Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex

Abstract

The asymptotics of some spin foam amplitudes for a quantum 4-simplex is known to display rapid oscillations whose frequency is the Regge action. In this note, we reformulate this result through a difference equation, asymptotically satisfied by these models, and whose semi-classical solutions are precisely the sine and the cosine of the Regge action. This equation is then interpreted as coming from the canonical quantization of a simple constraint in Regge calculus. This suggests to lift and generalize this constraint to the phase space of loop quantum gravity parametrized by twisted geometries. The result is a reformulation of the flat model for topological BF theory from the Hamiltonian perspective. The Wheeler-de-Witt equation in the spin network basis gives difference equations which are exactly recursion relations on the 15j-symbol. Moreover, the semi-classical limit is investigated using coherent states, and produces the expected results. It mimics the classical constraint with quantized areas, and for Regge geometries it reduces to the semi-classical equation which has been introduced in the beginning.

16 pages, the new title is that of the published version (initial title: A taste of Hamiltonian constraint in spin foam models)

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Cited by in corpus (27)

• The Spin Foam Approach to Quantum Gravity
• Spinor Representation for Loop Quantum Gravity
• Spin Foams and Canonical Quantization
• A new vacuum for Loop Quantum Gravity
• Path integral measure and triangulation independence in discrete gravity
• Canonical simplicial gravity
• Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
• The Hamiltonian constraint in 3d Riemannian loop quantum gravity
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Deformation Operators of Spin Networks and Coarse-Graining
• Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
• Dirac's discrete hypersurface deformation algebras
• Bubble divergences and gauge symmetries in spin foams
• Classical Setting and Effective Dynamics for Spinfoam Cosmology
• A new Hamiltonian for the Topological BF phase with spinor networks
• Super-Group Field Cosmology
• Generating Functions for Coherent Intertwiners
• Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces
• Noether's Charge in the Super-Group Field Cosmology
• Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators
• Topological lattice field theories from intertwiner dynamics
• Asymptotics of Wigner 3nj-symbols with Small and Large Angular Momenta: an Elementary Method
• Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity
• The new spin foam models and quantum gravity
• A New Recursion Relation for the 6j-Symbol
• Static four-charge squashed black hole in five-dimensional $STU-W^{2}U$ supergravity and its thermodynamics
• Loop quantum gravity - a short review