In this work, we construct several diffeomorphism invariant observables in AdS_3 gravity in the context of the entanglement wedge cross section. The set of observables that we construct includes the entanglement wedge cross section and four extra half geodesics' lengths. We try to study the properties of these observables in Hamiltonian formalism including: the system's evolutions generated by these observables and the brackets between these observables. We use two approaches to study these questions, by the brackets of the Brown-York tensor’s components in the covariant phase space formalism and by canonical formalism, which give consistent results.
With the two methods, we get several interesting results related to these questions. We find that the entanglement wedge cross section generates a novel behavior in the system's evolution, which include a split at the HRT surface. We compute the brackets between these observables, and the only non-zero ones are the brackets of the entanglement wedge cross section with the four half geodesics. Inspired from this set-up, we also construct a slightly different geodesic network where all of the geodesics' lengths commute with each other.
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