First-order quantum correction in coherent state expectation value of loop-quantum-gravity Hamiltonian

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