Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

TL;DR

Reality conditions for Ashtekar gravity are recast as second-class constraints within a Lorentz-covariant canonical framework. The authors show the equivalence to selfdual Ashtekar gravity at $\beta=i$ and construct a beta-independent Dirac-bracket structure on an extended phase space that includes both selfdual and anti-selfdual variables, enabling a consistent complex-conjugation operation. By implementing the reality conditions through this extended Dirac structure and Gauss constraints, they provide a viable route to quantization, including a triad-like representation where anti-selfdual variables are Hermitian conjugates of the selfdual ones. The work thus offers a covariant, Lorentz-symmetric approach to the long-standing reality-condition problem in Ashtekar gravity with implications for loop-quantization strategies.

Abstract

We show the equivalence of the Lorentz-covariant canonical formulation considered for the Immirzi parameter $β=i$ 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.