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

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 $β=0$ and leaving the lapse and shift functions unconstrained. The $β= 0$ 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.

• The paper establishes a complete Lagrangian canonical analysis of the Holst model with β=0, emphasizing a gauge-unfixed approach with unconstrained lapse and shift.
• It rigorously decomposes the field equations into differential, algebraic, and evolution constraints, aligning them with standard ADM formulations.
• The study extends the canonical framework to arbitrary dimensions, setting a foundation for LQG-like quantization beyond the conventional 3+1 setting.

Lagrangian Canonical Analysis of the Holst Model for $β = 0$

Introduction and Motivation

This work formulates a complete Lagrangian canonical analysis of the Holst model for General Relativity (GR), specifically targeting the regime where the Barbero parameter $\beta$ is set to zero. Standard formulations of Loop Quantum Gravity (LQG) and related canonical quantum gravity programs rely on the Holst action, a variant of the frame-affine (vielbein + independent spin connection) general relativity action, supplemented with a so-called Holst term controlled by a dynamical parameter $\gamma$. The introduction of Ashtekar-Barbero-Immirzi (ABI) variables, pivotal for LQG in $3+1$ dimensions, typically relates this $\gamma$ to the Barbero parameter, whereas this analysis strictly sets $\beta=0$ regardless of $\gamma$.

The decision to focus on $\beta=0$ is technically and conceptually significant. The authors emphasize that, unlike the Holst parameter $\gamma$, the Barbero parameter is kinematical rather than dynamical. Moreover, while $\gamma$ is meaningful mostly in four spacetime dimensions, setting $\beta$0 extends the reach of this formalism to arbitrary dimension and establishes a promising foundation for performing LQG-like canonical analysis beyond $\beta$1 dimensions.

Geometric and Topological Framework

The work is situated on a four-dimensional spin manifold $\beta$2 equipped with a principal $\beta$3 bundle. Fields consist of a spin frame $\beta$4 and a spin connection $\beta$5, defined with rigorous attention to global/topological constraints (vanishing Stiefel-Whitney classes, existence of spin structure). The Holst action is then

$\beta$6

The canonical transformation to ABI variables is performed on spacetime, not space. The reduction of the structure group to the compact $\beta$7 subgroup (the usual move in LQG) and the associated reductive splitting are handled rigorously, with full awareness of their topological conditions. In particular, for dimensions greater than $\beta$8, only a “zero” (i.e., $\beta$9) splitting is available, cementing the necessity of the present treatment for future dimension-agnostic quantum gravity approaches.

Field Equations in ABI Variables for $\gamma$0

By executing the change $\gamma$1, the spin connection splits into an $\gamma$2 Barbero-Immirzi connection $\gamma$3 and an Immirzi field $\gamma$4, both defined globally on spacetime (not only on spatial slices). The curvature and auxiliary objects are decomposed accordingly, and all equations are re-expressed in terms of these fields.

The resulting system is a set of 40 field equations for 40 variables, after accounting for gauge-fixing of 3 Lorentz boosts in the frame.

$\gamma$5

A rigorous projection of these equations with respect to the canonical spacetime foliation (lapse and shift functions left unconstrained) and complete 3+1 decomposition are carried out, forming the technical heart of the work.

Canonical Decomposition, Constraints, and Evolution Equations

A full canonical analysis is performed, sorting the projected field equations into three categories:

• Differential constraints: These consist of 10 equations, including the Gauss law ($\gamma$6), vector/momentum constraints, and the Hamiltonian constraint (scalar constraint):

  $\gamma$7

where $\gamma$8 is the extrinsic curvature, and $\gamma$9 is the Ricci scalar on the hypersurface.

• Algebraic constraints: Twenty-one equations ensure the auxiliary variables (the differences between the connection fields and their Levi-Civita projections) vanish, along with various symmetry conditions. For instance, the auxiliary fields $3+1$0, $3+1$1 (expressing deviations from the Levi-Civita connection in ABI variables) are fixed algebraically.
• Evolution equations: Six first-order partial differential equations for the extrinsic curvature, matching precisely the number of dynamical degrees of freedom once all constraints and gauge-fixings are accounted for. These are compatible with the evolution part of the standard Einstein equations.

An important technical finding is that three equations, which are usually automatically satisfied (especially when lapse and shift are gauge-fixed as in many traditional treatments), appear as differential constraints when the gauge is left free. Their triviality (or lack thereof) thus depends explicitly on the gauge—a result made plain by the Lagrangian, fully gauge-unfixed analysis.

Implications and Extensions

The analysis confirms that the Holst model with $3+1$2 admits a regular, fully determined initial value formulation, with direct correspondence to the standard ADM (3+1) constraints and evolution equations of Einstein gravity. Importantly, the lapse $3+1$3 and shift $3+1$4 are not fixed, differentiating this work from most canonical treatments which often fix the gauge at the outset. This insight is relevant for quantization procedures where preservation of general covariance (including time reparametrizations and spatial diffeomorphisms) is essential.

Most crucially, the framework is explicitly applicable in any dimension, sidestepping special properties tied to the Holst parameter in $3+1$5 dimensions. This is vital for ongoing efforts to generalize LQG and similar programs beyond four spacetime dimensions, an open theoretical direction with implications for supergravity, string-theoretic extensions, and higher-dimensional gravity.

On a theoretical level, the work points toward an algorithmic, model-independent procedure for canonical decomposition in any metric-affine gravity theory, steadily clarifying the algebraic and differential structure of constraints directly from the Lagrangian, without Hamiltonian manipulations or prior gauge reduction.

Conclusion

This analysis demonstrates that the Holst model with $3+1$6 yields a closed, consistent system of canonical equations: a complete set of constraints and evolution equations in fully Lagrangian form, valid for arbitrary dimensions and unconstrained lapse and shift. All results are consistent with standard $3+1$7 ADM constraints and extend them in a formally precise, gauge-unfixed setting. The treatment exposes the underlying structure necessary for the extension of ABI variables and LQG-like frameworks outside $3+1$8 dimensions and establishes a procedural standard for future studies aiming at the quantization of gravity in this general context.

The question of compatibility between constraint preservation and evolution, well-understood in standard GR via the Bianchi identities, is highlighted as open for this specific Holst-Lagrangian setup—a direction for additional research, especially with regard to generalized conservation laws and their role in constrained evolution. The adoption of rigorous bundle-topological language supplies a foundation for genuinely covariant treatments in both classical and quantum theories.