Spin Foam Perturbation Theory

TL;DR

This paper develops a perturbation framework for spin foam models on triangulated manifolds, enabling efficient computation of perturbed evolution operators and exploring their limits, especially for topological quantum field theories.

Contribution

It introduces a convergent power series approach for perturbing spin foam models and explicitly sums these series in the dilute gas limit for topological models.

Findings

Power series for perturbed models can be efficiently computed.

Explicit summation of series in the dilute gas limit yields triangulation-independent models.

Models are mostly trivial outside of 2D in the dilute gas limit.

Abstract

We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the evolution operators of the perturbed model as convergent power series in the coupling constant governing the perturbation. The terms in the power series can be efficiently computed when the unperturbed model is a topological quantum field theory. Moreover, in this case we can explicitly sum the whole power series in the limit where the number of top-dimensional simplices goes to infinity while the coupling constant is suitably renormalized. This `dilute gas limit' gives spin foam models that are triangulation-independent but not topological quantum field theories. However, we show that models of this sort are rather trivial except in dimension 2.