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

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 paper, we reformulate this result through a difference equation, asymptotically satisfied by these models, and whose semiclassical 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-DeWitt equation in the spin network basis gives difference equations which are exactly recursion relations on the 15j-symbol. Moreover, the semiclassical 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 semiclassical equation which has been introduced in the beginning.