Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

TL;DR

This paper demonstrates the equivalence between Lorentz-covariant canonical formulation with Immirzi parameter and selfdual Ashtekar gravity, proposing a new approach to reality conditions using Dirac brackets on an extended phase space.

Contribution

It establishes the equivalence for the case = i and introduces a novel method to handle reality conditions via Dirac brackets in an extended phase space.

Findings

Lorentz-covariant formulation is equivalent to selfdual Ashtekar gravity for = i

Proposes a new method to impose reality conditions using Dirac brackets

Extends phase space to include selfdual and anti-selfdual variables

Abstract

We show the equivalence of the Lorentz-covariant canonical formulation considered for the Immirzi parameter $\beta=i$ to the selfdual Ashtekar gravity. We also propose to deal with the reality conditions in terms of Dirac brackets derived from the covariant formulation and defined on an extended phase space which involves, besides the selfdual variables, also their anti-selfdual counterparts.