From Lagrangian to Hamiltonian formulations of the Palatini action

TL;DR

This paper explores the Lagrangian and Hamiltonian formulations of the Palatini action in General Relativity, highlighting the assumptions needed for each approach and implications for quantization.

Contribution

It clarifies the assumptions required in both formulations and discusses their relevance for quantizing General Relativity.

Findings

Lagrangian formulation requires metric compatibility or tetrad compatibility.

Hamiltonian formulation yields Einstein's equations with tetrad compatibility.

Hamiltonian approach with metric compatibility and zero torsion is relevant for quantization.

Abstract

We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.