The Ponzano–Regge cylinder and propagator for 3d quantum gravity

Abstract

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano–Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes subspaces with different total recoupled spin. This may select the vanishing overall spin sector at late time depending on the chosen cylinder boundary data, leading to an emergent symmetry scenario in the continuum limit. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.

Spin network entanglement: coarse-graining

“…In principle, the scalar product for H ∂Γ can be also defined by the scalar product for H Γ by considering boundary holonomies [55]. The advantage with using bulk-boundary maps method is that we analyze the boundary holonomies and bulk holonomies separately.…”

“…In this sense, spin network wave-function is referred to coarse-grainer. We refer the interested reader to [55] for the gluing operation for time boundaries.…”