Holonomy Operator and Quantization Ambiguities on Spinor Space
aa r X i v : . [ g r- q c ] M a r Holonomy Operator and Quantization Ambiguities on Spinor Space
Etera R. Livine , Johannes Tambornino Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 All´ee d’Italie, Lyon 69007, France
We construct the holonomy-flux operator algebra in the recently developed spinor formulationof loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiargrasping and Wilson loop operators are written as composite operators built from the gauge-invariant‘generalized ladder operators’ recently introduced in the U( N ) approach to intertwiners and spinnetworks. We comment on quantization ambiguities that appear in the definition of the holonomyoperator and use these ambiguities as a toy model to test a class of quantization ambiguities whichis present in the standard regularization and definition of the Hamiltonian constraint operator inloop quantum gravity. I. INTRODUCTION
The recent development of spinor techniques for loop (quantum) gravity has led to various interesting applications,from a better understanding of the discrete geometries underlying spin network states to the derivation of Hamiltonianconstraint operators encoding the dynamics prescribed by spinfoam models (see [1] for a review).To start with, the introduction of spinor variables allowed a compact reformulation of the loop gravity phase space[2–6], with a clear geometrical interpretation as “twisted geometries” generalizing the discrete Regge geometries. Thisbecame particularly relevant for the construction and interpretation of spinfoam models when analyzing the hierarchyof constraints to impose on arbitrary discrete space-time geometries in order to implement a proper quantum version ofgeneral relativity [7]. Furthermore, following the generalization of these spinor variables to twistor networks allowingto describe a Lorentz connection [8, 9], these spinor techniques (or actually upgraded to twistor techniques) allowed toexplore and better understand the phase space structure underlying the discrete path integral defining the spinfoamamplitudes [10–12]. Following a parallel but different line of research, this parametrization of the loop gravity phasespace in terms of spinors naturally led to the definition of coherent states [13–16], which allow a slight modificationand a convenient re-writing of the spinfoam amplitudes. These techniques led to some exact results for the evaluationspinfoam amplitudes [17, 18] or for the dynamics of spinfoam cosmology [19]. More generally, it is possible to describeand define the spinfoam amplitudes and 3nj symbols of the recoupling theory of spins (SU(2) representations) directlyin terms of spinors and their quantization. At a semi-classical level, this allows to derive and study their asymptoticalbehavior (at large spins) (e.g. [20]). At the full quantum level, this spinor techniques lead to differential equations andrecursion relations satisfies by the spinfoam amplitudes, which are interpreted as the Hamiltonian constraints encodingthe dynamics of the spin network states for the quantum geometry. This was indeed done explicitly for the BF theoryspinfoam amplitudes [21] (following the approach of [22] for 3d quantum gravity) and for the spinfoam cosmologyamplitudes on the 2-vertex graph [19]. Finally, the use of spinor variables to parameterize the loop gravity phasespace allowed a systematic study of the gauge invariant observables at the discrete level. This led to the identificationof observables generating the basic SU(2)-invariant deformations of intertwiners. We identified in particular u ( N )-subalgebra of observables (where N is the valency of the intertwiner), which turned out powerful in the study ofthe intertwiner spaces [23] and the construction of appropriate coherent intertwiners [13–15] and in implementingsymmetry reductions on fixed graph in order to define mini-superspace models for loop quantum gravity hopefullyrelevant for cosmology [19, 24].The quantization of the spinor phase space is a priori rather straightforward since the complex variables arequantized as harmonic oscillators. The equivalence of this quantization scheme with the standard loop quantumgravity using spin network states was proved at the level of the Hilbert space on a fixed graph in [4, 5]. Nevertheless,all the relevant observables have not been consistently studied. Indeed, we have well studied the geometric observables(such as areas and angles) constructed from the triad classically and thus from the s u (2) generators at the quantumlevel. However the holonomy operator has not yet been explicitly constructed in this context. The goal of the presentshort paper is to remedy this shortcoming.We first remind the reader of the loop gravity phase space and its parametrization in terms of spinors. Then wedescribe the unambiguous quantization of the holonomy-flux algebra in terms of harmonic oscillators and holomorphicfunctions. This leads to a complete description of the grasping and holonomy operators of loop quantum gravity. Inthe spinorial picture these emerge both as composite operators built from some generalized ladder operators, ˆ E ’sand ˆ F ’s, which provide a complete set of SU(2)-invariant operators living on each vertex and acting on interwiners[4, 13, 23, 24]. This extends the work done for the classical phase space [4] to the quantum realm. Furthermore,we analyze some quantization- and operator-ordering ambiguities which are encountered in the definition of theholonomy operator on spinor space if we quantize it using the same technique as Thiemann’s trick for the definitionof the Hamiltonian constraint operator in loop quantum gravity [25]. We show that it leads to an anomaly and wecomment on the choice of quantization scheme. II. LOOP GRAVITY WITH SPINORS
The Hilbert space of loop quantum gravity on a given oriented graph Γ with E edges is defined as the space of L functions over E copies of the SU(2) Lie group provided with the Haar measure, that is H Γ := L (SU(2) E , d E g ). . Thiscan be understood as a quantization of ( T ∗ SU(2)) E , namely one copy of the cotangent bundle T ∗ SU(2) ≃ SU(2) × s u (2)for each edge e of the graph, which is usually parameterized with the couple ( g, J ), where g ∈ SU(2) is the holonomyof the Ashtekar-Barbero connection along the edge e and J = ~J · ~σ ∈ s u (2) ∼ R is related to the flux of the densitizedtriad through a surface dual to that edge. Furthermore, if J is assumed to live on the target vertex of e one can canuse the group element g to parallel-transport it to the source vertex of e and obtain J = − g ˜ Jg − . The full Poissonalgebra, attached to the edge e , is then given by { J i , J j } = ǫ ijk J k , { ˜ J i , ˜ J k } = ǫ ijk ˜ J k , { ~J, g AB } = + i ~σg ) AB , { ~ ˜ J, g AB } = − i g~σ ) AB , { g AB , g CD } = 0 , { J, ˜ J } = 0 , (1)where J and ˜ J are considered as 3-vectors and the SU(2) group element g defined in the fundamental representationas a 2 × A, B = 0 , FIG. 1: On the left, an oriented edge of a spinor network carrying the holonomy g which maps the spinor ˜ z at its source vertexonto the spinor z at its target vertex. On the right, a vertex v of a spinor network with each of the attached edges carrying aspinor z i The next step is to impose SU(2) gauge invariance at the vertices v of the graph. This is implemented by the (firstclass) closure constraints, G v ≡ X e ∋ v J ev = 0 , (2)which generates SU(2) transformation on both J ’s and g ’s. At the quantum level, it is taken into account by goingfrom the Hilbert space H Γ = L (SU(2) E ) to the gauge-invariant Hilbert space H o Γ = L (SU(2) E / SU(2) V ) where wehave quotiented by the SU(2)-action at all vertices. The final step would be to implement the Hamiltonian constraints(corresponding to the invariance under space-time diffeomorphisms in the continuum theory). This issue is still-openand we will not discuss it here. We will focus on the kinematical structures of the theory.Focusing on a single edge of the graph, it was shown in [2–6] that an alternative parameterization of T ∗ SU(2) ispossible, in terms of two spinors | z i , | ˜ z i ∈ C . The spinors are interpreted as living at the source and target verticesof the edge, as illustrated on fig.1. A spinor | z i is an element of C and has components z A , A = 0 ,
1. We denote itsconjugate by h z | = (¯ z , ¯ z ) and its dual by | z ] = ǫ | ¯ z i , ǫ = − iσ . The space C has the standard positive definiteinner product h z | w i := ¯ z w + ¯ z w . To be precise, the Hilbert space of loop quantum gravity arises as a sum over the individual graph-Hilbert spaces, or more preciselyas an inductive limit, H LQG := ∪ Γ H Γ / ∼ , where ∼ denotes an equivalence relation between states living on different graphs and thecompletion in an appropriate topology is taken. See [26] for details. We further impose a matching constraint, M := h z | z i − h ˜ z | ˜ z i enforcing the norms of the two spinors to be equaland generating a U(1) gauge invariance (under multiplication of the two spinors by opposite phases). It can be shownthat C × C , the space of two spinors | z i and | ˜ z i , reduces to T ∗ SU(2) by symplectic reduction with M (see [2, 3, 5]for details). The group and Lie-algebra variables are explicitly reconstructed as ~J = 12 h z | ~σ | z i , ˜ J = 12 h ˜ z | ~σ | ˜ z i ,g = | z i [˜ z | − | z ] h ˜ z | p h z | z ih ˜ z | ˜ z i . (3)This relation obviously implies that g | ˜ z ] = | z i and thus reproduce the constraint J = − g ˜ Jg − . Endowing C withthe symplectic structure { z A , ¯ z B } = − iδ AB , one easily recovers the phase space (1). In that sense, the spinors canbe understood as (complex) Darboux-like coordinates for the space T ∗ SU(2) in which the symplectic structure (1) istrivial.Here, let us stress a subtlety about the orientation of the edge. Indeed the group element introduced above maps onespinor onto the other, as g | ˜ z ] = | z i and g | ˜ z i = −| z ], so that its inverse is defined as g − | z i = | ˜ z ] and g − | z ] = −| ˜ z i .This is slightly different from the change of orientation which would lead to a group element ˜ g satisfying ˜ g | z ] = | ˜ z i and ˜ g | z i = −| ˜ z ] as we follow the definitions given above. The difference is actually just a sign flip: g − = | ˜ z ] h z | − | ˜ z i [ z | p h z | z ih ˜ z | ˜ z i , ˜ g = | ˜ z i [ z | − | ˜ z ] h z | p h z | z ih ˜ z | ˜ z i = − g − . This sign flip is not so important, but it is to be kept in mind. This sign ambiguity can be traced to the change fromvector variables to spinor variables. Indeed, the Poisson algebra of the variables ( g, J ) is unchanged under the change g ↔ − g , while the Poisson algebra of the variables ( g, z ) will be affected (by a mere sign). This is simply because a3-vector does not see the difference between the 3d rotations (in SO(3)) induced by g and − g , while these two SU(2)transformations act differently on a spinor.This spinor parametrization provides a direct link between spin network states and discrete geometries and providesan interesting new perspective on loop quantum gravity [1, 5, 6], spinfoam models [11, 12, 14–16], quantum spinfoamcosmology [4, 19, 24], topological BF-theory [21] and group field theory [27].Now, turning the focus to a given vertex v of the graph, one has one spinor variable z e per edge attached to v (here we do not distinguish between the z ’s and ˜ z ’s), as illustrated on fig.1. One can easily identify a complete set ofSU(2)-invariant observables, i.e commuting with the constraints G v : E ve e := h z e | z e i , F ve e := [ z e | z e i , F ve e := h z e | z e ] . (4)These scalar products between spinors interestingly form a closed algebra [4, 13, 23] and in particular the E -observableform a u ( N ) algebra (where N is the number of edges attached to the vertex v ) which is interpreted as generatingthe deformation of the intertwiner for fixed boundary area [13, 23, 28]. These observables then serve as basic buildingblocks for all gauge-invariant observables on a given graph Γ and in particular allow to decompose at the classicallevel the holonomy observable into basic deformations of the spin(or) network [4]. Here we will generalize this to thequantum level. III. THE HOLOMORPHIC REPRESENTATION
The standard quantization scheme for T ∗ SU(2) is to consider the Hilbert space L (SU(2)), with the usual orthogonalbasis given by the Wigner matrices D jmn ( g ) := h j, m | g | j, n i and with the holonomy g acting by multiplication andthe vector X acting by derivation as the s u (2) generators. Here the choice of spinor variables to parameterize thephase space leads to a different polarization, but which has been shown to be unitarily equivalent to the standardquantization (e.g. [5]). The quantization of the canonical Poisson bracket { z A , ¯ z B } = { ˜ z A , ¯˜ z B } = − iδ AB leadsnaturally to two copies of the Bargmann space F := L ( C , dµ ) of holomorphic, square integrable functions with anormalized Gaussian measure, where the spinors are naturally represented as the raising- and lowering operators ofharmonic oscllators. Taking into account the (area) matching constraint ˆ M = 0 one is led to the space H spin := F ⊗ F / U(1) . (5)A natural orthonormal basis of H spin is given by holomorphic polynomials in two spinors, P jmn ( z, ˜ z ) := ( z ) j + m ( z ) j − m p ( j + m )!( j − m )! ( − j + n (˜ z ) j + n (˜ z ) j − n p ( j + n )!( j − n )! , (6)where the spin j ∈ N / − j ≤ . . . m, n · · · ≤ + j . These polynomialshave a simple interpretation in terms of SU(2) representations: P jmn ( z, ˜ z ) = h j, m | j, z i [ j, ˜ z | j, n i = h j, m | j, z i h j, n | j, ˜ z ] , (7)where | j, z i is the SU(2) coherent state labeled by the spinor | z i while | j, ˜ z ] denotes the SU(2) coherent state labeledby the dual spinor | ˜ z ] = ǫ | ¯ z i (see e.g. [13–15] for more details). Orthonormality and completeness read as Z dµ ( z ) dµ (˜ z ) P jmn ( z, ˜ z ) P j ′ m ′ n ′ ( z, ˜ z ) = δ jj ′ δ mm ′ δ nn ′ , X j ∈ N / j X m,n = − j P jmn ( z , ˜ z ) P jmn ( z , ˜ z ) = I (2 h z | z ih ˜ z | ˜ z i ) , where dµ is the Gaussian measure and I ( x ) is the zeroth modified Bessel function of first kind which plays the roleof the delta-distribution on H spin .As explained in [5, 6], these holomorphic polynomials P jmn are unitarily related to the standard Wigner matricesin L (SU(2)), i.e. there exists a unitary map T : L (SU(2)) → H spin mapping D jmn ( g ) = h j, m | g | j, n i 7−→ ( T D jmn )( z, ˜ z ) := 1 √ j + 1 P jmn ( z, ˜ z ) = 1 √ j + 1 h j, m | j, z i [ j, ˜ z | j, n i . (8)Considering the formula (3) for the holonomy g in terms of the spinors z, ˜ z , it appears that this map T extracts theholomorphic part of the group element g . The pre-factor in √ j + 1 then ensures that this map still conserves thenorms and scalar products, i.e. that it is unitary. Details on this map and its properties can be found in [5].The elementary operators on H spin are the ladder-operators [ a, a † ] = 1 which directly correspond to the classicalspinors { z, ¯ z } = − i . Holonomies g and fluxes J then emerge as composite operators. Let us start with the basis(6) which decomposes as P jmn ( z, ˜ z ) = e jm ( z ) ⊗ ˜ e jn (˜ z ) = e jm ( z ) ⊗ e jn ( | ˜ z ]), where e jm ( z ) := ( z ) j + m ( z ) j − m √ ( j + m )!( j − m )! is the wellknown orthonormal Fock basis in F . Each of the z and ˜ z spinors gets quantized into a set of two harmonic oscillatoroperators [ˆ a , (ˆ a ) † ] = [ˆ a , (ˆ a ) † ] = 1 , [ b ˜ a , ( b ˜ a ) † ] = [ b ˜ a , ( b ˜ a ) † ] = 1 . (9)These act as the usual raising- and lowering-operators on F respectively:¯ z −→ (ˆ a ) † e jm ( z ) := z e jm ( z ) = p j + m + 1 e j + m + ( z )¯ z −→ (ˆ a ) † e jm ( z ) := z e jm ( z ) = p j − m + 1 e j + m − ( z ) z −→ ˆ a e jm ( z ) := ∂ z e jm ( z ) = p j + m e j − m − ( z ) z −→ ˆ a e jm ( z ) := ∂ z e jm ( z ) = p j − m e j − m + ( z ) . (10)At first, it might seem awkward that ¯ z be quantized as the multiplication by z while z get becomes the differentiation ∂ z . To make it more normal, one should instead consider anti-holomorphic polynomials in the spinors z and ˜ z , then¯ z would be the multiplication by ¯ z and z the differentiation ∂ ¯ z . This detail does not truly matter. What’s importantis the action of the operators on the basis states e jm and how they shift the j and m .The quantization of the ˜ z -sector is carried out exactly as above for the z -sector. But since we act on slightlydifferent wave-functions, the ˜ e jn (˜ z ) = e jn ( | ˜ z ]) instead of the e jm ( z ), we will get different pre-factors and shifts in j and n : ˜ z −→ ( b ˜ a ) † ˜ e jn (˜ z ) := ˜ z ˜ e jn (˜ z ) = p j − n + 1 ˜ e j + n − (˜ z )˜ z −→ ( b ˜ a ) † ˜ e jn (˜ z ) := ˜ z ˜ e jn (˜ z ) = − p j + n + 1 ˜ e j + n + (˜ z )˜ z −→ b ˜ a ˜ e jn (˜ z ) := ∂ ˜ z ˜ e jn (˜ z ) = p j − n ˜ e j − n + (˜ z )˜ z −→ b ˜ a ˜ e jn (˜ z ) := ∂ ˜ z ˜ e jn (˜ z ) = − p j + n ˜ e j − n − (˜ z ) . (11)The matching constraint is quantized as ˆ M = (ˆ a ) † ˆ a + (ˆ a ) † ˆ a − (ˆ˜ a ) † ˆ˜ a − (ˆ˜ a ) † ˆ˜ a , where we use the obviousnotation a, ˜ a (corresponding to z, ˜ z ) to denote ladder-operators acting in the first and second copy of F respectively.This generates a U(1)-invariant on polynomials of z and ˜ z as expected and we check that ˆ MP jmn = 0 for all basiselements of H spin . Then we require operators on the space H spin , built from these ladder-operators on F , to commutewith the U(1)-constraint ˆ M . Such invariant operators are the flux and holonomy operators, as we develop in the nextsection. IV. QUANTIZING THE HOLONOMY-FLUX ALGEBRA
Let us start with the flux-operators, corresponding to the quantization of the 3-vectors J :ˆ J + = (ˆ a ) † ˆ a , ˆ J − = (ˆ a ) † ˆ a ˆ J = 12 [(ˆ a ) † ˆ a − (ˆ a ) † ˆ a ] . (12)The commutators are an exact representation of the classical Poisson brackets, i.e the ˆ J form an s u (2)-algebra:[ ˆ J , ˆ J ± ] = ± ˆ J ± , [ ˆ J + , ˆ J − ] = 2 ˆ J . (13)This is the standard Schwinger representation for the s u (2) Lie-algebra. Their action on holomorphic polynomials iseasily computed and reproduces the well-known action of the s u (2) generators on basis states:ˆ J + P jmn = p ( j − m )( j + m + 1) P jm +1 ,n , ˆ J − P jmn = p ( j + m )( j − m + 1) P jm − ,n , ˆ J P jmn = m P jmn . (14)Another essential operator on H spin is the area-operator ˆ E which arises as a quantization of the norm of the 3-vector | ~J | = h z | z i , ˆ E = 12 [(ˆ a ) † ˆ a + (ˆ a ) † ˆ a ] (15)which is diagonalized in the standard basis (6) such thatˆ E P jmn = j P jmn (16)and commutes with ˆ J ± , ˆ J . The geometric interpretation of this operator ˆ E is that it gives the area carried by thatedge. The set { ˆ J ± , ˆ J , ˆ E } forms a u (2)-algebra.The ˜ z -sector differs slightly from the formulas above. The quantization of the 3-vectors is carried out exactly thesame way and the expression of the operators ˜ J in terms of the raising and lowering operators remains the same.Nevertheless the action on basis states gets a sign flip: c ˜ J + P jmn = − p ( j + n )( j − n + 1) P jm,n − , c ˜ J − P jmn = − p ( j − n )( j + n + 1) P jm,n +1 , c ˜ J P jmn = − n P jmn , b ˜ E P jmn = j P jmn = ˆ E P jmn . (17)Nevertheless the operators b ˜ J still form the expected s u (2) algebra without any sign flip (this is actually the complexconjugate representation of s u (2) compared to the z -sector).To derive the action of the holonomy operator, we start by its action on the Wigner matrices in L (SU(2)) and pullback to H spin using the unitary map T as in (8). Indeed, on L (SU(2)) we know that the holonomy operators ˆ g AB (here taken as the matrix elements of the group element g in the fundamental representation of SU(2)) acts simplyby multiplication, that is ˆ g AB ψ ( g ) = g AB ψ ( g ) ∀ ψ ∈ L (SU(2)) . (18)Using the SU(2)-recoupling theory, the action of the holonomy is easily computed. It is convenient to switch from theindices A, B = 0 , α, β = ± . We get after the pull-back:ˆ g αβ P jmn = 4 αβ j + 1 p ( j + 2 αm + 1)( j + 2 βn + 1) P j + m + α,n + β + 12 j + 1 p ( j − αm )( j − βn ) P j − m + α,n + β . (19)Note that the precise pre-factor j +1 is crucial to ensure that the classical Poisson algebra relations (1) are correctlyimplemented on L (SU(2)). In particular, one can apply this formula to the special case when we act with thecharacter χ on the character χ j : \ χ ( g ) χ j = X alpha = β ˆ g αβ X m = n P jmn = X m P j + mm + P j − mm = χ j + + χ j − , (20)as expected. Instead of using the formulas from SU(2) recoupling, we can follow our quantization rules from theclassical expression (3) of the holonomy: g = √ h z | z ih ˜ z | ˜ z i (cid:18) ¯ z ˜ z − z ˜ z z ˜ z + ¯ z ˜ z − ¯ z ˜ z − z ˜ z z ˜ z − ¯ z ˜ z (cid:19) −→ ˆ g = (cid:18) g −− g − + g + − g ++ (cid:19) = (ˆ a ) † ( b ˜ a ) † − ˆ a b ˜ a ˆ a ˆ˜ a + (ˆ a ) † ( b ˜ a ) † − ˆ a ˆ˜ a − (ˆ a ) † ( b ˜ a ) † ˆ a b ˜ a − (ˆ a ) † ( b ˜ a ) † !
12 ˆ E +1 . (21)And one finds the exact same action as above when computed using the recoupling of SU(2) representations.The polynomial part is straightforwardly quantized and the only subtlety is the pre-factor √ h z | z ih ˜ z | ˜ z i that getsregularized and quantized as (2 ˆ E + 1) − . This regularization, which makes the non-polynomial part a well-definedoperator on all of H spin (the a priori expression (2 ˆ E ) − diverges on the state P mn for j = 0), seems a bit ad hoc atthe first glance. However, there are two reasons which justify this choice: first, this is the only regularization whichensures that ˆ g acts on L (SU(2)) as required (after acting with the map T ). Second, even without knowing about theunitary mapping between L (SU(2)) and H spin , one would have constructed the same operator ˆ g by starting with anarbitrary regularization f ( ˆ E ) of the non-polynomial part of the group element such that it is well-defined on all statesin H spin and then demand that the commutator [ˆ g AB , ˆ g CD ] = 0 is implemented with no-anomaly. This conditionselects the regularization chosen here.Now that we have constructed the operators ˆ J, ˆ˜ J and ˆ g AB , we still have to check the final commutation relationsbetween them in order to conclude that the Poisson algebra (1) is represented correctly on H spin . The Poisson bracketsbetween J ’s and the g ’s can be re-written explicitly as: { J , g α,β } = − i α g α,β , { J ± , g α,β } = i ( 12 ∓ α ) g − α,β , (22)which gets quantized as: [ ˆ J , ˆ g α,β ] = α ˆ g α,β , [ ˆ J ± , ˆ g α,β } = − ( 12 ∓ α ) ˆ g − α,β . (23)It is straightforward that these commutators are satisfied by the operators as we have defined above. V. QUANTIZATION AMBIGUITY AND THIEMANN’S TRICK
As stated above, the ordering ambiguities in the holonomy-operator on H spin are strongly restricted by demandingthat the classical Poisson algebra (1) be represented non-anomalously. A conceptually similar, but mathematicallymuch more involved problem occurs in the definition of the Hamiltonian constraint, which generates the quantumdynamics in loop quantum gravity. The issue is that the classical Hamiltonian constraint is polynomial but for apre-factor given by he inverse square-root of the determinant of the triad, which does not have a clear unambiguousquantization. Thiemann [25] was the first to propose a mathematically well-defined quantum operator corresponding tothe classical Hamiltonian constraint. His construction relies strongly on some classical Poisson-identities, reexpressingthe inverse square root of det E which appears in the classical Hamiltonian constraints as ǫ abc ǫ jkl E bk E cl p | det E| = { A ja , V } (24)where V = R Σ dσ p | det E| is the total volume of the spatial manifold Σ. The basic variables are the Ashtekar-connection A ja and the densitized triad E aj , which is related to the (inverse) spatial metric as q ab = E aj E bj / | det E| . ThisPoisson-identity is used to reformulate the (Euclidean part of the) classical Hamiltonian constraint as H Eucl = T r ( F ab E a E b ) p | det E| = T r ( F ∧ { A, V } ) . (25)and the corresponding quantum operator is then defined as ˆ H Eucl := i ~ T r ( ˆ F ∧ [ ˆ A, ˆ V ]). This definition, using thePoisson-identity (24) as a way to regulate a possibly diverging non-polynomial expression is rather non-standard andits physical and mathematical consistency has not been checked intensively so far. Whether the classical hypersurfacedeformation algebra, encoding general relativity’s invariance under diffeomorphisms, is represented non-anomalouslyremains an open issue [29, 30]. To analyze the fate of the hypersurface deformation algebra in the full theory is ratherdifficult conceptually and technically. However, the spinorial formalism described in this article can be used to modela quantization based on such Poisson-identities in a much simpler setting. Indeed the holonomy g contains a similarpre-factor, given by the inverse square-root of the product of the norms of the spinors. It is possible to use a similartrick to re-absorb this pre-factor and generate it through a Poisson bracket. But in our (much) simpler framework,we know the exact quantization of the holonomy ˆ g , so we can test if such Poisson-identities lead more or less to thecorrect quantization or not.More precisely, we consider the Poisson-bracket of a spinor | z i with the square-root of the total area E = h z | z i ,which allows to generate an inverse square-root of the norm of z : {| z i , √ E } = 12 √ E {| z i , E } = − i √ | z i p h z | z i , {| z ] , √ E } = + i √ | z ] p h z | z i , (26)Using the same Poisson-identities for the spinor ˜ z , we can therefore write the classical group element (3) as g = − (cid:8)p ˜ E , (cid:8) √ E , | z i [˜ z | − | z ] h ˜ z | (cid:9)(cid:9) . (27)Similarly to the definition of the Hamilton constraint operator in loop quantum gravity we can now promote thisidentity to the definition of a holonomy operator by replacing the Poisson brackets by commutators, {· , ·} → − i [ · , · ].We simply substitute the classical phase space functions √ E , p ˜ E and | z i [˜ z | − | z ] h ˜ z | with the corresponding well-defined operators on H spin . As a result we obtain the new definition: b g ′ ≡ (cid:2)d √ E , (cid:2) d √ E , | ˆ a † i [ˆ˜ a | − | ˆ a † ] h ˆ˜ a | (cid:3)(cid:3) . (28)It is straightforward to compute its action on basis states: b g ′ αβ P jmn = 8( r j + 12 − p j ) p ( j + 2 αm + 1)( j + 2 βn + 1) P j + m + α,n + β +4 αβ p j − r j −
12 ) p ( j − αm )( j + 2 βn ) P j − m + α,n + β . (29)Note that the action of this operator is very similar to (19), but for the different pre-factors in j in front of each term.In the asymptotic limit of large spins j , we do recover that the pre-factors above give back 1 / (2 j + 1) as expected atleading order. But for small j ’s, the pre-factors differ, which means that the quantization of the holonomy is clearlydifferent for small spins, i.e. close to the Planck scale. Moreover, as stated above, the exact form of the combinatorial There are some more steps to be taken, a regularization is chosen adapted to the graph, connection A and curvature F , which do notexist as operators on the Hilbert space of loop gravity, are approximated by holonomies, etc. However, a key ingredient in the definitionof the constraint, which amounts to choosing a particular operator ordering, is the Poisson identity (24). The main argument in favorof this operator ordering is that the classical volume functional V has a well defined quantum analogue in loop gravity. ˆ E is a well defined, positive operator (16). Its spectrum is bounded from below by 0, therefore the operator d √ E is well-defined on H spin and acts on the basis P jmn as d √ E P jmn = √ j P jmn . Furthermore, because all functions in H spin are U(1)-invariant so that ˆ˜ E = ˆ E asoperators on H spin . pre-factors in (19) is important in order to obtain a non-anomalous representation of the classical Poisson-algebra(1). With the latter choice derived from quantizing the Poisson-identities,[ b g ′ αβ , b g ′ γδ ] P jmn = 0 , (30)which contradicts the fact that any two holonomy operators should Poisson-commute.In general, if we define the holonomy operator acting as: b G αβ P jmn ≡ αβ f + ( j ) p ( j + 2 αm + 1)( j + 2 βn + 1) P j + m + α,n + β + f − ( j ) p ( j − αm )( j − βn ) P j − m + α,n + β , (31)where f + ( j ) and f − ( j ) are the j -dependent pre-factors corresponding to the quantization of the inverse square-root ofthe norms of the spinors, we can compute the resulting commutator. We only give the commutator between b G ++ and b G −− for the sake of simplicity (to avoid a mess with the indices), but all the commutators can be computed similarly:[ b G ++ , b G −− ] P jmn = 2( m + n ) (cid:20) j f − ( j ) f + ( j −
12 ) − ( j + 1) f + ( j ) f − ( j + 12 ) (cid:21) P jmn . (32)It is fairly easy to check that this factor vanishes for our quantization ˆ g , when f − = f + = (2 j + 1) − . On the otherhand, for the quantization using the Thiemann-like trick, we get for [ b g ′ ++ , b g ′−− ]: (cid:20) j f − ( j ) f + ( j −
12 ) − ( j + 1) f + ( j ) f − ( j + 12 ) (cid:21) ∼ j →∞
18 1 j . (33)Therefore we conclude that a quantization based on the Poisson-identity (26) does not give the desired result andleads to an anomaly in the algebra at the quantum level. Defining an operator via Poisson-identities of the kind (26)amounts to a specific choice of operator ordering in the quantum theory. In the simple test case considered here, wehave shown that this particular operator ordering does not lead to a proper quantum representation of the classicalPoisson algebra.This is a standard with the quantization of non-polynomial observables. Having a closed algebra of observables atthe classical level guides us here to choose the correct quantization and operator ordering.The quantum dynamics of loop quantum gravity (and loop quantum cosmology, which is often used as a finitedimensional toy model) relies substantially on the Poisson-identity (24) which is, at least in spirit, very similar tothe one tested here. Extrapolating from our results to the case of loop quantum gravity, it would not be surprisingif a similar inconsistency would show up in the quantization of the Hamiltonian constraint. To clarify this issue wethink it could be helpful to study further toy models based on Poisson identities of the type (24) and check them forinternal consistency. VI. GRASPINGS AND VOLUME OPERATOR
So far we have restricted our attention to operators that were defined on a single edge of a spin network, namelythe holonomy- and flux-operators (12) and (21) acting on H spin . Now we would like to discuss operators acting on thefull spin network, and thus taking into account the SU(2) gauge invariance at the vertices of the graph. The Hilbertspace on an arbitrary graph Γ is H Γ = L (SU(2) × E / SU(2) × V ) in terms of the total number of edges E and the totalnumber of vertices V . It can be recast in the spinor framework as [4, 5]: H Γ = L (SU(2) × E / SU(2) × V ) = (cid:18) E ⊗ e =1 H spin e (cid:19) / SU(2) V = E ⊗ e =1 ( F ⊗ F ) / U(1) E / SU(2) V = (cid:20) V ⊗ v =1 (cid:18) N v ⊗ i =1 F (cid:19) / SU(2) (cid:21) / U(1) E (34)where i = 1 , .., N v labels the edges attached to a given vertex v . The initial definition focuses on degrees of freedomattached to the edges of the graph -the group element g e - up to the SU(2) gauge invariance at the vertices. The spinorframework allows to break the degrees of freedom on each edge into two pieces -the two spinors z e and ˜ z e - attachedto its source and target vertices, up to a U(1) gauge invariance along the edge allowing to glue the two spinors (byimposing that their norm be equal). In the end, this allows to factorize the Hilbert spaces around each vertex and tore-write the full Hilbert space as encoding degrees of freedom attached to each vertex up to the U(1) gauge invarianceon all edges. This is natural from the point of view that spin networks are made of intertwiners living at each vertexof the graph, or equivalently at the classical level that spinor networks can be interpreted geometrically as polyhedradual to each vertex and glued together along the edges [2–4, 23]. This is at the heart of the U( N ) framework forspin networks [4, 13–15, 23], where the space of intertwiners at each vertex v is identified as living in an irreduciblerepresentation of the unitary group U( N v ) (which depends on the total area around the vertex v ).Indeed, now focusing on a vertex v of the graph, we have N edges attached to it (we dropped the index v off N v ),and thus N spinors z i . As we have seen earlier in section II, we have a set of SU(2)-invariant observables at theclassical level given by the scalar between those spinors and their dual: E vij := h z i | z j i , F vij := [ z i | z j i , F vij := h z j | z i ] = −h z i | z j ] , where the E -matrix is Hermitian, E ij = E ji and the E -matrix anti-symmetric, F ij = − F ji At the quantum level, wehave working on the Hilbert space H v := (cid:18) N ⊗ i =1 F (cid:19) / SU(2). The quantization of those observables is straightforward:ˆ E ij := (ˆ a i ) † ˆ a j + (ˆ a i ) † ˆ a j , ˆ F ij := ˆ a i ˆ a j − ˆ a i ˆ a j ˆ F ij † := ˆ a i † ˆ a j † − ˆ a i † ˆ a j † . (35)These operators are now the basic building blocks for all SU(2)-invariant operators acting on spin networks. TheU( N ) formalism is based on the fact that the ˆ E ij operators form a closed u ( N ) algebra [23, 28].Now we are interested in the SU(2)-invariant version of the flux and holonomy operators, studied in the previoussection. We construct in the present section the grasping operators around a given vertex v , as polynomials in theoperators E v and F v . In the next section, we will deal with the holonomy operators, defined around closed loopsof the graph. They will involve polynomials of the operators E v and F v (for all vertices v around the loop), up tothe same norm factors that appeared for the quantization of the group element on a single edge. We will pay specialattention to those factors and associated operator ordering.Coming back to a single vertex, the operators ˆ E vij and ˆ F vij acts on the couple of edges i and j attached to the vertex v . The ˆ E -operators shift a quantum of area from one edge to another, while the ˆ F - and ( ˆ F † )-operators respectivelyannihilate and create a quantum of area on both edges. In this sense these operators can be regarded as generalizedcreation and annihilation operators. They are invariant under the action of SU(2) at the vertex. They are not howeverinvariant under the U(1)-symmetry on the edges and thus do not qualify as operators on the fully gauge invariantHilbert space H γ .On the other hand, the natural observables invariant under both U(1) and SU(2) symmetries are the scalar combi-nation of the 3-vectors, i.e the scalar product ~J i · ~J j and higher order combinations involving vector products such as( ~J i ∧ ~J j ) · ~J k . All these observables can actually be written in terms of the E and F observables as polynomials (ofthe same in E, F as in the J ’s). For instance, starting with the scalar product observable on a single edge, this givesthe squared area carried by that edge, which is easily translated in the E -observables: | ~J i | = ~J i · ~J i = E ii . (36)This relation also holds at the quantum level, except for a correction term accounting for the quantum ordering[23, 28]: ~ ˆ J i · ~ ˆ J i = ˆ E ii ( ˆ E ii + 1) . (37)Such correction terms lead to ordering ambiguities in the area spectrum of loop quantum gravity. For instance, onecan define the area directly as the operator ˆ E ii , with spectrum j , or as the squareroot of the SU(2) Casimir operator q ~ ˆ J i , with spectrum p j ( j + 1). Such ambiguities appear crucial in solving some of the (second class) constraints inloop gravity or spinfoam, for instance in the construction of the EPRL spinfoam model [31]. Notice nevertheless thatthe operators ˆ E ii and q ~ ˆ J i are different and do not have the same commutation relation with the other observables.Therefore selecting the particular Poisson bracket that we want to keep (without anomaly) as commutators at thequantum level would select one particular ordering over all others. The same is easily done for the scalar productobservables between two different edges [23, 24]: ~J i · ~J j = − | F ij | + 14 E ii E jj = 12 | E ij | − E ii E jj −→ ~ ˆ J i · ~ ˆ J j = −
12 ˆ F † ij ˆ F ij + 14 E ii E jj = 12 ˆ E ij ˆ E ji −
14 ˆ E ii ˆ E jj −
12 ˆ E ii . (38)0We also look at the cubic operator ˆ U ijk ≡ − i [ ~ ˆ J i · ~ ˆ J j , ~ ˆ J i · ~ ˆ J k ] = ǫ abc ˆ J ai ˆ J bj ˆ J ck . For a 3-valent vertex, this operator vanishesdue to the SU(2) gauge-invariance. It is non-trivial for a 4-valent vertex and actually defines the squared volumeoperator (up to a numerical factor) (see e.g. [32] for the geometrical interpretation or [33, 34] for a study of thisoperator in loop quantum gravity and more recently [35]). The peculiarity of this operator is that it is not positive(its spectrum is real but symmetric under change of sign), so it is non-trivial to define its square-root. One can verynaively take its absolute value in a basis which diagonalize it, but this seems an ad hoc definition weaken by the factthat we do not know explicitly the exact spectrum and eigenstates of the operator U . Nevertheless, it is the onlywell-defined proposal for a volume operator. It would seem better suited to identify the positive and negative modesof the operator but this turns out more complicated and it is not yet achieved . For vertices with valency larger orequal to 5, the squared volume operator is defined by adding the operators U ijk over all (oriented) triplets of edgesattached to the vertex. This operator turns out to be easily written in terms of the operators E or F . After a littlealgebra, we obtain: ˆ U ijk = − i (cid:16) ˆ E ij ˆ E jk ˆ E ki − ˆ E ik ˆ E kj ˆ E ji (cid:17) = − i (cid:16) ˆ F † ij ˆ E kj ˆ F ik − ˆ F † ik ˆ E jk ˆ F ij (cid:17) . (39)It might be interesting to study the action and spectrum of each of these combinations of E and F operators separately.One can also generalize the construction of such gauge-invariant grasping operators of higher order. The operators E and F are already SU(2) invariant, so we only have to deal with enforcing the U(1) invariance. This is achieved byrequiring that matching the indices i.e requiring that each edge appear the same number of times through creationoperators and annihilation operators. VII. SPINORIAL REPRESENTATION OF THE WILSON LOOP
The second important class of operators in loop quantum gravity are the Wilson loop operators ˆ W L which, incontrast to the grasping operators, capture non-local information about the states in H Γ . In terms of holonomies theWilson loop operators are defined as ˆ W L := T r ( → Y e ⊂L ˆ g e ) (40)that is the trace over the oriented product of holonomies ordered along all edges e part of a (oriented) loop L (i.e.we take the inverse of a group element if the edge is oriented in the opposite direction than the loop). An expressionfor classical Wilson loops in terms of the classical observables E and F was given in [4]. It was however unclear howto quantize these expressions due to the ambiguity in regularizing the inverse norm factors in the holonomies at hequantum level. Having obtained an explicit form of the holonomy operator on a single edge (21) in section IV, weare now able to provide an explicit formula for the Wilson loop operators in terms of the generalized ladder operators( ˆ E, ˆ F ). One issue in order to extract a square-root of the squared volume operator ˆ U in the spinorial framework is that it is of order 6 in the spinorcomponents i.e in the creation and annihilation operators a Ai , and that there does not exist any cubic gauge-invariant combinations ofthese operators. It might thus be interesting to identify another operator from which to extract the volume (squared or not) of order 8for example, from which we could maybe extract a square-root of order 4 in the spinor components. ✶ v v v v v v v v v n − v n e e e e e e e n − e n FIG. 2: The loop L = { e , e , .., e n } on the graph Γ. Writing the group elements g e in terms of the spinor variables as in eqn.(3), we use our operator ordering (21)and we regroup the creation and annihilation operations around the vertices along the loop as was done in [4] at theclassical level. This gives:ˆ W L = X r i =0 , ( − P i r i W { r i }L × Y i [2 ˆ E ii + 1] − , (41) W { r i }L ≡ Y i r i − r i ˆ F ii,i − + (1 − r i − ) r i ˆ E ii − ,i + r i − (1 − r i ) ˆ E ii,i − + (1 − r i − )(1 − r i )( ˆ F ii,i − ) † . (42)The index i labels the edge around the loop (from an arbitrary origin vertex), as illustrated on figure 2. Here wehave chosen all the edges oriented in the same direction along the loop for the sake of simplicity. In the general case,we would get an overall sign for each edge oriented in the opposite direction. For given r i ’s, we call the operator W { r i }L the generalized holonomy operator (following the classical nomenclature introduced in [4]). It is a polynomialoperator in the operators ˆ E and ˆ F . Its action is fairly simple despite its seemingly complicated structure. It raises thespin by on the edge i if r i = 0 and lowers it by one half if r i = 1. The non-polynomial part, given classically by theinverse norm factors, is regularized by a contribution of [2 ˆ E + 1] − per edge, all of them ordered to the right . This isthe simplest scenario, instead of the possibility of these inverse norm factors entering the generalized holonomies andmessing up their structure. The slight difference from the ordering conjectured in [4], with the inverse norm factorsplit as a square-root on the right and one of the left, is actually due to the non-trivial factor 1 / √ j + 1 in the map T given in eqn.(8) between the Wigner matrices D jmn ( g ) and their holomorphic counterpart.This operator, expressed in terms of generalized ladder operators, is unitarily equivalent to the standard Wilsonloops of loop quantum gravity. To understand its structure and action, let us give some examples. In the simplestcase, the graph is just given by one loop with a single vertex (see figure 3). There are four sets of ladder operators, (cid:0)(cid:1) PSfrag replacements Γ a, b ˜ a, ˜ b ev FIG. 3: The simplest case: Γ contains just one vertex v and a loop attached to it. The associated Hilbert space H Γ consists of holomorphic square integrable functions in two spinors | z i and | z i . Thus, there are four sets of ladder operators[ a , ¯ a ] = [ b , ¯ b ] = [ a , ¯ a ] = [ b , ¯ b ] = 1 out of which Grasping - and Wilson loop-operators, which provide a complete set ofSU(2) invariants, are constructed. one doublet for each end of the edge or equivalently one doublet on each leg around the single vertex. Let us denote2 PSfrag replacements γ v wa , b ˜ a , ˜ b ˜ a , ˜ b a , b e e FIG. 4: A loop within Γ that goes through only two vertices. The doublet of harmonic oscillator operators ( a, b ) is attachedto the vertex v while The doublet of harmonic oscillator operators (˜ a, ˜ b ) is attached to the vertex w . these by a, b and ˜ a, ˜ b . The Wilson loop operator ˆ W for this loop can be decomposed into ˆ E - and ˆ F -operators as :ˆ W = ( ˆ F † + ˆ F ) h E + 1 i − , (43)with ˆ E = a † a + b † b = ˜ a † ˜ a + ˜ b † ˜ b, ˆ F = b ˜ a − a ˜ b, ˆ F † = b † ˜ a † − a † ˜ b † . The operator ˆ E gives the spin on the single edge, i.e. the area carried by that edge, while the operators ˆ F and ˆ F † actat the vertex and respectively decreases and increases the spin on the edge by one half. We can compute the actionof these operators on our Hilbert space. Since we have a single loop, an orthogonal basis is given by the characters χ j = P m P jmm , and we get :ˆ E χ j = j χ j , ˆ F χ j = (2 j + 1) χ j − , ˆ F † χ j = (2 j + 1) χ j + . (44)One easily check that we indeed recover the expected action of the holonomy operator due to the factor (2 j + 1) − ,as given in eqn.(20): ˆ W χ j = χ j − + χ j + = \ χ ( g ) χ j . (45)One can go further and check that indeed ˆ F † + ˆ F = ˆ g −− + ˆ g ++ with the group element operators given earlier ineqn.(21).Beyond this consistency check on the single loop, a more generic example is given by the following situation: considera loop L within a graph Γ that goes through only 2 vertices (see figure 4). Computing the Wilson loop operatoraround the loop indicated in figure we obtain :ˆ W = h ˆ E ˆ˜ E + ( ˆ E ) † ( ˆ˜ E ) † + ( ˆ F ) † ( ˆ˜ F ) † + ˆ F ˆ˜ F i h E + 1 i − h E + 1 i − . (46) This is the quantization of the classical formula:T rg = T r | z i [˜ z | − | z ] h ˜ z | p h z | z ih ˜ z | ˜ z i = [˜ z | z i − h ˜ z | z ] p h z | z ih ˜ z | ˜ z i = F + F p h z | z ih ˜ z | ˜ z i . In order to check that ˆ F † is indeed the adjoint operator to ˆ F , one must keep in mind that the P mn are orthonormal, so that the states χ j are not normalized but such that h χ j | χ j i = (2 j + 1). This might seem awkward since the characters are normalized with respect tothe Haar measure on the SU(2)-group, R dg χ j ( g ) = 1. However, this is actually the reason for the 1 / p (2 j + 1) factor in the T -mapgiven in eqn.(8) between the Wigner matrices D jmn ( g ) and their holomorphic counterpart. This is the quantization of the classical formula:T rg g − = T r ( | z i [˜ z | − | z ] h ˜ z | ) ( | ˜ z ] h z | − | ˜ z i [ z | ) p h z | z ih ˜ z | ˜ z i p h z | z ih ˜ z | ˜ z i = E ˜ E + E ˜ E − F ˜ F − ¯ F ˜ F p h z | z ih ˜ z | ˜ z i p h z | z ih ˜ z | ˜ z i . L allow to split the full holonomy operator intosmaller polynomial operators which act by simple shifts on all edges of the loop. In some simple cases, they havealready been used to generate recursion relations on spin network evaluations (and more particularly on the 6j-symbolof the recoupling theory of spins) and to generate the action of the Hamiltonian constraints in 2+1 Riemanniangravity [21]. We hope that this reformulation of all loop quantum gravity gauge-invariant operators in terms of theladder operators ˆ E and ˆ F will somewhat allow a more systematic approach to the study of gauge-invariant operatorsentering the Hamiltonian (constraint) for loop quantum gravity in 3+1 dimensions. VIII. OUTLOOK AND CONCLUSION
In this short note, we have constructed the holonomy-flux operators in the spinor representation of loop quantumgravity. Holonomies and fluxes emerge as composite operator built from a set of harmonic oscillator operators whichare considered to be the more elementary operators in this picture. Because of this compositeness, there are certainoperator ordering ambiguities which need to be investigated. Here we showed that an operator ordering is selectedby the requirement that the classical holonomy-flux algebra be represented non-anomalously on the spinorial Hilbertspace H spin . This guarantees that this representation of the holonomy-flux algebra is unitarily equivalent to thestandard one on L (SU(2)).Taking SU(2)-gauge invariance at the nodes of Γ into account we constructed the familiar grasping and Wilsonloop operators on H spin . They can be written in terms of the generalized ladder operators ˆ E, ˆ F introduced in theU( N )-formalism [4, 13, 14, 23, 24], capturing the gauge-invariant content of the individual intertwiner spaces. Aninteresting point to note is that in the spinor formalism the distinction between Wilson loops on the one hand andgrasping operators on the other side becomes blurry: both are gauge invariant combinations of the same elementaryˆ E, ˆ F , the only difference being that the grasping operators are localized around a vertex of Γ whereas the Wilson loopoperators contain non-local information on the spin network state.An interesting side results of this note is the observation that the spinor formalism can be used as a simple toymodel to test the quantization procedure leading to the Hamiltonian constraint operator in loop quantum gravity. Thisquantization procedure rests on a peculiar Poisson-identity, which is used to get rid of potentially diverging operatorsand can be modeled by a similar (at least in spirit) Poisson-identity in the spinor formalism. Here we showed thata quantization of the holonomy operator based on that Poisson-identity leads to an anomalous representation of theholonomy flux algebra on H spin . While this calculation does not allow a direct conclusion for the full theory, wesuggest that more toy models of this kind should be considered to collect (counter-?) evidence for an anomaly-freeimplementation of the Dirac algebra in loop quantum gravity. Acknowledgments
EL and JT acknowledge support from the Programme Blanc LQG-09 from the ANR (France). [1] M. Dupuis, S. Speziale and J. Tambornino,
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U(N) tools for Loop Quantum Gravity: The Return of the Spinor ,Class.Quant.Grav.28 (2011) 055005 [arXiv:1010.5451][5] E.R. Livine and J. Tambornino,
Spinor Representation for Loop Quantum Gravity , J. Math. Phys. 53, 012503 (2012)[arXiv:1105.3385][6] E.R. Livine and J. Tambornino,
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Discrete Gravity Models and Loop Quantum Gravity: a Short Review , SIGMA 8(2012) 052 [arXiv:1204.5394][8] M. Dupuis, L. Freidel, E.R. Livine and S. Speziale,
Holomorphic Lorentzian Simplicity Constraints , J. Math. Phys. 53(2012) 032502 [arXiv:1107.5274] [9] E.R. Livine, S. Speziale and J. Tambornino, Twistor Networks and Covariant Twisted Geometries , Phys. Rev. D 85 (2012)064002 [arXiv:1108.0369][10] W.M. Wieland,
Twistorial phase space for complex Ashtekar variables , Class. Quantum Grav. 29 (2012) 045007[arXiv:1107.5002][11] S. Speziale and W.M. Wieland,
The twistorial structure of loop-gravity transition amplitudes , arXiv:1207.6348[12] W.M. Wieland,
Hamiltonian spinfoam gravity , arXiv:1301.5859[13] L. Freidel and E.R. Livine,
U(N) Coherent States for Loop Quantum Gravity , J.Math.Phys.52 (2011) 052502[arXiv:1005.2090][14] M. Dupuis and E.R. Livine,
Revisiting the Simplicity Constraints and Coherent Intertwiners , Class.Quant.Grav. 28 (2011)085001 [arXiv:1006.5666][15] M. Dupuis and E.R. Livine,
Holomorphic Simplicity Constraints for 4d Spinfoam Models , Class.Quant.Grav. 28 (2011)215022 [arXiv:1104.3683][16] M. Dupuis and E.R. Livine,
Holomorphic Simplicity Constraints for 4d Riemannian Spinfoam Models , Conference Pro-ceedings of Loops ’11 (Madrid, Spain, 2011), to appear in Journal of Physics: Conference Series (JPCS) [arXiv:1111.1125][17] L. Freidel and J. Hnybida,
On the exact evaluation of spin networks , arXiv:1201.3613[18] V. Bonzom and E.R. Livine,
Generating Functions for Coherent Intertwiners , arXiv:1205.5677[19] E.R. Livine and M. Mart´ın-Benito,
Classical Setting and Effective Dynamics for Spinfoam Cosmology , arXiv:1111.2867[20] L. Yu and R.G. Littlejohn,
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The Hamiltonian constraint in 3d Riemannian loop quantum gravity , Class.Quant.Grav.28(2011) 195006 [arXiv:1101.3524][23] L. Freidel and E.R. Livine,
The Fine Structure of SU(2) Intertwiners from U(N) Representations , J.Math.Phys. 51 (2010)082502 [arXiv:0911.3553][24] E.F. Borja, J. Diaz-Polo, I. Garay and E.R. Livine,
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Modern canonical quantum general relativity , Cambridge University Press, Cambridge UK, 2007[27] M. Dupuis, F. Girelli and E.R. Livine,
Spinors and Voros star-product for Group Field Theory: First Contact ,arXiv:1107.5693[28] F. Girelli and E.R. Livine,
Reconstructing Quantum Geometry from Quantum Information: Spin Networks as HarmonicOscillators , Class.Quant.Grav. 22 (2005) 3295-3314 [arXiv:gr-qc/0501075][29] J. Lewandowski and D. Marolf,
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