Loop Quantum Cosmology and Spin Foams

TL;DR

This paper demonstrates how loop quantum cosmology provides concrete support for spin foam models by showing the equivalence of physical inner products and transition amplitudes, and introduces a sum-over-histories approach without continuum limits.

Contribution

It establishes a direct link between LQC and SFMs, showing the physical inner product equals a sum over discrete geometries and can be expanded perturbatively similar to group field theory.

Findings

Physical inner product equals transition amplitude in LQC.

Vertex expansion involves only volume transitions, no time reference.

Exact inner product obtained without continuum limit.

Abstract

Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs). Specifically, it is shown that: i) the physical inner product in the timeless framework equals the transition amplitude in the deparameterized theory; ii) this quantity admits a %convergent vertex expansion a la SFMs in which the $M$-th term refers just to $M$ volume transitions, without any reference to the time at which the transition takes place; iii) the exact physical inner product is obtained by summing over just the discrete geometries; no `continuum limit' is involved; and, iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of group field theory. This sum over histories reformulation of LQC also addresses certain other issues which are briefly summarized.