Semiclassical limit of new path integral formulation from reduced phase space loop quantum gravity

Abstract

Recently, a new path integral formulation of loop quantum gravity (LQG) has been derived in M. Han and H. Liu, Phys. Rev. D 101, 046003 (2020). from the reduced phase space formulation of the canonical LQG. This paper focuses on the semiclassical analysis of this path integral formulation. We show that dominant contributions of the path integral come from solutions of semiclassical equations of motion (EOMs), which reduce to Hamilton's equations of holonomies and fluxes $h(e)$, ${p}^{a}(e)$ in the reduced phase space ${\mathcal{P}}_{\ensuremath{\gamma}}$ of the cubic lattice $\ensuremath{\gamma}$: $\frac{\mathrm{d}h(e)}{\mathrm{d}\ensuremath{\tau}}={h(e),\mathbf{H}}$, $\frac{\mathrm{d}{p}^{a}(e)}{\mathrm{d}\ensuremath{\tau}}={{p}^{a}(e),\mathbf{H}}$, where $\mathbf{H}$ is the discrete physical Hamiltonian. The semiclassical dynamics from the path integral becomes an initial value problem of Hamiltonian time evolution in ${\mathcal{P}}_{\ensuremath{\gamma}}$. Moreover when we take the continuum limit of the lattice $\ensuremath{\gamma}$, these Hamilton's equations reproduce correctly classical reduced phase space EOMs of gravity coupled to dust fields in the continuum, as far as initial and final states are semiclassical. Our result proves that the new path integral formulation has the correct semiclassical limit and indicates that the reduced phase space quantization in LQG is semiclassically consistent. Based on these results, we compare this path integral formulation and the spin foam formulation, and show that this formulation has several advantages including the finiteness, the relation with canonical LQG, and the freedom from cosine and flatness problems.