Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

We study the limit of the Lorentz-covariant canonical formulation where the Immirzi parameter approaches β = i. We show that, formulated in terms of a shifted spacetime connection, which also plays a crucial role in the covariant quantization, the limit is smooth and reproduces the canonical structure of the self-dual Ashtekar gravity. The reality conditions of Ashtekar gravity can be incorporated by means of the Dirac brackets derived from the covariant formulation and defined on an extended phase space which involves, besides the self-dual variables, also their anti-self-dual counterparts