2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions

TL;DR

This paper calculates Lorentzian spin foam amplitudes for dipole transitions, revealing power-law decay, non-trivial correlations, and providing computational tools and comparisons to simplified models, advancing understanding of non-simplicial quantum gravity structures.

Contribution

It introduces a method to compute Lorentzian amplitudes with up to two vertices, analyzes their behavior, and compares them with simplified models to assess reliability.

Findings

Amplitudes decrease as a power law with increasing spin.

Spin correlations emerge only with internal faces, decreasing with the Immirzi parameter.

The simplified EPRLs model approximates full amplitudes under certain conditions.

Abstract

We compute transition amplitudes between two spin networks with dipole graphs, using the Lorentzian EPRL model with up to two (non-simplicial) vertices. We find power-law decreasing amplitudes in the large spin limit, decreasing faster as the complexity of the foam increases. There are no oscillations nor asymptotic Regge actions at the order considered, nonetheless the amplitudes still induce non-trivial correlations. Spin correlations between the two dipoles appear only when one internal face is present in the foam. We compute them within a mini-superspace description, finding positive correlations, decreasing in value with the Immirzi parameter. The paper also provides an explicit guide to computing Lorentzian amplitudes using the factorisation property of SL(2,C) Clebsch-Gordan coefficients in terms of SU(2) ones. We discuss some of the difficulties of non-simplicial foams, and provide a specific criterion to partially limit the proliferation of diagrams. We systematically compare the results with the simplified EPRLs model, much faster to evaluate, to learn evidence on when it provides reliable approximations of the full amplitudes. Finally, we comment on implications of our results for the physics of non-simplicial spin foams and their resummation.