Plebanski Theory and Covariant Canonical Formulation

TL;DR

This paper demonstrates the equivalence between the Hamiltonian and covariant canonical formulations of general relativity using Plebanski and Hilbert-Palatini actions, and introduces a simplified shifted connection relevant for loop quantum gravity.

Contribution

It establishes a formal equivalence between two formulations of general relativity and constructs a simplified connection for covariant loop quantization within the Plebanski framework.

Findings

Confirmed the symplectic structure equivalence via Dirac brackets

Constructed a shifted connection with simplified brackets

Discussed implications for spin foam models

Abstract

We establish an equivalence between the Hamiltonian formulation of the Plebanski action for general relativity and the covariant canonical formulation of the Hilbert-Palatini action. This is done by comparing the symplectic structures of the two theories through the computation of Dirac brackets. We also construct a shifted connection with simplified Dirac brackets, playing an important role in the covariant loop quantization program, in the Plebanski framework. Implications for spin foam models are also discussed.