Classical Setting and Effective Dynamics for Spinfoam Cosmology

TL;DR

This paper derives effective cosmological dynamics from loop quantum gravity using spinfoam models on a simplified 2-vertex graph, linking canonical and covariant formalisms in a symmetry-reduced setting.

Contribution

It introduces a Hamiltonian for spinfoam cosmology on a 2-vertex graph and connects it with the transition amplitude in the covariant approach.

Findings

The transition amplitude satisfies an exact differential equation.

The Hamiltonian and covariant formalisms are explicitly linked.

The model reproduces features of homogeneous and isotropic cosmologies.

Abstract

We explore how to extract effective dynamics from loop quantum gravity and spinfoams truncated to a finite fixed graph, with the hope of modeling symmetry-reduced gravitational systems. We particularize our study to the 2-vertex graph with N links. We describe the canonical data using the recent formulation of the phase space in terms of spinors, and implement a symmetry-reduction to the homogeneous and isotropic sector. From the canonical point of view, we construct a consistent Hamiltonian for the model and discuss its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the dynamics from the spinfoam approach. We compute exactly the transition amplitude between initial and final coherent spin networks states with support on the 2-vertex graph, for the choice of the simplest two-complex (with a single space-time vertex). The transition amplitude verifies an exact differential equation that agrees with the Hamiltonian constructed previously. Thus, in our simple setting we clarify the link between the canonical and the covariant formalisms.