Semiclassical Analysis of Spinfoam Model with a Small Barbero-Immirzi Parameter

TL;DR

This paper analyzes the semiclassical behavior of the Lorentzian EPRL spinfoam model considering the sum over spins with a small Barbero-Immirzi parameter, revealing two distinct regimes with different degrees of freedom and geometric constraints.

Contribution

It introduces a novel asymptotic analysis of the spinfoam model with independent large spin and small Barbero-Immirzi parameters, identifying two regimes with different geometric properties.

Findings

In one regime, the model yields a discrete Einstein equation via Regge calculus.

In the other regime, only small deficit angles are allowed, constraining the geometry.

Large spins lead to configurations with small or zero deficit angles, indicating semiclassical geometries.

Abstract

We study the semiclassical behavior of Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model, by taking into account of the sum over spins in the large spin regime. The large spin parameter \lambda and small Barbero-Immirzi parameter \gamma are treated as two independent parameters for the asymptotic expansion of spinfoam state-sum (such an idea was firstly pointed out in arXiv:1105.0216). Interestingly, there are two different spin regimes: 1<<\gamma^{-1}<<\lambda<<\gamma^{-2} and \lambda>\gamma^{-2}. The model in two spin regimes has dramatically different number of effective degrees of freedom. In 1<<\gamma^{-1}<<\lambda<<\gamma^{-2}, the model produces in the leading order a functional integration of Regge action, which gives the discrete Einstein equation for the leading contribution. There is no restriction of Lorentzian deficit angle in this regime. In the other regime \lambda>\gamma^{-2}, only small deficit angle is allowed |\Theta_f|<<\gamma^{-1}\lambda^{1/2}$ mod 4\pi Z. When spins go even larger, only zero deficit angle mod 4\pi Z is allowed asymptotically. In the transition of the two regimes, only the configurations with small deficit angle can contribute, which means one need a large triangulation in order to have oscillatory behavior of the spinfoam amplitude.