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

Abstract

Given the Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian $\widehat{H[N]}$, the coherent state expectation value $⟨\widehat{H[N]}⟩$ admits an semiclassical expansion in ${\ell }_{\mathrm{p}}^2$. In this paper, we compute explicitly the expansion of $⟨\widehat{H[N]}⟩$ on the cubic graph to the linear order in ${\ell }_{\mathrm{p}}^2$, 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 $⟨\widehat{H[N]}⟩$. In particular, some key innovations in our algorithm substantially reduce the computational complexity in the Lorentzian part of $⟨\widehat{H[N]}⟩$. 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 ${\ell }_{\mathrm{p}}^2$.

24 pages