  In this paper we show that the instanton representation of Plebanski gravity
exhibits a Hilbert space of harmonic oscillator-like coherent states. We put in
place the formalism and carry out the construction of the states, and we
elucidate on their physical interpretation. Additionally, we provide an
invertible map between the Ashtekar variables and this Hilbert space of states,
via the instanton representation. Finally, we compare and constrast our
formalism and some of our results with the corresponding ones in loop quantum
gravity.
