Matrix Elements of Thiemann's Hamiltonian Constraint in Loop Quantum Gravity

TL;DR

This paper explicitly computes the matrix elements of Thiemann's Hamiltonian constraint operator in loop quantum gravity using graphical recoupling techniques, providing a detailed algebraic formulation for its action on spin network states.

Contribution

It offers the first explicit calculation of matrix elements of Thiemann's Hamiltonian constraint on trivalent states in loop quantum gravity using graphical methods.

Findings

Matrix elements expressed in compact algebraic form

Action on trivalent states explicitly computed

Graphical recoupling techniques employed

Abstract

We present an explicit computation of matrix elements of the hamiltonian constraint operator in non-perturbative quantum gravity. In particular, we consider 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. We exhibit the matrix elements of the hamiltonian constraint operator in the spin network basis in compact algebraic form.