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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