Is Barbero's Hamiltonian formulation a Gauge Theory of Lorentzian Gravity?

TL;DR

This paper critiques Barbero's Hamiltonian formulation of General Relativity, arguing it does not qualify as a gauge theory of gravity like Ashtekar's formulation, due to issues with its interpretation as a space-time gauge field.

Contribution

The paper provides a critical analysis of Barbero's real connection formulation, highlighting its limitations in representing gravity as a gauge theory compared to Ashtekar's complex connection approach.

Findings

Barbero's real SO(3) connection lacks a space-time gauge field interpretation.

The formulation is not diffeomorphism invariant when interpreted as a space-time gauge theory.

It concludes that Barbero's approach does not constitute a gauge theory of gravity.

Abstract

This letter is a critique of Barbero's constrained Hamiltonian formulation of General Relativity on which current work in Loop Quantum Gravity is based. While we do not dispute the correctness of Barbero's formulation of general relativity, we offer some criticisms of an aesthetic nature. We point out that unlike Ashtekar's complex SU(2) connection, Barbero's real SO(3) connection does not admit an interpretation as a space-time gauge field. We show that if one tries to interpret Barbero's real SO(3) connection as a space-time gauge field, the theory is not diffeomorphism invariant. We conclude that Barbero's formulation is not a gauge theory of gravity in the sense that Ashtekar's Hamiltonian formulation is. The advantages of Barbero's real connection formulation have been bought at the price of giving up the description of gravity as a gauge field.