Casting Loop Quantum Cosmology in the Spin Foam Paradigm

TL;DR

This work demonstrates that loop quantum cosmology provides a controlled arena in which the timeless group-averaged inner product and the deparameterized Schrödinger transition amplitude admit convergent vertex expansions analogous to spin foam models. It shows a clear factorization of the constraint into matter and gravity sectors, a history-by-history reorganization by the number of volume transitions, and a perturbative expansion in a coupling λ that mirrors group field theory methods. A key result is the precise link between the GFT coupling λ and the cosmological constant Λ via Λ = (3/(2 γ^2 ℓ_0^2))(1 − λ), offering a potential route to the observed small positive Λ. The findings illuminate orientation and reality questions in spin foams, demonstrate the emergence of cosine-type classical limits, and suggest how discrete quantum geometry at the Planck scale shapes the vertex expansions foundational to SFMs and GFT.

Abstract

The goal of spin foam models is to provide a viable path integral formulation of quantum gravity. Because of background independence, their underlying framework has certain novel features that are not shared by path integral formulations of familiar field theories in Minkowski space. As a simple viability test, these features were recently examined through the lens of loop quantum cosmology (LQC). Results of that analysis, reported in a brief communication [1], turned out to provide concrete arguments in support of the spin foam paradigm. We now present detailed proofs of those results. Since the quantum theory of LQC models is well understood, this analysis also serves to shed new light on some long standing issues in the spin foam and group field theory literature. In particular, it suggests an intriguing possibility for addressing the question of why the cosmological constant is positive and small.