Plebanski Theory and Covariant Canonical Formulation

TL;DR

This work completes the canonical analysis of the Plebanski action for general relativity by deriving the Dirac brackets for the enlarged phase space and showing their equivalence to the Lorentz-covariant canonical formulation used in covariant loop quantum gravity. It introduces a shifted connection that yields simplified Dirac brackets and diagonalizable area operators, establishing a direct bridge between the Plebanski (canonical) and CLQG formalisms. The results clarify how the B-field relates to gauge generators within the constrained theory and illuminate the role of the Immirzi parameter, arguing it does not affect the covariant symplectic structure. The findings have significant implications for spin foam models, particularly in guiding the construction of covariant, spin-foam approaches compatible with canonical quantization. Overall, the paper sets a concrete groundwork for deriving spin foam models from a canonical Plebanski framework and for reconciling CLQG with Barrett–Crane-type 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.