Lorentzian spinfoam gravity path integral and geometrical area-law entanglement entropy

This paper investigates entanglement entropy in 3+1 dimensional Lorentzian covariant Loop Quantum Gravity (LQG). We compute the entanglement entropy for a spatial region from states dynamically generated by a spinfoam path integral that sums over a family of 2-complexes. The resulting entropy exhibits a geometric area law, $S \simeq \beta a$, where the area $a$ of the entangling surface is determined by the LQG area spectrum and the leading coefficient $\beta>0$ is independent of the underlying 2-complexes. By relating the coupling constant of the sum over 2-complexes to the Barbero-Immirzi parameter $\gamma$, we reproduce the Bekenstein-Hawking formula for the range $0 < \gamma \lesssim 1/2$. This work provides a Lorentzian path integral approach to gravitational entropy without the need for contour prescriptions.