Poincare-Plebanski formulation of GR and dual simplicity constraints

TL;DR

This paper revises the classical continuum formulation of spin foam quantum gravity, emphasizing the role of extended configuration space and dual simplicity constraints to better capture the geometric content of General Relativity.

Contribution

It introduces a tetradic version of simplicity constraints and extends the configuration space to include tetrads, improving the geometric interpretation of spin foam models.

Findings

Identified the tension with volume simplicity constraints in current models.

Proposed switching from normal 3-forms to tetrads for better geometric representation.

Characterized the extended Poincare BF theory at Lagrangian and Hamiltonian levels.

Abstract

We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the 'volume' part of simplicity constraints, required for the passage from the topological BF theory to gravity. The crucial role, played by 4d normals in the linear version of constraints, necessitates the extension of the configuration space, and we argue to switch from normal 3-forms directly to tetrads. The requirement of vanishing torsion leads to consider first an unconstrained extended Poincare BF theory, which we characterize fully both at the Lagrangian and Hamiltonian levels, paying special attention to its gauge symmetries. The simplicity constraints are introduced naturally, in the spirit of Plebanski formulation, and we give their tetradic version, dual to that of using 3-forms. This brings us much closer to the geometric content of General Relativity.