Degenerate Plebanski Sector and Spin Foam Quantization

TL;DR

This paper demonstrates that the degenerate sector of Spin(4) Plebanski gravity is exactly solvable, equivalent to SU(2) BF theory, and proposes a new spin foam quantization approach incorporating secondary second class constraints.

Contribution

It provides an exact solution for the degenerate sector, clarifies the relation to SU(2) BF theory, and introduces a novel quantization method using secondary constraints.

Findings

Degenerate sector is solvable and equivalent to SU(2) BF theory.

Standard representation restrictions are insufficient for correct vertex amplitudes.

New quantization formula incorporates secondary second class constraints.

Abstract

We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact ensures that its spin foam quantization is given by the SU(2) Crane-Yetter model and allows to test various approaches of imposing the simplicity constraints. Our analysis strongly suggests that restricting representations and intertwiners in the state sum for Spin(4) BF theory is not sufficient to get the correct vertex amplitude. Instead, for a general theory of Plebanski type, we propose a quantization procedure which is by construction equivalent to the canonical path integral quantization and, being applied to our model, reproduces the SU(2) Crane-Yetter state sum. A characteristic feature of this procedure is the use of secondary second class constraints on an equal footing with the primary simplicity constraints, which leads to a new formula for the vertex amplitude.