Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex
aa r X i v : . [ g r- q c ] A ug A taste of Hamiltonian constraint in spin foam models
Valentin Bonzom ∗ Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada (Dated: September 12, 2018)The asymptotics of some spin foam amplitudes for a quantum 4-simplex is known to display rapidoscillations whose frequency is the Regge action. In this note, we reformulate this result througha difference equation, asymptotically satisfied by these models, and whose semi-classical solutionsare precisely the sine and the cosine of the Regge action. This equation is then interpreted ascoming from the canonical quantization of a simple constraint in Regge calculus. This suggeststo lift and generalize this constraint to the phase space of loop quantum gravity parametrizedby twisted geometries. The result is a reformulation of the flat model for topological BF theoryfrom the Hamiltonian perspective. The Wheeler-de-Witt equation in the spin network basis givesdifference equations which are exactly recursion relations on the 15j-symbol. Moreover, the semi-classical limit is investigated using coherent states, and produces the expected results. It mimics theclassical constraint with quantized areas, and for Regge geometries it reduces to the semi-classicalequation which has been introduced in the beginning.
INTRODUCTIONAsymptotics of spin foam amplitudes from semi-classical Hamiltonian dynamics
A good spin foam model for quantum gravity is (often) expected to reproduce Regge calculus (a large distanceapproximation of general relativity) in the classical limit. The idea goes back to Ponzano and Regge, [18], whomade the key observation that the Wigner 6j-symbol, an object from the theory of representations of SU(2), behavesfor large spins as the cosine of the Regge action for a tetrahedron, with the spins as edge lengths. This gave amodel for quantum gravity in three dimensions, where spins are interpreted as quantized lengths. The achievementof Loop quantum gravity (LQG) then gave a new birth and justification to the idea that quantum gravity can beformulated from algebraic objects, coming from the representation theory of a Lie group, attached to chunks ofspacetime (simplices, polyhedra).Still, it is not so obvious and straightforward to imagine why the semi-classical limit of spin foams would have tobe expressed in terms of the Regge approximation to general relativity (remember spin foams are initially designed toprovide transition amplitudes between the kinematical states of LQG, based on cylindrical functionals of the Ashtekar-Barbero connection, [17]). This idea was suggested in [19]. In particular, it is based on the fact that LQG supportsa discrete area spectrum (built from the Casimir of SU(2)), very similar to the Ponzano-Regge ansatz for quantizedlengths in three dimensions.So when a new model is proposed, the natural thing that is to be done is to check its semi-classical limit, whereby “checking“ it is usually meant chasing after the Regge action. However, it turns out that the models that haveso far attracted the most attention all have such Regge contributions, in particular models that are known not todescribe quantum gravity, like the Ooguri model (a model for the topological BF theory in four dimensions, [16]) andthe Barrett-Crane model (though the oscillatory part involving the Regge action is only a subleading term [13]).In a series of papers (see for instance [4, 6]), the asymptotics of the 15j-symbol (for the Ooguri model) and forthe Euclidean and Lorentzian EPRL 4-simplex have been precisely studied. It appears that different behaviours are ∗ [email protected] observed according to whether or not the boundary data determine a Regge metric on the 4-simplex, so that oscillationswith the Regge action do occur or do not.So we would like to get a criterion which would tell us whenever a model has such oscillations with the Regge action,say at the leading order. We obtain such a criterion as a difference equation of second order on the 4-simplex amplitude,(3). This equation is actually well-known from the three-dimensional case. Indeed the 6j-symbol is fully characterizedby a second order recursion relation (coming from the Biendenharn-Elliott, or pentagon identity), which, althoughgenerally complicated, simplifies in the semi-classical limit where it allows to determine the asymptotics, [12, 21].Since the asymtotics provides a regime where the spin foam amplitude may be approximated by some quantumRegge calculus, it is natural to look for an interpretation, or even better, for a derivation of this difference equation asthe quantization of a constraint in Regge calculus. It turned out to be very simple, and natural. The correspondingconstraint is a sort of flatness constraint which enables to built a flat 4-simplex from its boundary. It states that themomenta conjugated to the triangle areas have to be the dihedral angles between adjacent tetrahedra, computed fromthe areas like in a flat 4-simplex. This gives a first link between a classical constraint and the asymptotics of spinfoams.So far the relation between spin foam models and the Hamiltonian constraint of general relativity has been partic-ularly evasive (see open problem (14) in [20]). Our result gives a taster for this relation. To go further and ultimatelysavour it, we need to derive the semi-classical constraint from the quantization of a Hamiltonian operator in LQG.We will perform this task in the case of the Ooguri model. The analysis is a simple extension of results to appearfrom a collaboration with L. Freidel, [8]. There it is shown in the 3d case that a projection of the curvature onto thecomponents of the triad, thus taking the form of the Hamiltonian constraint EEF , can be quantized in LQG. In thesimplest situation, on the boundary of a tetrahedron, the Wheeler-de-Witt equation is a difference equation which is exactly the recursion relation defining the 6j-symbol. In 4d, the physical (flat) state on the boundary of a 4-simplexis the 15j-symbol. One can lift the Hamiltonian used in 3d to 4d, and using the methods of [8], we claim that theWheeler-de-Witt equation reproduces the recursion relations satisfied by the 15j-symbol which were derived in [9].The organization is as follows. In the section I: • we exhibit a difference equation whose solutions are the exponentials of ± i times the Regge action of the 4-simplex. • we derive this equation as the quantization of a classical constraint in area Regge calculus. The constraint statesthat a point on the phase space is given by a set of ten areas and ten dihedral angles which are those of a flat4-simplex, determined by the areas.The section II focuses on the Wheeler-de-Witt equation for the Ooguri model. • We define a classical constraint, attached to a node and a cycle of a spin network graph, by projecting thecurvature onto some components of the gravitational field, II A. • This Hamiltonian is rewritten in terms of twisted geometries [14], a nice parametrization of the LQG phase spaceon a single graph. It appears as a generalization of the above constraint for Regge calculus to the whole LQGphase space (including non-Regge boundary data), II B. • The corresponding Wheeler-de-Witt equation is studied, more particularly in the large spin limit, in the coherentstate basis. Using the WKB approximation, it reduces to the classical Hamiltonian on twisted geometries, withquantized areas.In particular, in the Regge sector of boundary data, it reproduces the semi-classical equation asymptotically satisfied byspin foam models. The natural variables are areas and normals of triangles. Thus, it strengthens from the Hamiltonianpoint of view the result that quantum area-angle calculus is the semi-classical limit of quantized geometries (in theRegge sector).We will also argue that such difference equations obtained through canonical quantization in the LQG frameworklead to a position where the same analysis as that of [6] can be done and used to extract the asymptotics (like in 3dactually, [8, 21]).In the section III, we discuss additional interesting difference equations. One is derived from our main semi-classicalequation (3), and shown to probe the closure of the simplex (it was already introduced and precisely described in [9],though from a quite different path). We also sketch the possibility of introducing more speculative constraints, whoseasymptotical behaviour exhibits oscillations with the Regge action.All technical details are skipped in the main text to ease a fluent reading, and are reported in appendix.
I. A NEW LOOK AT THE SEMI-CLASSICAL BEHAVIOUR OF 4-SIMPLEX SPIN FOAMAMPLITUDES
Crucial references on the asymptotics of spin foam models are [3–6]. There it is shown that several spin foam modelsget in the large area limit rapid oscillations with the Regge action. We want to track back this phenomenon to thefact that they satisfy in this regime the same equation, solved by exponentials of i times the Regge action.Consider a 4-simplex, with five tetrahedra on its boundary labelled by a = 1 , . . . ,
5. The semi-classical regimecorresponds to large values of the quantum numbers of triangle areas ( j ab ∈ N ) a
A. Recursion relations in the asymptotics
Our first point is that linear combinations of exponentials of ± i times the Regge action are the solutions to thefollowing difference equation of the second order: h ∆ ab + 2 (cid:0) − cos Θ ab (cid:1)i V ( A ab ) = 0 , (3)when solved via the semi-classical approximation for large λ . Here ∆ ab is the discrete second derivative with respectto the area variable A ab (may it be j ab or γj ab ): ∆ f ( x ) = f ( x + 1) + f ( x − − f ( x ). An equivalent form (which willbe that naturally coming out in the next sections) is obtained by defining some ladder operators which shift an areaby ± δ + ab V ( A ab ) = V ( A ab + 1) , δ − ab V ( A ab ) = V ( A ab − . (4)Then, the semi-classical equation becomes: h (cid:0) δ + ab + δ − ab (cid:1) − cos Θ ab i V ( A ab ) = 0 . (5)We look for solving the equation `a la WKB , when all spins are rescaled by λ ≫
1, and with the ansatz: ψ ( λA ab ) = Φ( A ab ) e iS ( λA ab ) . (6)We assume Φ does not scale with λ , while S scales linearly, so the idea is as usual: a slowly varying amplitude, witha rapidly oscillatory phase. To zeroth order, ψ ( λA ab ± ≃ ψ ( λA ab ) e ± iS ′ ( λA ab ) , (7)where S ′ is the derivative of S seen as a function on the real line. The equation (3) becomes:cos S ′ ( A ab ) − cos Θ ab = 0 , (8)or: S ′ ( A ab ) = ± Θ ab . So one has to integrate the dihedral angle with respect to the area. The result is known to bethe Regge action, since when varying it with respect to A ab , the variations of the dihedral angles cancel thanks to theSchlaefli identity, P a
In the large spin limit, we can just evaluate cos( s ab ( X )) on each saddle point. If it is the same for all of them, thenit can be factorized from the amplitude. This is what happens in the SU(2) Ooguri model and the Lorentzian EPRLmodel, where the saddle points X ∗ gives: s ab ( X ∗ ) = ± Θ ab . This point is important and prevents the asymptotics fromgetting other frequencies than the Regge action itself. As an example, our reasoning does not apply to the geometricsector of the Euclidean EPRL model, since it receives oscillations from the Regge action and from γ − S R also, withthe same scaling. Then, the spin foam amplitude satisfies a higher order difference equation in the asymptotics, whichis simply the product of the difference operator (3) for both frequencies: h ∆ ab + 2 (cid:0) − cos Θ ab γ (cid:1)i h ∆ ab + 2 (cid:0) − cos Θ ab (cid:1)i V ( A ab ) = 0 . (11) B. Recursion relations as Wheeler-de-Witt equations in quantum area Regge calculus
Since semi-classical spin foams can be approximated with quantum Regge calculus, we now want to understand ourmain equation (3) in this framework. Here comes our second important point: this equation (3) has a nice geometricinterpretation as a quantization of a constraint in Regge calculus. Consider the set of areas ( A ab ) such that theyuniquely determine a genuine flat 4-simplex as the configuration space. Like in [10], we take the conjugated momentato be angles ( θ ab ), with the canonical brackets: { A ab , θ cd } = δ ( ab ) , ( cd ) . (12)This framework has been derived from a canonical discretization of the Plebanski’s action for gravity in [10] (seethere for the full details on the symplectic structure of the phase space). On this phase space, we also consider theconstraints: χ ab ≡ cos θ ab − cos Θ ab ( A ) = 0 . (13)This was already studied in the above reference, and there argued to form an Abelian algebra. It should be notedthat the authors of [10] were then interested in the gauge symmetry corresponding to the translation of a vertex ofthe simplex. Here, we would like instead to generate independent shifts of areas to produce our equation of interest(3). For that purpose, the constraint (13) is what we need. Furthermore, its geometric meaning is quite clear: themomenta ( θ ab ) are constrained to be the dihedral angles (Θ ab ) of the flat 4-simplex determined by its areas.Let us now proceed to the most naive quantization, using wave functions of the angles. They can be expanded ontothe Fourier components, ( e i P j ab θ ab ), where the integers ( j ab ) are the eigenvalues of the area operators ˆ A ab (they getdiscrete spectra since the variables θ ab live on a compact set). Then, periodic functions over ( θ ab ) act by multiplication,and in particular: \ e ± iθ ab h X { j cd } ψ ( j cd ) e i P j cd θ cd i = X { j cd } ψ ( j ab ∓ , j cd ) e i P j cd θ cd , (14)= δ ∓ ab ψ. (15)This simply means that on the Fourier coefficients ψ ( j cd ) of a state | ψ i , the operator \ e ± iθ ab acts by shifting thevariable j ab by ∓
1. Also, as the Fourier exponentials are the eigenfunctions of the area operators, we simply promotethe complicated functions Θ ab ( A ) to operators through: \ cos Θ ab ( A ) e i P j cd θ cd = cos Θ ab ( j ) e i P j cd θ cd , (16)as far as the set ( j cd ) allows to define the dihedral angles. Thus the classical constraint χ ab can be imposed at thequantum level, d χ ab | ψ i = 0 , (17)where it becomes exactly the difference equation we are looking for: h ∆ ab + 2 (cid:0) − cos Θ ab (cid:1)i ψ ( j ab ) = 0 . (18) II. QUANTUM DYNAMICS OF THE FLAT 4-SIMPLEX IN LOOP QUANTUM GRAVITY(THE OOGURI MODEL REVISITED)
The above approach has obvious limitations: • First, it only holds asymptotically, and we expect the full quantum gravity amplitude to satisfy a differenceequation with non-trivial coefficients, which would contain all information about the full asymptotic expansion. • Second, it is not clear what the role of the additional boundary data of spin foams (the normals to the triangles)can be here. However, they are part of the phase space of Loop Quantum Gravity (on a single graph). Inaddition, it has been shown in previous studies that spin foams are better understood in terms of area-angleRegge calculus, [7], instead of area calculus (well-known to suffer from several drawbacks), [11]. This leads tothe third point. • We have so far focused only on Regge geometries, since the asymptotic behaviour is different on the otherconfigurations. But from the LQG point of view, there is no specific reason to distinguish between Reggeand non-Regge geometries. Furthermore, the Ooguri model, which also shows up these different asymptoticbehaviours, is nevertheless built from a single constraint, namely the flatness of a gauge field, without regardsfor the amount of geometricity contained in the canonical momenta.This leads us to revisit the Ooguri model for SU(2) BF theory, with a new form, more geometric, of the Hamiltonianconstraint.
A. The proposal: projecting the curvature
So we now turn to the phase space inherited from LQG on the dual complex Γ to the boundary of a 4-simplex.The notation a = 1 , . . . , ab ) for triangles and ( abc ) for edges.By duality, they correspond on the complex Γ to nodes a = 1 , . . . ,
5, links ( ab ), and faces, also referred to as cycles( abc ). We will mainly use the terminology corresponding to cells of Γ, but also switch to the point of view of thetriangulation as soon as we find it relevant.The phase space is precisely the same as that of SU(2) Yang-Mills on such discretization, and the same as in thetopological SU(2) BF theory, up to a scaling of the fundamental brackets by the Immirzi parameter (that we will ignorein the following). The ten links of Γ carry SU(2) elements, ( g ab ), which represent parallel transport operators betweenthe source and target vertices of each link . The phase space is then simply the cotangent bundle over SU(2) L , with L = 10 and with its natural symplectic structure. The momentum to g ab is thus a 3-vector E ab , for a < b , with { E iab , g ab } = τ i g ab . (19)These momenta can be seen as smearings the triad field E αi of the continuum over the triangles (dual to the link of Γ),and will be therefore called triad variables. We have denoted ( τ i ) i =1 , , anti-hermitian generators of the Lie algebra.The standard interpretation is the following. Each tetrahedra of the boundary carries a local reference frame, and E ab We choose the notation so that g ab goes from b to a . Also: g ba = g − ab . is defined relatively to that of the tetrahedron a . Momenta E ba acting on the right of g ab can be defined by parallellytransporting E ab to the frame of the tetrahedron b using the adjoint action of the group: E ba = − Ad( g − ab ) E ab . (20)Like in lattice gauge theory, the gauge group is SU(2) V , where V = 5 is the number of nodes of the graph.We now consider a Hamiltonian constraint for the Ooguri model. It is usually taken to be: g ( abc ) ≡ g ab g bc g ca = , (21)That is to say: the parallel transport around the cycle ( abc ) made of the three links ( ab ) , ( bc ) , ( ca ) is trivial. But itis unlikely that its quantization will help to understand quantum gravity. So we would like a constraint which wouldlook like closer to the Hamiltonian constraint of general relativity, ǫ ijk E αi E βj F ( A ) kαβ = 0 (where α, β are indices onthe canonical surface, and A is a SU(2) gauge field of curvature F and conjugated momentum E ). First the curvatureis discretized around the 2d regions of the complex dual to the triangulation. More precisely, consider the regionbounded by the links ( ab ) , ( bc ) , ( ca ), then the component of F along these directions is regularized in LQG as: ǫ ijk F kαβ −→ δ ij − (cid:0) Ad( g ( abc ) ) (cid:1) ij . (22)Then, the idea is to project it at each node of the cycle along the two triad variables which meet there. Define: H abc = E ab · E ac − E ab · Ad( g ( abc ) ) E ac , (23)and the constraint H abc = 0. For each cycle, there are three such constraints (and for a generic triangulation, thereis one constraint for each node of a cycle). So there are enough constraints to enforce g ( abc ) = ± when the triadvariables around the cycle span the three dimensions . B. A Hamiltonian for twisted geometries
We are interested in the relation between H abc and the previous constraint χ ab , (13), at the classical level, andwith the semi-classical equation (3) after quantization. Classically, it is possible to define in the geometric sector thedihedral angles Θ ab via the triad variables, and like in [10], another notion of dihedral angles involving the groupelements. Then, the above constraint just states the equality of the two notions. Details will appear in a collaborationwith L. Freidel [8].Here, we prefer to translate the constraint into the language of twisted geometry [14]. This is a nice reparametrizationof the LQG phase space which makes clear the nature of the involved geometries. In particular, space is formed bygenuine polyhedra like tetrahedra, but their gluing does not lead to Regge metrics, since two adjacent polyhedra maydescribe their common boundary with different shapes. We take advantage of this fact and give an interpretation of H abc which also holds for non-Regge situations.The parametrization maps the set ( E ab , g ab ) to a new set:( E ab , g ab ) → ( A ab , ~N ab , ~N ba , ξ ab ) , (24)defined as follows. A ab is the norm of E ab (and equals that of E ba ), and ~N ab its direction: E ab = A ab ~N ab , (25) At least, it is not hard to see that there is no smooth deformation of this relation. However, there may be a finite set of possibilities that wehave not investigated. In particular, the solutions of the equation E · E − E · Ad( g ) E = 0 are: g = exp( t E ) exp( η ( E × E )) exp( t E ),where t , t are arbitrary. But η admits only a finite number of values, since there is a finite number of SO(3) rotations with axis E × E solving the equation. for all a, b . Then, since ~N ab and ~N ba are taken as independent, the equation (20): ~N ab = − Ad( g ab ) ~N ba has to besolved for g ab . Take a set of SU(2) rotations ( n ab ( ~N )) a,b such that n ab maps an axis of reference in R , say ˆ z , onto ~N ab . This leads to the introduction of the angles ξ ab through: g ab = n ab ǫ e ξ ab τ z n − ba . (26)(The matrix ǫ = ( − ) is there to account for the minus sign in the parallel transport relation (20), since ǫ maps thedirection ˆ z onto its opposite − ˆ z .) Obviously, the normals and the triad variables are unchanged when adding a phaseon the right of n ab like: n ab → n ab e λ ab τ z . (27)The invariance of g ab then requires to change ξ ab accordingly. In particular, for a given g ab , ξ ab can always bereabsorbed into the rotation n ab or n ba . The set of rotations ( n ab ) is very convenient, as we will see, and its useprefigures what happens at the quantum level. Indeed, the semi-classical coherent states we will later use are actuallylabelled by such rotations rather than only by the normals ( ~N ab ).The generator of gauge transformations on the vertex dual to the tetrahedron a is: X b = a A ab ~N ab = 0 , (28)This condition actually takes the form of a closure relation for the tetrahedron and hence leads to this nice interpre-tation: the variable A ab is the area of the triangle ( ab ), while ~N ab and ~N ba are respectively the normals to the sametriangle with respect to the frame of the tetrahedra a and b . So it guarantees that one can built a flat tetrahedron in R for each a .The areas and normals describe the intrinsic geometry of the canonical surface. In particular, in the gauge invariantsector, the dihedral angle φ abc between the triangles ( ab ) , ( ac ) is given by:cos φ abc = − E ab · E ac A ab A ac . (29)Another key quantity we will need, defined only in terms of the six normals around a cycle ( abc ), is:cos Θ ( a ) bc ( ~N ) = cos φ abc − cos φ bac cos φ cba sin φ bac sin φ cba . (30)If this quantity is independent of a (that is computing it from any cycle containing the link ( bc ) gives the same answer),then it is exactly the 4d angle Θ bc between the tetrahedra b and c , computed from the normals. This is exactly thecriterion that turns a set of variables satisfying (28) into a Regge metric on the triangulation [11]:cos Θ ( a ) bc ( ~N ) = cos Θ ( a ′ ) bc ( ~N ) . (31)The splitting between intrinsic and extrinsic geometries in the twisted parametrization, and in particular, theinformation about the extrinsic geometry contained in the set of normals ( ~N ab ) has been discussed in [14]. Here,we are able to go further on this issue by writing the constraint H abc in this new set of variables. To get a definiteexpression, we need the Euler decomposition of the product n − ab n ac : n − ab n ac = e α abc τ z e ( π − φ abc ) τ y e α acb τ z , (32)which defines the angles ( α abc ) a,b,c (notice however that they are changed under (27)).The Hamiltonian H abc then admits the following form on twisted geometries (see appendix): H abc = − A ab A ac (cid:16) cos φ abc − cos φ bac cos φ cba + sin φ bac sin φ cba cos (cid:0) ξ bc + α bca + α cba (cid:1)(cid:17) . (33)The surprise is that this is the form of the standard relation between the 3d and 4d dihedral angles within a flat4-simplex, (30), though it holds on the whole phase space. Now restrict attention to the Regge-geometric sector(where (31) is satisfied). As soon as the tetrahedra are non-degenerate (the 3d angles are neither 0 nor π ), the angle( ξ bc + α bca + α cba ) can be extracted from the constraint H abc = 0, to give the dihedral angle Θ bc . Moreover, since ξ bc is also independent of a obviously, then ( α bca + α cba ) also is. So we can use (27) to achieve a phase choice where thiscombination is π . This gives H abc ∝ cos Θ bc ( ~N ) − cos (cid:0) ξ bc (cid:1) = 0 . (34)So the classical constraint H abc = 0 really corresponds to the constraint χ ab = 0 in their common domain of applicability(the geometric sector), and is clearly related to building a flat 4-simplex out of its boundary tetrahedra. A key differenceis that the dihedral angles of the 4-simplex are rather computed from the set of normals ( ~N ab ) rather than from theareas. This suggests that the semi-classical limit of LQG is given by quantum area-angle Regge calculus. Further, itmakes the formula applicable on the whole phase space, by going back to (33) for H abc , since the latter still makes senseoutside of the geometric sector (when the gluing of the tetrahedra is not that of a 4-simplex), and also for degeneratetetrahedra (when some 3d angles are such that sin φ abc = 0).We would like to mention that at this stage it is possible to discuss the solutions of the constraints (33) in a waywhich is fully parallel to the analysis of [3]. First, in the geometric sector, there are clearly two solutions, ξ bc = ± Θ bc ( ~N ) . (35)Then, assume there are at least two distinct solutions, ( ξ + bc , ξ − bc ). That leads to: E ba · Ad( g + bc ) E ca = E ba · Ad( g − bc ) E ca .This relation has been studied in [10] where it was coined edge-simplicity constraint, and shown to actually imply theRegge gluing relations (31). So in the non-geometric sector, the constraint has either one solution, or no solution. C. The Wheeler-de-Witt equation and its semi-classical regime
1. The Wheeler-de-Witt equation as recursion relations
When a theory has gauge symmetries, the latter turn into constraints in the Hamiltonian analysis. They canbe imposed either before or after quantizing. In general relativity and BF theory, the Hamiltonian itself is aconstraint, and the program of LQG is to quantize first and then constrain. The kinematical Hilbert space is H Γ = L (SU(2) L / SU(2) V ), spanned by the so-called spin network functions. These are just built from the Fouriermodes of the ten group elements, that are their matrix elements in the representation ( j ab ) a
1) + A ( j ) ψ ( j ) + A +1 ( j ) ψ ( j + 1) = 0 . (40)The coefficients A ± take the form: A +1 ( j ) = jE ( j + 1), and A − ( j ) = ( j − E ( j ), for E ( j ) = h(cid:0) ( j + i + 1) − j (cid:1)(cid:0) j − ( j − i ) (cid:1)(cid:0) ( j + i + 1) − j (cid:1)(cid:0) j − ( j − i ) (cid:1)i , (41)and the coefficient A is given by: A ( j ) = (cid:0) j + 1 (cid:1)n (cid:2) j ( j + 1) i ( i + 1) + j ( j + 1) i ( i + 1) − j ( j + 1) i ( i + 1) (cid:3) − (cid:2) j ( j + 1) + i ( i + 1) − j ( j + 1) (cid:3)(cid:2) j ( j + 1) + i ( i + 1) − j ( j + 1) (cid:3)o . (42)If now on the node a = 1, the intertwiner ι a pairs ( j , j ) together to the virtual spin i , then the equation becomesmore complicated: X ǫ ,ǫ = − , , A ǫ ,ǫ ( j , i ) ψ ( j + ǫ , i + ǫ ) = 0 . (43)Such relations were derived by writing down explicitly a special invariance of the Ooguri model under a change oftriangulation. Thus, it was known that these relations encode the symmetries of the model at the quantum level. Butthey had so far never been derived from the quantization of a Hamiltonian constraint. The generic process is that the action of a triad variable inserts a generator, and then, the contraction of their vector indices producegraspings on the spin network in the spin 1 representation. After some recoupling, one can extract a special 6j-symbol with a spin 1 ateach node of the cycle (including the virtual spins of intertwiners).
2. The Wheeler-de-Witt equation in the semi-classical regime
The above equation is not really suitable for the semi-classical analysis however. It comes from the uncertaintyprinciple, that a tetrahedron is described quantum mechanically by only five quantum numbers (four areas ( j ab ), andone spin i a to specify the intertwiner) [2]. So to launch our Wheeler-de-Witt equation in the semi-classical limit, wego to an overcomplete basis of coherent intertwiners [15]. First build the usual SU(2) coherent state | j, n ( ~N ) i from aSU(2) rotation n ( ~N ) which maps the reference axis ˆ z onto a unit 3-vector ~N : | j, n ( ~N ) i = n ( ~N ) | j, j i . (44)It is important to keep in mind that the state is not fully determined by the direction ~N , but also by a choice ofphase. Indeed, changing n like in (27) does not affect the vector ~N , but multiplies the state by a phase. A coherentintertwiner ι a ( n ab ) on the tetrahedron a is labelled by four rotations ( n ab ) b = a corresponding to four unit vectors of R , ( ~N ab ) b = a . It is defined by a group averaging process: Z SU(2) dh a ⊗ b = a h a | j ab , n ab i . (45)It is shown in [15] that the norm of this intertwiner is peaked for large spins on vectors ( ~N ab ) which satisfy the closurecondition (28), but with quantum areas ( j ab ). Therefore, these vectors can be interpreted as normals to the trianglesof a (up to a global rotation). The spin network state in the basis of coherent intertwiners reads: s { j ab ,n ab } ( g ab ) = Z SU(2) Y a =1 dh a Y a
1. These shifts extend to the coherent states, to | j bc + η, n bc i , h j bc + η, n cb ǫ | in the large spin limit, via the operators E ba , E ca . We thus get: (cid:16) \ E ba · Ad( g bc ) E ca (cid:17) s { j ab ,n ab } ≃ cos φ bac cos φ cba s j bc −
12 sin φ bac sin φ cba (cid:16) e − i ( α bca + α cba ) δ + bc + e i ( α bca + α cba ) δ − bc (cid:17) s j bc , (49)where the angles α are determined by: n − ab n ac = e α abc τ z e ( π − φ abc ) τ y e α acb τ z , (50)like for classical twisted geometries.2The final step is to rewrite the quantum condition: b H abc | ψ i = 0 , (51)for the coefficients of an arbitrary expansion in the coherent spin network basis. Again, we have to invoke the semi-classical, large spin limit, since coherent intertwiners generally have non-trivial overlap. They become orthogonal forlarge spins, and then the Wheeler-de-Witt equation for H abc reads: (cid:16) cos φ abc − cos φ bac cos φ cba (cid:17) ψ ( j bc ) + 12 sin φ bac sin φ cba (cid:16) e i ( α bca + α cba ) ψ ( j bc + 1) + e − i ( α bca + α cba ) ψ ( j bc − (cid:17) = 0 . (52)Here the dependence on other variables than j bc have been dropped. This is our key equation, which generalizesthe semi-classical equation satisfied by the exponential of the Regge action (3) to the whole phase space of twistedgeometries (with sufficiently large spins).The following is devoted to specializing (52) to Regge or non-Regge boundary data, and see that it reproduces theequations and results previously discussed. We look for solving the equation `a la WKB , when all spins are rescaledby λ ≫
1, and with the ansatz: ψ ( λj bc ) = Φ( j bc ) e iS ( λj bc ) . (53)We assume Φ does not scale with λ , while S grows linearly. To zeroth order, ψ ( λj bc ± ≃ ψ ( λj bc ) e ± iS ′ ( λj bc ) , where S ′ is the derivative of S seen as a function on the real line. So, the equation (52) becomes: (cid:16) cos φ abc − cos φ bac cos φ cba (cid:17) + sin φ bac sin φ cba cos (cid:0) S ′ ( j bc ) + α bca + α cba (cid:1) = 0 . (54)We can now make contact with the first semi-classical equation of the paper (3), via the classical constraint H abc writtenfor twisted geometries in (33). Indeed, the latter is the same as this, with S ′ ( j bc ) instead of the angle ξ bc .Assume non-degeneracy of the tetrahedra, so that the formula (30) for the 4d dihedral angles Θ ab as functionsof the normals ( ~N ab ) is well-defined. Further assume the boundary data satisfy the gluing constraints (31) and theclosure relation (28). Then, we know from the discussion above (34) that: (i) the quantity into brackets on the left of(54) is: sin φ bac sin φ cba cos Θ bc , (ii) the angle ( α bca + α cba ) can be set to π by a change of phase in the coherent states | j ab , n ab ( ~N ) i . Then, we get to (8), cos S ′ ( j bc ) − cos Θ bc = 0 , (55)solved by exponentials of ± i times the Regge action S R . Now assume that (54) has only one solution for S ′ . Thesame way as it is discussed in [3], the value of S ′ can be reabsorbed into a change of phase of the coherent states (thischange of phase depends on the boundary data ( j ab , ~N ab , ~N ba )). Such a choice cancels the oscillations on the wavefunction, S = 0, like in the asymptotic analysis of [6]. III. OUTLOOK
The equation (3) has been shown to come from a constraint stating that the momenta conjugated to the areas haveto be the dihedral angles of a flat 4-simplex with these values of area (in the geometric sector). So this equationencodes the full information on the reconstructed 4-simplex, and we can imagine deriving from it other equationscontaining a few less geometric content, but still relevant. We here give one interesting example, which is a differenceequation probing the closure of the simplex. Indeed, we can form the 5 × G ab ) = (cid:0) cos Θ ab (cid:1) . (56)3Since the angles are those of a flat 4-simplex (actually determined by the areas and normals ( A ab , ~N ab , ~N ba ) a
The author is indebted to Laurent Freidel and Etera Livine. Numerous ideas actually came from them. Note that the natural areas in the Barrett-Crane model are (2 j ab + 1), and not the spins themselves. So the operators δ ± ab should thenbe understood as generating shifts of ± in the spins (or equivalently, shifts of ± APPENDIX: TECHNICAL DETAILSOn the Hamiltonian for twisted geometries
Since the normals ~N ab can be defined by: ~N ab = Ad( n ab ) ˆ z, (60)we can easily use the rotations ( n ab ) to capture the geometric information. Let us form: n − ab n ac = e α abc τ z e ( π − φ abc ) τ y e α acb τ z . (61)This is the Euler decomposition of the product n − ab n ac . All that we need can be extracted from the matrix elementsof this product in the representation of spin 1. Indeed, one can check that φ abc as it appears above is the (3d) dihedralangle between ( ab ) , ( ac ): h , | n − ab n ac | , i = E ab · E ac A ab A ac = − cos φ abc . (62)Furthermore, the sine of the dihedral angle can also be directly extracted: h , ± | n − ab n ac | , i = ± √ e ∓ iα abc sin φ abc , h , | n − ab n ac | , ± i = ∓ √ e ∓ iα acb sin φ abc . (63)Let us now prove the formula (33). Similarly to (62), one has: E ba · Ad( g bc ) E ca = h , | n − ba g bc n ca | , i , (64)by definition of the different group elements here involved. Then, write g bc in terms of the variables of twistedgeometries, (26): E ba · Ad( g bc ) E ca = h , | ( n − ba n bc ) ǫ e ξ bc τ z ( n − cb n ca ) | , i . (65)The matrix elements of ǫ = e − πτ y in the representation of spin j are: h j, m | ǫ | j, m ′ i = ( − j − m ′ δ m, − m ′ . Hence,introducing a resolution of the identity: E ba · Ad( g bc ) E ca = X m = − , , h , | n − ba n bc | , m i ( − − m e imξ bc h , − m | n − cb n ca | , i . (66)So (33) comes from explicitly writing down the three terms in the sum and the matrix elements. Quantization of the Hamiltonian in the large spin limit
To evaluate the action of H abc on coherent spin network states, let us first collect a few results. The action of thegenerators of su (2) is what we expect from states with a nice semi-classical behaviour: ~τ | j, n ( ~N ) i = ( − i ) j ~N | j, n ( ~N ) i + o ( j ) . (67)It is natural to see the 3-vector ~N in the Lie algebra su (2). Then, its components on the spherical basis ( τ m ) m = − , , are: N m = h , m | n ( ~N ) | , i , m = − , , +1 . (68)5SU(2) invariant operators acting on a coherent intertwiner naturally commute with the group action in (45). Inparticular, these results can be used to compute the action of E ab · E ac . Each triad operator inserts a generator, sothat we have to consider the action of ( P i τ iab ⊗ τ iac ) on the coherent intertwiner. It commutes with the group action,and finally leads to: (cid:16) \ E ab · E ac (cid:17) s { j ab ,n ab } ≃ − j ab j ac (cid:0) ~N ab · ~N ac (cid:1) s { j ab ,n ab } , (69) ≃ j ab j ac cos φ abc s { j ab ,n ab } . (70)Acting with E ba · Ad( g bc ) E ca is a bit more involved, but there are no conceptual difficulties. First the adjoint actionof g bc is a Wigner matrix D (1) in the spin 1, which recouples to the matrix elements of g bc in the state, usingClebsch-Gordan coefficients: D ( j bc ) AB ( g bc ) D (1) kp ( g bc ) = X η = − , , C j bc A kj bc + ηA + k C j bc B pj bc + ηB + p D ( j bc + η ) A + k B + p ( g bc ) . (71)The question is then whether (and since the answer is yes, how) the coherent state | j bc , n bc i in the state also receives ashift by η = − , , E ba is recast so that the quantity of interest is a state in the tensor product of the representation spaces H j bc ⊗ H : | j bc , n bc i ⊗ n ba | , i = X m = − , , (cid:16) h , m | n − bc n ba | , i (cid:17) | j bc , n bc i ⊗ n bc | , m i . (72)On the right hand side, we have just introduced the identity on H as: P m n bc | , m ih , m | n − bc . The quantity intobrackets can then be evaluated thanks to the formula (32) which has already been used for twisted geometries. Quiteclearly, this introduces the cosine of the dihedral angle φ bac when m = 0, and its sine, with some phase exp( ∓ iα bca )when m = ±
1. Then, a key technical point which leads to the final expression is due to a careful inspection of thescaling properties of the Clebsch-gordan coefficients in H j ⊗ H for large j : | j, n ( ~N ) i ⊗ n ( ~N ) | , m i ≃ | j + m, n ( ~N ) i . (73)This is exact when m = 1, but only holds asymptotically for m = − ,
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