Lagrangian and Hamiltonian formulations of the Palatini action

TL;DR

This paper derives Ashtekar-Barbero variables in Loop Quantum Gravity from the Palatini action, highlighting the necessity of torsion-free or compatibility conditions to accurately describe General Relativity.

Contribution

It provides a derivation of key variables in LQG directly from the Palatini formulation, clarifying the conditions needed for a consistent description of gravity.

Findings

Ashtekar-Barbero variables derived from Palatini action

Torsion-free or compatibility condition required

Clarifies foundational aspects of LQG

Abstract

We derive the Ashtekar-Barbero variables in Loop Quantum Gravity(LQG) starting from the Palatini action. We find that we need either the torsion-free condition or the compatibility condition for the Palatini action to describe General Relativity.