Geometrical representation of Euclidean general relativity in the canonical formalism

TL;DR

This paper presents a covariant SU(2) formulation of Euclidean general relativity's constraints using Ashtekar variables, emphasizing a special covariant derivative and a jet space coordinate system to clarify physical and gauge degrees of freedom.

Contribution

It introduces a novel SU(2) covariant representation of the constraints and a coordinate framework that separates physical and gauge directions in the canonical formalism.

Findings

Algebraic form of constraints in terms of curvature and torsion

Classification of local invariant charges

Separation of physical and gauge directions

Abstract

We give an SU(2) covariant representation of the constraints of Euclidean general relativity in the Ashtekar variables. The guiding principle is the use of triads to transform all free spatial indices into SU(2) indices. A central role is played by a special covariant derivative. The Gauss, diffeomorphism and Hamiltonian constraints become purely algebraic restrictions on the curvature and the torsion associated with this connection. We introduce coordinates on the jet space of the dynamical fields which cleanly separate the constraint and gauge directions from the true physical directions. This leads to a classification of all local diffeomorphism and Gauss invariant charges.