First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian. (arXiv:2102.03591v3 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Zhang_C/0/1/0/all/0/1">Cong Zhang</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Song_S/0/1/0/all/0/1">Shicong Song</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Han_M/0/1/0/all/0/1">Muxin Han</a>

Given the non-graph-changing Hamiltonian $widehat{H[N]}$ in Loop Quantum
Gravity (LQG), $langlewidehat{H[N]}rangle$, 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
$langlewidehat{H[N]}rangle$ 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 $langlewidehat{H[N]}rangle$.
Additionally, some quantum correction effects resulting from
$langlewidehat{H[N]}rangle$ in cosmology are discussed at the end of this
paper.

Given the non-graph-changing Hamiltonian $widehat{H[N]}$ in Loop Quantum
Gravity (LQG), $langlewidehat{H[N]}rangle$, 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
$langlewidehat{H[N]}rangle$ 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 $langlewidehat{H[N]}rangle$.
Additionally, some quantum correction effects resulting from
$langlewidehat{H[N]}rangle$ in cosmology are discussed at the end of this
paper.

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