First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian: Overview and Results. (arXiv:2012.14242v4 [gr-qc] UPDATED)

First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian: Overview and Results. (arXiv:2012.14242v4 [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 Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian
$widehat{H[N]}$, the coherent state expectation value
$langlewidehat{H[N]}rangle$ admits an semiclassical expansion in
$ell^2_{rm p}$. In this paper, we compute explicitly the expansion of
$langlewidehat{H[N]}rangle$ 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
$langlewidehat{H[N]}rangle$. 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
$ell_{rm p}^2$.

Given the Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian
$widehat{H[N]}$, the coherent state expectation value
$langlewidehat{H[N]}rangle$ admits an semiclassical expansion in
$ell^2_{rm p}$. In this paper, we compute explicitly the expansion of
$langlewidehat{H[N]}rangle$ 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
$langlewidehat{H[N]}rangle$. 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
$ell_{rm p}^2$.

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