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

matrixelementsequ1a

matrixelementsequ1

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.

matrixelementsfig0

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:

matrixelementsequ6.3

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

matrixelementsequ6.4

where 

matrixelementsequ5.8

 

matrixelementsequ5.9and

matrixelementsequa17

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.

matrixelementsfig2

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

matrixelementsequ6.4

 

matrixelementsfig3

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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