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

TL;DR

This paper reformulates the asymptotics of spin foam amplitudes for a quantum 4-simplex as a difference equation, linking it to canonical quantization and loop quantum gravity, and investigates its semi-classical limit.

Contribution

It introduces a difference equation framework for spin foam models, connecting their asymptotics to canonical quantization and extending the approach to loop quantum gravity phase space.

Findings

The difference equation's semi-classical solutions are sine and cosine of the Regge action.

The Wheeler-de-Witt equation yields recursion relations on the 15j-symbol.

Semi-classical analysis with coherent states confirms expected classical limits.

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.