbourhood of z . Now suppose we only took the first order term in the Taylorexpansion of f . Then I λ ≈ ˆ dz ˆ U ( z ) d n z exp[ iλ ( f ( z ) + D i ( z )( z − z ) i )]= ˆ dz exp[ iλ ( f ( z ) + D i ( z ) z i )] ˆ U ( z ) d n z e iλD i ( z ) z i (28)If we further assume that the contribution away from a critical point is (after taking the Taylorapproximation) so small that the integral above can be extended to the whole z -space, the integralover z is directly related to the delta “function”: ˆ d n z e iλD i ( z ) z i = 12 πλ δ ( D i ( z )) (29)in this extremely crude approximation, divergences show up when D i ( z ) = 0 . While this points tothe necessity of refining the method, which happens by taking the Taylor expansion to second order(enough in most applications), it also serves as a very simple justification that the contributions ofpoints z satisfying D i ( z ) = 0 are dominant, justifying the definition of critical point above. Takingthe second order expansion of f , then, we get the more accurate formula I λ = ˆ d n z exp[ iλ ( f ( z ) + D i ( z ) z i )] ˆ d n z exp[ iλ ( D i ( z ) z i ) + H ij ( z )( z − z ) i ( z − z ) j ] (cid:89) i δ ( D i ( z ))= ˆ Σ C d n z e iλf ( z ) ˆ d n z e iλH ij ( z )( z − z ) i ( z − z ) j (30)where Σ C , the critical surface , is the hypersurface of z -space formed by all critical points. Usinganalytic continuation of the standard formula ´ d n x e − A αβ x α x β = (cid:113) (2 π ) n det A to complex A , we can solvethe integral over z : ˆ d n z e iλH ij ( z )( z − z ) i ( z − z ) j = (cid:18) πiλ (cid:19) n/ (cid:112) det H r ( z ) (31)where H r is the restriction of H to the orthogonal complement of its null space, as the conditionsimposed on the z constrain some degrees of freedom of H . In the context of state sum models the critical point equations can be interpreted as classical equationsof motion for the interior variables of the simplicial complex (boundary data is fixed). Considering the The critical surface is in fact a submanifold of z -space iff det H r ( z ) (cid:54) = 0 ∀ z ∈ Σ C . < γ < , the equations of motion are (cid:60) ( S I ) = R max (32) δ g ve S I = 0 (33) δ n ef S I = 0 (34) δ j fI S I = 0 (35)Or are they? (35) in particular has rarely been considered in existing literature. The main reason issimple - unlike the other spin foam variables in play, the j f ∈ N are discrete, and it is unclear whetherthere is an extension of the stationary phase method applying to sums over general discrete variables.The only work in this direction that we are aware of is Lachaud’s[27] results for sums over finite fields,which is in general not the case of the j f sums.The other equations of motion can be written explicitly, and are as follows: • (32) gives the gluing condition: R ( g ± ve ) (cid:126)n ef = − R ( g ± ve (cid:48) ) (cid:126)n e (cid:48) f , where R ( g ) is the rotation matrixassociated to g by the 2-1 surjective homomorphism SU(2) → SO(3); • (33) gives the closure condition: (cid:80) f ∈ e (cid:80) ± j ± f (cid:15) ef ( v ) R ( g ± ve ) (cid:126)n ef = 0 , where (cid:15) ef ( v ) is defined tobe 1 if the orientation of f agrees with the one induced from e according to (4), and -1 otherwise. (cid:15) ef ( v ) are also subject to the orientation conditions, (cid:15) ef ( v (cid:48) ) = − (cid:15) ef ( v ) = − (cid:15) e (cid:48) f ( v (cid:48) ) . • if the previous two conditions are met, (34) is automatically satisfied.The main existing result for EPRL asymptotics is the reconstruction theorem , proven originally byBarrett et al[14] for the case of one single 4-simplex, and more recently extended by Han and Zhang[11,12] for a general simplicial complex with boundary. Essentially, the reconstruction theorem states thatgiven a set of boundary data satisfying a number of conditions guaranteeing their geometricity, called“Regge-like”, and non-degenerate interior spin foam variables j f , g ve , n ef satisfying the equations ofmotion, then it is possible to construct a classical, non-degenerate geometry which matches them and isunique up to global symmetries. The proof is constructive and involves defining bivectors X ef ( j f , n ef ) which are interpreted as area bivectors of the discrete geometry, while the g ve are identified with thespin connection (in both cases up to sign factors). Additionally, the Regge deficit angles Θ f can beidentified within the bivector formalism, such that the semiclassical action is found to be S = (cid:88) f i(cid:15) [ j f N f π − γj f sign ( V )Θ f ] (36)where N f ∈ N and V is the 4-volume of the connected component of the discrete manifold thatcontains f , its sign depending on the orientation induced from spin foam variables. Since the firstterm is a half-integer times iπ and only gives a ± sign when exponentiated, it is mostly ignored, so this“classical” form for S bears an uncanny resemblance to the discrete Einstein-Hilbert action in Reggecalculus[21]: S Regge = (cid:88) f A f Θ f (37)where A f is the area of the triangle f , which coincides with γj f in the reconstructed geometry. Given (36), it is readily seen how the j-equation (35) was the original motivation to the “flatnessproblem” mentioned by Freidel and Conrady[23] and later Bonzom[20]. The result shows that theEPRL action (24) can be written as S = (cid:88) f I j f ˜Θ f ( g ve , n ef ) Han and Zhang developed their results for both the Euclidean and Lorentzian signature versions of the EPRL model.We will focus on Euclidean signature for this paper. ˜Θ f is a quantity that is proportional, in the semiclassical limit, to the Regge-like deficit angle, ˜Θ f → λ →∞ ± γ Θ f . If we were to ignore the discreteness of the j f and carry out the derivation as if it werecontinuous, the j-equation would be simply ˜Θ f = 0 , ∀ f , therefore showing that the classical geometriesreproduced by the model are restricted to be flat - a result that puts the model in question, since GRin four dimensions admits curved spacetime solutions. However, the applicability of this equation isquestionable, not only because of the issues with the discreteness of j f , but due to an ambiguity inthe way the semiclassical limit is taken - taking the limit of large j f , while at the same time summingover them. In the following we consider a slight reformulation.Assume that in the semiclassical limit the boundary face quantum numbers are given by j f B = λj ’ f B , ∀ f B where j (cid:48) f B ∈ N and λ → ∞ . Then, define new interior variables x f I = j fI λ ∈ N λ (and x ± f I accordingly). The partition function then takes the form Z ( λj ’ f B , g ve B , n ef B ) = (cid:88) x fI ˆ (cid:89) ve dg ve ˆ (cid:89) ef dn ef e iλ ( S I + S B ) (38)with S I = − i (cid:88) f I (cid:88) v ∈ f (cid:88) ± x ± f log (cid:104)J n ef | g ± ev g ± ve (cid:48) | n e (cid:48) f (cid:105) ≡ (cid:88) f I x f I ˜Θ f I ( g ve , n ef ) S B = − i (cid:88) f B (cid:88) v ∈ f (cid:88) ± j (cid:48) ± f log (cid:104)J n ef | g ± ev g ± ve (cid:48) | n e (cid:48) f (cid:105) (39)(we factor out i to explicit the fact that the argument of the exponential becomes pure imaginary whenthe gluing condition is satisfied). With this prescription, we don’t have to assume anything aboutthe x f I ’s, eliminating ambiguities, and the dependence of the partition function on λ is completelyexplicit. Additionally, we can propose a workaround to the discreteness issue, consisting of a continuumapproximation for the x f . Since the ∆ x f I = λ tend to zero for large λ , it makes sense to considerreplacing the sum over x f by an integral: x f I (cid:88) x fI f ( x f I )∆ x f I ≈ x f I ˆ ∞ f ( x f I ) dx f I (40)and therefore the “semiclassical” partition function would be Z SC ( λj ’ f B ) = (2 λ ) f I ˆ (cid:89) f I dx f ˆ (cid:89) ( ve ) I dg ve ˆ (cid:89) ( ef ) I dn ef e iλ ( S I + S B ) (41)Of course, one must be careful with the errors incurring from this approximation, which is essentiallythe rectangle method of numerical integration “done backwards”. It can be shown that the differencebetween the sum and the integral is of order λ , making the continuum approximation unreliable tocompute any quantum corrections to the zero-order, λ = ∞ results. It could still be argued that thatit can be used safely in the zero-order situation, but we will try to progress as much as possible withoutusing it. The problem is to estimate the integral (cid:88) j f µ ( j f ) ˆ dY e (cid:80) f iλx f ˜Θ f ( Y ) (42)where we used Y as short for the set of g ve , n ef integration variables. Using the stationary phasemethod for the integral over Y , we obtain ˆ dY e (cid:80) f iλx f ˜Θ f ( Y ) ≈ ˆ Σ C ( x f ) dY C (cid:89) f e iλx f ˜Θ f ( Y C ) (cid:18) − πiλ (cid:19) Y C / (cid:114) det (cid:104)(cid:80) f x f H fr ( Y C ) (cid:105) (43) Consider the difference ´ x +∆ xx f ( x ) dx − f ( x )∆ x. For ∆ x = 1 / λ the difference is of order /λ . In practicalsemiclassical calculations the integral will not extend to infinity because triangle inequalities limit the maximum valueof j . The cutoff will be of order λ , so the error in approximating the sum by an integral is of order 1/ λ . Y C are the critical points that solve the equations of motion, and Σ C the submanifold of Y -spacethey form. Ideally, if we use the continuum approximation, we could think of reversing order of integra-tion and doing the x integral first, but this is not possible for the general case because not only there isan x dependence on the determinant factor, which is a priori arbitrary, but due to the closure conditionthe critical surface Σ C also depends on x . This makes the integral seemingly intractable without fur-ther assumptions. There are some heuristic considerations that can be made on this form of Z that leadto something suggestive of the flatness problem, but the apparent “dead end” we reach here leads us toconsider a concrete example in which a full calculation is possible, the ∆ manifold studied in section 4.More recently, a different approach to asymptotics devised by Hellmann and Kaminski[30] deriveda result similar to the flatness problem. Their main idea is to introduce the concept of wavefront setsfor a distribution, which are designed with asymptotics in mind and represent the subspace of phasespace where the distribution is peaked in the limit of large λ . The wavefront sets of partition functionsof various models like BC and EPRL can be written using the holonomy (or operator) representationof spin foams[31] and their main result regarding asymptotics is an accidental curvature constraintacting on the deficit angles Θ f , γ Θ f = 0 mod 2 π, (44)which is not strictly flatness (the dependence on the Immirzi parameter is somewhat puzzling) but stilla worrying result in terms of the accuracy of the theory’s asymptotics in respect to Einstein theory.It is noteworthy that for the BC model, which can essentially be obtained form EPRL by taking thelimit γ → ∞ , the wavefront approach leads to an exact flatness constraint. ∆ In the following we will attempt to compute the asymptotic EPRL partition function for the case ofthe three 4-simplex manifold ∆ , which is represented in the figure below together with its 2-complexdual. This particular manifold is chosen as a simple example of a semiclassical calculation, sinceit has only one interior face f I . Therefore, assuming the boundary data are fixed, Regge-like, andnon-degenerate, the classical Euclidean geometry of ∆ is completely determined by the area j = λx and the deficit angle Θ of f I , two quantities that are easily seen to be completely determined by theboundary geometry. We will now define the EPRL model in this triangulation.Boundary faces are notated f vij , i, j ∈ { , ..., } where f vij is the triangle that does not contain the points i, j of the 4-simplex v it belongs to, and has the area variable x vij . Edges are labelled e vk , k ∈ { , ..., } and e vk is the tetrahedron that does not contain the point k of v . We will call the n ef as | n e,f (cid:105) v , v ∈{ A, B, C } for clarity, while the interior g ve are labelled g A , g A , g B , g B , g C , g C according to thefigure. The partition function is (proportional to, with extra pre-factors not being of importance inthe analysis) Z = (cid:88) x = j/λ µ ( λx ) x Y C ˆ Σ C ( x ) dY c e iλx ˜Θ( Y C ) (cid:112) det H r ( Y C ) (45)noting that the dimension Y of Y -space is that of 12 copies of S associated to the interior g ve andother 6 copies associated to the interior n ef . The dimension Y C of the critical surface is the numberof degrees of freedom unconstrained by the equations of motion.11 .1 Solving the equations of motion We will now study the equations of motion for ∆ . For starters, n vef and n v (cid:48) ef are related by the weakgluing equations (17): | n , (cid:105) A = J | n , (cid:105) C | n , (cid:105) C = J | n , (cid:105) B | n , (cid:105) B = J | n , (cid:105) A (46)We can choose a simpler notation for the interior n ef so that (46) reads | n AC (cid:105) = J | n CA (cid:105)| n CB (cid:105) = J | n BC (cid:105)| n BA (cid:105) = J | n AB (cid:105) (47)Stationary phase computation on the g, n integrals results in 6 interior gluing conditions, R ( g ± C ) (cid:46) (cid:126)n CA = − R ( g ± C ) (cid:46) (cid:126)n CB R ( g ± B ) (cid:46) (cid:126)n BC = − R ( g ± B ) (cid:46) (cid:126)n BA R ( g ± A ) (cid:46) (cid:126)n AB = − R ( g ± A ) (cid:46) (cid:126)n AC (48)36 interior-boundary gluing conditions, R ( g ± A ) (cid:46) (cid:126)n A ,i = − R ( g ± Ai ) (cid:46) (cid:126)n Ai,i R ( g ± A ) (cid:46) (cid:126)n A ,i = − R ( g ± Ai ) (cid:46) (cid:126)n Ai,i R ( g ± B ) (cid:46) (cid:126)n B ,i = − R ( g ± Bi ) (cid:46) (cid:126)n Bi,i R ( g ± B ) (cid:46) (cid:126)n B ,i = − R ( g ± Bi ) (cid:46) (cid:126)n Bi,i R ( g ± C ) (cid:46) (cid:126)n C ,i = − R ( g ± Ci ) (cid:46) (cid:126)n Ci,i R ( g ± C ) (cid:46) (cid:126)n C ,i = − R ( g ± Ci ) (cid:46) (cid:126)n Ci,i , i ∈ { , , } (49)and 6 closure conditions, x (cid:2) (1 + γ ) R ( g + C ) + (1 − γ ) R ( g − C ) (cid:3) (cid:46) (cid:126)n CA + b.t. ( C +) = 0 x (cid:2) (1 + γ ) R ( g + A ) + (1 − γ ) R ( g − A ) (cid:3) (cid:46) (cid:126)n AC + b.t. ( A +) = 0 x (cid:2) (1 + γ ) R ( g + B ) + (1 − γ ) R ( g − B ) (cid:3) (cid:46) (cid:126)n BC + b.t. ( B +) = 0 (50) − x (cid:2) (1 + γ ) R ( g + C ) + (1 − γ ) R ( g − C ) (cid:3) (cid:46) (cid:126)n CB + b.t. ( C − ) = 0 − x (cid:2) (1 + γ ) R ( g + A ) + (1 − γ ) R ( g − A ) (cid:3) (cid:46) (cid:126)n AB + b.t. ( A − ) = 0 − x (cid:2) (1 + γ ) R ( g + B ) + (1 − γ ) R ( g − B ) (cid:3) (cid:46) (cid:126)n BA + b.t. ( B − ) = 0 (51)where the b.t. represents terms depending exclusively on boundary variables. Indeed, the closure condi-tions contain sums over edges in each vertex, so each of them contains exactly one term correspondingto the interior edge, and the rest of the sum depends on the boundary edge variables. The boundaryterms are labelled by the edges they pertain to.First off, we will note that Eqs. (49) determine all the interior g ve uniquely in terms of boundarydata. Indeed, consider the first equation referring to g ± A . The only term in this equation that is nota boundary variable is R ( g ± A ) , and the indices 1,2,3 can be grouped in a matrix form equation: R ( g ± A ) (cid:46) (cid:2) (cid:126)n A , (cid:126)n A , (cid:126)n A , (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ≡ N A = − (cid:2) R ( g ± A ) (cid:46) (cid:126)n A , R ( g ± A ) (cid:46) (cid:126)n A , R ( g ± A ) (cid:46) (cid:126)n A , (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ≡ V ± A (52)Note that the non-degeneracy assumption on the boundary data implies that, since all tetrahedraare non-degenerate, any set of three out of the four (cid:126)n ef that define a tetrahedron must be linearly12ndependent. This means that N A is invertible in the equation above, which can then immediatelybe solved: R ( g ± A ) = − N − A V ± A (53)and similar solutions are derived for the remaining g ve . This result means that the purely interiorgluing conditions (48), if consistent (consistency should be guaranteed by the boundary data beingRegge-like), are redundant, however we will analyse them together with the closure conditions in thefollowing, as they have valuable physical content for the problem.It is possible to eliminate three of the closure equations by using the gluing ones: indeed, substi-tuting (48) on (51), we obtain (50) while being forced to impose that b.t. ( A +) = − b.t. ( A − ) (and similarfor the B ± and C ± boundary terms). Conditions on boundary variables are not problematic if theycan be related to the equations for Regge-like data. To elaborate on this and to properly solve theclosure conditions we need to specify the boundary data. The equations (50) in their full form are (cid:2) (1 + γ ) R ( g + C ) + (1 − γ ) R ( g − C ) (cid:3) (cid:46) (cid:0) x(cid:126)n CA + x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , (cid:1) = 0 (cid:2) (1 + γ ) R ( g + B ) + (1 − γ ) R ( g − B ) (cid:3) (cid:46) (cid:0) x(cid:126)n BC + x B (cid:126)n B , + x B (cid:126)n B , + x B (cid:126)n B , (cid:1) = 0 (cid:2) (1 + γ ) R ( g + A ) + (1 − γ ) R ( g − A ) (cid:3) (cid:46) (cid:0) x(cid:126)n AB + x A (cid:126)n A , + x A (cid:126)n A , + x A (cid:126)n A , (cid:1) = 0 (54)The solution of these equations is simple to obtain, noting that they are of the form M (cid:46) (cid:126)v = 0 , acondition satisfied if and only if (cid:126)v = 0 or M has a vanishing determinant. The second possibilitycan be ruled out, though, by proving that M = (1 + γ ) G + (1 − γ ) H has nonzero determinant for all G, H ∈ SO (3) and < γ < . Proof starts with noting that (det M ) = det( M t M ) . It is possible toget a general expression for det( M t M ) : M t M = (cid:2) (1 + γ ) G t + (1 − γ ) H t (cid:3) [(1 + γ ) G + (1 − γ ) H ]= 2(1 + γ ) + (1 − γ )( G t H + H t G )= 2(1 + γ ) + (1 − γ )( A + A t ) (55)defining A ≡ G t H ∈ SO (3) . We can compute the determinant in a basis where A + A t is diagonal - notethat the identity matrix is basis-invariant and A + A t is a symmetric real matrix, hence diagonalizable.To do so we need its eigenvalues, which can be found using one of the several possible parameterizationsof SO (3) . Here we use a parameterization by Janaki and Rangarajan[28]: A = cos θ cos θ sin θ cos θ − cos θ sin θ sin θ sin θ sin θ + cos θ sin θ cos θ − sin θ cos θ cos θ cos θ + sin θ sin θ sin θ cos θ sin θ − sin θ sin θ cos θ − sin θ − cos θ sin θ cos θ cos θ (56)where θ i ∈ [0 , π ] are angles for simple rotations. A + A t can then be diagonalized, being a symmetricreal matrix. There is a basis in which A + A t = a b c , where a = 2 b = c = sin θ sin θ sin θ + cos θ (cos θ + cos θ ) + cos θ cos θ − (57)are its eigenvalues. In this basis, M t M = 2(1 + γ ) + (1 − γ ) b b = γ ) + b (1 − γ ) 2(1 + γ ) + b (1 − γ ) (58)so that (det M ) = 4 (cid:2) γ ) + b (1 − γ ) (cid:3) . Therefore, det M = 0 ⇔ b = − γ − γ (59)13t is straightforward to verify that − ≤ b ≤ for all values of θ i , which makes the above conditionimpossible in the < γ < range we are working on. Hence, M is always invertible in the conditionsof our study, and the closure conditions are simplified: x(cid:126)n CA + x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , = 0 x(cid:126)n BC + x B (cid:126)n B , + x B (cid:126)n B , + x B (cid:126)n B , = 0 x(cid:126)n AB + x A (cid:126)n A , + x A (cid:126)n A , + x A (cid:126)n A , = 0 (60)Notice that these are precisely the necessary and sufficient conditions for the 3 tetrahedra of ∆ thatcontain the interior face f to be geometrical in the Euclidean sense, which shows that the large areaslimit for this manifold imposes a discrete classical geometry on it. Also, the partition function isconsiderably simplified, since x and all the interior (cid:126)n ef are fixed: x = (cid:12)(cid:12) x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , (cid:12)(cid:12) (cid:126)n BA = − x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , (cid:12)(cid:12) x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , (cid:12)(cid:12) (61)and similarly for (cid:126)n AC and (cid:126)n CB . In particular, note that the j-equation (35) seems not to apply in thisexample: x is fixed in terms of boundary data by the gluing/closure conditions, without need of anextra equation for it. Note that the other two closure conditions also give expressions for x , leadingto additional constraints on boundary data: (cid:12)(cid:12) x A (cid:126)n A , + x A (cid:126)n A , + x A (cid:126)n A , (cid:12)(cid:12) = (cid:12)(cid:12) x B (cid:126)n B , + x B (cid:126)n B , + x B (cid:126)n B , (cid:12)(cid:12) = (cid:12)(cid:12) x C (cid:126)n C , + x C (cid:126)n C , + x C (cid:126)n C , (cid:12)(cid:12) . (62)Additionally, the relations between (50) and (51) make it so that (cid:126)n CA = − (cid:126)n CB (cid:126)n BC = − (cid:126)n BA (cid:126)n AC = − (cid:126)n AB (63)and together with weak gluing, we obtain that (cid:126)n AB = (cid:126)n BC = (cid:126)n CA ≡ (cid:126)n . The partition function is nowreduced to Z = µ ( λx ) x ˆ Σ C dY c e iλx ˜Θ( Y C ) (cid:112) det H r ( Y C ) (64)where, with x and (cid:126)n ef fixed, the only integrations remaining are over group elements and the phases α ef , and the face amplitude µ becomes no more than a pre-factor. The critical surface Σ C in thisnew expression is S × U (1) , corresponding to the one free vector (cid:126)n ∈ S and the three free phases α AB , α BC , α CA necessary to define the respective coherent states. We will attempt to find a compact expression for the deficit angle ˜Θ using the new data. The “quantumdeficit angle” for ∆ is ˜Θ = ± i (cid:88) ± (1 ± γ ) (cid:20) log (cid:104)J n CA | (cid:16) g ± C (cid:17) † g ± C | n CB (cid:105) + log (cid:104)J n BC | (cid:16) g ± B (cid:17) † g ± B | n BA (cid:105) + log (cid:104)J n AB | (cid:16) g ± A (cid:17) † g ± A | n AC (cid:105) (cid:21) = ± i (cid:88) ± (1 ± γ ) (cid:20) log (cid:104) n AC | (cid:16) g ± C (cid:17) † g ± C | n CB (cid:105) + log (cid:104) n CB | (cid:16) g ± B (cid:17) † g ± B | n BA (cid:105) + log (cid:104) n BA | (cid:16) g ± A (cid:17) † g ± A | n AC (cid:105) (cid:21) (65) We will focus on the first of the three matrix elements in the above expression. The results for theother two can be easily extrapolated by symmetry. In order to perform the necessary computations,we will use the following parameterizations of SU (2) and the Hilbert space H / of spin states: • For the SU(2) variables, we use the decomposition ∀ g ∈ SU(2) , g = z α Σ α , (cid:0) z (cid:1) + (cid:0) z (cid:1) + (cid:0) z (cid:1) + (cid:0) z (cid:1) = 1 (66)14here Σ = and Σ i = iσ i for i = 1 , , ( σ i are the Pauli matrices). SU(2) is thereforediffeomorphic to S , and considering the change of variables z = cos γ cos β z = cos γ sin β z = sin γ cos β z = sin γ sin β , (67)with Jacobian sin(2 γ )2 , where < β i < π and < γ < π , it follows that a general SU(2) matrixcan be written as g = (cid:34) cos γe iβ i sin γe − iβ i sin γe iβ cos γe − iβ (cid:35) . (68) • For the H / variables, naively, one could parametrize them as follows: ∀ | n (cid:105) ∈ H / , | n (cid:105) = (cid:20) w + iw w + iw (cid:21) , (cid:0) w (cid:1) + (cid:0) w (cid:1) + (cid:0) w (cid:1) + (cid:0) w (cid:1) = 1 (69)obtaining ´ H / dn = ´ S dw . However, it is advantageous to consider a change of variables thatreflects the construction of a coherent state. Recall that | n (cid:105) = e iα G ( (cid:126)n ) | + (cid:105) (70)where (cid:126)n ∈ S , α is an undetermined phase and | + (cid:105) = (1 , is the eigenstate of J z with eigenvalue + . The SU(2) element G ( (cid:126)n ) is the rotation that takes (cid:126)z to (cid:126)n and is readily calculated. Considerthe parameterization of S in spherical coordinates (cid:126)n = (sin θ cos φ, sin θ sin φ, cos θ ) (71)To go from (cid:126)z to (cid:126)n we perform a rotation of angle θ around the axis (cid:126)n ⊥ = ( − sin φ, cos φ, . Fromthis we get G ( (cid:126)n ) = exp (cid:18) iθ (cid:126)σ · (cid:126)n ⊥ (cid:19) = exp (cid:18) iθ φσ y − sin φσ x ) (cid:19) = (cid:20) cos θ e − iφ sin θ − e iφ sin θ cos θ (cid:21) . (72)and therefore | n (cid:105) = e iα (cid:20) cos θ − e iφ sin θ (cid:21) . (73)The Jacobian of the change of coordinates from (cid:126)w to ( θ, φ, α ) is sin( θ )2 .Since the matrix element (cid:104) n AC | (cid:0) g ± C (cid:1) † g ± C | n CB (cid:105) is a scalar, it does not depend on the choice of basisin H / . Since the vector part for each of the coherent states present is the same, we will choose abasis in which (cid:126)n AB = (cid:126)n = (0 , , to carry out computations . This translates to | n i (cid:105) = e iα i (cid:20) (cid:21) , (74)for i ∈ { BA, CB, AC } . Notice that due to each of the coherent states appearing exactly once as a braand a ket in (79), the contribution of the phases α i will cancel out and we can just consider | n (cid:105) = (cid:20) (cid:21) from now on. With the coherent states taken care of, we can move on to g ± C and g ± C . We need to use There appears to be an ambiguity with this choice, coming from the parameterization of S in spherical coordinates- (cid:126)n = (0 , , is obtained when θ = 0 , which makes φ undefined. But it is evident from (72) that G (0 , ,
1) = . R ( g ) for g ∈ SU (2) . Westra gives us a parameterization for g = (cid:20) x y − ¯ y ¯ x (cid:21) , | x | + | y | = 1 : R ( g ) = (cid:60) ( x − y ) (cid:61) ( x + y ) − (cid:60) ( xy ) −(cid:61) ( x − y ) (cid:60) ( x + y ) 2 (cid:61) ( xy )2 (cid:60) ( x ¯ y ) 2 (cid:61) ( x ¯ y ) | x | − | y | (75)In our set of coordinates for SU(2), x = cos γe iβ and y = i sin γe − iβ , hence we can write R ( g ) = cos γ cos(2 β ) + sin γ cos(2 β ) cos γ sin(2 β ) + sin γ sin(2 β ) sin(2 γ ) sin( β − β ) − cos γ sin(2 β ) + sin γ sin(2 β ) cos γ cos(2 β ) − sin γ cos(2 β ) sin(2 γ ) cos( β − β )sin(2 γ ) sin( β + β ) − sin(2 γ ) cos( β + β ) cos(2 γ ) (76)While daunting at first, this expression becomes more tractable within the context of the gluingcondition and the basis choice we made for (cid:126)n AB . The gluing condition is reduced to sin(2 γ A ) sin( β A − β A )sin(2 γ A ) cos( β A − β A )cos(2 γ A ) = sin(2 γ B ) sin( β B − β B )sin(2 γ B ) cos( β B − β B )cos(2 γ B ) (77)where the variables labelled A pertain to g A and the ones labelled B pertain to g B , and we omitthe ± index for simplicity. It is clear that the gluing condition does not fix g A completely given g B ,since they only depend on the differences β A,B − β A,B ≡ δ A,B . Analysing the equations, • the third equation implies γ A = γ B = γ , since γ A,B ∈ [0 , π ] and the cosine function is injectivein this domain; • given that γ A = γ B , the first and second equations read sin δ A = sin δ B and cos δ A = cos δ B ,which for δ A,B ∈ [0 , π ] is enough to infer δ A = δ B .Hence, we have that, in our chosen basis for H / , if g ± C is given by the coordinates ( γ ± , β ± , β ± ) , then g ± C is given by ( γ ± , β ± + (cid:15) ± , β ± + (cid:15) ± ) where (cid:15) ± ∈ [0 , π [ . We can now compute (cid:104) n | (cid:0) g ± C (cid:1) † g ± C | n (cid:105) : (cid:104) n | (cid:0) g ± C (cid:1) † g ± C | n (cid:105) = (cid:2) (cid:3) (cid:34) cos γe − iβ − i sin γe − iβ − i sin γe iβ cos γe iβ (cid:35) (cid:34) cos γe i ( β + (cid:15) ) i sin γe − i ( β + (cid:15) ) i sin γe i ( β + (cid:15) ) cos γe − i ( β + (cid:15) ) (cid:35) (cid:20) (cid:21) . = (cid:104) cos γe − iβ sin γe − iβ (cid:105) (cid:34) cos γe i ( β + (cid:15) ) sin γe i ( β + (cid:15) ) (cid:35) = e i(cid:15) (78) Taking logarithms, we get simply i(cid:15) , and substituting (with proper labels) on the expression for ˜Θ andrepeating the process for the other two inner products in ˜Θ (we shall identify the variables pertainingto each of these terms with an index i ∈ { , , } ), we obtain ˜Θ = ± (cid:88) ± (1 ± γ ) (cid:88) i =1 (cid:15) ± i . (79)Remember that all g ve have been determined earlier using the interior-boundary conditions. Therefore,the (cid:15) ± i can be expressed in terms of the boundary data through some simple algebra. We give an ex-ample. R ( g ± A ) and R ( g ± A ) are known. Let’s call them A, B for simplicity. Using the parameterization(76), we want to find either β or β for each matrix, and take their difference to obtain (cid:15) . Step bystep: • γ is obtained through cos(2 γ ) = A . Since γ ∈ [0 , π ] , the cosine function is injective in thisdomain and we can write γ = cos − ( A ) . There will be three cases to consider due to thepossibility of sin(2 γ ) being zero. If < γ < π/ , it’s easy to extract the sine and cosine of β ± β through A , A and A , A respectively. The angles can then be obtained using the angle function A ( x, y ) ≡ − (cid:16) x y (cid:17) .The result for β is β = 12 (cid:34) A (cid:32) A (cid:112) − A , A (cid:112) − A (cid:33) + A (cid:32) A (cid:112) − A , A (cid:112) − A (cid:33)(cid:35) (80) • If γ = 0 , it is readily seen that R ( g ) does not depend on β but β has a simple expression β = 12 A ( A , A ) (81) • If γ = π/ , R ( g ) does not depend on β instead. β is found to be β = 12 A ( A , A ) (82)so we can combine the two extremal cases into one, as they give the same formal expression for (cid:15) .Why the emphasis on determining the (cid:15) ± i ? As seen in (79), the deficit angle ˜Θ has a very simpleexpression in terms of them, and they can be interpreted geometrically. Indeed, note that the expressionfor ˜Θ f in a general face can be written as a sum over vertices, ˜Θ f = (cid:80) v ∈ f ˜Θ vf . We know fromHan/Zhang’s work (among others) that the action is interpreted as a holonomy around a certain face,going through all the vertices it belongs to. And in the expression for ˜Θ vf , ˜Θ vf = (cid:88) ± ± γ ) log (cid:104)J n ef | g ± ev g ± ve (cid:48) | n e (cid:48) f (cid:105) (83) ∼ (cid:88) ± ± γ ) (cid:15) ± i (84)the inner product clearly illustrates the parallel transport between the two tetrahedra in v whichcontain f . Therefore ˜Θ vf can be associated to the internal angle ∠ ( e, e (cid:48) ) vf , as illustrated by the figurebelow, a two dimensional sketch of the geometric structure around a vertex.The sum of all internal angles is equal to π minus the deficit angle Θ Regge , while the sum of allthe ˜Θ vf should tend asymptotically to a sign factor times iγ Θ Regge . Hence, the correct identificationwhich relates the (cid:15) to the internal angles is ± iγ ˜Θ vf = ± iγ (cid:88) ± (cid:88) i (1 ± γ ) (cid:15) ± i ∼ ∠ ( e, e (cid:48) ) vf (85)The results obtained in this section seem positive towards the consistency of EPRL/FK asymptoticswith Regge calculus, in contradiction with the flatness problem, since we are able to obtain geomet-rically consistent values for the key quantities in this problem, the area γj and the deficit angle Θ of the only interior triangle in the manifold. In fact, a similar result has been claimed by Perini andMagliaro[29], although the paper in question does not treat the problem in detail and fails to addressone important difficulty which we will now mention: the behaviour of the state contributions when j
17s varied. This is a problem because j is discrete, and while we get equations of motion that guaranteethe nonexistence of a critical point when j is different from the unique value j found above, it hasnot been properly justified that the contribution from this point is dominant over certain non-criticalconfigurations with different values of j , since it is unclear how to vary the action over it. Additionally,the value of j that solves exactly the closure conditions will in general be a non-integer, therefore thereis some uncertainty in this calculation which is important to address. The closure conditions will, ingeneral, not be exactly satisfied, because of the discreteness feature. j To address the issue, we will use results from Chapter 7 of [26] related to the stationary phase method.In particular we are interested in the following theorem about the study of the stationary phase integralwhen the functions that define it depend on free parameters.Theorem:
Let f ( x, y ) be a complex valued C ∞ function in a neighbourhood K of (0 , ∈ R n + m ,such that (cid:61) ( f ) ≥ , (cid:61) ( f (0 , , D x f (0 ,
0) = 0 and det D x f (0 , (cid:54) = 0 . Let u be a C ∞ functionwith compact support in K . Then ˆ u ( x, y ) e iλf ( x,y ) dx ∼ λ →∞ e iλf (cid:18) πiλ (cid:19) n/ (cid:115) D x f (0 , y ) (86) where the superscript 0 in front of the determinant signals that the corresponding function is specifiedmodulo the ideal I of functions generated by the derivatives D x f ( x, y ) . Essentially, what the theorem states is that if x = 0 is a critical point of f when the free param-eter y is zero, then when y is non-zero the point is “moved”, and is in general not a critical pointany more, but its contribution to the full integral is approximated by the formula above. The keypoint is that if f has an imaginary part, this contribution is suppressed by a factor e − λ (cid:61) ( f ) . We areinterested in this suppression factor for the integral we are studying, where the free parameter y istaken to be x − x , x being the critical value of x . But what is f ? The proof of the theorem aboveuses the Malgrange preparation theorem, also explained in Chapter 7 of [26]. Basically, one can choosea set of functions X i ( y ) satisfying X i (0) = 0 such that the ideal I of functions generated by ∂f∂x i isalso generated by { x i − X i ( y ) } i , and using the Malgrange preparation theorem it is possible to writethe following expansion for f ( x, y ) near the critical point: f ( x, y ) ≈ (cid:88) | α | 0) + ∂f∂x i (cid:12)(cid:12)(cid:12)(cid:12) (0 , (cid:124) (cid:123)(cid:122) (cid:125) =0 x i + ∂f∂y (cid:12)(cid:12)(cid:12)(cid:12) (0 , (cid:124) (cid:123)(cid:122) (cid:125) ≡ δ y (88) + 12 ∂ f∂y∂x i (cid:12)(cid:12)(cid:12)(cid:12) (0 , (cid:124) (cid:123)(cid:122) (cid:125) ≡ K i yx i + 12 ∂ f∂y (cid:12)(cid:12)(cid:12)(cid:12) (0 , (cid:124) (cid:123)(cid:122) (cid:125) ≡ δ y + 12 ∂ f∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) (0 , (cid:124) (cid:123)(cid:122) (cid:125) ≡ H ij x i x j (89)The second order Malgrange expansion for f ( x, y ) is (setting f = 0 ) f ( x, y ) ≈ f ( y ) + 12 f ij ( y )( x i − X i ( y ))( x j − X j ( y )) (90)18quating both expansions and gathering terms independent, linear and quadratic in x , we get f (0 , 0) + δ y + 12 δ y = f + 12 f ij X i X j K i x i y = − (cid:0) f ij + f ji (cid:1) x i X j H ij x i x j = 12 f ij x i x j (91)which we solve to obtain ( H ij is the inverse matrix of H ij . Remember we assumed det H (cid:54) = 0 ) f = f (0 , 0) + δ y + 12 δ y − K i H ij K j y − H ij K i y = X j f ij = H ij (92)Applying to the ∆ case, remembering that we chose y = x − x , we see that f (0 , is the action atthe critical point S C , δ = − i ˜Θ C ∼ ± γ Θ Regge and δ = 0 . Note that δ is real. We are only interestedin the imaginary part of f , which is quadratic in ( x − x ) , and gives us the suppressing factor as exp (cid:18) λ (cid:61) (cid:0) K i H ij K j (cid:1) ( x − x ) (cid:19) (93)Note that the variation of x has to be discrete. We would set j = j + n , n ∈ Z , so that x − x = n λ .This allows us to write the partition function as a sum over n in terms of the term corresponding to n = 0 , the critical term: Z = Z C (cid:88) n exp (cid:18) − A λ n (cid:19) (94)where A = −(cid:61) (cid:0) K i H ij K j (cid:1) . If x is thought of as an approximately continuous variable, the distribu-tion of x values follows a Gaussian curve with standard deviation σ = (cid:113) λA . This is a sufficientlysmall deviation, assuming A finite, to conclude that the distribution of the ( j f , g ve , n ef ) variables issufficiently peaked around the critical surface. Since A does not have any λ dependence, the positiveresult should be guaranteed simply by A (cid:54) = 0 . However, the most rigorous approach to this problemis to compute the sum of the series in (94) and obtain the statistics of the discrete variable n (note,in particular, that j as given by the closure equations might not be a semi-integer, so the dominantcontribution would come from the semi-integer closest to it). The EPRL/FK action S = − i (cid:88) f (cid:88) v ∈ f (cid:88) ± j f (1 ± γ ) log (cid:104)J n ef | (cid:0) g ± ve (cid:1) † g ± ve (cid:48) | n e (cid:48) f (cid:105) (95)can be interpreted in terms of this stationary phase method by setting j f ≡ y as the free parameter,and x i ≡ (cid:0) { g ve } a , { n ef } b (cid:1) as the dependent variables, where a, b signal an appropriate coordinatesystem in which to express the interior g ve , n ef (which can be, for example, the parameterizationsof SU(2) and H / specified in section 4.2). The quantities necessary to compute the approximatepartition function (94) are K i = ∂ S∂j f ∂x i (cid:12)(cid:12)(cid:12)(cid:12) critical = ∂ ˜Θ f ∂x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) critical (96) H ij = ∂ S∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) critical (97)where “critical” means the derivatives are computed at the unique critical point for ∆ determined insection 4.1, and K i is simplified due to the action being linear in j , being reduced to first derivativesof the quantum deficit angle of the interior face ˜Θ f . The conditions of theorem (86) require that det H (cid:54) = 0 for the stationary phase method to be applicable. However, explicit computation of thisdeterminant, even using algebraic computation software, proves to be a bit too cumbersome becauseof the dependence of the derivatives in question on a high number of a priori arbitrary boundary19ariables, { g ve , n ef } B - even though it is possible to compute det H explicitly in terms of them, andobtain a numeric answer if numeric data are introduced for the EPRL variables, it is not clear atthe moment whether, for example, it is nonzero for all their possible values. For that reason, we willanalyse the determination of EPRL boundary data from geometric constructions, in order to obtainvalues for H in concrete cases.While showing consistency of the EPRL behaviour with Einstein theory in such examples is in noway a proof for the general case even within ∆ , it would nevertheless be an interesting result, andon the flipside, an inconsistency would be a significant result on its own, albeit a negative one. Tosummarize the possible outcomes: • det H = 0 : then the stationary phase method is not valid (in particular the quantity A is notdefined), and we must find a different method to evaluate the asymptotics; • det H (cid:54) = 0 and A = 0 : in that case the Gaussian distribution (94) has infinite standard deviationand as such will not specify the semiclassical value of x , failing to reproduce the expected classicalresult; • det H (cid:54) = 0 and A (cid:54) = 0 : the Gaussian distribution around the semiclassical value of x shouldguarantee reproduction of the expected geometric values. In particular, if one can verify this tohappen for a certain boundary configuration, continuity conditions assure that the EPRL asymp-totics match the expected classic solutions in a certain open neighbourhood of that configuration,which would give us some confidence that the semiclassical limit is correct for a significant rangeof boundary data. It does not, however, discard the possibility of there existing isolated pointsin the critical surface for which one of the two situations above happen, and it is unclear howthis would affect the overall statistics. To obtain the EPRL spin foam variables g ve , (cid:126)n ef , j f for a given example, we need to essentially carrythe procedure of the reconstruction theorem backwards and determine how they are related to the ge-ometrical data which defines the classical triangulation ∆ . Obtaining the spins j f is straightforward.Indeed, it has already been established that j f are directly related to the triangle areas via A f = γj f (within our semiclassical approximation of j being large).The Livine-Speziale coherent states | n ef (cid:105) are expressed in terms of (cid:126)n ef ∈ S , the normal vectorsto the tetrahedron faces’ Euclidean images in the tangent spaces T e ∆ ≈ R , and phases α ef ∈ U(1)which can be consistently defined by imposing Regge boundary conditions but are of no consequenceto the dynamics of the model, and can therefore safely be ignored. The one difficulty in correctly iden-tifying the (cid:126)n ef is that computing the norms of the geometrical tetrahedra in R does not immediatelytell you which n ef is which within a certain tetrahedron. A solution to this issue is to consider gluingmatrices . Indeed, considering a gluing equation R ( g ± ve ) (cid:126)n ef = − R ( g ± ve (cid:48) ) (cid:126)n e (cid:48) f , (98)notice that the + and − equations contained in it can both be manipulated to give the value of (cid:126)n e (cid:48) f ,and therefore (cid:0) R − ( g + ve (cid:48) ) R ( g + ve ) − R − ( g − ve (cid:48) ) R ( g − ve ) (cid:1) (cid:126)n ef = 0 . (99)Defining the matrix in brackets as the gluing matrix between two tetrahedra, R ee (cid:48) , (cid:126)n ef must lie inits null space, and furthermore, if the tetrahedron is non-degenerate (which we are assuming it is)such null space must have dimension 1. Comparing the resultant null spaces with the normals of thegeometric tetrahedra then gives the correct answer for (cid:126)n ef .Obtaining the g ve is somewhat less trivial. The first step is to identify what they represent geo-metrically. Indeed, g ve are SO(4) group elements related to the triangulated equivalent of the spinconnection, which in the geometrical setup translates to mapping the geometrical tetrahedron e ∈ v to its image in the tangent space T e ∆ . We have to define what this means, though. It is still necessary to consider the geometric tetrahedra with this procedure since simply solving (99) gives thecorrect normals up to a minus sign, which must be fixed in accordance with geometric consistency. v ∈ ∆ and a tetrahedron e ∈ v defined by points p , ..., p . Note that for ageneral triangulation each 4-simplex lives on its own copy of R : if the entire triangulation can beembedded isometrically in R that implies all the deficit angles are zero and the triangulation is flat.We will define the tetrahedron’s geometric matrix M ve and projected matrix M (3) ve : • to construct M ve , consider an oriented trivector τ ve = (cid:8) τ ve , τ ve , τ ve (cid:9) consisting of the three edgevectors coming out of a previously defined pivot point. For example, if p is chosen as the pivot,a possible trivector is { p − p , p − p , p − p } . If e is non-degenerate, the trivector definesa (non-orthonormal) basis of the 3-dimensional hyperplane e lives on, which can be equated to T e ∆ . Compute the normal to this hyperplane, N ve , which is the normal to the tetrahedron. Notethat there are two possible orientations for this normal, so we will establish as a convention thatthe orientation to choose is the one that makes det M ve > . The full matrix is then M ve = (cid:8) N ve , τ ve , τ ve , τ ve (cid:9) . (100)Note that this matrix is, by construction, invertible, since its 4 columns are linearly independent. • for M (3) ve , write down an orthonormal basis of T e ∆ as defined above, for example using the Gram-Schmidt orthonormalization algorithm, and determine the coordinates of the vectors in τ ve onthat basis. Call them τ (3) ve . We will regard T e ∆ as a subspace of R normal to (1 , , , , since itwill help with decomposing g ve into its SU(2) components g ± ve . The projected tetrahedron matrixis then M (3) ve = (cid:0) τ ve (cid:1) (3) (cid:0) τ ve (cid:1) (3) (cid:0) τ ve (cid:1) (3) (101)which is also invertible by the same reasons as above.Note that M ve is not unique to a tetrahedron, but the g ve rotation will be well defined provided thatthe orientations of both are consistent with respect to the considerations of section 2, that is, derivingthe orientation of each tetrahedron from the 4-simplex v by (4) and permuting the edge vectors in τ ve to guarantee the same sign for all M ve associated with v . With these definitions in place, g ve is theSO(4) matrix that rotates the projected matrix into the geometric matrix, i.e. g ve · M (3) ve = M ve ⇔ g ve = M ve (cid:16) M (3) ve (cid:17) − (102)Next step is to find g ve ’s SU(2) components. To do this we will use a result of van Elfrinkhof[32] whichgives an algorithm for decomposition of a SO(4) rotation into left- and right-isoclinic rotations, whichcan each be associated to SU(2) elements. Given a matrix g ∈ SO(4), define the associate matrix