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

TL;DR

This work identifies a universal second-order difference equation for large-spin spin-foam amplitudes of a quantum 4-simplex, whose semiclassical solutions are exponentials of the Regge action and which can be interpreted as a quantized flatness constraint in area Regge calculus. By recasting the Ooguri SU(2) BF theory in twisted geometries and projecting curvature with a Hamiltonian constraint H^a_{bc}, the authors derive a Wheeler–DeWitt equation that recovers the expected Regge-limit dynamics and yields recursion relations for the 15j-symbol. Through coherent-state analyses, the semiclassical regime shows the difference equation reduces to the classical Regge relation S′ = Θ, reproducing the known semiclassical behavior of spin-foam amplitudes. The results forge a link between canonical LQG, spin-foam asymptotics, and Regge calculus, offering a path to understanding full amplitudes and non-Regge boundary data in a Hamiltonian framework.

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.