Derivation of Ashtekar variables from tetrad gravity

The new gravitational variables introduced recently by Ashtekar are derived from the standard tetrad gravity formalism with full local Lorentz invariance. This is done by a succession of canonical transformations, which involve as a first step the transition from arbitrary tetrads to three-dimensional triads and pure gauge boost variables. It is then shown that the weighted contravariant triads and mixed components of the extrinsic curvature are also canonically conjugate. Finally, the new variables are explicitly derived and proved to be canonical because of a remarkable identity obeyed by the spatial spin connection in three dimensions.