Effective action and semiclassical limit of spin foam models

TL;DR

This paper develops an effective action framework for spin foam models of quantum gravity, demonstrating how to recover the classical Einstein-Hilbert action in the semiclassical limit through modifications of vertex amplitudes.

Contribution

It introduces a method to derive an effective action for spin foam models, showing how to modify vertex amplitudes to obtain the correct classical limit and quantum corrections.

Findings

Regge action appears as leading term in semiclassical expansion

Modified vertex amplitudes yield finite models with Einstein-Hilbert classical limit

First and second-order quantum corrections are computed

Abstract

We define an effective action for spin foam models of quantum gravity by adapting the background field method from quantum field theory. We show that the Regge action is the leading term in the semi-classical expansion of the spin foam effective action if the vertex amplitude has the large-spin asymptotics which is proportional to an exponential function of the vertex Regge action. In the case of the known three-dimensional and four-dimensional spin foam models this amounts to modifying the vertex amplitude such that the exponential asymptotics is obtained. In particular, we show that the ELPR/FK model vertex amplitude can be modified such that the new model is finite and has the Einstein-Hilbert action as its classical limit. We also calculate the first-order and some of the second-order quantum corrections in the semi-classical expansion of the effective action.