Towards Spinfoam Cosmology

TL;DR

This paper calculates the transition amplitude in loop quantum gravity for homogeneous isotropic cosmologies, showing it reproduces the Friedmann equation in the classical limit, thus linking quantum gravity with classical cosmology.

Contribution

It introduces a method to compute transition amplitudes using the new vertex in loop quantum gravity, connecting quantum states to classical cosmological dynamics.

Findings

Amplitude lies in the kernel of a differential operator matching the Friedmann Hamiltonian.

Results indicate the new vertex reproduces classical Friedmann dynamics in the appropriate limit.

First-order approximations in vertex and graph expansions successfully recover classical cosmology.

Abstract

We compute the transition amplitude between coherent quantum-states of geometry peaked on homogeneous isotropic metrics. We use the holomorphic representations of loop quantum gravity and the Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at first order in the vertex expansion, second order in the graph (multipole) expansion, and first order in 1/volume. We show that the resulting amplitude is in the kernel of a differential operator whose classical limit is the canonical hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an indication that the dynamics of loop quantum gravity defined by the new vertex yields the Friedmann equation in the appropriate limit.