A Lagrangian framework for canonical analysis for the Holst model with $\beta = 0$

TL;DR

This paper conducts a canonical analysis of the Holst model for General Relativity with a zero Barbero parameter, enabling extensions of Loop Quantum Gravity beyond 3+1 dimensions while maintaining a consistent 3+1 decomposition.

Contribution

It introduces a novel canonical framework for the Holst model at β=0, leaving lapse and shift unconstrained, and clarifies the nature of certain differential constraints.

Findings

Derived 37 equations matching the 37 field components.

Identified three differential constraints dependent on gauge choices.

Framework remains consistent with standard 3+1 Einstein equations without fixing lapse and shift.

Abstract

We perform a canonical analysis of the Holst model for General Relativity, within the framework laid out in arXiv:2401.07307 and arXiv:2010.07725, distinguishing our approach by setting the Barbero parameter to $\beta=0$ and leaving the lapse and shift functions unconstrained. The $\beta = 0$ choice is of particular interest because it is viable across all dimensions, providing a necessary foundation for extending the Loop Quantum Gravity formalism beyond $3+1$ dimensions. Through field decomposition and the projection of the field equations, we derive a system of 37 equations (10 differential constraints, 21 algebraic constraints, and 6 evolution equations) exactly matching the 37 field components to be determined. Moreover, leaving the gauge unfixed reveals that three equations, which are typically identically satisfied under normal evolution, are actually differential constraints whose triviality depends on specific gauge choices. The resulting framework remains fully consistent with the standard $3+1$ decomposition of the Einstein equations without requiring any constraints on the lapse and shift functions.