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

Abstract

Given the non-graph-changing Hamiltonian $\widehat{H[N]}$ in Loop Quantum Gravity (LQG), $⟨\widehat{H[N]}⟩$, the coherent state expectation value of $\widehat{H[N]}$, admits an semiclassical expansion in ${\ell }_{\mathrm{p}}^2$. In this paper, as presenting the detailed derivations of our previous work arXiv:2012.14242, we explicitly compute the expansion of $⟨\widehat{H[N]}⟩$ to the linear order in ${\ell }_{\mathrm{p}}^2$ 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 $⟨\widehat{H[N]}⟩$. Particularly, some key innovations in our algorithm substantially reduce the complexity in computing the Lorentzian part of $⟨\widehat{H[N]}⟩$. Additionally, some quantum correction effects resulting from $⟨\widehat{H[N]}⟩$ in cosmology are discussed at the end of this paper.

Comments: 29+25 pages, 2 figures, presenting the detailed derivations of our previous work arXiv:2012.14242, adapted to the published version