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We study the Ashtekar formulation of linear gravity starting from the ADMfirst order action for the non linear theory, linearizing it, and performing acanonical transformation that coordinatizes the phase space in terms of thealready linearized Ashtekar variables. The results obtained in this way are inaccordance with those obtained through the standard method, in which, afterintroducing the Ashtekar variables for the full theory, a linearization aroundthe flat Abelian connection and its conjugate momentum is performed.

Abelian Ashtekar formulation from the ADM action