First-order quantum correction in coherent state expectation value of loop-quantum-gravity Hamiltonian
Cong Zhang, Shicong Song, Muxin Han
Abstract
Given the nongraph-changing Hamiltonian $\stackrel{^}{H[N]}$ in loop quantum gravity (LQG), $\stackrel{^}{H[N]}$, the coherent state expectation value of $\stackrel{^}{H[N]}$, admits a semiclassical expansion in ${\ensuremath{\ell}}_{\mathrm{p}}^{2}$. In this paper, we explicitly compute the expansion of $\stackrel{^}{H[N]}$ to the linear order in ${\ensuremath{\ell}}_{\mathrm{p}}^{2}$ 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, supported by rigorous proofs and several theorems, is developed to overcome the complexity in the computation of $\stackrel{^}{H[N]}$. Particularly, some key innovations in our algorithm substantially reduce the complexity in computing the Lorentzian part of $\stackrel{^}{H[N]}$. Moreover, with the algorithm developed in the present work, we can compute the expectation value of arbitrary monomial of holonomies and fluxes on one edge up to arbitrary order of ${\ensuremath{\ell}}_{\mathrm{p}}^{2}$. Finally, some quantum correction effects resulted from $\stackrel{^}{H[N]}$ in cosmology are discussed at the end of this paper.