Lorentzian spin foam amplitudes: graphical calculus and asymptotics

TL;DR

The paper develops a Lorentzian spin foam framework by extending a diagrammatic calculus to SL(2,$\mathbb{C}$) representations and defines a 4-simplex amplitude via a Lorentzian EPRL-like construction. Using coherent boundary states and stationary-phase methods, it derives an exact asymptotic expansion in the large-representation limit, showing that Lorentzian boundary data yield two oscillatory terms governed by the Lorentzian Regge action with the Immirzi parameter $\gamma$, while Euclidean boundary data contribute separate, non-suppressed Euclidean Regge terms; non-Regge-like data lead to vector-geometry contributions or suppression. The analysis connects critical-point geometry to bivector configurations, with p$_{ab}=\gamma k_{ab}$ arising as a necessary simplicity constraint, and provides explicit formulas for dihedral boosts, Regge phases, and Hessian factors. These results support the semiclassical consistency of Lorentzian spin foam models and sharpen the role of the Immirzi parameter in their asymptotics, offering a path toward extending the analysis to larger triangulations and observable computations. The work highlights how boundary data type (Lorentzian vs Euclidean) dictates which geometric sector dominates the amplitude and clarifies how parity-related configurations contribute to the overall semiclassical picture.

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.

Figures (8)

• Figure 1: A spin network vertex
• Figure 2: The maximum diagram
• Figure 3: The crossing diagram
• Figure 4: Reidemeister move I
• Figure 5: An example of a Lorentzian spin network

Theorems & Definitions (1)

• Lemma 1