GGeneralized Spinfoams
You Ding, Muxin Han,
1, 2 and Carlo Rovelli Centre de Physique Th´eorique ∗ , CNRS-Luminy Case 907, F-13288 Marseille, EU Max-Planck-Institut f¨ur Gravitationsphysik, Am M¨uhlenberg 1, D-14476 Golm, EU (Dated: November 2, 2018)We reconsider the spinfoam dynamics that has been recently introduced, in the generalizedKami´nski-Kisielowski-Lewandowski (KKL) version where the foam is not dual to a triangulation.We study the Euclidean as well as the Lorentzian case. We show that this theory can still be ob-tained as a constrained BF theory satisfying the simplicity constraint, now discretized on a generaloriented 2-cell complex. This constraint implies that boundary states admit a (quantum) geomet-rical interpretation in terms of polyhedra, generalizing the tetrahedral geometry of the simplicialcase. We also point out that the general solution to this constraint (imposed weakly) depends on aquantum number r f in addition to those of loop quantum gravity. We compute the vertex amplitudeand recover the KKL amplitude in the Euclidean theory when r f = 0. We comment on the eventualphysical relevance of r f , and the formal way to eliminate it. I. INTRODUCTION
The spinfoam formalism [1–3] offers a formulation of thedynamics of quantum gravity strictly related to loopquantum gravity (LQG)[4–6]. The precise relation be-tween the two approaches is well-understood in 3 dimen-sions [7], and under study in 4 dimensions [8].The spinfoam theory introduced in [9, 10] can be de-rived starting from the Plebanski formulation of GR [11](including the Barbero-Immirzi parameter γ ), and de-fined as a BF theory discretized on a simplicial cellu-lar complex and constrained by the so called simplic-ity constraint. The constraint can be imposed usingthe master-constraint technique [9, 12], or, more simply,using the Gupta–Bleuler procedure, namely asking thematrix elements of the constraint to vanish on physicalstates [13]. The resulting model has remarkable proper-ties: (i) the boundary states have a geometrical interpre-tation in terms of quantum tetrahedral geometry [3, 14];(ii) there are strong indications that the semiclassical be-havior of the theory matches classical general relativity[15–18], thus correcting difficulties of earlier models [19];and (iii) the boundary kinematics is strictly related tothat of LQG [9, 13].The relation with LQG, however, is limited by the factthat the simplicial-spinfoam boundary states include onlyfour-valent spin networks. This is a drastic reduction ofthe LQG state space. In [20], Kami´nski, Kisielowski,and Lewandowski (KKL) have considered a generaliza-tion of the spinfoam formalism to spin networks of arbi-trary valence, and have constructed a corresponding ver-tex amplitude. This generalization provides truncatedtransition amplitudes between any two LQG states [1],thus correcting the limitation of the relation between the ∗ Unit´e mixte de recherche (UMR 6207) du CNRS et des Universit´esde Provence (Aix-Marseille I), de la Meditarran´ee (Aix-MarseilleII) et du Sud (Toulon-Var); laboratoire affili´e `a la FRUMAM (FR2291). model and LQG. This generalization, on the other hand,gives rise to several questions. The KKL vertex is ob-tained via a “natural” mathematical generalization of thesimplicial Euclidean vertex amplitude. Is the resultingvertex amplitude still related to constrained BF theory(and therefore to GR)? In particular, do KKL states sat-isfy the simplicity constraint? Can we associate to thesestates a geometrical interpretation similar to the one ofthe simplicial case? Can the construction be extended tothe physically relevant Lorentzian case?Here we answer several of these questions. We showthat it is possible to start form a discretization of BFtheory on a general 2-cell complex, and impose the sameboundary constraints that one impose in the simplicialcase (simplicity and closure). Remarkably, on the onehand, they reduce the BF vertex amplitude to a (gener-alization of) the KKL vertex amplitude, in the Euclideancase studied by KKL. On the other hand, a theorem byMinkowski [21] garantees that these constraints are pre-cisely those needed to equip the classical limit of eachtruncation of the boundary state space to a finite graph,with a geometrical interpretation, which turns out to bein terms of polyedra [22].These results reinforce the overall coherence of the gen-eralized spinfoam formalism.Surprisingly, however, the state space defined by im-posing the simplicity constraint weakly is larger thanthe one of quantum gravity. It includes one additionaldegree of freedom, described by a new quantum num-ber r f . The quantum number r f affects non-triviallyboth the face amplitude and the vertex amplitude of themodel. The quantum number r f is frozen if in addition tothe weak imposition of the (linear) simplicity constraint, The enlargement is not an effect from the generalization to ar-bitrary 2-cell complexes. The Hilbert space is enlarged also inthe simplicial case, compared with the state space defined in[9]. This additional quantum number was first noticed by SergeiAlexandrov [26]. a r X i v : . [ g r- q c ] N ov we also impose strongly a diagonal quadratic constraint.With a suitable operator ordering of this constraint, thestate space can be reduced back down to the LQG statespace.Does the r f quantum number have physical relevance?If we take the principle that the quantum theory we areseeking has the same number of degrees of freedom asthe classical theory, then the answer is negative. Thisprinciple indicates that the appropriate way of imposingthe constraints is the one that gets rids of the extra states.However, we think it is nevertheless interesting to keepin mind the existence of these additional solutions to theweak simplicity constraints. We comment more on thisin the conclusion.An outline for the article is as follows. In Section II,we review the spinfoam representation of the BF parti-tion function on a general complex, and we discuss thestructure of the boundary Hilbert space of BF theory. InSection III, we implement the geometric constraint to theBF boundary Hilbert space. After solving the constraintweakly, two new boundary Hilbert space are constructedfor both the Euclidean and the Lorentzian theory. Wealso show that the new boundary Hilbert space carriesa representation of quantum polyhedral geometry. InSection IV, we derive the new spinfoam vertex ampli-tude and face amplitude from the new boundary Hilbertspace. In Section VI, we conclude and point out the openissues. We assume that the Barbero-Immirzi parameter γ is positive. FIG. 1: A generalized spinfoam vertex.
II. SPINFOAM REPRESENTATION OF BFTHEORY
We start with a brief review of the construction of theBF spinfoam partition function and the structure of itsboundary Hilbert space [23], which is the starting pointof the definition of the theory. The BF partition functionis formally defined by the path integral Z BF := (cid:90) DA DB exp (cid:0) i (cid:90) M tr( B ∧ F [ A ]) (cid:1) (1) where B is a 2-form field on the manifold M , with valuesin the Lie algebra g of a group G and F is the curvatureof the G -connection A . Here we take the internal gaugegroup G to be either G = Spin (4) (for the Euclideancase) or G = SL (2 , C ) (for the Lorentzian case). A formalintegration over B gives Z BF = (cid:90) DA (cid:89) x ∈ M δ ( F [ A ]) (2)which is an integration over the flat connections. In or-der to make sense of the formal path integral (2), wediscretize it. However, instead of discretizing the pathintegral on an oriented 2-complex dual to a simplicial de-composition of the manifold M as is usually done, weintroduce here an arbitrary oriented 2-complex K (as in[20]) with or without boundary.We take a combinatorial definition of an ori-ented 2-complex. An oriented 2-complex K :=( V ( K ) , E ( K ) , F ( K ) consists of sets of vertices v ∈ V ( K ), edges e ∈ E ( K ) and faces f ∈ F ( K ), equipped with a boundary relation ∂ associating an ordered pair of ver-tices ( s ( e ) , t ( e )) (“source” and “target”) to each edge e and a finite sequence of edges { e (cid:15) ekf k } k =1 ,...,n to each face f , with t ( e k ) = s ( e k +1 ), t ( e n ) = s ( e ) and (cid:15) ef = ±
1; herewe call e − the edge with reversed order of e . We let ∂f denote the cyclically ordered set of edges that bound theface f , or (if it is clear from the context) the cyclicallyordered set of vertices that bound the boundary edges of f . We also write ∂v to indicate the set of edges boundedby v , and of faces that have v in their boundary. Simi-larly, we write ∂e to indicate the set of the faces boundedby e . When e ∈ ∂f , we define (cid:15) ef = 1 if the orientationof e is consistent with the one induced by the face f and (cid:15) ef = − γ = ∂ K is a 1-cell subcomplex of K . An edge e ∈ E ( K ) is an edge of the boundary graph γ if and only if it is contained in only one face, otherwise itis an internal edge. A vertex v ∈ V ( K ) is a vertex of theboundary graph γ if and only if it is contained in exactlyone internal edge of K , otherwise it is an internal vertex of K . We assume boundary vertices and boundary edges toform a graph, which is the boundary of the two-complex.We introduce also the notion of the boundary graph γ v of a single vertex v . This is the graph whose nodesare the edges e in ∂v and whose links are the faces f in ∂v . The boundary relation defining the graph is therelation e ∈ ∂f and the orientation of the links is the oneinduced by the faces. The graph γ v can be visualizedas the intersection between the two complex and a smallsphere surrounding the vertex.We discretize the BF partition function on the oriented2-cell complex K , by replacing the continuous field A with the assignment of an element of G to each edge. Byconvention, g e − := g − e . Then equation (2) becomes Z BF ( K ) = (cid:90) dg e (cid:89) f δ (cid:0) (cid:89) e ∈ ∂f g (cid:15) ef e (cid:1) , (3) FIG. 2: An oriented 2-cell complex K := ( F ( K ) , E ( K ) , V ( K )),where F ( K ) = { f , · · · , f } , E ( K ) = { e , · · · , e } , V ( K ) = { v , · · · , v } . v is internal vertex, and e , e , e , e are in-ternal edges, while all other edges and vertices belong to theboundary graph γ = ∂ K . where dg e is the product over all the edges of the Haarmeasure, the product over f is over all the faces of K and the product over e is the product over the edgesbounding the face f of the group element associated tothese edges, ordered by the orientation of the face. Thisis the partition function of BF theory.We now express this partition function as a sum overrepresentations and intertwiners. For this, it is conve-nient to treat the Euclidean and Lorentzian cases sepa-rately. A. Spin(4) BF Theory
Consider the Euclidean case G = Spin (4). The deltafunction on
Spin (4) can be expanded in irreducible rep-resentations δ ( U ) = (cid:88) ρ dim( ρ ) χ ρ ( U ) (4)where ρ = ( j + , j − ) labels the unitary irrep of Spin(4),dim( ρ ) = (2 j + + 1)(2 j − + 1) is the dimension of the rep-resentation space, and χ ρ is the character of the repre-sentation ρ . Irreducible representations can also be con-veniently labelled with the two half integers k = j + + j − and p = j + − j − . Expanding the delta function in representations, (2)becomes Z BF ( K ) = (cid:90) dg e (cid:89) f (cid:32)(cid:88) ρ dim( ρ ) χ ρ ( U f ) (cid:33) = (cid:88) ρ f (cid:90) dg e (cid:89) f dim( ρ f ) χ ρ f ( U f ) . (5)This is the expression for the spinfoam amplitude in thegroup element basis. Let us now translate this into themore common representations-intertwiners basis. This can be obtained by performing the integrals, pre-cisely as in the simplicial case. We have one integrationper edge, of the form K M , N = (cid:90) dg e (cid:89) f ∈ ∂e Π ρ f M f N f ( g (cid:15) ef e ) (6)where Π ρMN ( g ) is the matrix element of the Spin(4) repre-sentation ρ ; M = M f , ..., M f n is a multi-index; and theproduct is over the n faces bounded by e (including re-peated faces). In the case where K is dual to a simplicialcomplex, n = 4. It is immediate to see that K M , N is theoperator in the tensor product ( (cid:78) f out ρ f ) ⊗ ( (cid:78) f in ρ † f )of the ρ f representation spaces (where f in are the faceswith the same orientation as e and f out are the faceswith opposite orientation.) that projects on its invariantsubspace H e = Inv (cid:2) ( (cid:79) f out ρ f ) ⊗ ( (cid:79) f in ρ † f ) (cid:3) . (7)Let I label an orthonormal basis in H e . (These are calledintertwiners.) Then K M , N = (cid:88) I I M I † N . (8)For each internal edge e , the two intertwiners are asso-ciated to the two vertices bounding the edge (see Figure3), in the sense that their indices are contracted with theother intertwiners at the same vertex. The result of the FIG. 3: Assign I e to the begin point and assign I † e to the endpoint of an internal edge e . integration is therefore Z BF ( K ) = (cid:88) ρ f (cid:89) f dim( ρ f ) (cid:88) I e (cid:89) v A v ( ρ f , I e ) . (9)Here the sum over I e is over the assignment of one inter-twiner to each edge e of K . The product over v is overthe vertices of K . The vertex amplitude A v ( ρ f , I e ) is de-fined as follows. Say at the vertex v ∈ V ( K ) there are n outgoing edges e out and m incoming edges e in . Then A v ( ρ f , I e ) := tr (cid:32)(cid:79) e out I e out (cid:79) e in I † e in (cid:33) (10)The trace in eq.(10) is precisely the spinfoam trace de-fined in [20]. The contractions between the intertwinersin the spinfoam trace could be described by the follows:For each edge e each index M i is associated with a face f bounded by the edge e . The trace is defined by contract-ing the two indices associated with the same face of thetwo intertwiners corresponding to the two edges bound-ing f . This can be easily seen to give the character χ ρ of(5). In the special case when the complex K is dual to asimplicial complex, there are 5 internal edges joining at v and each pair of edges determines a 2-face, the spinfoamtrace is nothing but the Spin(4) 15-j symbol.Alternatively, the BF partition function can also beexpressed in the form [20] Z BF ( K ) = (cid:88) ρ f (cid:89) f dim( ρ f ) tr (cid:79) e ∈ E ( K ) P e (11)where P e := (cid:80) I e I e ⊗ I † e is understood as the projectionoperator projecting from the product of the representa-tions on the 2-faces bounded by e to its invariant sub-space. And the index contractions in tr (cid:0) ⊗ e ∈ E ( K ) P e (cid:1) arethe contractions between intertwiners, as above.All gravitational spinfoam theories have this samestructure. B. SL(2, C ) BF Theory Let now G = SL (2 , C ). The derivation of the spinfoamrepresentation of SL (2 , C ) is as above, with a few differ-ences. SL (2 , C ) unitary irreps (in the principle series)can be labelled by the same quantum numbers ( k, p ) asthe SO (4) ones, but now p is a real number [31]. Theunitary irreps of SL (2 , C ) are infinite dimensional andcan be decomposed into an infinite direct sum of SU(2)irreps, i.e. V ( k,p ) = ∞ (cid:77) j = k V ( k,p ) j (12)where V ( k,p ) j ∼ V j is the carrier space of the spin j repre-sentation of SU(2). This decomposition provides a con-venient basis | j, m > in V ( k,p ) , obtained diagonalizing L and L z of SU(2). In this basis, for g ∈ SL (2 , C ), we writethe representation matrices on V ( k,p ) as Π ( k,p ) jm,j (cid:48) m (cid:48) ( g )where j ∈ { k, k + 1 , · · · , ∞} and m ∈ {− j, · · · , j } . Asone might expect from the fact that p is a continuouslabel, the representation “matrix element” Π ( k,p ) jm,j (cid:48) m (cid:48) isdistributional on the Hilbert space L [ SL (2 , C )] definedby the Haar measure. These matrix elements form ageneralized orthonormal basis and define a Fourier-liketransform. That is, for any square integrable function f ( g ) on SL (2 , C ), f ( g ) = 18 π (cid:88) k (cid:90) + ∞−∞ d p ( k + p ) tr (cid:104) F ( k, p ) Π ( k,p ) ( g − ) (cid:105) F ( k, p ) = (cid:90) SL (2 , C ) f ( g ) Π ( k,p ) ( g ) d µ H ( g ) (13)which is known as Plancherel theorem [31]. Accordingly,we have an identity for Fourier decomposition of deltafunction on SL (2 , C ) δ ( g ) = 18 π (cid:88) k (cid:90) + ∞−∞ tr (cid:104) Π ( k,p ) ( g ) (cid:105) ( k + p ) d p (14)in analogy with eq.(4). Proceeding as in the Euclideancase, we find Z BF ( K ) = (cid:90) (cid:89) e d g e (cid:89) f δ ( U f ) (15)= (cid:88) k f (cid:90) d p f (cid:89) f ( k f + p f ) (cid:90) d g e (cid:89) f tr (cid:104) Π ( k f ,p f ) ( U f ) (cid:105) As in the euclidean case, each g e integral is of the form K jm , j (cid:48) m (cid:48) = (cid:90) d g e (cid:89) f ∈ ∂e Π ( k f ,p f ) j f m f ,j (cid:48) f m (cid:48) f (cid:0) g (cid:15) ef e (cid:1) . (16)Formally, this is still a projector on the invariant com-ponent of the tensor product of n irreducibles. However,since now one of the two Casimirs has continuous spec-trum p , then the trivial representation p = k = 0 is nota proper subspace of the tensor product, but only a gen-eralized subspace. This does not forbids us to introducean orthonormal basis of intertwiners I in this subspace,as we did in the Euclidean case, and write K jm , j (cid:48) m (cid:48) = (cid:88) I I jm I † j (cid:48) m (cid:48) (17)but we have to remember that the intertwiners are gener-alized vectors. Using this, we can formulate the spinfoamrepresentation of SL (2 , C ) BF theory in the same way aswe did for Spin(4) theory. • The Fourier decomposition of the SL (2 , C ) deltafunction assigns an SL (2 , C ) irrep labeled by( k f , p f ) to each face f . • Eq.(16) assigns an SL (2 , C ) intertwiner I e to eachsource of each edge e , and a dual intertwiner I e † toits target. • At each vertex v with n outgoing edges e out , · · · , e outn and m incoming edges e in , · · · , e inm ,the intertwiners I e out and I e in † are contracting ontheir j , m and j (cid:48) , m (cid:48) indices, according to how thefaces neighboring the vertex are bounded by theedges. The result of this contraction gives the spin-foam vertex amplitude A v (cid:16) ( k, p ) f , I e (cid:17) := tr (cid:32)(cid:32)(cid:79) e out I e (cid:33) ⊗ (cid:32)(cid:79) e in I † e (cid:33)(cid:33) (18) • Finally the partition function of SL (2 , C ) BF the-ory is Z BF = (cid:88) k f I e (cid:90) d p f (cid:89) f ( k f + p f ) (cid:89) v A v (cid:16) ( k, p ) f , I e (cid:17) (19)This expression, however, is ill defined, due to the factthat the intertwiners are generalized vectors, and thetrace (18) may diverge. This issue is addressed and an-swered in [32], where it is shown that the source o f thedivergence is a redundant integral over SL (2 , C ) in thedefinition of A v . It is then immediate to regularize A v by removing one SL (2 , C ) integration per each vertex.The resulting amplitude is proven in [32] to be finite, ex-cept for some particular pathological vertices, which weexclude here for simplicity. In what follows we alwaysassume that the vertex amplitude is so renormalized. C. Boundary Hilbert Space
Let us rewrite the partition function (3) in a slightlydifferent form. Split each edge e bounded by the vertices v and v (cid:48) into two half edges ( ev ) and ( ev (cid:48) ), and associatea group element g ev to each half edge (oriented towardsthe vertex). Then replace each integral dg e with the twointegrals dg ev , dg ev (cid:48) . This gives Z BF ( K ) = (cid:90) dg ev (cid:89) f δ (cid:0)(cid:89) e ∈ ∂g ( g − ev g ev (cid:48) ) (cid:15) ef (cid:1) , (20)where there is one integration per each couplevertex/adjacent-edge. Next, let v be a vertex in theboundary of the face f . For each such couple fv , in-troduce a group variable g fv . Then (20) can be rewrittenin the form Z BF ( K ) = (cid:90) dg fv dg ev (cid:89) f δ ( (cid:89) v ∈ ∂f g fv ) (cid:89) fv δ ( g − fv g ev g − e (cid:48) v ) (21)where e and e (cid:48) are the two edges in the boundary of f that meet at v , ordered by the orientation of f . This canbe rewritten in the form Z BF ( K ) = (cid:90) dg fv (cid:89) f δ (cid:0) (cid:89) v ∈ ∂f g fv (cid:1) (cid:89) v A v ( g fv ) (22)where the vertex amplitude A v ( g f ) is defined by A v ( g f ) = (cid:90) (cid:89) e ∈ ∂v dg e (cid:89) f ∈ ∂v δ ( g e f g f g − e (cid:48) f ) (23) is a function of one group element for each face in theboundary of v . Here the integral is over one group ele-ment per each edge in the boundary of the vertex v and,as before, e and e (cid:48) are the two edges in the boundary of f that meet at v . This is the “holonomy” form of thepartition function [30].Let | F v | be the number of links f of the graph γ v ,namely the number of faces f in ∂v . The vertex ampli-tude (23) is a function in H γ v = L [ G | F γ | ] . (24)We call this the (non-gauge invariant) boundary Hilbertspace of the vertex v . It is easy to se that the vertex am-plitude (23) is an element of this space. More precisely,it is an element of the (possibly generalized) subspace K γ v = L [ G | F γ | /G | E γ | ] (25)where | E γ | is the number of nodes of γ v , namely thenumber of edges in ∂v , formed by the states invariantthe gauge transformation ψ ( g e ) = ψ (Λ s e g e Λ t e ) (26)where Λ ∈ G and s e and t e are the source and target of e . A moment of reflection shows also that (10) and (18)are simply the amplitude (23) expressed in the standardspin network basis of K γ v . Let us now study the bound-ary space H γ v in more detail. (It is convenient to con-sider the non-gauge-invariant Hilbert space H γ v , besidesthe gauge invariant one because the expressions of geo-metric constraints will not be gauge invariant, thus theycan only be represented as operators on H γ v .)The natural derivative operator defined on the Hilbertspace L [ G ] is the left invariant derivative that generatesthe right G action: J IJ ψ ( g ) = dd α ψ ( e αT IJ g ) (cid:12)(cid:12)(cid:12) α =0 (27)where T IJ ( I, J = 0 , · · · ,
3) is a standard Lie algebragenerator of
Lie ( G ).Fix an SU(2) subgroup of G , and choose a basis in Lie ( G ) such that the direction I = 0 is preserved by SU (2). Then we can split the six generators T IJ of Lie ( G ) into 3 rotation generators and 3 boost genera-tors. Accordingly, we define ( i, j, k = 1 , , L i := 12 (cid:15) ijk J jk , K i := J i (28)which have the standard commutation relations (cid:2) L i , L j (cid:3) = (cid:15) ijk L k , (29) (cid:2) K i , K j (cid:3) = s(cid:15) ijk L k , (30) (cid:2) K i , L j (cid:3) = (cid:15) ijk K k (31)where s = +1 for Spin (4) and s = − SL (2 , C ).We denote by J IJf the left invariant derivative operatoracting on the variable g f of ψ ( g f ) ∈ H γ v . Notice that theright invariant vector field R IJ ψ ( g ) = dd α ψ ( ge αT IJ ) (cid:12)(cid:12)(cid:12) α =0 (32)satisfies R IJ ψ ( g ) = J IJ ψ ( g − ). Therefore J IJf − = R IJf . (33)The bivector operators J IJf have a physical interpreta-tion in terms of the BF theory we started from. They arethe quantum operators that quantize the discretized ver-sion of the 2-form field B IJ , restricted to a 3-dimensionalboundary. The reason for this is the follows: Classicallythe Hamiltonian analysis of BF theory can be carried out[33]. The resulting non-vanishing Poisson bracket reads (cid:110) (cid:15) abc B abIJ ( x ) , A KLd ( x (cid:48) ) (cid:111) = δ cd δ K [ I δ LJ ] δ ( x, x (cid:48) ) (34)where a, b, c = 1 , , x and x (cid:48) belong to a 3-dimensionalspatial manifold S . These canonical conjugate variablescan be discretized in analogy with Hamiltonian latticegauge theory. Given a graph γ imbedded in S , there ex-ists a 2-cell complex dual to the graph γ , such that givena link f in the graph there is a unique 2-face S f dualto the link f . This 2-cell complex defines a polyhedraldecomposition of the spatial manifold σ . With this set-ting, we associate a group variable g f ∈ G to each link f , and associate a Lie algebra variable B IJf to each S f (the Lie algebra variables are also labeled by f becauseof the 1-to-1 correspondence between links and 2-faces).The Poisson algebra of these discretized variables has thefollowing standard expression (cid:110) g f , g f (cid:48) (cid:111) = 0 (cid:110) B IJf , g f (cid:48) (cid:111) = δ ff (cid:48) T IJ g f (cid:110) B IJf , B
KLf (cid:48) (cid:111) = δ ff (cid:48) f IJ,KLMN B MNf (35)where f IJ,KLMN denotes the structure constant of
Lie ( G ). In our case, if we consider our boundary graph γ v and abstractly define the above Poisson algebra on γ v ,we find that the bivector operator J IJf for each orientedlink f (as a right invariant vector) is the quantum op-erator representing the Lie algebra variable B IJf (up to − i (cid:126) ), because of the commutation relation between J IJf and g f on the boundary Hilbert space. III. BOUNDARY QUANTUM GEOMETRY
We now consider a modification of BF theory. Themodification is obtained by restricting the boundaryspace H γ v by imposing a certain constraint. Let us firstdefine this constraint and then discuss the consequencesand the motivation of imposing it. A. Geometric Constraints
Consider a vertex v and its boundary graph γ v . Foreach link f , consider the Lie algebra element Σ given by B f = ∗ Σ f + 1 γ Σ f (36)where the star indicates the Hodge dual in the Lie alge-bra. Consider a node e of the boundary graph γ v , andlet f ∈ ∂e be all oriented away from e . Then define
1. Simplicity Constraint:
There exists a unit vector( n e ) I for each e such that, for all f ∈ ∂e ( n e ) I ∗ Σ IJf = 0 . (37)
2. Closure Constraint: (cid:88) f ∈ ∂e Σ IJf = 0 . (38)These are the two constraints on which we focus. Themain motivation for considering these constraints is thefact that the action of general relativity in the Holst for-mulation can be written in the form S GR [ e, ω ] = (cid:90) B ∧ F [ ω ] (39)where ω is an SL (2 , C ) connection, B = ∗ Σ + 1 γ Σ (40)and Σ IJ = e I ∧ e J (41)where e I is the tetrad one form. The restriction Σ IJf (cid:12)(cid:12)(cid:12) B ofΣ to any space-like boundary B satisfies the conditions: n I Σ IJ (cid:12)(cid:12) B = 0 (42)where n I is the normal to the boundary and d Σ = 0 . (43)Equations (36), (37) and (38) can be seen as a discreteconsequence of equations (40), (42) and (43). Here, how-ever, we take the discretized equations (36), (37) and (38)as our starting point, and study their consequences. Afull discussion on the relation of these equations with con-tinuum general relativity will be considered elsewhere. The Plebanski simplicity constraint implies the constraints givenhere. However the reverse is not true in general, unless “shape-matching” conditions [22] are imposed on each face shared by twopolyhedra. We do not demand such shape-matching conditionshere. There is some evidences from the large- j behavior of thegeneralized spinfoam model that non-shape-matching amplitudesare suppressed in the large- j asymptotic [25]. The key consequences of these constraints is that theyallow Σ to determine a classical polyhedral geometry ateach node e of the boundary graph γ v . (See also [22].)This follows from the following Theorem III.1.
Given an F-valent node e in γ v , let F bivectors Σ f satisfy (37) and (38). Then there exists a(possibly degenerate) flat convex polyhedron in R with F faces, whose face area bivectors coincide with Σ IJf . Theresulting polyhedron is unique up to rotation and trans-lation.
Proof:
Without loss of generality, we fix the unitvector ( n e ) I = (1 , , ,
0) (we call this the time-gauge).The simplicity constraint eq.(37) reduces toΣ if = 0 . (44)Hence the surviving components of Σ IJf are Σ ijf . Wedenote these nonvanishing components simply by Σ if = (cid:15) ijk B jk or (cid:126) Σ f , in terms of which the closure constraint(38) reads (cid:88) f (cid:126) Σ f = 0 . (45)Consider (cid:126) Σ f as vectors in R . Call | Σ f | the length ofthe 3-vector (cid:126) Σ f , and let (cid:126)n f := (cid:126) Σ f / | Σ f | . We first supposethe unit vectors (cid:126)n f are non-coplanar. Then we recallMinkowski’s Theorem [21], which states that wheneverthere are F non-coplanar unit 3-vectors (cid:126)n f and F positivenumbers A f satisfying the condition (cid:88) f A f (cid:126)n f = 0 , (46)then there exists a convex polyhedron in R , whose faceshave outward normals (cid:126)n f and areas A f . And the result-ing polyhedron is unique up to rotation and translation. When we apply Minkowski’s theorem to our case, wesee that the existence of the unit 3-vectors (cid:126)n f and thelengths | Σ f | , as well as the closure constraint eq.(45),together imply that there is a convex polyhedron in R ,unique up to translation and rotation, such that each (cid:126)n f is an outward normal of a face and each | Σ f | is an area ofa face. Such a polyhedron can be concretely constructedvia Lasserres reconstruction algorithm [34]. Let e i thenatural triad in R , then the 3-vector (cid:126) Σ f can be expressedas an oriented area:Σ ijf = (cid:90) f e i ∧ e j . (47) Imagine the polyhedron immersed in a homogeneous fluid.Eq.(46) multiplied by the pressure is the sum of the pressureforces acting on the faces, which obviously vanishes. K boundary graph γ v boundary 3d geometry e edge node polyhedron f face link face of polyhedronTABLE I: The different geometrical interpretations of the la-bels e and f . Finally, the case of coplanar unit 3-vectors (cid:126)n f canbe obtained as a limit of non-coplanar case, yieldingdegenerate polyhedra. (cid:3) This geometrical interpretation equips the variables e and f with a further new meaning: they represent, re-spectively, polyhedra in a 4d space and faces of thesepolyedra. See Table 1.The geometrical interpretation in terms of tetrahedra(and now polyhedra) has raised a lively discussion and itis sometimes unpalatable to the more canonical-orientedpart of the community. Part of this discussion is basedon misunderstanding. The precise claim here is that if wetake the diff-invariant Hilbert space of the theory and we truncate it to a finite graph (so that the observable alge-bra is also truncated), then the truncated Hilbert space(with its observables algebra) has a classical limit, andthis classical limit can be naturally interpreted as de-scribing a collection of polyhedra. This is well consistentwith classical general relativity, because classical generalrelativity as well admits truncations where the geome-try is discretized. Also, this is not inconsistent with thecontinuous picture for the same reason for which the factthat the truncation of Fock space to an n particle Hilbertspace describes discrete particles, is not inconsistent withthe fact that Fock space itself describes a (quantized)field.Let us now see how the constraints translate on thevariable B given in (36). We have easily: Simplicity Constraint: C Jf = n I (cid:18) ∗ B IJf − sγ B IJf (cid:19) = 0 , (48) Closure Constraint: G IJe = (cid:88) f ∈ e B IJf = 0 , (49)where s = +1 for Spin (4) and s = − SL (2 , C ).Consider a single polyhedron e , with the time-gauge( n e ) I = (1 , , , L jf := (cid:15) jkl B klf and boost K jf := B jf components of B IJf . Thenthe simplicity constraint (48) becomes simply (cid:126)K f = sγ (cid:126)L f ; (50)the rotation generators are proportional to the boost gen-erators. The closure constraint (49) can be written as (cid:88) f ∈ ∂e (cid:126)L f = 0 , (51a)and (cid:88) f ∈ ∂e (cid:126)K f = 0 . (51b)where the second, eq.(51b), is redundand, by eq.(50).Let us now move to the quantum theory, and imposethe two constraints (50) and (51a) weakly [9, 13] on thequantum states. This gives Simplicity Constraint: (cid:68) ψ, (cid:126)K f ψ (cid:48) (cid:69) = sγ (cid:68) ψ, (cid:126)L f ψ (cid:48) (cid:69) . (52) Closure Constraint: (cid:88) f ∈ ∂e (cid:68) ψ, (cid:126)L f ψ (cid:48) (cid:69) = 0 (cid:88) f ∈ ∂e (cid:68) ψ, (cid:126)K f ψ (cid:48) (cid:69) = 0 , (53)These equations define a subspace H Eγ v (respectively H Lγ v in Lorentzian case) of the boundary Hilbert space H γ v of BF theory, where the constraints hold weakly. Thatis, we define H Eγ v as the subspace where these equationshold for any two states ψ and ψ (cid:48) in a dense domain, forall nodes s of γ v . B. New Boundary Hilbert Space:Euclidean Theory
Let us now construct H Eγ v . Here we first define H Eγ v and then prove that it solves the geometric constraint.We begin with some preliminaries on the structure of theBF boundary Hilbert space. In the Euclidean theory, thisspace has the following decomposition H γ v = (cid:79) f L [ Spin (4)] = (cid:79) f (cid:77) ρ f V ρ f ⊗ V ∗ ρ f . (54)where V ρ denote the representation space for the Spin(4)irrep ρ and V ∗ ρ is the representation space for the Spin(4)adjoint irrep ρ ∗ . For each face f , V ρ f and V ∗ ρ f transformsin a gauge transformation (26) under the action of Λ s f Λ t f , where s f and t f are the initial and final points ofthe link f . By regrouping all representations space thattransform under the action of the same Λ e , namely byregrouping the representation spaces associated to thesame vertex e of γ v we can rewrite the decomposition inthe form H γ v = (cid:77) { ρ f } (cid:79) e (cid:79) f ∈ ∂e V ( e,f ) ρ f (55) where V ( s f ,f ) ρ f := V ρ f V ( t f f ) ρ f := V ∗ ρ f (56)Therefore the sum over colorings ρ f associates a repre-sentation space (cid:79) f ∈ ∂e V ( e,f ) ρ f (57)to each vertex e . This space can be seen as the quanti-zation of the shapes of a polyhedron with faces havingfixed areas, determined by the coloring ρ f [22].Since Spin (4) ∼ SU (2) + × SU (2) − , a unitary irrepof Spin (4) is given by a tensor product of two SU(2)irreps. V ρ = V j + ⊗ V j − with spins j + and j − . We cancharacterize ρ by ρ = ( p, k ), where p = j + + j − , k = j + − j − . (58)The SU (2) ± subgroups of Spin (4) are its canonical self-dual and antiself dual components, generated by (cid:126)L ± (cid:126)K ,and should not be confused with the (non-canonical)SU(2) subgroup generated by (cid:126)L , used to pick a timegauge. If we decompose V ρ = V p,k in irreducibles ofSU(2), we have V p,k = V j + ⊗ V j − = p (cid:77) j = | k | V p,kj . (59)We now define H Eγ v . In the representation space V p,k ,pick the V p,kj subspace (in the decomposition above),where j is defined by p = j + r (60) k = γj − r (61)By doing so, we obtain the subspace V γj + r,j − rj in each V p,k . By restricting in this manner all the V ρ f subspacesin (54) we obtain a subspace of H γ v . We define the non-gauge-invariant new boundary space to be this subspace.That is (cid:77) { j f ,r f } (cid:79) e (cid:79) f ∈ e ( V j f + r f ,γj f − r f j f ) ( e,f ) (62)where the sum is over non-negative half-integers j f and r f . The possible coloring in H Eγ v are labelled by the twonon-negative half-integer quantum numbers j f and r f .The quantum number j f characterizes the SU(2) spin ofthe representation and is easily identified with the corre-sponding LQG quantum number which is associated toeach link of the graph. r f is a new quantum number, alsoassociated to each link of the graph.Notice also that (60) restricts also the possible valuesof j and r to those for which p = γj + r is half integer.This awkward feature of the Euclidean case disappearsin the Lorentzian theory.We can translate all this in terms of the ( j + , j − ) no-tation. This gives j + = 1 + γ j and j − = 1 − γ j + r (63)and the modified γ -simplicity relation (1 − γ ) j + = (1 + γ )( j − − r ) . (67)Next, we define the gauge invariant new boundaryspace. Consider the diagonal actions of h ∈ SU (2) oneach product representation space eq.(57) at each e . Wedenote the invariant subspaces under this actions by I { j f } e = Inv SU (2) (cid:79) f ∈ e ( V j f + r f ,γj f − r f j f ) ( e,f ) (68)The gauge invariant new boundary Hilbert space is de-fined by H Eγ v := (cid:77) { j f ,r f } (cid:79) e I { j f } e . (69)An orthonormal basis in H Eγ v can be constructed asfollows. Given a polyhedron e with F faces, we assign at e an F -valent SU (2) intertwiner i A ··· A F e associated with F SU (2) irreps j f , f = 1 , · · · F . An orthonormal basis isthen defined by the following functions on [ Spin (4)] | E ( γ v ) | T Eγ v ,j f ,r f ,i e ( g f ) = (cid:89) f (cid:113) [(1+ γ ) j f + 1][(1 − γ ) j f + 2 r f + 1] (cid:89) e (cid:20) i A e ··· A eF e (cid:89) f ∈ e C m + ef m − ef A e (cid:21) (cid:89) f (cid:20) (cid:15) n + ef n + e (cid:48) f (cid:15) n − ef n − e (cid:48) f (cid:21)(cid:89) ( e,f ) (cid:20) D γ j f m + ef n + ef ( g + ef ) D − γ j f + r f m − ef n − ef ( g − ef ) (cid:21) (70)here g ef = ( g + ef , g − ef ) ∈ Spin (4), D j ( g ) is the representa-tion matrix of the SU (2) irrep j , and C m + ef m − ef A ef denotes The
Spin (4) irreps for a given Barbero-Immirzi parameter γ ,should be such that r = (1 + γ ) j − − (1 − γ ) j + γ (64)is a non-negative integer, and satisfy0 (cid:54) r (cid:54) j + + j − − | j + − j − | (65)implying | − γ | γ j + (cid:54) j − (cid:54) j + or j + (cid:54) j − (cid:54) γ γ j + . (66) the Clebsch-Gordan coefficient ( A f = − k f , · · · , k f ) (cid:28) γ j f , − γ j f + r f ; j f , A ef (cid:12)(cid:12)(cid:12) (71) (cid:12)(cid:12)(cid:12) γ j f , m + ef ; 1 − γ j f + r ef , m − ef (cid:29) .(cid:15) n ± ef n ± e (cid:48) f are the unique 2-valent SU (2) intertwiners withrepresentations j + f = γ j f and j − = − γ j f + r f respec-tively. Thus T E ( γ v ,j f ,r f ,i e ) is essentially a function over g f = g ef g fe (cid:48) . Note that if we ask the quantum numbers r f to be some fixed integers, then the spin-network func-tions T E ( γ v ,j f ,r f ,i e ) can be equivalently considered as an SU (2) spin-network functions, thus the boundary Hilbertspace is spanned by SU (2) spin-networks, as the case ofLQG kinematical Hilbert space.We are now ready to prove our first main result. Theorem III.2.
The Hilbert space H Eγ v solves the geo-metric constraint (52-53), with s = 1 . Proof:
The closure constraint (53) follows im-mediately since the states in H Eγ v is invariant underthe diagonal SU (2 ∂ gauge transformation ( g + ef , g − ef ) (cid:55)→ ( h e g + ef , h e g − ef ) at each e (the constraint is even solvedstrongly). The nontrivial proof is for the simplicity con-straint (52). Define the self-dual/anti-self-dual operators: (cid:126)J ± f := 12 ( (cid:126)L f ± (cid:126)K f ) (72)then (52) reads(1 − γ ) (cid:68) ψ, (cid:126)J + f ψ (cid:48) (cid:69) E − (1 + γ ) (cid:68) f, (cid:126)J − ψ ψ (cid:48) (cid:69) = 0 . (73)The operators (cid:126)J ± f on L ( Spin (4)) act on individual V ( e,f ) ρ f (see, e.g. Sec.32.2 of [4]). Therefore we only need to showthat in each Clebsch-Gordan subspace V ρ =( j + ,j − ) j , with j + ≡ γ and j − ≡ − γ k + r , the following relation holdsfor all pairs Φ , Ψ of vectors(1 − γ ) (cid:104) Ψ | (cid:126)J + | Φ (cid:105) − (1 + γ ) (cid:104) Ψ | (cid:126)J − | Φ (cid:105) = 0 (74)where (cid:104) | (cid:105) is the Hermitian inner product on the Spin (4)irrep V ρ =( j + ,j − ) .To evaluate these matrix elements, we use the explicitrepresentation of the vectors as multi-spinors. The vec-tors in the SU (2) irrep V j can be represented as totallysymmetric spinorial tensors with 2 j spinor indices. Thegenerators of SU (2) are then Pauli matrices σ Ai B actingon each index, followed by a sum. A state | Φ (cid:105) in H j inthe Clebsch-Gordan subspace V j + ,j − j ⊂ V j + ⊗ V j − can beexpressed by ( A i , B i = 1 , A ...A j + ,B ...B j − = (75) (cid:15) A B ...(cid:15) A r B r φ A r +1 ...A j + ,B r +1 ...B j − , A , ...A j + ) indicesunderstood and the same for the ( B , ..., B j − ) indices. The action of (cid:126)J − on the state Φ in (75), can then becomputed explictly, giving J − i Φ ( A ...A j + )( B ...B j − ) = j − (cid:88) p =1 σ B p i (cid:101) B p Φ ( A ...A j + )( B ... (cid:101) B p ...B j − ) (76)= r (cid:88) p =1 σ B p i (cid:101) B p (cid:15) A B ...(cid:15) A p (cid:101) B p ...(cid:15) A r B r φ ( A r +1 ...A j + B r +1 ...B j − ) + j − (cid:88) p = r +1 σ B p i (cid:101) B p (cid:15) A B ...(cid:15) A r B r φ ( A r +1 ...A j + B r +1 ... (cid:101) B p ...B j − ) = − r (cid:88) p =1 σ A p i (cid:101) A p (cid:15) A B ...(cid:15) (cid:101) A p B p ...(cid:15) A r B r φ ( A r +1 ...A j + B r +1 ...B j − ) + j − (cid:88) p = r +1 σ B p i (cid:101) B p (cid:15) A B ...(cid:15) A r B r φ ( A r +1 ...A j + B r +1 ... (cid:101) B p ...B j − ) where in the third step, we use the identity σ Bi (cid:101) B (cid:15) A (cid:101) B = − σ Ai (cid:101) A (cid:15) (cid:101) AB coming from the SL (2 , C ) invariance of (cid:15) AB . Thenthe matrix elements of (cid:126)J − are (cid:104) Ψ | J − i | Φ (cid:105) = − r (cid:88) p =1 σ A p i (cid:101) A p Ψ ( A ...A p ...A j + )( B ...B j − ) Φ ( A ... (cid:101) A p ...A j + )( B ...B j − ) + j − (cid:88) p = r +1 σ B p i (cid:101) B p (cid:15) A B ...(cid:15) A r B r ψ ( A r +1 ...A j + B r +1 ...B p ...B j − ) (cid:15) A B ...(cid:15) A r B r φ ( A r +1 ...A j + B r +1 ... (cid:101) B p ...B j − ) = ( − r ) σ A j + i (cid:101) A j + Ψ ( A ...A j + )( B ...B j − ) Φ ( A ... (cid:101) A j + )( B ...B j − ) +(2 j − − r ) σ A j + i (cid:101) A j + (cid:15) A B ...(cid:15) A r B r ψ ( A r +1 ...A j + B r +1 ...B j − ) (cid:15) A B ...(cid:15) A r B r φ ( A r +1 ... (cid:101) A j + B r +1 ...B j − ) = 2( j − − r ) σ A j + i (cid:101) A j + Ψ ( A ...A j + )( B ...B j − ) Φ ( A ... (cid:101) A j + )( B ...B j − ) (77)Similarly, (cid:104) Ψ | J + i | Φ (cid:105) = 2 j + σ A j + i (cid:101) A j + Ψ ( A ...A j + )( B ...B j − ) Φ ( A ... (cid:101) A j + )( B ...B j − ) . (78)Then eq.(74) follows immediately(1 − γ ) (cid:104) Ψ | J (+) i | Φ (cid:105) − (1 + γ ) (cid:104) Ψ | J ( − ) i | Φ (cid:105) = 2 (cid:2) (1 − γ ) j + − (1 + γ )( j − − r ) (cid:3) σ A j + i (cid:101) A j + Ψ ( A ...A j + )( B ...B j − ) Φ ( A ... (cid:101) A j + )( B ...B j − ) = 0 (79)which proves the simplicity constraint eq.(52). (cid:3) C. New Boundary Hilbert Space:Lorentzian Theory
Now we turn to the case of G = SL (2 , C ). In this casethe decomposition of the Hilbert space reads H γ v = (cid:79) f L (cid:16) SL (2 , C ) , d µ H (cid:17) (80)= (cid:79) f (cid:77) k f = N / (cid:90) ⊕ R d p f (cid:0) p f + k f (cid:1) V ( k f ,p f ) ⊗ V ∗ ( k f ,p f ) where k f are still non-negative half-integers but p f ∈ R is now a real number. Here (cid:82) ⊕ denotes a direct integraldecomposition [35] (see also Chapter 30 of [4]). V ( k,p ) denotes the unitary irrep of SL (2 , C ) in the principal se-ries, and V ∗ ( k,p ) denotes the adjoint irrep. We can thenproceede as in the EUclidean theory. The BF boundaryHilbert space reads H γ v = (cid:77) { k f } (cid:89) f (cid:90) ⊕ R d p f (cid:89) f (cid:0) p f + k f (cid:1) (cid:79) e (cid:79) f ∈ e V ( e,f )( k f ,p f ) (81)1The representation space V ( k,p ) is infinite-dimensionaland can be decomposed into SU (2) irreps (irreps of thesubgroup generated by (cid:126)L ), i.e. V ( k,p ) = ∞ (cid:77) j = k V k,pj . (82)This time we introduce the two parameters j and r by p = γj j + 1 j − r , (83) k = j − r. (84)and we define the new boundary space by restricting each V ( k,p ) to its V k,pj subspace satisfying (83). This time p does not need to be half-integer, therefore (83) can besolved for any j . The new quantum numbers associatedto each face are j f and r f , each being a nonnegative halfinteger.As before, we consider the diagonal SU (2) action ateach e for all h e ∈ SU (2). The invariant subspace underthis action is I j f e = Inv SU (2) (cid:79) f ∈ e (cid:18) V γjf ( jf +1) jf − rf ,j f − r f (cid:19) ( e,f ) (85)The new boundary Hilbert space is defined by a prod-uct of these invariant subspaces over all the polyhedra e ,followed by a sum over all the possible j f and r f : H Lγ v := (cid:77) { r f ,j f } (cid:79) e I j f e (86)where j f and k f are non-negative half-integers with con-straints (1) j f ≥ r f . H Lγ v is a direct sum over a set ofsubspaces contained in the fiber Hilbert spaces of H γ v (see eq.(80)), thus has well-defined inner product.An orthonormal basis is constructed as follows. Con-sider the oriented boundary graph γ v . Given a F -valentvertex/polyhedron e , we assign it an intertwiner i A ··· A F e associated with F spins j f , f = 1 , · · · , Fi e ∈ Inv (cid:79) −−−→ ( e,f ) outgoing V j f (cid:79) −−−→ ( e,f ) incoming V ∗ j f (87)An orthogonal basis in H Lγ v is given by the following func-tions (distributions) on SL (2 , C ) T L ( γ v ,j f ,r f ,i e ) ( g f ) = (88) (cid:89) e i A e ··· A eF e (cid:89) ( e,e (cid:48) ) Π ( γjf ( jf +1) jf − rf ,j f − r f ) j f A ef ,j f A e (cid:48) f ( g f )here Π ( p,k ) denotes the representation matrix in SL (2 , C )irrep labeled by ( p, k ). All the A ef indices of the repre-sentation matrices are contracted with the A ef indices ofthe intertwiners. The new boundary Hilbert space H Lγ v is not a sub-space of the BF boundary Hilbert space H γ v , because T L ( γ v ,j f ,r f ,i e ) are constructed by Π ( k,p ) which are distri-butions. In order to check the geometric constraintsEqs.(52) and (53) on H Lγ v , we have to compute the(dual) action of the bivector operator on the distributions T L ( γ v ,j f ,k f ,i e ) . Fortunately the Hilbert space L (cid:0) SL (2 , C ))has the structure of direct integral decomposition (seeeq.(80)). Then the (dual) action of the bivector opera-tors (cid:126) ˆ K and (cid:126) ˆ L gives the actions of Lie algebra generators (cid:126)L and (cid:126)K on each fiber Hilbert space V ( k,p ) .We are now ready to proove our second main result Theorem III.3.
The Hilbert space H Lγ v solves the geo-metric constraint (52,53), with s = − . Proof:
Closure constraint follows immediately andstrongly by the diagonal SU (2) invariance at each poly-hedron e . We only need to consider a single irrep V ( k,p ) ( p = γj ( j +1) k ) because (cid:126)L and (cid:126)K leave it invariant and,different ( p, k )’s label orthogonal subspaces in H Lγ v .A canonical basis in V ( p,k ) is obtained diagonalizing theCasimir operators J · J, ∗ J · J, L · L and L . The basiscan be denoted | ( p, k ); j, m (cid:105) or simply as | j, m (cid:105) since weonly consider a single irrep. On this canonical basis, thegenerators act in the following way [36]: L | j, m (cid:105) = m | j, m (cid:105) ,L + | j, m (cid:105) = (cid:112) ( j + m + 1)( j − m ) | j, m + 1 (cid:105) ,L − | j, m (cid:105) = (cid:112) ( j + m )( j − m + 1) | j, m − (cid:105) ,K | j, m (cid:105) = − α ( j ) (cid:112) j − m | j − , m (cid:105) − β ( j ) m | j, m (cid:105) + α ( j +1) (cid:112) ( j + 1) − m | j + 1 , m (cid:105) ,K + | j, m (cid:105) = − α ( j ) (cid:112) ( j − m )( j − m − | j − , m + 1 (cid:105)− β ( j ) (cid:112) ( j − m )( j + m + 1) | j, m + 1 (cid:105)− α ( j +1) (cid:112) ( j + m + 1)( j + m + 2) | j + 1 , m + 1 (cid:105) ,K − | j, m (cid:105) = α ( j ) (cid:112) ( j + m )( j + m − | j − , m − (cid:105)− β ( j ) (cid:112) ( j + m )( j − m + 1) | j, m − (cid:105) + α ( j +1) (cid:112) ( j − m + 1)( j − m + 2) | j + 1 , m − (cid:105) , where L ± = L ± iL , K ± = K ± iK (89)and α ( j ) = ij (cid:113) ( j − k )( j + p )4 j − , β ( j ) = kpj ( j +1) (90)Using these equations, one can check directly that (cid:104) j, m (cid:48) | (cid:0) K i + β ( j ) L i (cid:1) | j, m (cid:105) = 0 . (91)which is nothing but (cid:104) j, m (cid:48) | (cid:0) K i + γL i (cid:1) | j, m (cid:105) = 0 . (92)because pk = γj ( j + 1). (cid:3) D. Quantum Polyhedral Geometry
In this section we show that the boundary Hilbertspace H Eγ v and H Lγ v carries a representation of quantumpolyhedral geometry, consistent with the classical poly-hedral geometry that we have discussed in Section III A.Recall that we defined two different bivectors J IJef andΣ
IJef related by B IJf = (cid:18) ∗ Σ f + 1 γ Σ f (cid:19) IJef (93)Theorem III.1 states that classically, the geometric con-straint of B IJf implies that B IJf is the area bivector of aface f of a polyhedron e . On the BF boundary Hilbertspace H γ v the bivector B IJf is quantized to be the leftinvariant vector field J IJf . Inverting the above equation,we can write the quantum operator corresponding to Σ(which we indicate with the same symbol) asΣ
IJf := γ γ − s (cid:18) ∗ J IJef − γ J IJef (cid:19) (94)Give a polyhedron/vertex e of the boundary, if we choosethe unit vector ( n e ) I = (1 , , , jf for each face f .That is, the matrix elements of the operators Σ if vanishon H Eγ v and H Lγ v , thus we consider them as vanishingoperators on H Eγ v or H Lγ v . The nontrivial operator on H Eγ v and H Lγ v isΣ if ≡ (cid:15) ijk Σ jkf = γ γ − s (cid:18) ˆ K if − γ ˆ L if (cid:19) (95)Because of the quantum simplicity constraint (52), wecan identify ˆ K ief with sγ(cid:126)L ef on the dense domain of thenew boundary Hilbert space, as far as the matrix ele-ments of the operators are concerned. Thus, in the senseof their matrix element (cid:126) Σ f = sγ (cid:126)L f (96)By the SU (2) gauge invariance, then (cid:88) f ∈ ∂e ˆΣ f = 0 (97)(with all f ’s oriented out of e .) Consider now a family ofcoherent states that makes the spread of these operatorssmall. These coherent states are then characterized byeigenvalues of (cid:126) Σ f that satisfy the equation above. ByMinkowski theorem, they determine a polyhedron e ateach vertex. (cid:126) Σ ef represents the normal to face area ofthe polyhedron e , normalized so that its norm is the areaof the face [22]. The area operator for a face f (in unitsthat 8 π(cid:96) p = 1 [1]) is thenˆ A f = γ (cid:113) ˆ L ief ˆ L ief = γ (cid:113) j f ( j f + 1) . (98) It is clear that the area operator doesn’t depend on theorientation of the face. Thus the two areas of the twofaces of the two polyhedra e and e (cid:48) that are determinedby the same face f are equal. (Recall that the one ofthe two is determined by the left invariant vector field J and the other by the right invariant vector field R , since R f = J f − .)At fixed values of the areas, the shapes of the polyhedrais described by the intertwiner spaces at each e . We recallthat an over-complete basis in these spaces is formed bythe Livine-Speziale coherent intertwiners [10] || (cid:126)j, (cid:126)n (cid:105) := (cid:90) SU (2) d µ H ( g ) (cid:89) f ⊂ e D j f ( g ) | j f , n f (cid:105) (99)These can be labeled [16] by the elements in × f S /SL (2 , C ). Thinking of S as the compactified com-plex plane of z f , a coherent intertwiner is determined by F quantum area j f and F − (cid:126)ZZ k = ( z k +3 − z )( z − z )( z k +3 − z )( z − z ) (100)which are invariants of SL (2 , C ). The space of thesecross-ratio × f S /SL (2 , C ) can be identified [38] with theKapovich and Millson phase space S F [39], which is also the space of shapes of polyhedra at fixed areas j f . Thus,we can label the coherent intertwiner by || (cid:126)j, (cid:126)Z (cid:105) , in vari-ables that relate directly to the shape of the polyhedron.The resolution of identity in the intertwiner space canbe expressed as a integral over the Kapovich and Millsonphase space S F , i.e. I ( (cid:126)j ) = (cid:90) S F d µ ( (cid:126)Z ) || (cid:126)j, (cid:126)Z (cid:105) (cid:104) (cid:126)j, (cid:126)Z || (101)where the explicit expression of the measure d µ ( (cid:126)Z ) isgiven in [16]. Finally the volume operator for a polyhe-dron can be defined as in [22], in terms of the classicalvolume of a polyhedron and the coherent intertwiner.Notice that the quantum polyhedral geometry doesn’tdepend on the quantum numbers r f . The quantumnumbers r f don’t affect the quantum 3-geometry on theboundary. IV. AMPLITUDESA. Vertex Amplitude: Euclidean theory
If we take BF theory and restrict all vertex-boundaryspaces to H Eγ v (or H Lγ v ) we obtain a new dynamical model.Here we give explicitly its vertex and face amplitude.Let’s start with the Euclidean case. The BF vertex am-plitude can be written in the holonomy representation:(each edge joining at v is uniquely determined by a ver-3tex/polyhedron e on the boundary) reads A v ( g f ) = (cid:88) j ± f ,i ± e (cid:89) f (cid:113) j + f + 1 (cid:113) j − f + 1 A v ( j + f , j − f ; i + e , i − e ) T BFγ v ,j ± f ,i ± e ( g f ) (102)Here A v ( j + f , j − f ; i + e , i − e ) = tr (cid:32)(cid:79) e ∈ v I † e (cid:33) (103)where I = ( i + , i − ) and we assume the valence of v is n . T BF ( γ v ,j ± f ,i ± e ) ∈ H γ v is a Spin (4) spin-network function onthe boundary graph γ v T BFγ v ,j ± f ,i ± e ( g f ) := T γ v ,j + f ,i + e ( g + f ) T γ v ,j − f ,i − e ( g − f ) (104)where T γ v ,j f ,i e ( g f ) = (cid:89) f (cid:112) j f + 1 (cid:89) e (cid:20) ( i e ) { m ef } (cid:21) (105) (cid:89) ( e,f ) (cid:20) D j f m ef n ef ( g ef ) (cid:21) (cid:79) f (cid:20) (cid:15) n ef n e (cid:48) f (cid:21) The vertex amplitude eq.(102) is a distribution of theboundary Hilbert space H γ v , i.e. there is a dense domainof H γ v spanned by the spin-network functions T BF ( γ v ,j ± f ,i ± e ) ,such that A v ( g ee (cid:48) ) lives in the algebraic dual of this densedomain. After imposing the geometric constraint, we re-strict ourself to the subspace H Eγ v . Such a restrictionresults in a (dual) projection of the vertex amplitude A v ,i.e. we obtain A Ev ( g f ) = (cid:88) j f ,r f ,i e (cid:10) T γ v ,j f ,r f ,i e , A v (cid:11) T Eγ v ,j f ,r f ,i e ( g f ) (106)where T γ v ,j f ,r f ,i e is a orthonormal basis of H Eγ v (recalleq.(70)), and (cid:104) , (cid:105) is the inner product of the BF bound-ary Hilbert space H γ v . The evaluation of A Ev is straight-forward: A Ev ( g f ) = (cid:88) j f ,r f ,i e (cid:89) f (cid:113) j + f + 1 (cid:113) j − f + 1 (107) (cid:88) i + e ,i − e A v (cid:16) j + f , j − f ; i + e , i − e (cid:17) (cid:89) e f i e i + e ,i − e T Eγ v ,j f ,r f ,i e ( g ee (cid:48) )where we write j + ≡ γ j and j − ≡ − γ j + r and foreach F -valent boundary polyhedron/vertex f i e i + e ,i − e = i A e ··· A eF e C m + e m − e A e · · · C m + eF m − eF A eF ( i + e ) m + e ··· m + eF ( i − e ) m − e ··· m − eF (108)Then in the ( j f , r f , i e )-spin-network representation, thevertex amplitude is A Ev ( j f , r f , i e ) = (cid:88) i + e ,i − e A v (cid:16) j + f , j − f ; i + e , i − e (cid:17) (cid:89) e f i e i + e ,i − e (109) which nontrivially depends on the quantum numbers r f via the definition of j − f .There is another way to write this vertex amplitudein ( j f , r f , i e )-spin-network representation. Define a map I { r f } E from SU (2) intertwiners to Spin (4) intertwiners,depending on the quantum numbers r f . Given an F -valent SU (2) intertwiner i e with spins k , · · · , k F , let I r f E : i e (cid:55)→ I r f E ( i e ) = i A e ··· A eF e C n + e n − e A e · · · C n + eF n − eF A eF (cid:90) d g + d g − (cid:89) f ∈ e D γ j k m + ef n + ef ( g + ) D − γ j f + r f m − ef n − ef ( g − ) (110)Given an edge e ∈ E ( K ), we associate an intertwiner I { r f } E ( i e ) to the inital point of the edge e , and a dualintertwiner I { r f } E ( i e ) † to the final point of e . Then thevertex amplitude A Ev can be written a spinfoam trace ofthe intertwiners I { r f } E ( i e ) A Ev ( k f , r f , i e ) = tr (cid:32)(cid:79) e ∈ v I { r f } E ( i e ) † (cid:33) (111)where we have again assumed that all the edges joiningat v are oriented towards v . B. Vertex Amplitude: Lorentzian theory
The Lorentzian vertex amplitude can be defined in thesame manner. The SL (2 , C ) BF vertex amplitude is ex-pressed in the holonomy representation as a distribution A v ( g f ) = (cid:88) k f ,I e (cid:90) (cid:89) f d p f (cid:89) f (cid:0) k f + p f (cid:1) (112) A v (cid:16) p f , k f ; I e (cid:17) T BFγ v , ( k,p ) f , ( l , n ) e ( g f )where A v (cid:16) p f , k f ; I e (cid:17) = tr (cid:32)(cid:79) e I † e (cid:33) (113)and T BFγ v ,p f ,k f ,I e ( g f ) = (cid:89) e I { j ef } , { m ef } ; I e (cid:89) f Π p f ,k f j ef m ef ,j e (cid:48) f m e (cid:48) f ( g f )Recall that we always assume the vertex amplitude isassociated with an integrable spin-network graph, thusis finite after regularization [32].We can project A v on the new boundary Hilbert space H Lγ v , in the same way as the Euclidean case A Lv ( g f ) = (cid:88) j f ,r f ,i e (cid:89) f (cid:32) γ j f ( j f + 1) ( j f − r f ) + ( j f − r f ) (cid:33)(cid:68) T Lγ v ,j f ,r f ,i e , A v (cid:69) T Lγ v ,j f ,r f ,i e ( g ee (cid:48) ) (114)4where (cid:104) , (cid:105) is the inner product on the BF boundaryHilbert space. The states T Lγ v ,j f ,r f ,i e ( g f ) = (cid:89) e i A e ··· A eF e (cid:89) ( e,e (cid:48) ) Π γjf ( jf +1) jf − rf ,j f − r f j f A ef ,j f A e (cid:48) f ( g f )form an orthogonal basis in H Lγ v . By using the orthogo-nality relation (cid:90) SL (2 , C ) d g Π ( p,k ) jm,ln ( g ) Π ( p (cid:48) ,k (cid:48) ) j (cid:48) m (cid:48) ,l (cid:48) n (cid:48) ( g ) =1 k + p δ kk (cid:48) δ ( p − p (cid:48) ) δ jj (cid:48) δ ll (cid:48) δ mm (cid:48) δ nn (cid:48) (115)it is straightforward to show that in the ( j f , r f , i e )-spin-network representation, the resulting vertex amplitudereads A Lv ( j f , k f , i e ) = (cid:68) T L ( γ v ,j f ,k f ,i e ) , A v (cid:69) (116)= (cid:88) I e A v (cid:16) ( γj f ( j f + 1) j f − r f , j f − r f ); I e (cid:17) (cid:89) e f i e I e where f i e I e := i { A ef } e I { j f } , { A ef } I e (cid:18) γj f ( j f + 1) j f − r f , j f − r f (cid:19) (117)As expected, the vertex amplitude A Lv obtained in thismanner is divergent, and we need a regularization proce-dure. To this aim, rewrite the vertex amplitude in termsof spinfoam trace as we did for the Euclidean theory. Wedefine a formal map I r f L from SU (2) intertwiners into SL (2 , C ) intertwiners, depending on the quantum num-bers r f I r f L ( i e ) { j (cid:48) f } , { A (cid:48) f } = (cid:90) d g (cid:89) f ⊂ e Π ( γjf ( jf +1) jf − rf ,j f − r f ) j (cid:48) f A (cid:48) ef ,j f A ef ( g ) · i { A ef } e which gives A Lv by a spinfoam trace A Lv ( j f , r f , i e ) = tr (cid:79) f ∈ e I { r f } L ( i e f ) † (118)To regularize the vertex amplitude A Lv it is sufficient toremoving one of the dg integration (which is reduntand)at each vertex. With this, the vertex amplitude A Lv isfinite. C. Face Amplitude and Partition Function
It is argued in [29] that the face amplitude of a spin-foam model is determined by three inputs: (a) the choiceof the boundary Hilbert space, (b) the requirement thatthe composition law holds when gluing two complexes K and K (cid:48) , (c) a particular locality requirement (see [29] for the details of the three assumptions). These require-ments are implemented if the partition function has theform (22). By inserting the vertex amplitudes that wehave defined into this expression, we complete the defi-nition of an Euclidean and a Lorentzian model.Expanding the delta function in representation, we ob-tain Z E,L ( K ) = (cid:88) j f ,r f ,i e (cid:89) f d E,L ( j f , r f ) (cid:89) v A E,Lv ( j f , r f , i e )where the Euclidean face amplitude is d E ( j f , r f ) = (cid:104) (1 + γ ) j f + 1 (cid:105)(cid:104) (1 − γ ) j f + 2 r f + 1 (cid:105) (119)the Lorentzian one is d L ( j f , r f ) = γ j f ( j f + 1) ( j f − r f ) + ( j f − r f ) . (120)where the dimension factors A Ef := (cid:104) (1 + γ ) k f + 1 (cid:105)(cid:104) (1 − γ ) k f + 2 r f + 1 (cid:105) and A Lf := (cid:104) k f + γ j f ( j f + 1) /k f (cid:105) are the face amplitudesfor the Euclidean and Lorentzian theories. In theEuclidean case, the face amplitudes is different fromthe one obtained in [29] and coincide with the onesdeduced from the BF partition function. In [29] the faceamplitude obtained is the dimension of SU (2) unitaryirrep i.e. 2 j f + 1. The origin of the difference is thedifference in the boundary Hilbert space. The one here, H Eγ v or H Lγ v , has additional degree of freedom withrespect to the space L ( SU (2) L ) of [29]. V. THE NEW DEGREE OF FREEDOM ANDRELATION TO QUANTUM GR
Does the new degree of freedom of the theory definedabove, which is captured by the quantum number r f , hasa physical interpretation relevant for quantum gravity?There are some reasons to suspect a negative answer. Letus consider the Euclidean theory for simplicity.First, we have seen that r f does not affect the bound-ary geometry. We expect all gravitational degrees of free-dom to be captured by the geometry. More precisely,in the classical theory we have the well known (“leftarea=right area”) relation | Σ + | = | Σ − | , (121)which implies | − γ | j + = | γ | j − (122)which in turns implies r j = 0. We can still obtain statescompatible with GR in the classical limit by demanding5that lim j ± →∞ rj − = 0 for 0 < γ < j ± →∞ rj − = 2 for γ > j asymptotic regime. But this begins to bea bit artificial.Furthermore, in the classical theory the area of a facecan be equally computed in the time gauge as A = (cid:112) (Σ f ) IJ (Σ f ) IJ or as A = γ (cid:113) Σ if Σ if . Classically thetwo areas A and A are equal after the simplicity con-straint is imposed, and they indeed equal in the large- j limit after quantization [9]. Let us denote the condition A = A the consistency constraint . If we ask A and A to be equal as operators in the quantum level on theboundary Hilbert space (as in the case of [9]), then againthis fixes r f . The precise value of r f fixed depends onhow the operators corresponding to A and A are or-dered. In this sense the quantum numbers r f are relatedto the operator-ordering ambiguities of the consistencyconstraint. Once an order is chosen, there is no moreindependent quantum number r f in the theory. With asuitable ordering, we can fix r f = 0For these consideration, it may be reasonable to sus-pect that the weak imposition of the simplicity con-straints may in fact be too weak to properly define quan-tum general relativity, in the same sense in which thestrong imposition of these constraints in the old Barrett-Crane model was too strong. There is a simple way out,which is to impose the (non-commuting) simplicity con-straints weakly, and the diagonal simplicity constraint(for instance in the form (121)) strongly. With this choiceof constraints, properly ordered, we obtain r f = 0, pre-cisely the LQG state space in the boundary, and pre-cisely the new models amplitudes. Finally, the gluing conditions gives the SU(2) face amplitude. Thus, we re-cover precisely the quantum gravity theory described forinstance in [1].Note that one could also take the point of view thatthe quantum numbers r f label different possible defini-tions of the spin-foam models. In each of these spin-foammodels, the boundary Hilbert space solves the simplicityconstraint weakly. And for different choices of r f theboundary Hilbert spaces are isometric to each other. VI. CONCLUSION AND OUTLOOK
By imposing the simplicity constraints on a quantumBF theory defined on an arbitrary cellular complex, wehave obtained a theory which: (1) is well defined bothin the Euclidean and the Lorentzian context; (2) gener-alizes the existing spinfoam model to general 2-cell com-plexes, along the lines suggested by [20]; (3) has bound-ary state that have a natural interpretation in the semi-classical limit as a polyhedral geometry on the boundary.In particular, we have shown that the KKL extension ofthe spinfoam formalism still satisfies the simplicity con-ditions weakly.The weak simplicity constraint allow a space largerthan the one of LQG to emerge. The physical interpreta-tion of the additional degree of freedom is unclear. It canbe eliminated by imposing the non-commuting simplicityconstraints weakly and the diagonal one strongly.
Acknowledgments
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