Hamiltonian analysis of General relativity with the Immirzi parameter

TL;DR

The paper performs a full Hamiltonian analysis of General Relativity in the tetrad‑connection (Holst) action with an arbitrary Immirzi parameter $β$, without gauge fixing. It derives the Hamiltonian, analyzes the constraint algebra, identifies secondary constraints, and solves the resulting second‑class pairs, obtaining a formulation with a first‑class gauge/ diffeomorphism structure. After gauge fixing to the time gauge, it recovers Barbero’s real‑connection gravity with the canonical variables $(A_{ai},E^{ai})$, clarifying the role of the Immirzi parameter and confirming Holst’s result from the un‑gauge-fixed action. This work provides a clean bridge from the tetrad‑connection formalism to the Loop Quantum Gravity framework via Barbero’s variables, and highlights how the Immirzi parameter enters the classical canonical structure.

Abstract

Starting from a Lagrangian we perform the full constraint analysis of the Hamiltonian for General relativity in the tetrad-connection formulation for an arbitrary value of the Immirzi parameter and solve the second class constraints, presenting the theory with a Hamiltonian composed of first class constraints which are the generators of the gauge symmetries of the action. In the time gauge we then recover Barbero's formulation of gravity.