First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian: Overview and Results

TL;DR

This paper calculates the first-order quantum correction to the expectation value of the LQG Hamiltonian in coherent states, providing a new algorithm to handle complex computations relevant for semiclassical cosmology.

Contribution

It introduces an efficient algorithm for computing the expectation value of the LQG Hamiltonian to linear order in Planck length squared, especially simplifying Lorentzian contributions.

Findings

Explicit first-order correction in semiclassical expansion

Development of a computationally efficient algorithm

Ability to compute arbitrary monomials of holonomies and fluxes

Abstract

Given the Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian $\widehat{H[N]}$, the coherent state expectation value $\langle\widehat{H[N]}\rangle$ admits an semiclassical expansion in $\ell^2_{\rm p}$. In this paper, we compute explicitly the expansion of $\langle\widehat{H[N]}\rangle$ on the cubic graph to the linear order in $\ell^2_{\rm p}$, when the coherent state is peaked at the homogeneous and isotropic data of cosmology. In our computation, a powerful algorithm is developed to overcome the complexity in computing $\langle \widehat{H[N]} \rangle$. In particular, some key innovations in our algorithm substantially reduce the computational complexity in the Lorentzian part of $\langle\widehat{H[N]}\rangle$. Moreover, the algorithm developed in the present work makes it possible to compute the expectation value of arbitrary monomial of holonomies and fluxes on one edge up to arbitrary order of $\ell_{\rm p}^2$.