Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint by Alesci, Thiemann, and Zipfel

This week I have been continuing my work on the Hamiltonian constraint in Loop Quantum Gravity,  The main paper I’ve been studying this week is ‘Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint’. Fortunately enough linking   covariant and canonical LQG was also the topic of a recent seminar by Zipfel in the ilqgs spring program.

The authors of this paper emphasize that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a test  the authors analyze the one-vertex expansion of a simple Euclidean spin-foam. They find that for fixed Barbero-Immirzi parameter γ= 1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for γ = 1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, they fi nd that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.

To circumvent the problems of the canonical theory, a
covariant formulation of Quantum Gravity, the so-called spin-foam model was introduced. This model is mainly based on the observation that the Holst action for GR  de fines  a constrained BF-theory. The strategy is first to quantize discrete BF-theory and then to implement the so called simplicity constraints. The main building block of the model is a linear two-complex  κ embedded into 4-dimensional space-time M whose boundary is given by an initial and final gauge invariant spin-network, Ψi respectively Ψf , living on the initial respectively final spatial hyper surface of a
foliation of M. The physical information is encoded in the spin-foam amplitude.


where Af , Ae and Av are the amplitudes associated to the internal faces, edges and vertices of  κ and B contains the boundary amplitudes.

Each spin-foam can be thought of as generalized
Feynman diagram contributing to the transition amplitude from an ingoing spin-network to an outgoing spin-network. By summing over all possible two-complexes one obtains the complete transition amplitude between ψi and ψf .

The main idea in this paper is that if f spin-foams provide a rigging map  the physical inner product would be given by


and the rigging map would correspond  to


Since all constraints are satis ed in Hphys the physical scalar product must obey


for all ψout, ψn ∈ Hkin.

As a test  the authors consider an easy spin-foam amplitude and show that


where  κ is a two-complex with only one internal vertex such that  Φ is a spin-network induced on the boundary of  κ and Hn is the Hamiltonian constraint acting on the node n.

 Hamiltonian constraint

The classical Hamiltonian constraint is





The constraint can be split into its Euclidean part H = Tr[F∧e] and Lorentzian part HL = C- H.



where V is the volume of an arbitrary region  ∑ containing the point x. Smearing the constraints with lapse function N(x) gives


This expression requires a regularization in order to obtain a well-de fined operator on Hkin. Using a triangulation T of the manifold  into elementary tetrahedra with analytic links adapted to the graph Γ of an arbitrary spin-network.


Three non-planar links de fine a tetrahedron . Now  decompose H[N] into a sum of one term per each tetrahedron of the triangulation,


To define  the classical regularized Hamiltonian constraint as,


The connection A  and the curvature are regularized  by the holonomy h in SU (2), where in the fundamental representation m = ½. This gives,


which converges to the Hamiltonian constraint if the triangulation is sufficiently fine.

As seen in the post

This can be generalized with a trace in an arbitrary irreducible representation m leading to


this converges to H[N] as well.


The important properties of the Euclidean Hamiltonian constraint are;

when acting on a spin-network state, the operator reduces to a sum over terms each acting on individual nodes. Acting on nodes of valence n the operator gives


The Hamiltonian constraint on di ffeomorphism invariant states is independent from the refi nement of the triangulation.


Since the Ashtekar-Lewandowsk volume operator annihilates coplanar nodes and gauge invariant nodes of valence three H does not act on the new nodes – the so called extraordinary nodes.

Action on a trivalent node

To ompute the action of the operator Hmdeltaon a trivalent node where all links are outgoing, denote a trivalent node by |n(ji,jj , jk)> ≡ |n3>, whereas ji, jj ,jk are the spins of the adjacent links ei, ej , ek:



To quantize  Hmdelta[N] the holonomies and the volume are replaced by their corresponding operators and the Poisson bracket is replaced by a commutator. Since the volume operator vanishes on a gauge invariant trivalent node  only need to compute;


so, h(m) creates a free index in the m-representation located at the node , making it non-gauge invariant and a new node on the link ek:


so we get,


where the range of the sums over a, b is determined by the Clebsch-Gordan conditions and


The complete action of the operator on a trivalent state |n(ji, jj , jk)> can be obtained by contracting the trace part with εijk. So, H projects on a linear combination of three spin networks which differ by exactly one new link labelled by m between each couple of the oldlinks at the node.

Action on a 4-valent node

The computation for a 4-valent node |n4> is


where i labels the intertwiner – inner link.

The holonomy h(m) changes the valency of the node and the Volume therefore acts on the 5-valent non-gauge invariant node. Graphically this corresponds to


and finally we get:


This can be simpli ed to;



Using  the de finition of an Euclidean spin-foam models as suggested by Kaminski, Kisielowski and Lewandowski (KKL) and since we,re   only interested in the evaluation of a spin-foam amplitude we choose a combinatorial de finition of the model:

Consider an oriented two-complex  de fined as the union of the set of faces (2-cells) F, edges (1-cells) E and vertices (0-cells) V such that every edge e is a 1-face of at least one face f (e ∈ ∂f) and every vertex V is a 0-face of at least one edge e (v ∈∂e).

We call edges which are contained in more than one face f internal and denote the set of all internal edges by Eint.  All vertices adjacent to more than one internal edge are also called internal and denote the set of these vertices by Vint. The boundary ∂κ is the union of all external vertices (-nodes) n ∉ Vint and external edges (links) l ∉Eint.

A spin-foam is a triple (κ, ρ; I) consisting of a proper foam whose faces are labelled by irreducible representations of a Lie-group G, in this case here SO(4) and whose internal edges are labeled by intertwiners I. This induces a spin-network structure ∂(κ, ρ; I) on the boundary of κ.

The BF partition function can be rewritten as


Av de fines an SO (4) invariant function on the graph Γ induced on the boundary of the vertex v


In the EPRL model  the simplicity constraint is imposed weakly so,


It follows immediately that


defi nes the EPRL vertex amplitude with


Expanding the delta function in terms of spin-network function and integrating over the group elements gives


In order to evaluate the fusion coefficients by graphical calculus it is convenient to work with 3j-symbols instead of Clebsch-Gordan coefficients. When replacing the Clebsch-Gordan coefficients we have to multiply by an overall factor cc.

Spin-foam projector

Given any couple of ingoing and outgoing kinematical states ψout, ψin, the Physical scalar product can be
formally de fined by


where η is a projector – Rigging map onto the Kernel of the Hamiltonian constraint.

Suppose that the transition amplitude Z


can be expressed in terms of a sum of spin-foams, then


this can be interpreted as a function on the boundary graph ∂κ;


in the EPRL sector.

Restricting the boundary elements h ∈ SU (2)  ⊂ SO (4) then;


where |S>N is a normalized spin-network function on SU(2). This fi nally implies;





Compute new solutions to the Euclidean Hamiltonian constraint by employing spin-foam methods. Show thatlinkingequ3.15


in the Euclidean sector with γ= 1 and s = 1, where κ is a 2-complex with only one internal vertex.

Trivalent nodes

Consider the simplest possible case given by an initial and final state  |Θ>, characterized by two trivalent nodes joined by three links:



the only states produced by the HamiltonianHm acting on a node, are given by a linear combination of spin-networks that diff er from the original one by the presence of an extraordinary (new) link. In particular the term sHwill be non vanishing only if |s> is of the kind:


The simplest two-complex κ(Θ,s) with only one internal vertex de fining a cobordism between   |Θ> and |s> is a tube  Θ x[0,1] with an additional face between the internal vertex and the new link m,

Since the space of three-valent intertwiners is one-dimensional and all labelings jf are fixed by the states   |Θ> , |s>  the fi rst sum  is trivial.


Γv= s and therefore we have,


where the sign factor is due to the orientation of s, The fusion coefficients contribute four 9j symbols since,


The full amplitude is




This yields,


The last two terms are equivalent to the first term when exchanging jk↔ jj. The EPRL spin-foam reduces just to the SU(2) BF amplitude that is just the single 6j  in the first line.

Now using the defi nition of a 9j in terms of three 6j’s,


The 9j’s involved in this expression can be reordered using the permutation symmetries  giving,


the statessphysare solutions of the Euclidean Hamiltonian constraint

The spin-foam amplitude selects only those terms which depend on the diagonal elements on the volume. This  simplifies the calculation since we do not have to evaluate the volume explicitly.

Four valent nodes

The case with ψ in = ψout = |n4>where


The matrix element <s| Hm|n4>  is non-vanishing if |s>is of the form


Choose again a complex  of the form


with one additional face jl.

The vertex trace  can be evaluated by graphical calculus


The fusion coefficients give two 9j symbols for the two trivalent edges and two 15j- symbols for the two four-valent edges.The fusion coefficients reduce to 1 when γ=1.Taking the scalar product  we obtain,


Taking the scalar product with the Hamiltonian gives,


Summing over a and using the orthogonality relation gives,


The three 6j’s in the two terms defi ne a 9j ,summing over the indexes and b respectively gives:


the final result is,


As for the trivalent vertex the spin-foam amplitude just takes those elements into account which depend on the diagonal Volume elements.


LQG is grounded on two parallel constructions; the canonical and the covariant ones. In this paper the authors  construct a simple spin-foam amplitude which annihilates the Hamiltonian constraint .They found that in the euclidean sector with signature s = 1 and Barbero-Immirzi parameter γ = 1 the Euclidean Hamiltonian constraint is annihilated by a spin-foam amplitude Z for a simple two-complex with only one internal vertex. The one vertex amplitudes of BF theory are explicit analytic solutions of the Hamiltonian theory.

Also the 6j symbol associated to every face is annihilated by the Euclidean scalar constraint. This is a generalization of the work by Bonzom-Freidel in the context of 3d gravity where they found that the 6jis annihilated by a suitable quantization of the 3d scalar constraint F = 0 . The spin-foam amplitude diagonalizes the Volume.


Related articles

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A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements by Gaul and Rovelli

This week I have been studying a paper which presents a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity.

I have been continuing my collaborative work on the matrix elements of the Hamiltonian Constraint in LQG, as seen the posts:

The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken.

This leads to a quantization ambiguity and to a family of operators with the same classical limit. The authors calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory.


Loop quantum gravity is a canonical approach to the quantization of general relativity. One of the issues in this approach is the identification of the physically correct . Hamiltonian constraint operator (HCO), encoding the dynamics of classical general relativity. In this paper, the authors analyze one of the quantization ambiguities entering the definition of the HCO, and a corresponding variant of Thiemann’s HCO. In particular, they study a family of operators Hm labelled by irreducible SU(2) representations  m ( = 2j), all having the same classical limit, namely the classical Hamiltonian constraint of general relativity. Thiemann’s HCO corresponds to the fundamental representation m = 1. In a nutshell, the HCO requires a gauge invariant point-splitting-like regularization, which is obtained by using the trace of the holonomy of the gravitational connection. Theimann’s  HCO  operator H¹ adds a link of colour 1 to the nodes of the spin network states. The operators Hm  for arbitrary m act on the spin network states by adding a link of color m.

The generalized Hamiltonian

Classical Theory

The construction of Thiemann’s HCO starts from the classical Lorentzian Hamiltonian constraint C of density weight one. Using real Ashtekar-Barbero variables, this can be written as: 1106equ1 Here a, b are tensorial indices on the compact spatial manifold ∑ , and i, j, k are su(2) indices. Square brackets denote antisymmetrization. The inverse densitized triad (an su(2) valued vector density) has components defined by 1106equ1a, where eai is the triad on ∑, and the real SU(2) Ashtekar-Barbero connection is 1106equ1b On  ∑ we have the induced  metric qab, whose inverse satisfies 1106equ1c , as well as the extrinsic curvature kab. Using the triad, one obtains kia  by transforming one spatial index into an internal one. lambdaiais the spin connection compatible with the triad. The AE form a canonically conjugate pair, whose fundamental Poisson brackets arevariables AEpoission, G being G with Newtons constant Gn. Finally, the components of the curvature of the connection are given by Fabk.

The Hamiltonian constraint can then be written as


A more convenient starting point for the quantization is given
by the polynomial expression for the densitized Hamiltonian constraint



where V is the volume of  ∑,


K is the integrated trace of the densitized extrinsic curvature of ∑ ,


Denote the Euclidean Hamiltonian constraint with H, and the Lorentzian term with T.

Using the relations,




because the integrated extrinsic curvature is the time derivative of the volume, i.e. it can be written as the Poisson bracket of volume and (Euclidean) Hamiltonian constraint at lapse equal to one – the Hamiltonian constraint can be entirely expressed in terms of the volume and the connection.

In this paper the authors restrict themselves to the study of the Euclidean constraint. Its smeared form it is,


where N(x) is the lapse function.

The regularization of H[N] is obtained by approximating Fab and Ac with holonomies of the connection around small loops.

Fix an arbitrary triangulation T of the manifold ∑ into elementary tetrahedra with analytic edges. Consider a tetrahedron , and a vertex v of this tetrahedron.

1106efig1Decompose the smeared Euclidean constraint  into a sum of one term per each tetrahedron of the triangulation,


Finally, consider the holonomy he of the connection along edges e, and  define the classical regularized Euclidean Hamiltonian constraint as




A straightforward calculation, using the expansions,




shows that for a fixed value of the connection and triad, the expression converges to the Hamiltonian constraint.  Write this as,


The expression 1106equ12 can immediately be transformed into a quantum operator, yielding the regularized HCO.


The authors now introduce an alternative regularization. Given an irreducible representation of spin j and colour m = 2j, can write,1106equ17

where R(m) is the matrix representing U in the representation m. Replacing the trace Tr with the trace Trm in the regularization of the constraint, gives,


using again the expansions of the holonomy,


and using


then if,




converges to the Hamiltonian constraint for a sufficiently fine triangulation, that is,


Quantum Theory

The HCO operator is defined by adapting the triangulation T to the
graph γ of the basis state ψ on which the operator is going to act,

By replacing the classical quantities with quantum operators, and the Poisson brackets with commutators, can  obtain the operator associated to a single tetrahedron:


Further  manipulations  give the generalized HCO as,


p is one, whenever  is a tetrahedron having three edges coinciding with three edges of the spin network state, that meet at the
vertex v. In the other cases p equals zero.

The action ofHm on trivalent vertices

In the following the authors calculate explicitly the action of generalized Euclidean Hamiltonian constraint operator on trivalent vertices.

Calculations are performed in the spin network basis. Graphical techniques –  Penrose’ graphical binor calculus are introduced in the connection representation, for example to represent spin network states or operators. This method is equivalent to the graphical description in the loop representation, which satisfies the basic axioms of the tangle-theoretic ormulation of Temperley-Lieb recoupling theory.


The Hamiltonian constraint operator Hmacts independently on
single vertices . Look at its  action on a single trivalent vertex for an arbitrary but fixed color m ,


The trivalent vertex is denoted by |v(p, q, r)i ≡ |vi,
where  p, q and r are the colours of the adjacent edges ei, ei, ek.

Proceed by applying the operators  successively, performing the summation over i, j, k at the end.

The operator hsattaches an open colour-m loop segment to the edge ek, creating a new vertex on it, and altering the colour between the two vertices, i.e.


The action of  V

In the next step, the volume operator acts on the state hs. In the generalization of the HCO we adopt the Ashtekar-Lewandowski volume operator. Its action on function ψ, it is given by,


The first sum extends over the set V(γ) of vertices of the underlying graph, while the sum in  extends over all triples (eI , eJ , eK) of edges adjacent to a vertex.

The essential part of the volume operator  is given by W [ijk] – the ‘square’ of the volume. Its action is described in terms of the grasping of any three distinct edges eI , eJ and eK adjacent to vn.


Regarding the computations of the matrix elements for the generalized HCO, the authors obtain,


The action of W on a non-gauge-invariant 3-vertex can generally be expressed as,




This basis is realized by a rescaling, or vertex normalization respectively. The virtual internal edge is multiplied by √Δ, and each of the two virtual nodes is divided by an appropriate √ Δ , giving


With this normalization, we have,


Evaluating this by using the grasping operation and closing
the open network with itself, gives,


The closed network in this expression is simplified by applying the reduction formula to the upper right three triangle-like vertices,               {(β, c,m), (c, 2, c), (α, c,m)}, reducing it to a 9j−symbol.

Evaluating further until only fundamental, or ‘minimal’ closed networks remain, gives


Defining t = (β+α)/2 and e = (β − α)/2 = ±1,
the non-zero matrix elements are


Taking the absolute value and the square root, the double base transformation needs to be reverted to return to the original
basis in which the whole calculation is performed. In all, this is written as


or explicitly in terms of the matrix elements,


The action of the volume operator is in general not diagonal. Unfortunately, the complexity of the problem for arbitrary m and colourings  of the vertex, prevents an  explicit calculation  . Nevertheless, as soon as specific colouring  of a vertex are chosen, the complete calculation can be performed. It is
just the general expression that is lacking at present.

The relation between the vertex operator ˆV and the
square root of the local grasp √i W reads in the trivalent case

using this we obtain for the non-diagonal action of the volume operator,


Completing the action of Hm

To finish the computation of the action of the generalized Euclidean HCO HmΔ , the relevant part of the operator




Performing this computation diagrammatically,gives



Calculating the trace part of the generalized HCO we  get:





The full action of the generalized HCO on |v(p, q, r)>  is given (up to constant factors) by contracting the trace part with εijk. This leads in the complete action of Hmdelta to a sum of three terms. They are distinguished from each other by the assignment of colour-m segments between mutually distinct pairs of edges adjacent to the vertex. The corresponding amplitudes are determined  by cyclic permutations of argument pairs. For a
generic 3-vertex |v(p, q, r)>,


The compact final form for the action of the generalized local HCO Hmdelta  on a 3-valent vertex is,





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Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity

This week I have been continuing my collaborative work on the matrix elements of the Hamiltonian Constraint in LQG, see the posts:

So I have been reviewing knot theory especially skein relations, SU(2) recoupling theory and Lieb-Temperley algebra. And reviewing my old sagemath calculations which I’ve written up as a paper and will post as an eprint on arXiv.

The calculations I have  done this week are based on the paper ‘Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity by Borissov , De Pietri and Rovelli‘.

In this paper the authors present an explicit computation of matrix elements of the hamiltonian constraint operator in non-perturbative quantum gravity. They look at the euclidean term of Thiemann’s version of the constraint and compute its action on trivalent states, for all its natural orderings. The calculation is performed using
graphical techniques from the recoupling theory of colored knots and links. They  give the matrix elements of the Hamiltonian constraint operator in the spin network basis in compact algebraic form.

Thiemann’s Hamiltonian constraint operator.
The Hamiltonian constraint of general relativity can be written as



The first term in the right hand side of  defines
the hamiltonian constraint for gravity with euclidean signature – accordingly, he is called the euclidean hamiltonian constraint.


For trivalent gauge-invariant vertices,  (r, p, q) denotes the
colours associated to the edges I, J,K of a vertex v and we have:


where i = 1, 2 for the two orderings  studied. The explicit values of the matrix elements  a1 and a2are:







Using Sage Mathematics Software I coded these the trivalent vertex matrix elements :matrixelementsfig1

And then checked the resulting calculations against the values tabulated in the paper.


here we can see  A1 and A2 values corresponding to the values of :




My Sage Mathematics Software  code produces precisely the numerical values of A1 and A2 as indicated in the figure above.

For example, for (p,q,r)  = (1,0,1) I have A1 =0.000 and A2 = 0,07433 which is to 4 digit accuracy the tabulated value of 101.

From the point of view of my own research this is great and I can be confident in the robustness of my coding techniques. I will continue working through papers on the matrix elements of the LQG Hamiltonian constraints coding up the analyses working towards coding the full hamiltonian constraint and curvature reviewed in the posts:

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Matrix Elements of Lorentzian Hamiltonian Constraint in LQG by Alesci,Liegener and Zipfel

This week I have been studying a number of papers about the Hamiltonian constraint operator in Loop Quantum Gravity. I’ll be reviewing these papers over the next few posts.

I’m actually working on coding the  numerical implementation of these matrix elements, so I have included more than the usual detail of detail in this post.

The first paper I’ll look at is about the matrix elements of the Hamiltonian constraint. In this paper the authors look at the Hamiltonian constraint in loop Quantum Gravity. the Hamiltonian constraint is the key ele ment of the canonical formulation of LQG coding its dynamics. In Ashtekar-Barbero variables it naturally splits into the so called Euclidean and Lorentzian parts. Due to the high complexity of this operator, only the matrix elements of the Euclidean part have been considered so far.

In this paper the authors  evaluate the action of the full constraint, including the Lorentzian part. The computation requires the use of SU(2) recoupling theory and   n-j symbol identities. These identities, together with the graphical calculus used to derive them, also simplify the Euclidean constraint and are of general interest in LQG computations.

In the Hamiltonian formulation General Relativity  is completely governed by the Diffeomorphism and Hamiltonian constraints. Ashtekar  suggested to replacing  metric variables by connections
and tetrads. In these variables GR resembles other gauge theories, like Yang-Mills theory, and there is an SU(2)-Gauss constraint in addition to Diffeomorphism and Hamiltonian constraints.This formulation serves as the classical starting point of Loop Quantum Gravity (LQG). LQG follows the Dirac quantization program for constrained systems.

A major obstacle for completing the canonical quantization program in LQG is the implementation of the Hamiltonian constraint S. The difficulties are mainly caused by the non-polynomial structure of S
and the weight factor 1/√det(q) determined by the intrinsic metric    q := qab on the initial hypersurface

Thiemann  discovered that both problems, the non-polynomiality and the appearance of the weight factor, can be solved by expressing the inverse triads through the Poisson bracket of volume and connections. This made it possible to construct a finite, anomaly-free
operator that corresponds to the non-rescaled Hamiltonian constraint and acts by changing the underlying graph of the spin networks. The formal solution  to this constraint are superpositions of s-knot states with dressed nodes., that are nodes with a spider-web like structure. This construction is at the heart of many other approaches within canonical LQG, as the master constraint program, Algebraic Quantum Gravity (AQG) and most recently models with matter and symmetry reduced models like Loop Quantum Cosmology. Also the covariant approach i.e. spin foam models are motivated by the idea of realizing the time-evolution generated by a graph-changing Hamiltonian .

Despite it’s central role for LQG the action of S has been analysed explicitly in only very few examples and these are confined to the Euclidean part of the constraint only. This is mainly due to two reasons: first the presence of the volume operator and second the non trivial recoupling of SU(2) irreducible representations. The volume operator in LQG has been studied intensively – see post:

In this paper the authors explicitly compute the matrix elements of the full Lorentzian constraint in the Thiemann prescription for trivalent nodes. The final result still depends on the matrix elements of the volume which are unknown in closed form, but in principle computable.  The resulting compact formula presented in this paper opens the possibility of  testing the implementation of the constraint by simulations oranalyse the behaviour of S in a large j-limit .

The paper  is organized as follows:

  • Thiemann’s construction for the Euclidean and Lorentzian term
  • Main recoupling identities
  • Euclidean constraint – new and  compact expression
  •  Matrix elements of the Lorentzian part.


Classical constraint

Letei be a triad on a smooth, spatial hypersurface  defining the intrinsic metric qab. Here, a; b = 1; 2; 3 are tensorial and i; j = 1; 2; 3 are su(2)-indices.

In the following, lambdadenotes the spin connection associated to eiand Kab the extrinsic curvature. Givenkequalssgn it can be shown that the densitized inverse triad eequalsdetand aequalslambdaform a canonical conjugated pair,




If Fab denotes the curvature of A and s the signature
of the space-time metric then the classical Hamiltonian constraint is of the form

the square root in can be absorbed by using Poisson brackets of the connection A with the Volume,
and the integrated curvature,






where S was split into an Euclidean part H and remaining constraint T, which vanishes for γ² = 1 and s = 1. The second part above can be further modified by expressing K through the ’time’ derivative of the

hamitonianequ8So S is completely determined by the connection A, the curvature F and the volume V all of which have well-defined operator analogous in LQG.


The connection and curvature of the integrated constraint                       S[N] :=N(x)S(x) are replaced by holonomies along edges and loops respectively.

Since the Euclidean constraint H[N] depends linearly on the volume and the volume operator is acting locally on the nodes it suffices to construct a regularization in the neighborhood of a node n in a given
graph Γ and then extend it to all of Σ. Any three (non-planar) edges eI, eJ and eK incident at n constitute an elementary tetrahedron ΔIJK. Starting with ΔIJK one can now construct seven additional tetrahedra , so that the eight tetrahedra including IJK cover
a neighborhood of n. Afterwards this is extended to a full triangulation of Σ and H[N] is decomposed into ∑HΔ[N].


On the elementary tetrahedron ΔIJK the connection A and the curvature F are regularized as by smearing along sK and IJ respectively so that the regularized constraint is defined by


where hs is the holonomy alongside s and N(n) is the value of the lapse function N(x) at n. Generalizing this by considering holonomies in an arbitrary irrep m  gives,

hamitonianequ10with the normalization factor hamitonianequ10a Its action on a cylindrical function Ts on a spin net s with underlying graph Γ is


The first sum is running over all nodes of Γand the second over all elementary tetrahedra. lp is the Planck
length. Because every vertex is surrounded by 8 tetrahedra the smearing of H in the neighborhood of n yields eight times the same factor. Apart from that, at an m-valent vertex n there are hamitonianequ11aelementary tetrahedra each of which determines an adapted triangulation. Therefore we need to divide by E(n) to avoid over counting. A huge advantage of the operator defined above is that it is anomaly free.


The remaining part of the constraint T[N] can be quantized similarly. However, the regularization is a bit more involved because the extrinsic curvature must be regularized separately. In principle T adds two new links each of which is created by one operator p6equ

The holonomies in T can be regulated as above such that finally:


In these equations the triangulation T serves as a regulator. This regularization dependence can be removed by a suitable operator topology.


In this section the authors introduce the  tools for computing  matrix elements of the Hamiltonian constraint and some identities are proven.

 Graphical calculus

The evaluation of the  constraint is mainly based on recoupling theory of SU(2). In this context it is useful to work with graphical methods.

Latin letters represent irreducible representations j … ∈½Ν
Greek letters represent α…= -j,-j + 1,. j, are magnetic numbers.
A state |j,α> in the Hilbert space Hj is visualized by diaja                               The adjoint |j> is drawn as  diajadjoint

A representation matrix matrixR with g ∈SU(2) is pictured asdiaajb

Two lines carrying the same representation can be either connected by taking the scalar product, hamitonianequ13aaor by contraction with;



This two-valent intertwiner defines an isometry between the vector representation (tip, tip) and the adjoint representation (feather , feather) where the transformations.


are graphically encrypted in


The second identity can be derived from the first one using


Using hamitonianequ13 and the properties of Wigner matrices can prove the following identity:



The basic building block of recoupling theory are the 3j-symbols


All other intertwiners can be obtained from these ones. For example a node which has in- as well as outgoing links is constructed by multiplying with:hamitonianequ13

to give:


A four-valent node arises from the contraction of two 3j-symbols:


Different bases are related by 6j-symbols:


Action of the volume

Even though the volume can not be computed analytically for generic configurations it is comparatively easy to calculate it for gauge variant trivalent vertices transforming in a low spin. Even though the volume can not be computed analytically for generic configurations it is comparatively easy to calculate it for gauge variant trivalent vertices transforming in a low spin.This is exactly the type of nodes that are of interest in this paper.

Non-invariant trivalent intertwiners can be treated as four-valent (invariant) ones, e.g.


where one leg, here m, does not correspond to a true edge but indicates in which representation the node is transforming. In a slight abuse of the conventions these open ends m are drawn as solid rather than dashed lines for better visibility. Whether a leg at the intertwiner corresponds to a part of a true edge or
is indicating gauge variance follows from the context.
In general V is altering the intertwiner structure, so that



The action of the Euclidean constraint on trivalent nodes was determined  using Temperely-Lieb algebras and has also been recalculated by graphical methods.

Action on gauge-invariant trivalent vertices

For a single trivalent vertex with adjacent edges ei,ej ,ek the Euclidean constraint :


reduces to


where the Lapse N(n) was set to one and the subscript m in Trm indicates that the holonomies have spin m. This further reduces to


To begin with, the holonomy h‾¹sk   has to be coupled to the edge ek, that leads to a gauge variant vertex because the magnetic indices are not contracted yet. This gives,


The action of the remaining holonomies  can be simplified by using the Loop Trick theorem:


where ε(i,j,k) = 1 if the edges ei, ej ,ek are ordered anti-clockwise, otherwise it equals -1.

The full action of H is:



Action on gauge-variant nodes

To calculate the extrinsic curvature of gauge-variant nodes  we have to evaluate the action of H on nodes of this kind. The full amplitude is then given by:


This concludes the discussion of the action of the Euclidean constraint on trivalent vertices, invariant as well as variant ones. The matrix elements for higher valent nodes can be in principle obtained by analogous methods.


Now proceed to calculate the matrix elements of hamitonianequ41a

Because the volume and therefore the extrinsic curvature p6equ of a gauge-invariant trivalent node is zero, the only non-vanishing contribution of T on such nodes is proportional to:


The only unknown part in this expression is the extrinsic curvature K.

Extrinsic curvature of gauge-variant trivalent nodes

As the extrinsic curvature is linear in H its action on a trivalent  vertex decomposes into a sum,


and the full extrinsic curvature is given by




Matrix elements oftrm



To write down the complete action of T on trivalent invariant nodes we need the contributions from tmsj and tmsk.

The evaluation of the action of the first trace gives:


Computation of tmsk

The evaluation of the action of the second trace gives:



Contraction with ε

To obtain the full action of T on a trivalent (invariant) node both trace contributions as computed above must be summed up and contracted with the ε-tensor.

For the first trace this gives:

contracted first trace gives


In this  article the authors have derived for the first time an explicit formula for the matrix elements of the full Hamiltonian constraint in LQG including the Lorentzian part.  This constraint plays a major role in any canonical quantization program for GR based on real Ashtekar-Barbero variables so that the methods developed in the course of the calculation are also of interest in these approaches.

By exploiting several new recoupling identities, we significantly  the matrix element is simplified so that the recoupling part is totally captured in 6j and 9j symbols for which symmetry properties and explicit formulas are well known. The final expression still depends on the volume but can be implemented on a computer for further investigations.

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Numerical work with Sagemath 17: Exploring curvature

This post is just a bit of informal exploration of the LQG curvature operator which I’ve been doing in preparation for a more formal attack during the next week.


which produces results like those below:



Informally exploring some ideas in sagemath using work already done on length, angle and volume operators.

See posts





This informal work produces reasonable results in terms of magnitudes and form and I’ll refine this over time.

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