A finiteness bound for the EPRL/FK spin foam model

TL;DR

This paper establishes a convergence criterion for the EPRL/FK spin foam model of quantum gravity, showing that dividing the vertex amplitude by a certain power ensures an absolutely convergent partition function.

Contribution

It provides a finiteness bound for the EPRL/FK spin foam model by identifying a specific power of the vertex spins' product that guarantees convergence.

Findings

Partition function converges for p > 2.

Convergence depends on the large-spin asymptotics of the vertex amplitude.

The power p is independent of the spin foam 2-complex.

Abstract

We show that the EPRL/FK spin foam model of quantum gravity has an absolutely convergent partition function if the vertex amplitude is divided by an appropriate power $p$ of the product of dimensions of the vertex spins. This power is independent of the spin foam 2-complex and we find that $p>2$ insures the convergence of the state sum. Determining the convergence of the state sum for the values $0 \le p \le 2$ requires the knowledge of the large-spin asymptotics of the vertex amplitude in the cases when some of the vertex spins are large and other are small.