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Closed formula for the matrix elements of the volume operator in canonical quantum gravity

We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four spacetime dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state independent factor.


Introduction
The volume functional V for Lorentzian canonical quantum gravity plays a quite important role in the quantization process : if one follows the approach advertized in [1] and chooses a representation in which traces of the holonomy along closed loops for a certain SU(2) A i a connection can be promoted to basic configuration operators, then it becomes technically very hard to define physically interesting operators such as the Euclidean and Lorentzian Hamiltonian constraint operators, an operator corresponding to the length of a curve, an operator corresponding to the ADM energy, matter Hamiltonian operators and so forth. The common reason is that all these operators involve only the co-triad e i a of the intrinsic metric of the initial data hypersurface Σ polynomially which, however, cannot be written as a polynomial in the momentum conjugate to A i a and therefore does not seem to make any sense as a quantum operator.
Fortunately, there is a trick available : as proved in [2,3], the total volume of Σ is the generating functional of the co-triad meaning that one gets e i a from V essentially by functional derivation. Therefore, if we write δtṡ a e i a ∝ {h s , V } + o(δt 2 ) where s is an edge with parameter length δt, h s is the holonomy along s and {., .} denotes Poisson brackets then it would be sufficient to quantize V in such a way that (gauge-invariant) functions of a finite number of holonomies form a dense domain. Such a satisfactory quantization of V which has precisely this property was indeed given already in the literature [4,5]. Therefore, the strategy outlined above and in [2,3] can be implemented and, in particular, was successfully employed to construct various operators listed above [2,3,6,7,8]. However, the explicit computation of the spectrum of most of those operators was not performed so far. The reason is that the spectra of all these operators are clearly largely determined by the spectrum of the volume operator whose precise spectrum is fairly unknown except for the most simple situations [9,10].
The present paper is devoted to the derivation of the complete set of matrix elements of this operator in closed form from which the spectrum can be obtained straightforwardly.
In section 3, after introducing the notation in section 2, we develop a novel technique to regularize the volume operator which is state dependent very much in the same way as the regularization displayed in the second reference of [5]. This part of the paper is a continuation of the work on the point splitting regularizations of the volume and area operators as displayed in [1,5]. The derivation is rather short and gives a diffeomorphism covariant end result. We are unambiguously led to the Ashtekar-Lewandowski volume operator defined in [18] up to a factor of 27/8.
In section 4 we repeat the argument given in [5,9] which shows that the spectrum is discrete and that its computation reduces to linear algebra, in the sense that we just need to determine separately the eigenvalues of an infinite number of finite-dimensional, positive semi-definite and symmetric real-valued matrices.
Finally, in section 5 we derive the explicit formula for an arbitrary matrix element in a spin-network basis and compute the eigenvalues of the associated matrices in a number of simple cases. It should be noted that the formulae obtained, albeit quite complicated, are the starting point for a numerical evaluation of the spectrum and are in particular useful to determine the large spin behaviour of the volume operator.

Notation
Let us first introduce the notation. We consider the Hamiltonian formulation of the Palatini action of four-dimensional Lorentzian vacuum gravity. Denote by q ab the intrinsic metric on the initial data hypersurface Σ and let e i a be its co-triad where i, j, k, .. denote su (2) indices. The densitized triad E a i := det((e j b ))e a i multiplied by 1/κ, κ being Newton's constant, is the momentum conjugate to K i a := sgn(det((e j c )))K ab e b i where K ab is the extrinsic curvature of Σ. Upon performing a canonical point transformation [11,12] on the phase space coordinatized by (K i a , E a i /κ) we arrive at the chart (A i a /κ := (Γ i a + K i a )/κ, E a i ) where Γ i a is the spin-connection of e i a . It is easy to see that the configuration variable A i a is a su(2) connection. The virtue of casting general relativity into a connection dynamics formulation is as follows.
1) First of all, the phase space of general relativity is now embedded into that of an su(2) gauge theory, thereby opening access to a wealth of techniques that have been proving useful in quantizing those theories. Probably the most important technique is the use of traces of the holonomy along piecewise analytic loops in Σ, so-called Wilson-loop functionals, to coordinatize the space of smooth connections modulo gauge transformations A/G [13,14].
2) Using those Wilson loop functionals one can construct [15,16,17] a quantum configuration space of generalized connection modulo gauge transformations A/G and find natural, σ-additive, diffeomorphism invariant, regular Borel probability measures µ 0 thereon which equips us with a Hilbert space structure H := L ( A/G, dµ 0 ). Of course, for this to make sense we have to make the following assumtion : Wilson loop functionals continue to make sense as operators in the quantum theory. We will adopt this viewpoint in the sequel in the hope to capture one of the physically interesting phases of quantum general relativity. The interested reader is referred to [1] and references therein for further details.
The following paragraph is supposed to equip the reader with a working knowledge of the tools associated with A/G. We begin by defining cylindrical functions. Since the gauge invariant information contained in a connection is captured by finite linear sums of products of Wilson loop functionals we may label gauge invariant functions by piecewise analytic graphs γ which are just the union of the loops involved. Such a graph consists of maximally analytic pieces, called the edges e of γ and the edges meet in the vertices v of γ. A function f on A/G is said to be cylindrical with respect to a graph γ iff it can be written as ., h en (A)) and where e 1 , .., e n are the edges of γ.
Here h e (A) is the holonomy along e evaluated at A ∈ A/G and f γ is a complex valued function on SU(2) n . Since a function cylindrical with respect to a graph γ is automatically cylindrical with respect to any graph bigger than γ, a cylindrical function is actually given by a whole equivalence class of functions f γ . In the sequel we will not distinguish between this equivalence class and one of its representants. We say that a function cylindrical with respect to a graph γ is of class C n if it is of class C n with respect to the standard differentiable structure of SU(2) n and denote this class of functions by Cyl n γ (A/G), Cyl n (A/G) := ∪ γ∈Γ Cyl n γ (A/G) where Γ is the set of piecewise analytic graphs in Σ. Thus we have a differential calculus [18] on A/G. Finally, the measure µ 0 is the faithful projective limit [19,20] of the self-consistent projective family (µ 0,γ ) γ∈Γ defined by where g I := h e I (A). This equips us with an integral calculus on A/G.

Regularization of the volume operator
In this section we present a new regularization of the volume operator which uses the methods derived by Abhay Ashtekar and Jurek Lewandowski in [18,5,9]. Most of the issues involved in the regularization as presented here are borrowed from their work. The advantage of our procedure is that it is rather short and compact.
Let R ⊂ Σ be an open, connected region of Σ. We have the identity and can thus write the volume of the region R as measured by the metric q ab as follows The next step is to smear the fields E a i . Let χ ∆ (p, x) be the characteristic function in the coordinate x of a cube with center p spanned by the three vectors ∆ i = ∆ i n i (∆) where n i is a normal vector in the frame under consideration and which has coordinate volume vol = ∆ 1 ∆ 2 ∆ 3 det( n 1 , n 2 , n 3 ) (we assume the three normal vectors to be right oriented). In other words, , . > is the standard Euclidean inner product and θ(y) = 1 for y > 0 and zero otherwise. We consider the smeared quantity Notice that if we take the limits ∆ i , ∆ ′ i , ∆ ′′ i → 0 in any combination and in any rate with respect to each other then we get back to ( [1]) evaluated at the point p. This holds for any choice of linearly independent normal vectors n i , n ′ i , n ′′ i . The strange argument (x + y + z)/3 will turn out to be very crucial in obtaining a manifestly diffeomorphism covariant result. We will see this in a moment. Then it is easy to see that the classical identity holds. We could introduce absolute values because we have the classical identity det(q) = | det(q)| and the modulus is necessary for the regulated quantity (3.3) is not non-negative anymore.
The virtue of introducing the quantities (3.3) is that they can be promoted to quantum operators which have the dense domain Cyl 1 (A/G). To see this, notice that due to the canonical brackets {A i a (x), E b j (y)} = κδ b a δ i j δ (3) (x, y) we are naturally led to represent the operator corresponding to E a i byÊ a i (x) = −iℓ 2 p δ/δA i a (x) where ℓ p = √h κ is the Planck length. Now let be given a graph γ. In order to simplify the notation, we subdivide each edge e with endpoints v, v ′ which are vertices of γ into two segments s, s ′ where e = s • (s ′ ) −1 and s has an orientation such that it is outgoing at v while s ′ has an orientation such that it is outgoing at v ′ . This introduces new vertices s ∩ s ′ which we will call pseudo-vertices because they are not points of non-analyticity of the graph. Let E(γ) be the set of these segments of γ but V (γ) the set of true (as opposed to pseudo) vertices of γ. Let us now evaluate the action ofÊ a i (p, ∆) : on a function f = p * γ f γ cylindrical with respect to γ. We find (e : [0, 1] → Σ; t → e(t) being a parametrization of the edge e)Ê   Here we have used 1) the fact that a cylindrical function is already determined by its values on A/G rather than A/G so that it makes sense to take the functional derivative, 2) the definition of the holonomy as the path ordered exponential of e A with the smallest parameter value to the left, 3) A = dx a A i a τ i /2 where SU(2) ∋ τ j = −iσ j , σ i being the usual Pauli matrices, so that [τ i /2, τ j /2] = ǫ ijk τ k /2 and 4) we have defined tr(h∂/∂g) = h AB ∂/∂g AB , A, B, C, .. being SU(2) indices. We now wish to evaluate the whole operatorÊ(p, ∆, ∆ ′ , ∆ ′′ ) on f . It is clear that we obtain three types of terms, the first type comes from all three functional derivatives acting on f only, the second type comes from two functional derivatives acting on f and the remaining one acting on the trace appearing in (3.5) and finally the third type comes from only one derivative acting on f γ and the remaining two acting on the trace. Explicitly we find (we mean by θ(t, t ′ ) the theta function which is unity if 0 < t < t ′ < 1, 1/2 if t = 0 < t ′ < 1, 0 < t < t ′ = 0 and 1/4 if t ≤ t ′ ∈ {0, 1} and zero otherwise.
Finally m(t, t ′ , t ′′ ) = 1/2 k where k = 0, 1, 2, 3 is the possible number of arguments that equal 0, 1) )tr(h e (0, t)τ i h e (t, 1) ∂ ∂h e (0, 1) ) ∂ ∂h e ′′ (0, 1) )]tr(h e (0, t)τ i h e (t, 1) ∂ ∂h e (0, 1) ) The fact that the integrand of the terms involved inÔ 12,3 ,Ô 1,23 ,Ô 2,31 ,Ô 123 vanishes if either of the cases 0 < t = t ′ < 1, 0 < t ′ = t ′′ < 1, 0 < t = t ′ = t ′′ < 1 occurs is due to the fact that in this case inÔ 12,3 ,Ô 1,23 ,Ô 2,31 ,Ô 123 we get a trace which contains τ (i τ j) , τ (j τ k) , τ (k τ i) contracted with ǫ ijk which vanishes (to see this recall that the functional derivative is (one sided derivatives and δ distributions). This expression is also correct if x is an endpoint of e (in which case there is only one term which survives in (3.7)). which results in the case that we consider h e 1 τ j h e 2 instead of h e , e = e 1 • e 2 , x = e 1 ∩ e 2 a point of analyticity, in a term involving τ (i τ j) ). Given a triple e, e ′ , e ′′ of (not necessarily distinct) edges of γ, consider the functions This function has the interesting property that the Jacobian is given by which is precisely the form of the factor which enters all the integrals in (3.6). This is why we have introduced the strange argument (x + y + z)/3. We now consider the limit Lemma 3.1 For each triple of edges e, e ′ , e ′′ there exists a choice of vectors n i , n ′ i , n ′′ i and a way to guide the limit Here we have denoted byÔ ee ′ e ′′ (t, t ′ , t ′′ ) the trace(s) involved in the various terms of (3.6).
If at least one of e, e ′ , e ′′ does not intersect p then, if we choose ∆ i etc. smaller than some finite number ∆ 0 , (3.9) vanishes identically since the support of the characteristic functions is in a neighbourhood around p which shrinks to zero with the ∆ i etc. So let us assume that all of e, e ′ , e ′′ intersect p at parameter value t 0 , t ′ 0 , t ′′ 0 (this value is unique because the edges are not self-intersecting). Then we can write e(t) = p + c(t − t 0 ) where c is analytic and vanishes at τ = t − t 0 = 0. We have the case subdivision : ∂(t,t ′ ,t ′′ ) )) t 0 = 0. Notice that all vectorsċ(0),ċ ′ (0),ċ ′′ (0) are non-vanishing by the definition of a curve.
We consider first case I). We exclude the trivial case that all three curves lie in a coordinate plane or line such that the determinant already vanishes for all finite values of the ∆'s. Therefore there exist linearly independent unit vectors u, v, w (not necessarily orthogonal) in terms of which we may express c, c ′ , c ′′ . In case Ia) we have an expansion of the form where a, b, c, a ′ , b ′ , c ′ , a ′′ , b ′′ , c ′′ are real numbers with aa ′ a ′′ = 0 and at least one of the b's and c's being different from zero (also not for instance The characteristic functions have support in coordinate cubes spanned by the vectors n i , n ′ i , n ′′ i . Now, since u, v, w are linearly independent we may simply choose, for instance, n i := u, n ′ i := v, n ′′ i := w. It follows then and from the fact that 0 ≤ χ ≤ 1 that From the explicit expansions of c, c ′ , c ′′ we conclude that (3.11) has the bound for some sufficently large numbers δ 1 , δ ′ 1 , δ ′′ 1 . On the other hand we also see from the explicit expansion of | det( The prescription of how to guide the limit in case Ia) is then to synchronize ∆ 1 = ∆ ′ 1 = ∆ ′′ 1 = ∆ and to take the limit ∆ → 0 first. The integral is at least of order ∆ 5 while we divide only by an order of ∆ 3 so that the result vanishes. In case Ib) we have an expansion of the form (let w.l.g. c, c ′ have co-linear tangents) (3.13) We now argue as above and find that the product of the characteristic functions can be estimated by while the determinant can be estimated as above just that k is now given by k = min(m, m ′ , m ′′ , n, n ′ , n ′′ ) − 1 ≥ 1. The prescription is now ∆ 1 = ∆ ′ 1 = ∆ ′′ 2 =: ∆ → 0 first and we conclude that the integral is at least of order ∆ 4 while we divide again only by ∆ 3 such that the limit vanishes. In case Ic) finally we have an expansion of the form (3.14) This time we estimate the product of the characteristic functions for instance by while the determinant can be estimated as above and k is given by k = min(m, m ′ , n, n ′ , n ′′ )− 1 ≥ 1 so that we have actually the same situation as in case Ib) upon synchronizing this time ∆ 1 = ∆ ′ 2 = ∆ ′′ 2 =: ∆ → 0.
As for case II) we observe that the nonvanishing of the functional determinant at p implies that the map x ee ′ e ′′ is actually invertible in a neighbourhood of p by the inverse function theorem. In other words, there is only one Moreover, since the determinant is non-vanishing at p, all three edges must be distinct form each other. It follows now from our choice of edges that p must be a vertex v = e ∩ e ′ ∩ e ′′ of γ in order that he result is non-vanishing and thus from the choice of parametrization t 0 = t ′ 0 = t ′′ 0 = 0. Therefore, if we take the limit ∆ ′′ i → 0 first in any order then the condition χ ∆ ′′ (p, x ee ′ e ′′ ) = 1 will actually imply χ ∆ (p, e) = χ ∆ ′ (p, (e + e ′ )/2) = 1 for small enough ∆ ′′ i so that we can take these characteristic functions out of the integral and replace them by 1 if p is a common vertex of all three edges. Also we can replace the operatorÔ ee ′ e ′′ (t, t ′ , t ′′ ) bŷ O ee ′ e ′′ (v). This holds only if the triple intersects in p.
If not all of e, e ′ , e ′′ intersect in p then the limit will vanish anyway if we take a suitable limit of the ∆ i as we have shown before. We can account for that case by replacing Here v is the common vertex at which the distinct e, e ′ , e ′′ must be incident otherwise they could not even pass through a small enough neighbourhood of p. We can also assume that all three edges have linearly independent tangents at v and expand still around t = 0. The remaining integral divided by The factor 1/8 is due to the fact that in the limit ∆ ′′ → 0 we obtain an integral over C(e, e ′ , e ′′ ), the cone based at p and spanned byė(0),ė ′ (0),ė ′′ (0) where the orientation is taken to be positive. This integral just equals R 3 + d 3 tδ(0, t)δ(0, t ′ )δ(0, t ′′ ) = 1/8 as one can easily check. This furnishes the proof. ✷ We conclude that (3.6) reduces to (in particular, the operatorsÔ 12,3Ô1,23Ô2,31Ô123 drop out) where v on the right hand side is the intersection point of the triple of edges and it is understood that we only sum over such triples of edges which are incident at a common vertex. Moreover, is a right invariant vector field 2 in the τ i direction of su (2), that is, X(hg) = X(h). The factor of 1/8 in (3.17) comes from the fact that at t = t ′ = t ′′ = 0 we get only one-sided functional derivatives of the various edges involved (m(0, 0, 0) = 1/8, see above). We also have extended the values of the sign function to include 0 which takes care of the possibility that one has triples of edges with linearly dependent tangents. The final step consists in choosing ∆ = ∆ ′ and taking the square root of the modulus. We replace the sum over all triples incident a common vertex e,e ′ ,e ′′ by a sum over all vertices followed by a sum over all triples incident at the same vertex v∈V (γ) e∩e ′ ∩e ′′ =v . Now, for small enough ∆ and given p, at most one vertex contributes, that is, at most one of χ ∆ (v, p) = 0 because all vertices have finite separation. Then we can take the relevant χ ∆ (p, v) = χ ∆ (p, v) 2 out of the square root and take the limit which results in where we could switch the order of the X's because a triple contributes only if the corresponding edges are distinct and so the X's commute. Expression (3.18) is the final expression for the volume operator and coincides up to a factor 27/8 with the expression found in [5,9] while it is genuinely different from the one found in [4] as pointed out in [9]. Note that the final expression is manifestly diffeomorphism covariant. Although the procedure of adapting the limiting to a given triple of edges is somewhat non-standard there is an argument in favour of such a procedure : the discussion in lemma (3.1) reveals that any other regularization which would result in a finite contribution for the case where s(e, e ′ e ′′ ) is zero would necessarily depend on the higher order intersection characteristics of a triple of edges. However, since such a quantity is not diffeomorphism covariant which is inacceptable, the dependence must be trivial, that is, a constant (this is precisely what happens in (3.4) according to (3.11). While there is no a priori kinematical reason to prefer one operator over the other, there is a dynamical reason : the volume operator can be seen as an essential ingredient in the regularization of the Lorentzian Wheeler-DeWitt constraint operator [2,3]. The fact that (3.18) contains s(e, e ′ , e ′′ ) is the precise reason for this operator to be anomaly-free ! Therefore, the regularization leading to (3.18) is the one singled out by the dynamics of the theory !

Spectral Analysis
In this section we summarize a number of results due to Abhay Ashtekar and Jurek Lewandowski obtained in the series of papers [18,5,9]. We have included them here for the sake of self-containedness of the present paper. More specifically, the cylindrical consistency of the family of volume operatorsV γ as defined in [18] and in the present paper, the self-adjointness of the corresponding projective limit and the fact that the spectrum of the volume operator is discrete was first observed in [9]. The proof of these properties given in [9] uses methods developed in [18,1,5,9].
This section is subdivided into three parts. First we prove that the family of operators derived in (3.18) defines a linear unbounded operator on H. Next we show that the operator is symmetric, positive semi-definite and admits self-adjoint extensions and finally we show that its spectrum is discrete and that the operator so defined is anomaly-free.

Cylindrical Consistency
What we have obtained in (3.18) is a family of operators (V (R) γ , D γ ) γ∈Γ . That is not enough to show that this family of cylindrical projections "comes from" a linear operator on H. As proved in [18], for this to be the case we need to check that whenever γ ⊂ γ ′ then 1) p * γγ ′ D γ ⊂ D γ ′ where p γγ ′ is the restriction from γ ′ to γ. This condition makes sure that the operator defined on bigger graphs can be applied to functions defined on smaller graphs. 2) (V (R) γ ′ ) |γ =V (R) γ , this is the condition of cylindrical consistency and says that the operator on bigger graphs equals the operator on smaller graphs when restricted to functions thereon. A graph γ ⊂ γ ′ can be obtained from a bigger graph γ ′ by a finite series of steps consisting of the following basic ones : i) remove an edge from γ ′ , ii) join two edges e ′ , e ′′ , such that e ′ ∩e ′′ is a point of analyticity, to a new edge e = e ′ •(e ′′ ) −1 iii) reverse the orientation of an edge. Clearly, a dense domain forV (R) γ is given by D γ := Cyl 3 γ (A/G). This choice trivially satisfies requirement 1) since functions which just do not depend on some arguments or only on special combinations h e = h e ′ h e ′′ , h e ′ = h −1 e are still thrice continuously differentiable if the original function was (here we have used the fact that SU(2) is a Lie group, that is, group multiplication and taking inverses is an analytic map). Next, let us check cylindrical consistency. Consider first the case i) that γ does not depend on an edge e on which γ ′ does. Then clearly X i e f γ = 0 for any function cylindrical with respect to γ and so in the sum over triples over vertices in (3.18) the terms involving e drop out. Next consider the case ii). If e = e ′ • (e ′′ ) −1 is an edge of γ and e ′ , e ′′ are edges of γ ′ where v := e ′ ∩ e ′′ is a point of analyticity for γ while for γ ′ it is not, then while v is a vertex for γ ′ it is only a pseudo-vertex for γ and so inV (D) γ there is no term corresponding to v. On the other hand, since the vertex v is a pseudo vertex for γ it is in particular only two-valent and so the corresponding term inV (D) γ ′ drops out. Likewise, if v is a vertex for γ at which the outgoing edge e is incident, then from right invariance of the vector field we have X e = X e ′ •(e ′′ ) −1 = X e ′ and so at vertices that belong to both γ and γ ′ the corresponding vertex operators coincide. Finally, case iii) is actually excluded by our unambiguous choice of orientation. We conclude that there exists an operator (V (R), D) on H which is densely defined on D = Cyl 3 (A/G).

Symmetry, Positivity and Self-Adjointness
Notice that the vector field iX e is symmetric on H γ , the completion of Cyl 1 γ (A/G) with respect to µ 0,γ , e an edge of γ, because the Haar measure is right invariant. It follows from the explicit expression (3.18) in terms of the iX e that all the projectionsV (R) γ are symmetric. In this special case (namely, the volume operator leaves the space D γ invariant) this is enough to show thatV (R) is symmetric on D. Furthermore, allV (R) γ are positive semi-definite by inspection so thatV (R), D is a densely defined, positive semidefinite and symmetric operator. It follows that it has selfadjoint extensions, for instance its Friedrich extension.

Discreteness and Anomaly-freeness
The operatorV (D) has the important property that it leaves the dense subset Cyl ∞ γ )(A/G) ⊂ H invariant, separately for each γ ∈ Γ. Moreover, recall that H possesses a convenient basis, called the spin-network basis in the sequel [21,22,23]. Let us recall the most important features of that basis. Given a graph γ, let us associate with each edge e ∈ E(γ) an spin quantum number j e > 0. Also, with each vertex v ∈ V (γ) we associate a certain contraction matrix c v as follows : if π j is the irreducible representation of SU(2) corresponding to spin j then form the tensor product ⊗ e∈E(γ) π je (h e ) and for each vertex v contract the indices A of the matrices π je (h e ) AB for all e incident at v in a gauge invariant fashion. We obtain a gauge invariant function T γ, j, c (A), called a spin-network function. It turns out that given γ, j there are only a finite number of linearly independent c compatible with γ, j. Moreover, one can show that spin-network functions defined on different graphs are orthogonal and even on the same graph they are orthogonal whenever the vectors j don't equal each other. It follows that there is an orthonormal basis Now it is obvious that the operatorV (R) leaves the finite dimensional vector space U γ, j spanned by spin-network states compatible with γ, j invariant. The matrix is therefore finite-dimensional, positive semi-definite and symmetric. The task of computing its eigenvalues therefore becomes a problem in linear algebra ! Next, since from (3.18)V (R) γ = ℓ 3 and sinceV v,γ involves only those e ∈ E(γ) with v ∈ e, we find thatV v,γ can only change the entry c v in c. In other words, [V v,γ ,V v ′ ,γ ] = 0 and eachV v,γ can be diagonalized separately. Finally, since the spins j e only take discrete values it follows that H γ has a countable basis and the spectrum thatV (R) attains on D γ is therefore pure point. Let us check whether this is the complete spectrum. Assume it were not and letP the spectral projection on the rest of the spectrum (the existence of the spectral projections relies on the fact that V (R) is self-adjoint and not only symmetric). It follows that u =P v is orthogonal to D γ where v is any vector in H γ . But D γ is dense in H γ and so we find for every ǫ > 0 a φ ∈ D γ with ||u − φ|| < ǫ. Now we have from orthogonality ǫ 2 > ||u − φ|| 2 = ||u|| 2 + ||φ|| 2 > ||u|| 2 and so u = 0. This shows that the complete spectrum is already attained on D γ . It is purely discrete as well 3 . Last, we wish to show that the volume operators are anomaly free (given the fact that we have largely adapted our regularization to a graph, this statement is far from trivial). By this we mean the following : given any two open sets R 1 , R 2 ⊂ Σ we have vanishing Poisson brackets {V (R 1 ), V (R 2 )} = 0 because the functionals V (R) depend on the momentum variable E a i (x) only. Now, given a function f cylindrical with respect to a graph γ, it is not at all obvious any more that [V (R 1 ),V (R 2 )]f = 0 for any such f . Fortunately, given the above characterization of the spectrum, the commutator can be easily proved to vanish on cylindrical functions. To see this, note that the above results imply that if we choose any region R(γ) such that γ ⊂ R(γ) then there exists an eigenbasis of H γ ofV (R(γ)). Now consider any region R. Since all regions are open by construction, all regions fall into equivalence classes with respect to γ : R, R ′ are equivalent if they contain the same vertices of γ (any vertex either has a neighbourhood which lies completely inside R or it lies outside). Therefore any twoV (R),V (R ′ ) differ at most by some of theV v,γ all of which are contained in the expression forV (R(γ)). Since theV v,γ commute the eigenbasis ofV (R(γ) is a simultaneous eigenbasis of allV v,γ for all v ∈ V (γ) and so this eigenbasis is a simultaneous eigenbasis of allV (R) γ . Since all H γ are orthogonal, we have a simultaneous eigenbasis for allV (R). While it is in general not enough to verify that two self-adjoint, unbounded operators commute on a dense domain (rather, by definition, we have to check that the associated spectral projections commute) in our case we are done because the spectral projections are the projections on the various D γ because the point spectrum is already the complete spectrum. Thus we have verified that the commutator algebra mirrors the classical Poisson algebra.

The complete set of matrix elements
In this final section we are going to compute (4.1) in a spin-network basis for each γ, j, c, c ′ , R. The advantage of our approach as compared to the ones proposed already in the literature [10] will be obvious : in [10] one works with an overcomplete and nonorthogonal set of states. This prevents one from giving a unified treatment as it is done here and only allowed one to find a formula for the eigenvalues in a few simple cases. More generally one has to check all the time the linear independence of the states and can compute the matrix elements only in a case by case analysis. In the next subsection we show in more detail a result which follows immediately from [1,5,9] : namely, that the problem of computing the matrix elements of the volume operator reduces to that of computing the matrix elements of a homogenous polynomial of degree three of spin operators for a spin system. This opens access to powerful techniques well-known from the quantum theory of angular momentum for spin systems. Then we will show how the actual computation can be done by viewing the problem as a task in spin recoupling theory and that all the matrix elements can be written as algebraic functions of 6j symbols for which a closed formula exists (Racah formula). Finally we will diagonalize the volume operator for an infinite number of (very special) graphs.

The left regular representation and the spin-network representation
Consider a spin-network state T γ, j, c (A). This is a state for the connection representation L 2 (A/G, dµ 0 ) in the sense that the (equivalence classes modulo gauge transformations of) connection operators are diagonal in this representation. Notice that a spin-network state is gauge invariant so that it can be viewed alternatively as a function of [A], the gauge equivalence class of A. Here we will use the fact that dµ 0 extends to A [20] and reduces to the measure indicated in the introduction when integrating gauge invariant functions. Consider the usual Dirac (generalized) states δ A defined by δ A (A ′ ) := δ µ 0 (A, A ′ ) where δ µ 0 is the δ distribution with respect to µ 0 [23] and A, A ′ ∈ A. We define the spin-network representation by So far this is only notation. The advantage of this notion will become transparent upon proving that the states |γ, j, c > are nothing else than the usual angular momentum states associated with an abstract spin system. To see this, let v be a vertex of γ and e 1 , .., e n the edges of γ incident at v and we have chosen orientations such that all of them are outgoing at v. The spin-network state T γ, j, c (A) can be written where c v is the vertex contractor corresponding to v which contracts the indices corresponding to the starting points of the e i and M v contracts the indices corresponding to the endpoints (recall that the holonomy is a path ordered exponential with the smallest parameter value to the left). Note that c v is C−valued while M v still depends on the rest of the graph. Consider a local gauge transformation at v, that is, h e i → gh e i for some g ∈ SU(2). Then In order for this to be gauge invariant, notice that the tensor product representation ⊗ i π je i (g) can be written as a direct sum of orthogonal irreducible representations π j (g), and so we just need to choose c v to be proportional to one of the various equivalent (but orthogonal) trivial representations π 0 (g) = π 0 (1) contained in that decomposition (see [23] for more details) in order for (5.2) to be gauge invariant. This is precisely what a spin-network state is. The point is now that |m i , j i ; k 1 , .., k m > of an abstract spin system of n spins j 1 , .., j n under rotations (see, e.g., [24]). This is because |(m, j); k > transforms asÛ(g)|(m, j); k >= π j (g) m,m ′ |(m ′ , j); k > by definition, whereÛ(g) is a unitary representation of the rotation group. Here k 1 , .., k m are some additional quantum numbers of observables commuting with the operatorsĴ i which in our case coincide with the m ′ 1 , .., m ′ n . In other words, the left regular representation L g f (h e ) = f (gh e ) that we are dealing with in the connection representation can be seen to be equivalent with the representationÛ (g) of the spin-network representation. Let us now couple the spins j 1 , .., j n to resulting spin j = 0 and denote a particular state with zero angular momentum by |0 >. We claim that up to a constant where the inner product is now to be understood in the angular momentum Hilbert space of the abstract spin system. In other words, the c v are just usual Clebsh-Gordan coefficients. The proof is easy, we have where we have used unitarity of the group and of the representation and thatÛ (g)|0 >= |0 > is rotation invariant. The factor of proportionality is seen to equal unity because c 2 v = 1 and since m 1 ..mn |(m 1 , j 1 )..(m n , j n ); k 1 ..k m >< (m 1 , j 1 )..(m n , j n ); k 1 ..k m | = 1 due to completeness and Wigner-Eckart theorem [24]. Next, consider the self-adjoint right invariant vector fields Y e := −i/2X e . We have [X i , X j ] = −2ǫ ijk X k and so [Y i , Y j ] = iǫ ijk Y k which is the usual angular momentum algebra. It follows that (we use a dual interpretation of the SU(2) indices A, B, C, .. which we may choose to take values ±1/2 and so can be interpreted to be the eigenvalues of the 3-component of angular momentum) and so by multilinearity of the tensor product representation we find due to self-adjointness where j v comprises the components of j which correspond to the edges incident at v. Since spin-network states are thus characterized by the various zero-angular momentum eigenstates |0 > of the abstract spin system, there is a one-to-one correspondence between the possible choices of c v and |0 >. So let |0 > correspond to c v and |0 ′ > to c ′ v . Let p( Y ) be any polynomial of the Y i I corresponding to the vertex v then we have for its matrix elements Here we have used (5.6) in the first step, the orthonormality of the spin-network states in the second step and in the last step we have again recalled that m | j v , m >< j v , m| = 1 is the identity on the abstract angular momentum Hilbert space labelled by the spins j v (completeness, notice that we sum over m in (5.7)). Expression (5.7) is a huge simplification : in order to compute the matrix elements of the volume operator it is enough to compute the matrix elements of the angular momentum operator polynomial ǫ ijk J i e J j e ′ J k e ′′ between states that have zero angular momentum coming from j v . This task is devoted to the next subsection.

Recoupling Theory for n angular momenta
The first task is to label all the linearly independent states |0 > which correspond to the coupling of n angular momenta j 1 , .., j n to zero angular momentum. This is a well known problem in the quantum theory of angular momentum and runs under the name recoupling theory. We review here what we need for our computation. The states | j, m > are a complete basis for a spin system consisting of n angular momenta. In this basis the 2n mutually commuting operators (J i I ) 2 , J 3 I are diagonal with eigenvalues j I (j I + 1), m I . We are obviously interested in a basis in which the square of the operator of total angular momentum J := J 1 + .. + J n and its 3− component are diagonal with eigenvalues j(j + 1), m. In order to label that basis we need 2(n − 1) more quantum numbers of mutually commuting operators and commuting with J 2 , J 3 . The n operators J 2 I are readily verified to satisfy these conditions but we need n − 2 more. In order to motivate our choice notice thatV v,γ is the square root of the modulus of i(3/2) 3 1 4 I<J<K ǫ(e I , e J , e K )q IJK where 1 4 where Y IJ := Y I + Y J . Here we have used total antisymmetry of ǫ(e I , e J , e K ),q IJK and the fact that [X I , X J ] = 0, I = J in order to get rid of 3! and in order to sum only over I < J < K. This observation 4 urges us to study the operator and thus it would be convenient if we could work in a basis in which the J IJ also were diagonal. Clearly we cannot achieve this for all n(n − 1)/2 operators J IJ but we can work in different bases adapted toq IJK for a particular triple I < J < K and expand them in terms of each other. We will develop a theory of recoupling schemes here which is somewhat different from the one encountered in the standard literature and we also consider arbitrary n. The following paragraphs will therefore be somewhat detailed.
the basis | j, m > which allows us to write the 3nj−symbol in terms of Clebsh-Gordan coefficients which are real [24]. It is understood that in (5.9) and in what follows the summation extends only over the allowed values of g etc. which are purely fixed by the values of j 1 , .., j n . The task left is to write the 3nj−symbols of the form < g(IJ)| g ′ > in terms of known quantities. We will derive below a formula which expresses it in terms of the 6j symbol for which a closed formula (the Racah formula [24]) exists. We will denote g(IJ) = (g 2 (j I , j J ), g 3 (g 2 , j 1 ), .., g I+1 (g I , j I−1 ), g I+2 (g I+1 , j I+1 ), .., g J (g J−1 , j J−1 ), g J+1 (g J , j J+1 ), ..g n−1 (g n−2 , j n−1) ) where the notation j ′′ (j, j ′ ) means that we couple J, J ′ to resulting spin j ′′ of J ′′ thereby keeping explicit track of the coupling scheme. The following two elementary lemmata will be crucial in computing the 3nj−symbols.
.δ g n−1 ,g ′ n−1 where on the right hand side we consider a spin system consisting of spins J 1 , .., J J and total spin quantum number g J .
Definition 5.2 An extended spin-network state is a state of the form T γ, j, c as before just that the vertex contractors now are Clebsh-Gordan coefficients of the form < j 1 m 1 , .., j n m n |j 1 , .., j n , g 2 , .., g n−1 , j > where j maybe different from zero. This is important for applications in quantum gravity [2,3] and for the length operator [6] where we need the action of the volume operator on non-gauge invariant functions in an intermediate step.