Hamiltonian spinfoam gravity

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.

28 pages, 2 figures

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• Quantum Reference Frames at the Boundary of Spacetime
• A note on the secondary simplicity constraints in loop quantum gravity
• New action for simplicial gravity in four dimensions
• Spin foam models as energetic causal sets
• Timelike twisted geometries
• Dual loop quantizations of 3d gravity
• Spinfoams: Foundations
• The thermodynamics of quantum spacetime histories
• A new realization of quantum geometry
• A one-dimensional action for simplicial gravity in three dimensions
• Loop quantum gravity, twistors, and some perspectives on the problem of time