Hamiltonian spinfoam gravity

TL;DR

This paper develops a Hamiltonian formulation of spinfoam gravity, enabling canonical quantisation and revealing potential missing constraints related to torsion, with transition amplitudes aligning with the EPRL model.

Contribution

It introduces a Hamiltonian approach to spinfoam gravity, analyzes constraints, and connects the canonical quantisation to the EPRL model with new insights on torsion constraints.

Findings

No secondary constraints found, only restrictions on Lagrange multipliers.

Transition amplitudes match the EPRL model.

Potential missing torsional constraint affecting vertex amplitude.

Abstract

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.