New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis

TL;DR

This paper rederives a connection formulation of General Relativity using the Dirac algorithm on the Palatini action, confirming the consistency of the approach and clarifying the role of constraints in all dimensions.

Contribution

It provides a detailed Lagrangian analysis and demonstrates the equivalence of the Palatini and ADM formulations through gauge unfixing and constraint treatment.

Findings

Hamiltonian constraint matches ADM when constraints are satisfied

Second class constraints are converted to first class via gauge unfixing

Consistency of the new connection formulation is confirmed

Abstract

We rederive the results of our companion paper, for matching spacetime and internal signature, by applying in detail the Dirac algorithm to the Palatini action. While the constraint set of the Palatini action contains second class constraints, by an appeal to the method of gauge unfixing, we map the second class system to an equivalent first class system which turns out to be identical to the first class constraint system obtained via the extension of the ADM phase space performed in our companion paper. Central to our analysis is again the appropriate treatment of the simplicity constraint. Remarkably, the simplicity constraint invariant extension of the Hamiltonian constraint, that is a necessary step in the gauge unfixing procedure, involves a correction term which is precisely the one found in the companion paper and which makes sure that the Hamiltonian constraint derived from the Palatini Lagrangian coincides with the ADM Hamiltonian constraint when Gauss and simplicity constraints are satisfied. We therefore have rederived our new connection formulation of General Relativity from an independent starting point, thus confirming the consistency of this framework.