Hamiltonian spinfoam gravity

This paper presents a Hamiltonian formulation of spinfoam-gravity, which leads to a straight-forward canonical quantisation. To begin with, we derive a continuum action adapted to the simplicial decomposition. The equations of motion admit a Hamiltonian formulation, allowing us to perform the constraint analysis. We do not find any secondary constraints, but only get restrictions on the Lagrange multipliers enforcing the reality conditions. This comes as a surprise. In the continuum theory, the reality conditions are preserved in time, only if the torsionless condition (a secondary constraint) holds true. Studying an additional conservation law for each spinfoam vertex, we discuss the issue of torsion and argue that spinfoam gravity may indeed miss an additional constraint. Next, we canonically quantise. Transition amplitudes match the EPRL (Engle–Pereira–Rovelli–Livine) model, the only difference being the additional torsional constraint affecting the vertex amplitude. I. THE PROBLEM Loop quantum gravity [1–5] comes in two versions. The historically first of which provides a canonical quan-tisation of general relativity, and seeks to solve the Wheeler–De Witt equation [1, 3, 6]. The second, we call it spinfoam gravity [5, 7], proposes a covariant path integral. Both approaches share [8, 9] their kinematical structure—the Hilbert space with observables representing, area, angles, volume and parallel transport [10–12]