Algorithmic approach to Cosmological Coherent State Expectation Values in LQG

TL;DR

The paper introduces an analytical algorithm for calculating expectation values of coherent states in Loop Quantum Gravity on a lattice, simplifying computations especially for cosmological models, and assesses the impact of higher-order corrections on effective dynamics.

Contribution

It provides a new step-by-step analytical algorithm for expectation values in LQG coherent states, facilitating easier and more precise calculations in cosmological contexts.

Findings

The algorithm simplifies expectation value computations in LQG.

First-order corrections are included in the analytical expressions.

Next-to-leading order corrections are not suitable as effective Hamiltonian modifications.

Abstract

In the lattice approach to Loop Quantum Gravity on a fixed graph computations tend to be involved and are rarely analytically manageable. But, when interested in the expectation values of coherent states on the lattice which are sharply peaked on isotropic, flat cosmology several simplifications are possible which reduce the computational effort. We present a step-by-step algorithm resulting in an analytical expression including up to first order corrections in the spread of the state. The algorithm is developed in such a way that it makes the computation straightforward and easy to be implementable in programming languages such as Mathematica. Exemplarily, we demonstrate how the algorithm streamlines the road to obtain the expectation value of the euclidean part of the scalar constraint and as a consistency check perform the analytic computation as well. To showcase further applications of the algorithm, we investigate the fate of the effective dynamics program custom in Loop Cosmology and find that the next-to-leading order corrections can {\it not} be used as corrections for an effective Hamiltonian.