Flipped spinfoam vertex and loop gravity

TL;DR

This paper introduces a new vertex amplitude for 4d loop quantum gravity derived from a Regge discretization, improving the Barrett-Crane model and linking covariant and canonical formalisms.

Contribution

It presents a flipped spinfoam vertex that weakly imposes simplicity constraints, aligning boundary states with canonical loop quantum gravity and rectifying previous model limitations.

Findings

Boundary states match SO(3) loop quantum gravity

Vertex amplitude is SO(3) and SO(4)-covariant

Provides an independent derivation of quantum geometry

Abstract

We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett-Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett-Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO(3) loop quantum gravity -- the two can be identified as eigenstates of the same physical quantities -- providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. The vertex amplitude is SO(3) and SO(4)-covariant. It rectifies the triviality of the intertwiner dependence of the Barrett-Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.