Semiclassical behavior of spinfoam amplitude with small spins and entanglement entropy

Abstract

Spinfoam amplitudes with small spins can have interesting semiclassical behavior and relate to semiclassical gravity and geometry in four dimensions. We study the generalized spinfoam model [spinfoams for all loop quantum gravity (LQG) [Kaminski et al., Spin-foams for all loop quantum gravity, Classical Quantum Gravity 27, 095006 (2010); Erratum, Classical Quantum Gravity 29, 049502(E) (2012), Ding et al., Generalized spinfoams, Phys. Rev. D 83, 124020 (2011)] with small spins $j$ but a large number of spin degrees of freedom (d.o.f.), and find that it relates to the simplicial Engle-Pereira-Rovelli-Livine-Freidel-Krasnov model with large spins and Regge calculus by coarse-graining spin d.o.f. $\mathrm{Small}\text{\ensuremath{-}}j$ generalized spinfoam amplitudes can be employed to define semiclassical states in the LQG kinematical Hilbert space. Each of these semiclassical states is determined by a four-dimensional Regge geometry. We compute the entanglement R\'enyi entropies of these semiclassical states. The entanglement entropy interestingly coarse grains spin d.o.f. in the generalized spinfoam model, and satisfies an analog of the thermodynamical first law. This result possibly relates to the quantum black hole thermodynamics in [Ghosh and Perez, Black Hole Entropy and Isolated Horizons Thermodynamics, Phys. Rev. Lett. 107, 241301 (2011); 108, 169901(E) (2012)].