A Covariant Approach To Ashtekar's Canonical Gravity

TL;DR

This paper develops a Lorentz and coordinate-invariant formulation of canonical gravity using Ashtekar's variables, addressing complexities arising from their complex nature to ensure a covariant phase space description.

Contribution

It introduces a covariant phase space approach to Ashtekar's canonical gravity, resolving issues related to the complex variables and ensuring invariance under Lorentz and coordinate transformations.

Findings

Constructed a Lorentz and coordinate-invariant symplectic form.

Resolved complexities associated with Ashtekar's complex variables.

Provided a covariant phase space framework for canonical gravity.

Abstract

A Lorentz and general co-ordinate co-variant form of canonical gravity, using Ashtekar's variables, is investigated. A co-variant treatment due to Crnkovic and Witten is used, in which a point in phase space represents a solution of the equations of motion and a symplectic functional two form is constructed which is Lorentz and general co-ordinate invariant. The subtleties and difficulties due to the complex nature of Ashtekar's variables are addressed and resolved.