# Revisiting the solution of the second-class constraints of the Holst action

## TL;DR

This paper critically revisits the canonical analysis of the Holst action with a cosmological constant, identifying and correcting incomplete solutions to second-class constraints, and establishing a complete, Lorentz-invariant formulation that connects to Ashtekar-Barbero variables.

## Contribution

It provides a complete solution to the second-class constraints in the Holst action, correcting previous incomplete approaches and deriving a Lorentz-invariant theory with a noncanonical symplectic structure.

## Key findings

- Corrected the solution of second-class constraints in the Holst action.
- Established a Lorentz-invariant formulation with a noncanonical symplectic structure.
- Connected the formulations to Ashtekar-Barbero variables in the time gauge.

## Abstract

In this paper we revisit the nonmanifestly Lorentz-covariant canonical analysis of the Holst action with a cosmological constant. We take a viewpoint close to that of F. Cianfrani and G. Montani [Phys. Rev. Lett. 102, 091301 (2009)] and realize that the solution of the second-class constraints that the authors provide is incomplete, thus not accounting for the correct local dynamics of general relativity. We then mend their approach by adding the missing degrees of freedom to the solution and give a complete description of the resulting theory, which preserves Lorentz invariance but turns out to be endowed with a noncanonical symplectic structure. Later on and without resorting to any gauge condition, we perform a Darboux transformation to bring this theory into a canonical form. Finally, we show that in the time gauge both formulations, namely the noncanonical and the canonical ones, lead to the Ashtekar-Barbero variables.