Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

The European Physical Journal C, Apr 2017

To adopt a practical method to calculate the action of geometrical operators on quantum states is a crucial task in loop quantum gravity. In this paper, the graphical calculus based on the original Brink graphical method is applied to loop quantum gravity along the line of previous work. The graphical method provides a very powerful technique for simplifying complicated calculations. The closed formula of the volume operator and the actions of the Euclidean Hamiltonian constraint operator and the so-called inverse volume operator on spin-network states with trivalent vertices are derived via the graphical method. By employing suitable and non-ambiguous graphs to represent the action of operators as well as the spin-network states, we use the simple rules of transforming graphs to obtain the resulting formula. Comparing with the complicated algebraic derivation in some literature, our procedure is more concise, intuitive and visual. The resulting matrix elements of the volume operator is compact and uniform, fitting for both gauge-invariant and gauge-variant spin-network states. Our results indicate some corrections to the existing results for the Hamiltonian operator and inverse volume operator in the literature.

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Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

Eur. Phys. J. C Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity Jinsong Yang 1 2 Yongge Ma 0 0 Department of Physics, Beijing Normal University , Beijing 100875 , China 1 Institute of Physics , Academia Sinica, Taipei 115 , Taiwan 2 Department of Physics, Guizhou University , Guiyang 550025 , China To adopt a practical method to calculate the action of geometrical operators on quantum states is a crucial task in loop quantum gravity. In this paper, the graphical calculus based on the original Brink graphical method is applied to loop quantum gravity along the line of previous work. The graphical method provides a very powerful technique for simplifying complicated calculations. The closed formula of the volume operator and the actions of the Euclidean Hamiltonian constraint operator and the so-called inverse volume operator on spin-network states with trivalent vertices are derived via the graphical method. By employing suitable and non-ambiguous graphs to represent the action of operators as well as the spin-network states, we use the simple rules of transforming graphs to obtain the resulting formula. Comparing with the complicated algebraic derivation in some literature, our procedure is more concise, intuitive and visual. The resulting matrix elements of the volume operator is compact and uniform, fitting for both gauge-invariant and gauge-variant spin-network states. Our results indicate some corrections to the existing results for the Hamiltonian operator and inverse volume operator in the literature. - second one, based on the ?internal? regularization, was firstly defined by Ashtekar and Lewandowski [10]. In [11], Thiemann presented a rather short and compact regularization procedure to re-derive the second version of the volume operator. Playing a crucial role in LQG, the spectra of the volume operator is pursued. Certain matrix elements of the volume operator were calculated in the framework of loop representation by using a graphical tangle-theoretic Temperley?Lieb formulation in [18]. Then they were also derived in connection representation by a rigorous but tedious algebraic method in [11,19], and their special case was re-derived using generalized Wigner?Eckart theory [20] as well as the graphical methods in [13,21?23]. Although those components of the volume operator are rigorously defined, the computation of their actions on spin-network states are difficult. The main reason is the following. The volume element operator at a vertex v of a graph ? reads Vv = |q?v|. Although the matrix elements of q?v can be calculated using recoupling theory, the matrix has no obvious symmetries and hence is difficult to diagonalize analytically for the case that the dimension of the matrix is bigger than nine. On the one hand, the derivation of closed formula in [19] is rigorous. But there is no universal formula with so tedious and abstract method. Thanks to the calculation of the matrix elements of the volume operator in certain special case, some matrix elements of Thiemann?s Hamiltonian constraint operator and its generalization were derived in [21,22]. Later on, the matrix elements was re-derived in [13], and then the formula in [13] was corrected by sign factors in [14,23] using graphical method. Matter coupling is also an important issue in LQG. In the case of gravity coupled to a scalar field, the whole Hamiltonian constraint operator was constructed [17,24]. The matter part of the whole Hamiltonian constraint operator usually contains the ?inverse volume operator?, which is defined by the co-triad operators. In the symmetric model of loop quantum cosmology (LQC) [25], the analog of the inverse volume operator is bounded above. This fact is sometimes thought of as a reason for the singularity resolution in LQC. In particular, it is shown in [26] that in spatially curved anisotropic models inverse volume effects may become important to bind expansion and shear scalars. However, it is shown in [27] that the inverse volume operator with certain ordering in full LQG is unbounded on the zero volume eigenstates (at a gauge-invariant trivalent vertex). This throws doubt on whether one can generalize the conclusions of LQC to LQG. To understand definitely the inverse volume operator in LQG and its relation to the analogs in certain symmetric models, it is necessary to calculate in detail its action on the quantum states in LQG. There is no doubt that a simple and practical calculation method is desirable to further understand the volume, inverse volume and Hamiltonian constraint operators. As a powerful tool for practical calculation, graphic calculus has been introduced in LQG in a few papers (see e.g., [13,14,18,21?23,28,29]). These graphical methods are based on the graphical methods developed by Yutsis in [30], Brink in [31], Varshalovich in [32], and Kauffman in [33]. In order to represent conveniently the Clebsch?Gordan coefficients, Brink slightly modified the Yutsis? graphical method by introducing a line with an arrow on it to represent ?metric tensor?. Comparing to the Yutsis? method, Brink?s graphical method is more convenient and has wider scope of application. Varshalovich?s method gives a way to represent the Dirac?s bra and ket notation by introducing a line with an arrow outgoing from a node to represent ?ket? (state vector) and a line with double arrow coming into a node to represent the ?bra? (dual state vector). The above three methods are usually used to deal with the coupling problem of angular momentum in quantum mechanics. Moreover, Kauffman introduced a graphical method for the Temperley?Lieb algebra. Kauffman?s graphical method was firstly used in LQG in [18,21,22]. It is worth noting that Kauffman?s method was in fact used in [13] while the graphical notations in its main text are similar to those in [22]. Brink?s graphical method was only recalled in the appendix of [13]. Then Varshalovich?s method was adopted in [14,23,28]. Brink?s graphical method was also taken to study the propagator of spinfoam models in [29], in which the graphical method was only used to calculate the action of the right-invariant vector field (the ?grasping operator?) on the intertwiners but not the action of holonomy operator. Graphical method was also introduced to quantum reduced gravity in [34]. In this paper, the graphical calculus based on the Brink graphical method [31] and its suitable extension1 will be employed to study the volume, inverse volume and Hamiltonian constraint operators in LQG. Our aim is in two folds. One is to show that the graphical method is suitable to calculate the actions of different kinds of operators on spin-network states. The other is to cross-check the results obtained in the literature, on which some important applications are based. This method consists of two ingredients, graphical representation and graphical calculation. The algebraic formula will be represented by corresponding graphical formula in an unique and unambiguous way. Then the graphical calculation will be performed following the simple rules of transforming graphs, corresponding uniquely to the algebraic manipulation of the formula. A central goal of this paper is to derivate the closed formula for the matrix element of the volume operator, which involves only the flux operator, based on the rigorous graphical method. Comparing to the algebraic method, our derivation is obviously more compact and simple. Our analysis shows that the formula of the matrix elements for certain cases in [19] is also valid for other cases and hence can be regarded as a general expression. Then we will consider the actions of the gravitational Hamiltonian constraint operator and the inverse volume operator on spin-network states in the graphical 1 A similar scheme is introduced independently at almost the same time by Alesci et al. [35]. method. Both operators depend also on the honolomies in addition to fluxes. Note that, besides the regularization introduced by Thiemann [16], other proposal for the regularization of the Hamiltonian constraint operator is also available [36]. On the contrary to the conclusion in [27], our calculation shows that the inverse volume operator is bounded (zero-valued) at a trivalent non-planar vertex of the gauge-invariant spin-network states, which is a non-trivial eigenstate with zero-eigenvalue of the volume operator. This result opens a possible way to lift the result of singularity resolution of LQC to LQG. This paper is organized as follows. Section 2 is devoted to a brief review of the elements of LQG. In Sect. 3, the graphical method to LQG will be introduced systematically. In Sect. 4, we will derive the closed formula for the matrix element of the volume operator by the graphical method. It is shown how the simple rules of transforming graphs tremendously simplify our calculation. In Sect. 5, the construction of Thiemann?s Euclidean Hamiltonian constraint operator will be briefly reviewed, and its action on gauge invariant trivalent spin-network states will be calculated by the graphical method. In Sect. 6, we will compute the action of the inverse volume operator appeared in the Hamiltonian constraint for gravity coupled to matter field. The results will be discussed in Sect. 7. In Appendix A, we will review the representation theory of SU (2) group, including the notation of intertwiners and basic components of Brink?s graphical representation and some rules of transforming graphs. The detailed proof of some identities and results in the main text will be presented in Appendix B and Appendix C separately. 2 Preliminaries In this section, we briefly summarize the elements of LQG to establish our notations and conventions (see [1?4] for details). The classical starting point of LQG is the Hamiltonian formalism of GR, formulated on a 3-dimensional manifold of arbitrary topology. With Ashtekar?Barbero variables [37,38], GR can be cast in the form of a dynamical theory of the connection with SU (2) gauge group. We denote spatial indices by a, b, c, . . . and internal indices by i, j, k, . . . = 1, 2, 3. The phase space consists of canonical pairs ( Aia , E?ia ) of fields on , where Aia is a connection 1-form which takes values in the Lie algebra su(2), and E?ia is a vector density of weight 1. The densitized triad E?ia is related to the co-triad eai by 1 abc i jk ebj eck sgn(det(edl )), where ?abc is the Levi-Civit? tensor density of weight 1, and sgn(det(edl )) denotes the E?ia := 2 ? sign of det(edl ). The 3-metric on is expressed in terms of co-triads through qab = eai ebj ?i j . The only non-trivial Poisson bracket reads The fundamental variables in LQG are the holonomy of the connection along a curve and the flux of densitized triad through a 2-surface. Given an edge e : [0, 1] ? , the holonomy he( A) of connection Aia along the edge e is he( A) := Pexp = I2 + tn?1 dtn A(e(t1)) ? ? ? A(e(tn)), where A(e(t )) := e?a (t ) Aia (e(t ))?i , with e?a (t ) being the tangent vector of e, and ?i := ?i ?i /2 (with ?i being the Pauli matrices), P denotes the path ordering which orders the smallest path parameter to the left. The holonomy he( A) is the unique solution he([0,t=1])( A) of the parallel transport equation = he([0,t])( A) A(e(t )) [e1 ? e2](t ) := A ? Ag = ?(dg)g?1 + g Ag?1. where ? = 8? G, and ? is the Barbero?Immirzi parameter. The behavior of the connection under finite gauge transformations is and the inversion of an edge as e?1(t ) := e(1 ? 2t ). The holonomy (2.3) has two key properties he1?e2 ( A) = he1 ( A)he2 ( A), he?1 ( A) = he( A)?1. where abc is the 3-dimensional Levi-Civit? tensor density of weight ?1. Consider a finite piecewise analytic graph ? in , which consists of analytic edges e incident at vertices v. We insert a pseudo-vertex v? into each edge e and split e into two segments se and le such that e = se ? le?1 and the orientations of se and le are all outgoing from the two endpoints of e. We call the new graph the standard graph obtained from the original graph by splitting edges and adding pseudo-vertices. Denote the standard graph by ? , the set of its edges by E (? ), and the set of vertices, containing the true vertices v and pseudo-vertices v?, by V (? ). Our following discussion is based on the standard graphs. To construct quantum kinematics, one has to extend the configuration space A of smooth connections to the space A? of distributional connections. A function f on A? is said to be cylindrical with respect to a graph ? if and only if it can be written as f = f? ? p? , where p? ( A) = (he1 ( A), . . . , hen ( A)) and e1, . . . , en are the edges of ? . Here he( A) is the holonomy along e evaluated at A ? A? and f? is a complex-valued function on SU (2)n. Since a function cylindrical with respect to a graph ? is automatically cylindrical with respect to any graph bigger than ? , a cylindrical function is actually given by a whole equivalence class of functions f? . We will henceforth not distinguish the functions in one equivalence class. The set of cylindrical functions is denoted by Cyl(A?). The space Cyl(A?) can be completed as the kinematical Hilbert space Hkin := L2(A?, d?o) with d?o being the Ashtekar?Lewandowski measure. Now let us consider the transformation behavior of the cylindrical function in order to understand the purpose of introducing the intertwiner. The cylindrical function can be decomposed by the representations ? je (he( A)) of he( A) as where ? stands for contracting operator. Under finite gauge transformations, the above equation changes to ? je (g(v)) ? ? je (he( A)) ? ? je (g(v?)?1 ? he( Ag) = g(b(e))he( A)g( f (e))?1, E?i (S) := The transformation behavior (2.2) of the connection A under a gauge transformation leads to the corresponding transformation behavior of holonomy as where b(e), f (e) denote the beginning and final points of e, respectively. The flux E?i (S) of densitized triad E?ia through a 2-surface S is defined by The orthogonality of CGCs ensures that the cylindrical function f? ({he( A)}e?E(? )) in (2.12) is gauge invariant for J = J = 0. Hence the tensor ivJ is also called the gauge-invariant (variant) intertwiner, associated to v, corresponding to J takes 0 (nonvanishing value). The above discussion means that the basis of Hkin is2 = (?1) j1? in=2 ji ?J k2,...,kn?1 where J M ; a | j1m1 j2m2 ? ? ? jnmn is the complex conjugate of generalized CGCs describing the coupling of n angular momenta j1, . . . , jn to a total angular momentum J in the standard coupling scheme (i.e., j1 is first coupled to j2 to give a resultant a2, and then a2 is coupled to j3 to give a3, and so on), and a ? {a2, . . . , an?1} denotes the set of the angular momenta appeared in the intermediate coupling. Notice that the intertwiner presented in Eq. (2.14), differing the factor (?1) j1? in=2 ji ?J from CGCs, is more convenient to be represented in graphical formula. The matrix elements of the conjugate intertwiner iv? associated to pseudo-vertex v? at which two incoming edges with the same spin j meet are given by ? := j n1 j n2|00 . = J M | j1m1 j2m2 . The assignment of intertwiners to the true vertices and conjugate intertwiners to the pseudo-vertices is compatible with the transformation behavior (2.8) of holonomy. The CGCs are usually chosen to be real so that It is, therefore, not necessary to sedulously distinguish intertwiner from its conjugate when we do calculation. The gaugeinvariant spin-network states correspond to the states whose intertwiners in (2.14) associated to true vertices are specially 2 See Eq. (3.25) for the orthonormal basis of Hkin. chosen such that the resulting angular momenta J = 0. The normalized gauge-invariant/variant states consist of the orthonormal basis of the gauge-invariant/variant Hilbert space [40]. Two elementary operators in LQG are the holonomy and the flux operators. The holonomy operator acts as multiplication: [h?eI ( A)]B C ? f? (he1 ( A), . . . , hen ( A)) := ?1/2(heI ( A)) Given a graph ? and an oriented 2-surface S with conormal naS, the edges of ? can be split into two halves at an interior point if necessary. Then one can get a graph ?S adapted to S such that the edges of ?S belong to the following four types: (i) e is up with respect to S if e?a (0)naS(e(0)) > 0; (ii) e is down with respect to S if e?a (0)naS(e(0)) < 0; (iii) e is inside with respect to S if e ? S = e; (iv) e is outside with respect to S if e ? S = ?. Then the flux operator acts on a cylindrical function f? with respect to the graph ? adapted to S as a derivative operator, 3 Graphical method for LQG 3.1 Algebraic formula Graphic calculus has been introduced in LQG in a few papers (see e.g., [13,14,18,21?23,28,29]). Here we focus on the original Brink?s graphical method and its suitable extension to LQG. In LQG, under different physical considerations, one needs to construct operators, e.g., the geometric operators and the Hamiltonian operators, corresponding to their classical quantities based on the two elementary operators h?e( A) and Jei . The action of those operators on a given spin-network state will involve the actions of the two elementary operators. The action of h?e( A) on the spin-network states involves essentially the decomposition of the tensor product representation of SU (2), which is well known as the Clebsch?Gordan series (i jJ1 j2 )?1 E??i (S) ? f? (he1 ( A), . . . , hen ( A)) := ?i h? f? (he1 ( A), . . . , hen ( A)), E?i (S)! = 2 JeiI ? f? (he1 ( A), . . . , heI ( A), . . . , hen ( A)) := ?i is the self-adjoint operator of the right-invariant vector field on the copy of SU (2) corresponding to the I th edge. (i jJ1 j2 )?1 = ivJ ; a [? j1 (he1 )]m1 n1 ? ? ? ?i [? jI (?i )]m I m I [? jI #heI $]m I nI ? ? ? [? jn (hen )]mn nn which indicates that JeiI leaves ? and j invariant, but does change the intertwiner associated to v by contracting matrix elements of the i th ? with the intertwiner in the following way: = ivJ ; a However, in practical calculation, it is not convenient to directly compute the contraction of matrix elements of ?i with an intertwiner. One usually introduces the irreducible tensor operators [41], or the spherical tensors of ?i , to replace the original ?i for a reason that will become clear in a moment. The spherical tensors ?? (? = 0, ?1), corresponding to ?i (i = 1, 2, 3), are defined by [h?eI ( A)]B C ? T?v, j,i ( A) = ivJ ; a = ivJ ; a 1 ?0 := ?3, ??1 := ? ? (?1 ? i ?2) . 2 Then the contraction of matrix elements of ?i with an intertwiner is transformed to that of their tensor operators with the intertwiner. The matrix elements [? j (??)]m m can be related to the 3 j -symbols (or CGCs) by (see Appendix B.1 for a proof) i [? j (??)]m m = 2 where C(mj)m := (?1) j+m ?m ,?m is the contravariant ?metric? tensor on the irreducible representation space H j of SU (2) with spin j (see Appendix A.1 for a detailed explaination for the C(mj)m ) [42]. The spherical tensor ?? generates the self-adjoint right-invariant operator Je?I defined by Je?I ? f? (he1 ( A), . . . , heI ( A), . . . , hen ( A)) := ?i = ivJ ; a Any gauge-invariant operator, e.g., the volume operator considered in this paper, defined by J i s can be expressed in terms of the corresponding J ?s. Hence its action on the spin-network states is essentially equivalent to contracting 3 j -symbols (or CGCs) with corresponding intertwiners. 3.2 Graphical representation and graphical calculation The basic components of the original Brink?s graphical representation and the simple rules of transforming graphs are presented in Appendix A.1. In graphical representation, the 3 j -symbol is represented by an oriented node with three lines, each of which represents a value of j , i.e., where ? and + denote the clockwise orientation and the anti-clockwise orientation, respectively. A rotation of the diagram does not change the cyclic order of lines, and the angles between two lines as well as their lengths at a node have no significance. The ?metric? tensor C m(j)m in Eq. (A.6) which occurs in the contraction of two 3 j -symbols with the same values of j is denoted by a line with an arrow on it, i.e., and its inverse in Eq. (A.7) can be expressed as Summation over the magnetic quantum numbers m is graphically represented by joining the free ends of the corresponding lines. The contraction of a 3 j -symbol with a ?metric? is represented by a node with one arrow, which provides a way to represent the CGC, e.g., To give a precise way of presenting the CGC as Eq. (3.13) is the main motivation for Brink to modify the original Yutsis scheme [30,31]. Hence the intertwiner #ivJ ; a $ m1m2???mn M in Eq. (2.14) associated to a true vertex v, from which n edges are outgoing, is represented in a graphical formula by Eq. (A.38) as (see Appendix A.2 for a detailed interpretation) Now we will extend Brink?s representation and propose a graphical representation for the unitary irreducible representation ? j of SU (2). The matrix element [? j (g)]m n is denoted by a blue line with a hollow arrow (triangle) in it as The orientation of the arrow is from its row index m to its column index n. For group elements such as the holonomies he ? he( A) of the connection A along an edge e, their matrix elements can simply be represented by The advantages of the above graphical representation are the following: (i) The edge and the irreducible representation of the holonomy along the edge have been represented by the elements e and j labeling the line; (ii) the orientation of e with respect to the vertices has been reflected by the orientation of the arrow on the line; (iii) the row index (the tensor index of H?j ) and the column index (the tensor index of H j ) have been represented by the two indices m and n, respectively, labeling the starting and the ending points of the line; (iv) the matrix element [? j (he)]m n and the ?metric? tensor C m(j)m in the graphical formula are distinguished by different colors (blue v.s. black) and elements (two v.s. one) of the lines; (v) the coupling rules of the representations of holonomies match Brink?s representations for the CGC (see Eq. (3.18)). By Eqs. (2.7) and (A.57), the matrix elements of the inverse of a holonomy can be represented by The Clebsch?Gordan series in (3.1) yield the coupling rules of representations of holonomies as3 The action of [h?eI ( A)]B C on the spin-network state T v ?, j,i [? jn (hen )]mn nn can be represented by ( A) = #ivJ ; a $ in (3.9) can be represented by 3 Similar calculus based on the Varshalovich method were used in LQG (see e.g. [14]). Up to now, based on the Brink original graphical representation and its suitable extension to the irreducible representation of holonomy, the two elementary operators in LQG, the holonomy operator and the flux operator (essentially the self-adjoint right-invariant operator), and the spin-network states of the kinematical Hilbert space have been uniquely represented by corresponding graphs. Hence, in the graphical method, the actions of any well-defined operators in the kinematical space, for instances, the volume operator, the Hamiltonian operator and the inverse operator considered in this paper, on a spin-network state can be derived by the simple rules of transforming graphs (see Appendix A.2). The starting point of our scheme is the so-called standard graph ?std, which is obtained from its original graph ?org by splitting edges and adding pseudo-vertices. We still need to show that the spin-network function associated to the original graph ?org is equivalent to the one associated to its corresponding standard graph ?std acting by an operator. Recall that the standard graph ?std is obtained from ?org by the following procedure. We insert a pseudo-vertex v? into each edge e of ?org and split e into two segments se and le, such that e = se ? le?1 and the orientations of se and le are all outgoing from the two endpoints of e. The standard graph ?std consists of the new segments se and le, the new adding pseudo-vertices v?, and the (old) vertices of ?org. We can transform the spin networks based on the original graph into those on its standard graph by explicit transformation rules, and then find their relation. Consider an edge e with representation j in ?org starting from v and ending at v , assigning the intertwiners iv and iv , respectively. We assume that the edge e in the original graph ?org is regarded as the kth edge and the k th edge in the set of edges which incident at v and v , respectively, i.e., b(e) = b(ek ), f (e) = f (ek ). The relevant ingredient of a spin-network state associated to the edge e takes the form [see (A.39) for the graphical representation of the intertwiner associated to v at which there are coming and outgoing edges, (3.16) and (3.17) for the graphical representation of the holonomy] where we have used (3.17) in the second step and the rule (A.41) to remove two arrows in the last step. Repeating the above procedure, we can transform the spin-network states associated to the origin graph into those corresponding to its standard graph. By this trick, we finally find out the corresponding relation of the spin-network states between the original graph ?org and its standard graph ?std. The intertwiners associated to the vertices of the origin graph is replaced by its standard formula, and adding the pseudo-vertex v? for each edge e with a divalent intertwiner is graphically represented by an arrow with an orientation opposed to that of the original edge.4 We now show by the graphical method that the spin-network states can become orthonormal to each other. The normalized spin-network state takes the form where ?? is any graph bigger than ? and ? , |E (?? )| denotes the number of the edges in ?? , and d?H (g) is the Haar measure on SU (2). If ? differs from ? , e.g., there is an edge e with spin je belonging to ? but not ? , then the orthogonality relation, The scalar product of the spin-network states is defined by d?H (he) T norm ( A) T norm( A), ? , j ,i ? , j,i d?H (g) [? j (g)]m n [? j (g)]m n = 2?j j+,j 1 ?m,m ?n,n , implies that the corresponding integration in (3.26) becomes d?H (he ) [? je (he )]m n = 0. d?H (he) T norm ( A) T norm( A) ? , j ,i ? , j ,i d?H (he) T norm ( A) T norm( A), ? , j ,i ? , j ,i where in the second step we have used the fact that the Haar measure is normalized. By integrating over all representation functions on the edges, one can obtain the contract of the complex conjugation of intertwiners with the corresponding intertwiners at vertices. Thus we have 4 The intertwiner, a line with an arrow, associated to the pseudo-vertex v?, is not normalizable since we adopt ? je (he( A)) rather that its normalized form ?2 je + 1 ? je (he( A)) in the spin-network function (see Eq. (2.13)). If the original spin network is normalized, the intertwiner associated to v? will automatically be normalized. Then it will be expressed as ?21j+1 times a line with a spin- j arrow in the graphical representation in Eq. (3.24). where in the second step we have used the fact that the CGs (and thus the GCGs) are real, in the fourth step we have used Eqs. (A.45) and (A.41). Note that the result of Eq. (3.30) is based on the premise that the intertwiners at the same vertex v involve the same coupling scheme. If different coupling schemes at the same vertex were chosen, certain additional multiplication of 6 j -symbols would appear in the result. If the spin-network functions are gauge invariant, corresponding to J = 0 and M = 0, the two Kronecker delta functions ?J,J and ?M,M will not appear in Eq. (3.30). 4 The volume operator One of the important achievements in LQG is that the theory itself predicts that some geometric operators, such as area operator and volume operator, have discretized spectra. Some volume operator was also constructed for spin-foam models [43,44]. There are two versions of volume operator in canonical LQG. We only consider the volume operator defined in [10,11], which passed the consistency check in the quantum kinematical framework and was used to define a Hamiltonian constraint operator in LQG [16,45,46]. In this section, we will briefly review the construction of the volume operator. Then the graphical method, introduced in Sect. 3, will be used to derive the matrix element of the volume operator. 4.1 A brief review of the construction of the volume operator Classically, the volume function for a given open region R reads V (R) := d3x | det(q)| = where det(q) denotes the determinant of the 3-metric qab. To quantize the volume function, a suitable regularization procedure is needed which involves smearing E?ia . We now introduce the regularization adopted by Ashtekar and Lewandows in [10]. For given R ? , we fix a global coordinates {xa , a = 1, 2, 3} in a neighborhood of R in and partition P of R into a family C of closed cubes C with coordinate volume 3. For each C , one arranges three 2-surface S1, S2, S3, defined by xa = consta , intersecting in the interior of C . One smears these three densitized triads on those three 2-surfaces for each cell C in a given partition to give a regularized version of (4.1) as [10] VAPL (R) := C?C It is easy to see that (4.2) reduces to (4.1) as ? 0. The above regularization procedure is called internal because triads are smeared over three surfaces passing the interior of the cell. It is straightforward to promote the regularized formula (4.2) to its quantum operator by replacing the fluxes by their operators. It is convenient to introduce the permissible partitions P? (for sufficiently small ) adapted to a given standard graph ? (see [10] for details). One then obtains the regulated operator eI ?eJ ?eK =v where (eI , eJ , eK ) := abc (eI , Sa ) (eJ , Sb) (eK , Sc). Notice that the action of the regulated operator only depends on the properties of these surfaces at v. Hence the result is unchanged as we refine the partition and shrink C to v and hence the limit ? 0 is trivial. However, the limiting volume operator carries the information of our choice of partitions through (eI , eJ , eK ), which depend on the background structure?the coordinates choice defining Sa . By suitable averaging over relevant background structures in (4.2), the well-defined, background-independent volume operator reads5 eI ?eJ ?eK =v where ? (eI , eJ , eK ) ? sgn(det(e?I (0), e?J (0), e?K (0))) takes the values of 0, +1 and ?1, corresponding to whether the determinant of the matrix formed by the tangents of the three edges at v in that sequence is zero, positive, or negative. 4.2 The matrix elements of the volume operator The volume operator acts on a spin-network state as I <J <K, eI ?eJ ?eK =v where in the second step we have used the following identity (see Appendix B.1 for a proof): ? [? jI (?i )]m I m I ?i j [? jJ (? j )]m J m J = [? jI (??)]m I m I C(?1?) [? jJ (?? )]m J m J . q?I J K := ?4i i jk JeiI JejJ JekK = 4 ,?i j JeiI JejJ , ?lk JelJ JeK k 16 ?lk JelJ JekK ?i j JeiI JejJ ? 16 ?i j JeiI JejJ ?lk JelJ JekK q?I<JJKK ;I J > ? q?I<JIKJ ;J K > . Here ?i j := ?2tr(?i ? j ) is the Cartan?Killing metric on SU (2). The action of the volume operator is local, in the sense that its action is a sum on independent vertices. Therefore, we can focus on its action on a single vertex. The fact that the pseudo-vertices are divalent and the self-adjoint operators JeiI act only at the beginning points of eI implies that the summation in Eq. (4.5) is only over the true vertices v of ? . Equation (3.4) reveals the fact that the operators q?I J K and thus V? only change the intertwiners i in T?, j,i ( A). The operators q?I J K acts on an intertwiner by contracting the corresponding matrix elements of ?i with the intertwiner. Note that = ivJ ; a = ivJ ; a ?[? jI (?i )]m I m I ?i j [? jJ (? j )]m J m J m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn = 16 C ?(1?) Je?J Je?K C ?(1?) Je?I Je?J ? ivJ ; a = 16 ivJ ; a m1???m I ???m J ???mK ???mn = 16 C ?(1?) Je?I Je?J C ?(1?) Je?J Je?K ? ivJ ; a = 16 ivJ ; a m1???m I ???m J ???mK ???mn mJ?1 jJ?1 mK?1 jK?1 mK jK m1???m I ???m J ???mK ???mn Hence the operator q?I J K <I J ;J K > in (4.6) can be represented in terms of J ? by <J K ;I J > and q?I J K m1???m I ???m J ???mK ???mn With the above preparation, we now turn to the action of q?I J K on the intertwiner #ivJ ; a $ m1???m I ???m J ???mK ???mn M associated to a true vertex v in the graphical method. We first consider the case I > 2 and K < n, where aI ?1 and aK will appear in the final result. The other special cases will be dealt with later. According to Eq. (4.9), the first term in the parentheses of Eq. (4.6) evaluated on the intertwiner (3.14) can be represented by the following graphical formula (we present only the parts of the graph of the intertwiner which closely connect to the key steps in the following calculations): where X ( j1, j2) ? 2 j1(2 j1 + 1)(2 j1 + 2)2 j2(2 j2 + 1)(2 j2 + 2). Similarly, the second term in the parentheses of Eq. (4.6) acting on the intertwiner can be expressed as following the simple and rigorous rules of transforming graphs presented in Appendix A.2. Of course, there are alternative ways, corresponding to different choices of coupling, to remove the two curves with spin 1. The results obtained from different ways are related by unitary transformations. In the following, we choose a way guided by the simplicity principle that the number of changed intermediate values ai is as little as possible and the final result is as simple as possible. The calculations of the action of q?I J K on a given intertwiner in the graphical method consist of four steps. We first consider the case that J > I +1 and K > J + 1 (the other cases will be handled later) and focus on the action of q?I<JJKK ;I J > on #ivJ ; a $ m1???m I ???m J ???mK ???mn M . The four steps of our calculation are as follows (see Sect. 4 in [47] for details): In the first step, we have been dragging the two endpoints of curves with spin 1 attached to lines with spins jI and jK , respectively, down to join with two horizontal lines denoted by spins aI and aK ?1 by the following recoupling identities (see Appendix B.2 for a proof): In the second step, we have moved the two points labeled by (aI , aI , 1) and (aK ?1, bK ?1, 1), step by step, to the right-hand side of (aJ ?2, jJ ?1, aJ ?1) and the left-hand side of (aJ , jJ +1, aJ +1), respectively, by repeatedly applying of the following identities (see Appendix B.2 for a proof): where l = I + 1, . . . , J ? 1 and m = K ? 1, . . . , J + 1. In the third step, the identities (4.15) and (4.14) were used again to drag the two endpoints of two curves with spin 1 attached to the line with spin jJ down to join with two horizontal lines denoted by spins aJ ?1 and aJ , respectively. In the fourth (last) step, the two curves with spin 1 were removed from intertwiners by the identity (see Appendix B.3 for a proof) intertwiner #ivJ ; a $ <J K ;I J > on the and we have summed over bJ ?1 and aJ and relabeled the indices b by a . Hence the action of the operator q?I J K M reads which is a linear combination of new intertwiners. The expression (4.19) can be simplified in two aspects. One is to get more symmetric factors in the two multi-products. The other is to simplify the exponents. Notice that the result (4.19) was obtained in the case of J > I + 1 and K > J + 1. Under this case, the product terms of (2a + 1) can be reduced to (2aK ?1 + 1) m=J +1 m=J +1 This expression enables us to write the multiple product over m as the formula which is closer to the multiple product over l in Eq. (4.19). By simplifying the exponents and properly adjusting the ordering of multi-products of ?2a + 1, we finally obtain the compact result m1???m I ???m J ???mK ???mn (?1)aI?1+ jI +aK + jK (?1)aI ?aI (?1) lJ=?I1+1 jl (?1)? mK=?1J+1 jm X ( jI , jJ ) 2 X ( jJ , jK ) 2 1 1 (aI ,...,aK?1) l=I +1 m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn (?1)aI?1+ jI +aK + jK (?1)aI ?aI (?1) lJ=?I1+1 jl (?1)? mK=?1J+1 jm X ( jI , jJ ) 2 X ( jJ , jK ) 2 1 1 ? .(2aI + 1)(2aI + 1).(2aJ + 1)(2aJ + 1) .(2al + 1)(2al + 1)(?1)al?1+al?1+1 jl al?1 al / 1 al al?1 (aI ,...,aK?1) m=J +1 (aI ,...,aK?1) m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn The final results for the special cases of J ? 1 < I + 1 and K ? 1 < J + 1 can be obtained from (4.22) by omitting the corresponding multi-products 0lJ=?I1+1 and 0mK=?1J+1 and summations lJ=?I1+1 and mK=?1J+1. Combining the results (4.21) with (4.22), for the case of I > 2 and K < n, the action of q?I J K in Eq. (4.6) on the intertwiner can be explicitly written down as (?1)aK + jK +aI?1+ jI (?1)aI ?aI (?1) lJ=?I1+1 jl (?1)? mK=?1J+1 jm X ( jI , jJ ) 2 X ( jJ , jK ) 2 1 1 ? .(2aI + 1)(2aI + 1).(2aJ + 1)(2aJ + 1) .(2al + 1)(2al + 1)(?1)al?1+al?1+1 jl al?1 al / 1 al al?1 (2am + 1)(2am + 1)(?1)am?1+am?1+1 jm am?1 am / 1 am am?1 d?H (he)T norm ( A) q?I J K ? T?n,ojr,mi( A) ? , j ,i d?H (he)T norm ( A) q?I J K ? T?n,ojr,mi( A) ?, j ,i Again, we expect that the result (4.23) is general and also suitable for the remaining special cases of 0 < I 2 and K = n when aI ?1 and aK do not exist. While aI ?1 and aK do not exist in the intertwiner, we can ?create? them via the formula (see Appendix B.3 for a proof) a |q?I J K |a ? (?1)aK + jK +aI?1+ jI (?1)aI ?aI (aI ,...,aK?1) ? (?1) lJ=?I1+1 jl (?1)? mK=?1J+1 jm X ( jI , jJ ) 2 X ( jJ , jK ) 21 .(2aI + 1)(2aI + 1).(2aJ + 1)(2aJ + 1) 1 l=I +1 m1???m I ???m J ???mK ???mn ? (?1)? mK=?1J+1 jm X ( jI , jJ ) 2 X ( jJ , jK ) 21 .(2aI + 1)(2aI + 1).(2aJ + 1)(2aJ + 1) 1 l=I +1 a |q?I J K |a = ? a|q?I J K |a . The matrix element formula (4.27) derived in graphical method is the same as the formula obtained from algebraic manipulation for the case of I > 1 and J > I + 1 in [19], although different ways were adopted to deal with the recoupling problem. Moreover, as shown above, the formula (4.27) is also valid for other cases and hence can be regarded as a general expression. Finally, we consider some special cases which usually appear. With the following values of the 6 j -symbols [41]: 2 and K < n 2 and K = n a b c ? 1/ = (?1)s % 2(s + 1)(s ? 2a)(s ? 2b)(s ? 2c + 1) &1/2 1 c b 2b(2b + 1)(2b + 2)(2c ? 1)2c(2c + 1) (I) I = 1, J = 2, K = 3 In this case, the general matrix element formula (4.27) reduces to for n = 3 Moreover, we can further simplify the result (4.32), since the triangular conditions on the 6 j -symbols will constrain the values of a in (4.32) as a2 ? {a2 ? 1, a2, a2 + 1}. Denoting |a2 ? |a2, a3, . . . and |a2 ? 1 ? |a2 ? 1, a3, . . . , we get [2(a2 ? 1) + 1](2a2 + 1) ? [(a2 ? 1)(a2 ? 1 + 1) ? a2(a2 + 1)] 1 = ? ?(2a2 ? 1)(2a2 + 1) [( j1 + j2 + a2 + 1)(? j1 + j2 + a2)( j1 ? j2 + a2)( j1 + j2 ? a2 + 1) for n = 4 ? .(2a2 + 1)(2a2 + 1).(2a3 + 1)(2a3 + 1) where we have used the fact that (?1)2 j1+2 j2+2a2 (?1)2 j3+2a3+2a2 = 1 due to the triangle condition for ( j1, j2, a2) and (a2, j3, a3). (II) I = 1, J = 2, K = 4 In this case, the general matrix element formula (4.27) reduces to for n = 4 (III) I = 1, J = 3, K = 4 In this case, the general matrix element formula (4.27) reduces to Now let us consider the action of ( 21 )e?1?(v) on T v,s ( A). Notice that the spherical tensors ?? can be represented as Eq. ?, j,i (3.21). Hence we have where in the second step we have used the result of Eq. (5.18), and in the fourth step used the identity (see B.5 for a proof) Taking account of Eq. (6.17) can be reduced to Q11 = ? 1 1 ( 2 )e?1?(v)T v,s,norm, ( 2 )e?1?(v)T v,s,norm ? , j,i ? , j,i = ? SU (2)3 I =1,2,3 d?H (hsI )T v,s,norm( A) T v,s,norm( A) ? , j,i ? , j,i tr i jJ1=,10,;j2a, ?j3{a2= j1,a3= j3} i J =0; a ?{a2= j1,a3= j3} ? j1,1, j2, j3 tr i J =0; a ?{a2= j1,a3= j3} i J =0; a ?{a2= j1,a3= j3} j1,1, j2, j3 ? j1,1, j2, j3 The intertwiner in Eq. (6.19) is normalized because of where tr( ) denotes contracting magnetic quantum numbers, we have integrated holonomies to give the contraction of the intertwiner with its complex conjugate in the fourth step and used the fact that the intertwiner is real in the fifth step, and the intertwiner is normalized in the last step. Similarly, the action of e?2?(v) on T v,s ( A) yields ? , j,i 1 V ( j2 = j2 ? 1/2, j3, j1) 2 , and in the second step we have used the following identity (see Appendix B.5 for a proof): The intertwiner in Eq. (6.23) is also normalized because of = ? (2a + 1)(2 j2 + 1) (?1) j1+ j2+ j3 (2 j1 + 1)(2 j2 + 1) (?1) j1+ j2+ j3 (2a + 1)(2 j2 + 1) s(?1) j1+ j2+ j3 (2 j1 + 1)(2 j2 + 1)(?1) j1+ j2+ j3+1 7 Summary and discussion In the previous sections, the graphical method developed by Yutsis and Brink and their extensions, which suit the requirement of representing the holonomies and the intertwiners, are applied to LQG. The algebraic formula is represented by its corresponding graphical formula in an unique and unambiguous way. Then the matrix elements of the operator q?I J K , which is the basic building block of the volume operator, are calculated via the simple rules of transforming graphs. Note that the calculations that we did by the graphical method can also be performed by conventional algebraic techniques. Also, corresponding to every graphical reduction, there is an algebraic reduction because of the correspondence between the graphical and algebraic formulas. However, it is obvious that the graphical method is more concise, intuitive and visual. Note that in our graphical representation, a gauge-invariant intertwiner associated to a vertex v of a standard graph at which n edges with spin j1, . . . , jn incident is represented by where, in the last step, we have used the following identity (see Appendix B.6 for a proof): Note that here one has ?1 0 +1 = 1 (see also Appendix B.6 for a proof). Notice that both [? j (??)]m m and ??? (given by a special 3 j -symbol) in Eq. (7.4) have corresponding graphical representations. The action of q?I J K , corresponding to (7.4), on an intertwiner is given by The first and the third forms (equalities) of the expression (7.3) were adopted as the starting points, respectively, in [18] and in [11,19], and their matrix elements are calculated by graphical and algebraic methods, respectively. In this paper, we considered the second expression (equality) of q?I J K and derived its matrix elements by the graphical method introduced in Sect. 3. In [18], to compute the closed formula, Pietri and Rovelli adopted the Kauffman?s graphical method to deal with recoupling problems. Note that the idea in [18] to employ the first equality of (7.3) to calculate the volume operator can also be carried out by the unique and unambiguous rule of graphical calculation. From (3.5), we have m1???m I ???m J ???mK ???mn = ?4i i jk JeiI JejJ JekK ? ivJ ; a = ivJ ; a = ivJ ; a m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn m1???m I ???m J ???mK ???mn (?4i ) ??? [? jI (??)]m I m I [? jJ (?? )]m J m J [? jK (?? )]mK mK , where X ( jI , jJ , jK ) ? 2 jI (2 jI + 1)(2 jI + 2)2 jJ (2 jJ + 1)(2 jJ + 2)2 jK (2 jK + 1)(2 jK + 2). The derivation of the action of q?I J K on the intertwiner in the graphical method is to remove the three curves with spin 1 in (7.7) by using the previous rules of transforming graphs. The identities in Eqs. (4.14), (4.15), (4.16) and (4.17) enable us to reduce the graphical formula (7.7) as where the factor F (a, a , a , a ) involves the intermediate momenta a, a , a and a identity in the intertwiner. By the graphical we can remove the three curves with spin 1 and obtain the final result, which coincides with (4.23). The closed formula of the volume operator was also derived by Brunnemann and Thiemann in [11, 19] using the algebraic techniques. The derivation process in [11, 19] is rigorous but rather abstract and awkward. Our graphical method is convenient and visual, and our result (4.27) coincides with the formula derived by the algebraic calculation for the case of I > 1 and J > I + 1 in [19]. Moreover, our analysis shows that Eq. (4.27) is also valid for other cases and hence can be regarded as a general expression. In principle, in the light of the matrix elements of q?I J K in Eq. (4.27) we can finally write down the action of the volume operator on the spin-network states. We denote q?v ? 8 ? 4 I <J <K , eI ?eJ ?eK =v With the matrix elements of q?I J K , we can get the eigenvalues and corresponding eigenstates of q?v as Then we can write down the action of V?v on the intertwiner |iv associated to v as V?v |iv = |q?v | |iv = Hamiltonian of matters will diverse at the big bang singularity (with zero volume). However, in LQC the inverse volume operator corresponding to the inverse of the scale factor is bounded above [25]. To see whether the boundedness of the inverse scale factor operator in LQC is maintained by the inverse volume operator in LQG, the expectation values of the inverse volume operator V ?1alt,v with respect to gauge-invariant states at a trivalent vertex, which is a non-trivial eigenstate with zero-eigenvalue, was calculated in LQG [27, 57]. The conclusion drawn in [27, 57] is that V ?1alt,v is unbounded. To crosscheck the algebraic calculation in [27], the same action of V ?1alt,v has been calculated by graphical method in this paper. Based on the gauge invariant operators q?I J (v), the inverse volume operator V ?1alt,v defined in (6.10) takes eigenvalues on the orthonormal spin network state T v,s,norm( A). The eigenvalues of V ?1alt,v , presented in (6.15), consist of the eigenvalues ? , j ,i Q I J of q?I J (v). The different conclusions between Ref. [27] and ours come from the different eigenvalues Q I J on the same state T v,s,norm. More concretely, there are two differences on Q I J : (i) a global sign and (ii) the coefficients of Q I J . It turns ? , j ,i out that there are two mistakes made in [27], which lead to the incorrect value of Q I J . First, a minus sign was missed in the second step of Eq. (4.5) in [27], namely, the right formula should be e?iI (v) ? = ?e?iI (v) rather than e?iI (v) ? = e?iI (v), which is the reason of (i). Second, the values of coefficients C J jK ( A, M, gN ?1) defined in Eq. (3.4) in [27] were incorrect, J? j?K namely, the factor (?1)2 jK there should be replaced by 1. Hence the values of Q I J in Eq. (4.17) of [27] should be corrected. By taking the above two corrections and taking account of the different definitions of ?i (differing for each other by the factor 2), the algebraic calculation would give the same results of Q I J as in this paper. There are similar corrections for other values of Q I J . These corrections lead to a significant change of the conclusion, namely, the eigenvalue I J K L M N Q I L Q J M Q K N of V ?1alt,v on T v,s ( A) is indeed zero. In other words, on the contrary to the conclusion in [27], our calculation shows ? , j ,i that the inverse volume operator V ?1alt,v is bounded (zero-valued) at a trivalent non-planar vertex of the gauge-invariant spin-network states. This conclusion coincides with the one made in [58], although different quantum versions of volume function are adopted. In principle, the graphical calculation method can be applied to the general cases, where the spin-network states are defined on arbitrarily valent vertices and the holonomies appearing in the two operators are expressed in an arbitrary representation of the gauge group. However, for those general cases, the volume operator lacks the explicit matrix elements formula. This prevents us from doing further calculations. For the same reason, the matrix elements of the Lorentzian part of the full gravitational Hamiltonian constraint operator have not been explicitly written down even on the trivalent vertices except for certain special cases [23]. Acknowledgements The authors would like to thank Antonia Zipfel for helpful discussions. J. Y. would also like to thank Chopin Soo and Hoi-Lai Yu for useful discussions. J. Y. is supported in part by NSFC Grant No. 11347006, by the Institute of Physics, Academia Sinica, Taiwan, and by the Natural Science Foundation of Guizhou University (Grant No. 47 in 2013). Y. M. is supported in part by the NSFC (Grants No. 11475023 and No. 11235003) and the Research Fund for the Doctoral Program of Higher Education of China. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: Elements of graphical representation and calculation A.1 Representations of SU(2), Clebsch?Gordan decomposition, and the intertwiner I j1 j2 #? j1 (g) ? ? j2 (g)$ (I j1 j2 )?1 = where I j1 j2 is called the intertwining operator in the representation theory of groups [39]. Given bases em(j11) ?em(j22) ? H j1 ?H j2 (or | j1m1 j2m2 ? | j1m1 ? | j2m2 ? H j1 ? H j2 ) and e(MJ ) ? HJ (or | J M ? HJ ), the components of I j1 j2 , as matrix elements, are given by C m(jn) := (?1) j?n?n,?m = (?1) j+m ?m,?n, C(mjn) ? (C ( j)?1)mn := (?1) j?m ?m,?n = (?1) j+n ?n,?m , Cn( mj)C(njm) = C(mj)nC m(jn) = (?1)2 j ?mm . Obviously, C m(jn) and C(mjn) satisfy C ( j) m m = (?1)2 j C m(jm) , C(mjm) = (?1)2 j C(mj)m , [? j1 (g)]m1 n1 [? j2 (g)]m2 n2 (I jJ1 j2 )?1 ,? j (g)?e(mj)- en(j) := e(mj) ? j (g?1)en(j) , = J M | j1m1 j2m2 , j1, j2 and J , we denote I jJ1 j2 m1m2 ? #I j1 j2 $m1m2 J M which projects the bases em(j11) ? em(j22) of H j1 ? H j2 onto e(MJ ) of HJ ? H j1 ? H j2 . The corresponding matrix elements of representations ? j1 ? ? j2 and ?J in the two bases, respectively, are related to each other by the so-called Clebsch?Gordan series which implies that C m(jn) and C(mjn) are symmetric for integer j , and anti-symmetric for half-odd integer j . The operator C ( j) and its inverse C ( j)?1 can be used to lower and raise indices of the tensors on H j . Hence C m(jn) behaves like a metric tensor. Equation (A.5) can be written in the form of its components as C m(jm) [? j (g)]m n C(njn) = [? j (g)?]m n. [? j (g?1)]n m = [? j (g)?1]n m = [? j (g)?]n m = [? j (g)]m n = [? j (g)?]m where the overline denotes complex conjugation, and we have used (A.4) and (A.11) in the last two steps. By the map C ( j) we can define a natural inner product on H?j as (C ( j) f, C ( j)g)H?j := (g, f )H j , ? f, g ? H j . Then the base transformation = #I j1 j2 $m1m2 Now let us consider the decomposition of the tensor product ? j1 ? ? ? ? ? ? jn of irreducible representations of SU (2) for n > 2. The composition involves n ? 1 decompositions of the tensor products of two representations as (A.1) and the choice of the decomposition schemes. Denote ai (i = 2, . . . , n ? 1) the irreducible representations that appeared in the i th decomposition for a given scheme. In the following, we consider the standard scheme where we firstly decompose ? j1 ? ? j2 into 4 ?a2 , and then decompose ?a2 ? ? j3 into 4 ?a3 , and so on. We also denote a ? {a2, . . . an?1}. For given j1 . . . , jn, allowable J , and compatible vector a ? {a2, . . . an?1}, the corresponding Clebsch?Gordan series reads m1,...,mn,n1,...,nn I J ; a [? j1 (g)]m1 n1 ? ? ? [? jn (g)]mn nn (I jJ1?;??ajn )?1 N m1,...,mn,n1,...,nn and its inverse reads k2,...,kn?1 Notice that the factor (?1) j1? in=2 ji ?J in Eq. (A.15) involves only the spins j1, . . . , jn and J , not the intermediate momenta a2, . . . , an?1. From now on, the intertwiners refer in particular to i jJ1?;?a?jn . The Clebsch?Gordan series (A.14) can be written [? j1 (g)]m1 n1 ? ? ? [? jn (g)]mn nn (i jJ1?;?a?jn )?1 N [? j1 (g)]m1 n1 ? ? ? [? jn (g)]mn nn = (i jJ1?;?a?jn )?1 It is easy to generalize the above results to the decomposition of the tensor product ? j1 ? ? ? ? ? ? jk (g) ? ? jk+1 (g)?1 ? ? ? ? ? ? jn (g)?1 of k representations and n ? k inverse representations into a direct sum of irreducible representations ?J on HJ . The corresponding Clebsch?Gordan series read m1,...,mn,n1,...,nn ? ? ? [? jn (g)?1]nn mn (i jJ1?;?a?jn )?1 [? j1 (g)]m1 n1 ? ? ? [? jk (g)]mk nk [? jk+1 (g)?1]nk+1 Equation (A.18) can be written as which, in the case of J = 0, reduces to [? jk+1 (g)?1]nk+1 mk+1 ? ? ? [? jn (g)?1]nn mn i jJ1?;?a?jn m1???mk The fact that the operator i J ; a j1??? jn is unitary and its matrix elements take real numbers results in (i jJ1?;?a?jn )?1 Given n angular momenta j1, . . . , jn, the intertwiner space H j1,..., jn consists of the intertwiners i J ; a j1??? jn m1???mn Hence the tensor i 0; a j1??? jn m1???mk following inner product: H j1,..., jn ? where the last step can be arrived by using the fact that the matrix i J ; a j1??? jn m1???mn (?1) j1? j2?m3 ?2 j3 + 1 j3m3| j1m1 j2m2 = (?1) j1? j2? j3 2 j3 + 1 A.2 The basic components of the graphical representation and simple rules of transforming graphs In this subsection, we introduce the basic components of the graphical representation and simple rules of transforming graphs [31]. A graphical representation for the matrix elements of irreducible representations of SU (2) is also proposed. A graphical representation is a correspondence between graphical and algebraic formulas. Each term in an algebraic formula is represented by a component of an appropriate graph in a unique and unambiguous way. The Wigner 3 j -symbol is associated with the coupling of three angular momenta to give a zero resultant. The 3 j -symbol has simple symmetric properties and hence is easier to handle than the CGC. The 3 j -symbol is defined in terms of the CGC by [41,42] j3 ? m3| j1m1 j2m2 = (?1) j1? j2? j3 ?2 j3 + 1 The 3 j -symbol takes non-vanishing value when the parameters of the upper row ( j1, j2, j3) satisfy the triangular condition (i.e., | j1 ? j2| j3 j1 + j2) and when the sum of the parameters of the lower row (m1, m2, m3) is zero. The parameters ji and mi are simultaneously integers or half-integers, such that each of the numbers ji ? mi , j1 + j2 + j3, ? j1 + j2 + j3, j1 ? j2 + j3, j1 + j2 ? j3, takes some integer. The 3 j -symbol has the following properties. An even permutation of the columns leaves the numerical value unchanged, while an odd permutation is equivalent to a multiplication by (?1) j1+ j2+ j3 , i.e., Moreover, the 3 j -symbol has the symmetric property = (?1) j1+ j2+ j3 = (?1) j1+ j2+ j3 which reflects the fact that the 3 j -symbol takes real numbers (dues to the real CGCs), i.e., The orthogonality relation for the 3 j -symbol is expressed as Furthermore, the 3 j -symbol is normalized as = 1. The property (A.27) of the 3 j -symbol implies and its inverse in Eq. (A.7) can be expressed as A line with no arrow represents the expression The graphical representation of the 3 j -symbol was collected by Yutsis in [30] and slightly modified by Brink in [31] for convenience. The 3 j -symbol is represented by an oriented node with three lines, which stand for three coupling angular momenta j1, j2, j3 incident at the node [31]. The orientation of the node is meant for the cyclic order of the lines. A clockwise orientation is denoted by a ? sign and an anti-clockwise orientation by a + sign. Rotation of the diagram does not change the cyclic order of lines, and the angles between two lines as well as their lengths at a node have no significance. Consequently, any geometrical deformation of the diagram which preserves the orientation of the node does not change the 3 j -symbol represented by the diagram. The 3 j -symbol can be written in the graphical form In a graphical representation, two lines representing the same angular momentum can be joined. Summation over a magnetic quantum number m is graphically represented by joining the free ends of the corresponding lines. Equation (A.25) implies that the CGC can be represented graphically by Therefore, the graphical representation of the intertwiner defined in Eq. (A.15) is and the generalized intertwiner in Eq. (A.19) can be repressed by Now we outline some useful rules of transforming graphs, which can simplify our calculation of the action of operators on the quantum states in LQG. The frequently used rules for adding or removing arrows in a graph read which correspond to the algebraic formulas in Eqs. (A.8), (A.9), (A.10) and (A.29). The rule to remove a closed loop in a graph reads which is related to the orthogonality relation for the 3 j -symbol in Eq. (A.30). Equation (A.31) implies The special 3 j -symbol with one zero-valued angular momentum is related to the ?metric? tensor by Coupling four angular momenta to a zero resultant will involve the j m-coefficients. The j m-coefficients corresponding to different coupling schemes are related by the 6 j -symbol. The 6 j -symbol is defined by ([41] p. 94) ? C(mj11)m1 C(mj22)m2 C(mj33)m3 C(mj44)m4 C(mj55)m5 C(mj66)m6 . Graphically, we can express the 6 j -symbol in Eq. (A.48) as m4,m5,m6,m4,m5,m6 Hence we have the following graphical identity: where in the last step we have used Eq. (A.44) to remove three arrows. Taking account of the definition of 6 j -symbol in Eq. (A.48) and the fact that the 3 j -symbol is normalized in Eq. (A.31), we can easily show the following identity (see [41], p. 95): C(mj44)m4 C(mj55)m5 C(mj66)m6 = The following algebraic relation between two different coupling schemes: (2 j6 + 1)(?1) j2+ j3+ j5+ j6 corresponds to the following rule of transforming graphs: Using Eq. (A.33) and the symmetric properties of 6 j -symbol, from Eq. (A.53), we can get Similarly, using Eqs. (A.33) and (A.54), we have Note that Brink?s graphical representation in [31] does not involve how to represent graphically the matrix element of the representation of SU (2). Here, we will extend the Brink representation and propose a graphical representation for the unitary irreducible representation ? j of SU (2). The matrix element [? j (g)]m n is denoted by a blue line with a hollow arrow (triangle) in it, Up to now, we have expressed the quantum states [the spin-network states in (2.13)], and the two elementary operators (the holonomy and flux operators in (2.17) and (2.19)) of LQG in the graphical form. The Clebsch?Gordan series in (A.22) can be represented by where, in the second step, we used (A.43) to flip the orientations of two arrows. Using Eqs. (A.57) and (A.58), we have Appendix B Proofs for some identities B.1 Proofs of algebraic identities in Eqs. (3.7) and (4.8) By the definition of the matrix element [? j (?i )]m m := ddt """"t=0 [? j (et?i )]m m , i [? j (?1)]m m = ? 2 1 [? j (?2)]m m = 2 [? j (?0)]m m = [? j (?3)]m m = ?i m ?m ,m , 1 [? j (?+1)]m m = ? ? Taking account of the specialized formulas for the CGCs, = +i = ?i i [? j (??)]m m = 2 By definition (3.6), we have i = 2 = ? [? jI (??)]m I m I C(?1)? [? jJ (?? )]m J m J . B.2 Proofs of graphical identities in Eqs. (4.14), (4.15), (4.16) and (4.17) Equation (4.14) can be proved by where Eq. (A.53) was used in the second step, and in the last step we used Eq. (A.43), the fact (?1)?4bK?1 = 1, and the symmetric properties of the 6 j -symbol. Equation (4.16) can be proved by where Eq. (A.33) was used in the second and fourth steps, (A.53) was used in the third step, and in the last step we used the symmetric properties of the 6 j -symbol and the exponents were simplified. B.3 Proofs of graphical identities in Eqs. (4.18) and (4.24) Equation (4.18) can be proved by where the identities in Eqs. (A.33), (A.44), (A.41), (A.42) and (A.45) were used from the first to last steps. Equation (4.24) can be proved by B.4 Proofs of graphical identities in Eqs. (5.21) and (5.22) The graph on the left-hand side of (A.51) can be transformed to The first graph on the right-hand side of (A.51) represents the 6 j -symbol, which can be transformed as where we have used the rules (A.41)?(A.44) of transforming graphs. Hence Eq. (A.51) is equal to the following graphical identity: Graphically, Eq. (5.21) can be proved by where we have used the rules (A.41)?(A.44), and (B.23) in the third and fifth steps, respectively. Similarly, Eq. (5.22) can be shown by where we have used (B.23) in the second and fourth steps, and used the fact that the allowed triple ( j2, j3, j1) satisfy the triangular condition in the fifth step. B.5 Proofs of graphical identities in Eqs. (6.18) and (6.25) The identity (6.18) can be proved by Equation (6.25) can be obtained from where in the second step we have used the identity (A.53). B.6 Proofs of algebraic identities in Eqs. (7.5) and (7.6) To prove Eq. (7.5), we denote [? jI (?i )]nI mI [? jJ (? j )]nJ m J [? jK (?k )]nK mK by [?i ? j ?k ]nI mI nJ m J nK mK . Taking account of (B.12), we have = [?1?2 ? ?2?1]nI mI nJ m J [?3]nK mK + [?3]nI mI [?1?2 ? ?2?1]nJ m J nK mK = ?i [?+1??1 ? ??1?+1]nI mI nJ m J [?0]nK mK ? i [?0]nI mI [?+1??1 ? ??1?+1]nJ m J nK mK ? (?i )[?+1?0??1 ? ??1?0?+1]nI mI nJ m J nK mK = ?i [ +1?1 0?+1??1?0 + ?1+1 0??1?+1?0 + 0 +1?1?0?+1??1 = ?i ??? [???? ?? ]nI mI nJ m J nK mK = ?i ??? [? jI (??)]nI mI [? jJ (?? )]nJ m J [? jK (?? )]nK mK , where ??? is the Levi-Civit? symbol defined by ?1 0 +1 = 1. To show Eq. (7.6), notice that Recalling the symmetric property of the 3 j -symbol in Appendix A.2, an even permutation of the columns leaves the numerical value unchanged, while an odd permutation will lead to a factor (?1)1+1+1 = ?1 for the 3 j -symbol in Eq. (A.33). These symmetries of the 3 j -symbol are the same as those of ??? . Hence we have ??? = ?6 ?1 ?1 ?1 . Appendix C The diagonalization of the volume operator in 2-d intertwiner space The matrix of the operator i q? j1 j2 j3 in the above two states reads The eigenvalues and corresponding (normalized) eigenvectors of i q? j1 j2 j3 are given by Hence we obtain k=1 k=1 Now we derive the value of |b|. Using the matrix elements of a |q?234|a in Eq. 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Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity, The European Physical Journal C, 2017, DOI: 10.1140/epjc/s10052-017-4713-0