Lecture Notes in Loop Quantum Gravity. LN3: Boundary equations for Ashtekar-Barbero-Immirzi model

TL;DR

This paper performs a canonical analysis of the Ashtekar-Barbero-Immirzi model, deriving boundary equations and constraints that align with standard Loop Quantum Gravity results, without fixing the Immirzi parameter.

Contribution

It introduces a method to derive boundary equations and constraints in the ABI model without fixing the Immirzi parameter, aligning with standard LQG results.

Findings

Constraint equations depend on the Immirzi parameter only

The model allows independent conjugate fields (A, E)

Constraint equations match standard LQG results

Abstract

We shall here perform the canonical analysis of field equations of ABI model in order to determine constraint equations. We shall show that one can use algebraic constraints in the covariant framework to fix $k^i$ as a function of the frame and obtain a model where $(A^i_a, E_i^a)$ is a pair of independent fields which are also a pair of conjugated fields. We shall not impose any relation on Immirzi parameter $\beta$ and Holst parameter $\gamma$, still constraint equations will depend on $\beta$ only and they agree with standard result of LQG which are obtained by a suitable canonical transformation on a leaf of the ADM foliation used to define a Hamiltonian framework. We eventually state the scheme for quantization that will be discussed in the following lecture notes.