Lorentzian spin foam amplitudes: graphical calculus and asymptotics

TL;DR

This paper develops a graphical calculus for Lorentzian spin foam amplitudes and analyzes their asymptotic behavior, revealing connections to Lorentzian and Euclidean geometries in quantum gravity.

Contribution

It introduces a graphical calculus for Lorentz group representations and derives asymptotic formulas linking spin foam amplitudes to Regge actions for different boundary data.

Findings

Asymptotic formula includes two terms with Lorentzian Regge action.

Euclidean boundary data contribute unexpected Euclidean signature terms.

Results connect spin foam amplitudes to classical geometries in quantum gravity.

Abstract

The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics.