Matrix Elements of Lorentzian Hamiltonian Constraint in LQG

TL;DR

This paper computes the matrix elements of the full Lorentzian Hamiltonian constraint in Loop Quantum Gravity, including the complex Lorentzian part, using advanced SU(2) recoupling techniques and graphical calculus.

Contribution

It provides the first explicit evaluation of the full Hamiltonian constraint matrix elements, incorporating the Lorentzian component, which was previously only computed for the Euclidean part.

Findings

Derived the matrix elements of the Lorentzian Hamiltonian constraint.

Simplified Euclidean constraint calculations using new identities.

Enhanced the understanding of the full Hamiltonian operator in LQG.

Abstract

The Hamiltonian constraint is the key element of the canonical formulation of LQG coding its dynamics. In Ashtekar-Barbero variables it naturally splits into the so called Euclidean and Lorentzian parts. However, due to the high complexity of this operator, only the matrix elements of the Euclidean part have been considered so far. Here we evaluate the action of the full constraint, including the Lorentzian part. The computation requires an heavy use of SU(2) recoupling theory and several tricky identities among n-j symbols are used to find the final result: these identities, together with the graphical calculus used to derive them, also simplify the Euclidean constraint and are of general interest in LQG computations.