圈-量子引力哈密顿量相干态期望值的一阶量子修正
First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian
摘要：
给定圈量子引力中的非图变哈密顿量$\widehat{H[N]}$，$\langle\widehat{H[N]}\rangle$（$\widehat{H[N]}$的相干态期望值）在$\ell^2_{\rm p}$中存在一个半经典展开.本文在arXiv：2012.14242的基础上，针对均匀各向同性宇宙学数据下的相干态，在三次图上计算了$\langle\widehat{H[N]}\rangle$在$\ell^2_{\rm p}$中的线性展开.在我们的计算中，开发了一个强大的算法，支持严格的证明和几个定理，以克服在$\langle \widehat{H[N]} \rangle$计算的复杂性。特别是，我们的算法中的一些关键创新大大降低了计算$\langle\widehat{H[N]}\rangle$的洛伦兹部分的复杂性。最后讨论了$\langle\widehat{H[N]}\rangle$在宇宙学中的量子修正效应。
英文摘要：
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.
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