Hamiltonian gravity in tetrad-connection variables

TL;DR

The paper develops a complete Hamiltonian treatment of Einstein–Cartan gravity with a Holst–type action, maintaining full Lorentz covariance by avoiding the time gauge and by carefully separating first- and second-class constraints. It introduces an extended phase space that yields 18 canonical pairs and a constraint algebra resembling an Einstein–Yang–Mills system with an $SO(1,3)$ subalgebra, while second-class constraints are handled either by solving them or via Dirac brackets. The work demonstrates that canonical gauge transformations correspond to spacetime diffeomorphisms plus Lorentz transformations on shell, and that covariance survives the second-class reduction, providing a robust framework for covariant canonical quantization. It also exposes tensions with loop quantum gravity’s standard, time-gauge–based quantization and discusses the Lorentz-invariant definitions of area and volume, suggesting novel, covariant spectral properties with potential implications for spinfoam models and the interpretation of geometric spectra.

Abstract

A systematic Hamiltonian formulation of the Einstein-Cartan system, based on the Hilbert-Palatini action with the Barbero-Immirzi and cosmological constants, is performed using the traditional ADM decomposition and without fixing the time gauge. This procedure results in a larger phase space compared to that of the Ashtekar-Barbero approach as well as a larger set of first-class constraints generating gauge transformations that are on-shell equivalent to spacetime diffeomorphisms and SO(1,3) transformations. The imbalance in the number of components between the tetrad and the connection is resolved by the identification of second-class constraints implied by the action, which can be implemented by use of Dirac brackets or by solving them directly. The Hamiltonian system remains well-defined off the second-class constraint surface in an extended phase space with additional degrees of freedom, implying a more general geometric theory. Implications for canonical quantum gravity are discussed.