Testing the imposition of the Spin Foam Simplicity Constraints

TL;DR

This paper develops a three-dimensional Plebanski model with quadratic simplicity constraints, performs a full spin foam quantization, and compares different constraint imposition methods, revealing limitations of the BC model and the effectiveness of the EPRL approach with secondary constraints.

Contribution

It introduces a 3D Plebanski action with quadratic simplicity constraints, performs complete spin foam quantization, and analyzes the impact of different constraint implementation methods.

Findings

The BC prescription does not produce the correct vertex amplitude.

The EPRL prescription can recover the expected results with secondary second class constraints.

The model provides a clear comparison between classical solutions and quantum implementations.

Abstract

We introduce a three-dimensional Plebanski action for the gauge group SO(4). In this model, the $B$ field satisfies quadratic simplicity constraints similar to that of the four-dimensional Plebanski theory, but with the difference that the $B$ field is now a one-form. We exhibit a natural notion of "simple one-form", and identify a gravitational sector, a topological sector and a degenerate sector in the space of solutions to the simplicity constraints. Classically, in the gravitational sector, the action is shown to be equivalent to that of three-dimensional first order Riemannian gravity. This enables us to perform the complete spin foam quantization of the theory once the simplicity constraints are solved at the classical level, and to compare this result with the various models that have been proposed for the implementation of the constraints after quantization. In particular, we impose the simplicity constraints following the prescriptions of the so-called BC and EPRL models. We observe that the BC prescription cannot lead to the proper vertex amplitude. The EPRL prescription allows to recover the expected result when, in this three-dimensional model, it is supplemented with additional secondary second class constraints.