Spin foam model from canonical quantization

TL;DR

This paper proposes a modified spin foam model for 4D Lorentzian gravity based on canonical quantization, introducing projected spin networks as boundary states, which generalize the Barrett-Crane intertwiner.

Contribution

It introduces a novel modification to the Barrett-Crane model inspired by canonical loop quantum gravity, utilizing projected spin networks and a new quantization approach for bi-vectors.

Findings

Derived a boundary state space as projected spin networks

Modified the Barrett-Crane quantization to incorporate projections

Compared the new model with existing proposals

Abstract

We suggest a modification of the Barrett-Crane spin foam model of 4-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrette-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the B field (bi-vectors) should be promoted not to generators of the gauge algebra, but to their certain projection. The modification is also justified by the canonical analysis of Plebanski formulation. Finally, we compare our construction with other proposals to modify the Barret-Crane model.