Matrix elements of Lorentzian Hamiltonian constraint in loop quantum gravity

Abstract

The Hamiltonian constraint is the key element of the canonical formulation of loop quantum gravity (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 heavy use of $SU(2)$ recoupling theory and several tricky identities among $n\mathrm{\text{\ensuremath{-}}}j$ 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.