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.

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 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

1106equ2

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

1106equ3

1106equ4

where V is the volume of  ∑,

1106equ5

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

1106equ6

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

Using the relations,

1106equ7

and

1106equ8

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,

1106equ9

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,

1106equ11

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

1106equ12

where,

1106equ13

A straightforward calculation, using the expansions,

1106equ14

and

1106equ15

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

1106equ16

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,

1106equ18

using again the expansions of the holonomy,

1106equ20

and using

1106equ21

then if,

1106equ23

then,

1106equ24

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

1106equ25

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:

1106equ27

Further  manipulations  give the generalized HCO as,

1106equ33

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.

1106efig2

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 ,

1106equ48

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.

1106equ50

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,

1106equ52

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.

1106efig4

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

1106equ54

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

1106equ55

or

1106equ56

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

1106equ57

With this normalization, we have,

1106equ58

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

1106equ60

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

1106equ61

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

1106equ62

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

1106equ63

or explicitly in terms of the matrix elements,

1106equ64

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

1106equ65
using this we obtain for the non-diagonal action of the volume operator,

1106equ66

Completing the action of Hm

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

1106equ48

is

1106equ67

Performing this computation diagrammatically,gives

1106equ68a

1106equ69

Calculating the trace part of the generalized HCO we  get:

1106equ72

 

where,

1106equ73

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)>,

1106equ74

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

1106equ75

where,

1106equ76

 

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4 thoughts on “A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements by Gaul and Rovelli”

  1. 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.

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