Abelian Ashtekar formulation from the ADM action

Abstract

We study the Ashtekar formulation of linear gravity starting from the Arnowitt–Deser–Misner (ADM) first-order action of the exact theory, linearizing it, and performing a canonical transformation that coordinatizes the phase-space in terms of the already linearized Ashtekar variables. A quantum geometrical representation, based on the realization of the transverse-traceless part of the linear canonical variables, is presented, and its relationship with previous attempts is discussed. A related geometric representation that realizes directly the observables of the Fierz–Pauli theory, without resorting to deal with the linearized Ashtekar variables, is also discussed.