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:

- Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity
- Matrix Elements of Lorentzian Hamiltonian Constraint in LQG
- A curvature operator for LQG

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.

**Introduction**

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 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 for arbitrary m act on the spin network states by adding a link of color m.

**The generalized Hamiltonian**

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: 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 , where is the triad on ∑, and the real SU(2) Ashtekar-Barbero connection is On ∑ we have the induced metric , whose inverse satisfies , as well as the extrinsic curvature . Using the triad, one obtains by transforming one spatial index into an internal one. is the spin connection compatible with the triad. The form a canonically conjugate pair, whose fundamental Poisson brackets arevariables , G being with Newtons constant Gn. Finally, the components of the curvature of the connection are given by .

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,

and

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.

Decompose 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

where,

A straightforward calculation, using the expansions,

and

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

The expression 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,

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

using again the expansions of the holonomy,

and using

then if,

then,

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 of 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 acts 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 attaches 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 . 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,

or

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 **

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

is

Performing this computation diagrammatically,gives

Calculating the trace part of the generalized HCO we get:

where,

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 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 on a 3-valent vertex is,

where,

###### Related articles

- Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity (quantumtetrahedron.wordpress.com)
- Matrix Elements of Lorentzian Hamiltonian Constraint in LQG by Alesci,Liegener and Zipfel (quantumtetrahedron.wordpress.com)

We have received your request of our paper “Evaluating the gravitational interaction between two photons” published in Optik. We shall be glad to send it, however we are not in researchgate so please provide us an e-mail where we can send the paper. Regards, M. A. and M. Grado-Caffaro.