First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian

TL;DR

This paper explicitly computes the first-order quantum correction in the semiclassical expectation value of the LQG Hamiltonian for cosmology, introducing an efficient algorithm to handle complex calculations and discussing quantum effects in cosmological models.

Contribution

The paper presents a detailed derivation of the first-order quantum correction in LQG Hamiltonian expectation values using a novel algorithm to simplify complex computations.

Findings

Explicit first-order correction in LQG Hamiltonian expectation value

Development of an efficient algorithm for complex quantum calculations

Discussion of quantum correction effects in cosmology

Abstract

Given the non-graph-changing Hamiltonian $\widehat{H[N]}$ in Loop Quantum Gravity (LQG), $\langle\widehat{H[N]}\rangle$, the coherent state expectation value of $\widehat{H[N]}$, admits an semiclassical expansion in $\ell^2_{\rm p}$. In this paper, as presenting the detailed derivations of our previous work arXiv:2012.14242, we explicitly compute the expansion of $\langle\widehat{H[N]}\rangle$ to the linear order in $\ell^2_{\rm p}$ on the cubic graph with respect to the coherent state peaked at the homogeneous and isotropic data of cosmology. In our computation, a powerful algorithm is developed, supported by rigorous proofs and several theorems, to overcome the complexity in the computation of $\langle \widehat{H[N]} \rangle$. Particularly, some key innovations in our algorithm substantially reduce the complexity in computing the Lorentzian part of $\langle\widehat{H[N]}\rangle$. Additionally, some quantum correction effects resulting from $\langle\widehat{H[N]}\rangle$ in cosmology are discussed at the end of this paper.