Improved ($\bar{\mu}$-Scheme) Effective Dynamics of Full Loop Quantum Gravity

Abstract

We propose a new derivation from the full Loop Quantum Gravity (LQG) to the Loop Quantum Cosmology (LQC) improved $\bar{\mu}$-scheme effective dynamics, based on the reduced phase space formulation of LQG and a proposal of effective Hamiltonian/action in the full LQG. A key step of our program is an improved regularization of the full LQG Hamiltonian on a cubic lattice. The improved Hamiltonian uses a set of "dressed holonomies" $h_\Delta(\mathfrak{s})$ which not only depend on the connection $A$ but also depend on the length of the curve $\mathfrak{s}$. With the improved Hamiltonian, we propose a quantum effective action and derive a new set effective equations of motion (EOMs) for the full LQG. Then we show that these new EOMs imply the $\bar{\mu}$-scheme effective dynamics for both the homogeneous-isotropic and Bianchi-I cosmology, and predict bounce and Planckian critical density. As a byproduct, although the model is defined on a cubic lattice, we find that the improved effective Hamiltonian of cosmology is invariant under the lattice refinement. The cosmological effective dynamics, predictions of bounce and critical density are results at the continuum limit.