分类: 天文学 >> 天文学 提交时间: 2023-02-19
摘要: Given the Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian $\widehat{H[N]}$, the coherent state expectation value $\langle\widehat{H[N]}\rangle$ admits an semiclassical expansion in $\ell^2_{\rm p}$. In this paper, we compute explicitly the expansion of $\langle\widehat{H[N]}\rangle$ on the cubic graph to the linear order in $\ell^2_{\rm p}$, 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 $\langle \widehat{H[N]} \rangle$. In particular, some key innovations in our algorithm substantially reduce the computational complexity in the Lorentzian part of $\langle\widehat{H[N]}\rangle$. 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 $\ell_{\rm p}^2$.
分类: 天文学 >> 天文学 提交时间: 2023-02-19
摘要: 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.