Loop Quantum Cosmology and Spin Foams

TL;DR

Loop quantum cosmology (LQC) provides a concrete, solvable setting to test the spin foam paradigm by showing that the physical inner product between kinematical states equals a deparameterized transition amplitude $A( u_f,\phi_f;\nu_i,0) = \langle \nu_f| e^{i H \varphi}|\nu_i\rangle$ and admits a vertex expansion over discrete geometries without taking a continuum limit. This vertex expansion can be derived both from a sum-over-histories approach and from a perturbative treatment in a group-field theory–like framework, yielding two inequivalent but complete expansions that agree when summed to all orders. The timeless (group-averaged) inner product can be related to the transition amplitude, but leads to a distinct vertex expansion when truncated, illustrating a robust connection between deparameterized and timeless formalisms. The results strengthen the spin foam program by demonstrating exact, nonperturbative realizations of its core ideas in a controlled quantum cosmology model and hint at deep links between coupling constants in group field theory and the cosmological constant.

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.