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

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

2011-01-08

arxiv.org/abs/1101.1615v2

Phys.Rev.D84:024009,2011

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

10.1103/PhysRevD.84.024009

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.

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